mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: case h₂
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
h : SecondCountableTopologyEither α β
s : Set (α × β)
hs : IsOpen[instTopologicalSpaceProd] s
hβ : SecondCountableTopology β
F : Set β → Set α := ⋯
A : ∀ (a : Set β), IsOpen[inst✝¹²] (F a)
⊢ ⋃ a ∈ countableBasis β, F a ×ˢ a ⊆ s

Tactic:
rintro ⟨y1, y2⟩ hy

Goal state after:
case h₂
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
h : SecondCountableTopologyEither α β
s : Set (α × β)
hs : IsOpen[instTopologicalSpaceProd] s
hβ : SecondCountableTopology β
F : Set β → Set α := fun a ↦ {y | ∃ b, IsOpen[inst✝¹²] b ∧ y ∈ b ∧ b ×ˢ a ⊆ s}
A : ∀ (a : Set β), IsOpen[inst✝¹²] (F a)
y1 : α
y2 : β
hy : (y1, y2) ∈ ⋃ a ∈ countableBasis β, F a ×ˢ a
⊢ (y1, y2) ∈ s
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: case refine_2
α : Type u
inst✝ : PseudoEMetricSpace α
s t : Set α
r : ℝ≥0∞
H1 : ∀ x ∈ s, ∃ y ∈ t, edist x y ≤ r
H2 : ∀ x ∈ t, ∃ y ∈ s, edist x y ≤ r
x : α
xt : x ∈ t
⊢ infEDist x s ≤ r

Tactic:
· rcases H2 x xt with ⟨y, ys, hy⟩
    exact le_trans (infEDist_le_edist_of_mem ys) hy

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case inl.inr
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
g g' : δ → α
hf : ∀ (i : ι), Measurable (f i)
s : Set δ
hs : MeasurableSet s
hg : ∀ b ∈ s, IsLUB {a | ∃ i, f i b = a} (g b)
hg' : EqOn g g' sᶜ
g'_meas : Measurable g'
hι : IsEmpty ι
x : δ
hx : x ∈ s
⊢ Measurable g

Tactic:
· have A : ∀ b ∈ s, IsBot (g b) := by simpa using hg
      have B : ∀ b ∈ s, g b = g x := by
        intro b hb
        apply le_antisymm (A b hb (g x)) (A x hx (g b))
      have : g = s.piecewise (fun _y ↦ g x) g' := by
        ext b
        by_cases hb : b ∈ s
        · simp [hb, B]
        · simp [hb, hg' hb]
      rw [this]
      exact Measurable.piecewise hs measurable_const g'_meas

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

Goal state before:
pp: α : Type u
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : PseudoEMetricSpace β
f : α → β
K : Set α
U : Set β
hK : IsCompact K
ho : IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] U
hf : ∀ x ∈ K, ContinuousAt f x
hKU : MapsTo f K U
x : α
hx : x ∈ K
this : {f x} ⊆ U
δ : ℝ
hδ : 0 < δ
hthick : thickening δ {f x} ⊆ U
V : Set α := ⋯
⊢ f x ∈ thickening (δ / 2) {f x}

Tactic:
exact (self_subset_thickening (by positivity) _) rfl

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_5
μ : Measure δ
s : Set ι
f : ι → δ → α
hs : s.Countable
hf : ∀ i ∈ s, AEMeasurable (f i) μ
g : ι → δ → α := fun i ↦ if hi : i ∈ s then mk (f i) ⋯ else fun _b ↦ sSup ∅
this : ∀ i ∈ s, Measurable (g i)
⊢ (fun b ↦ ⨆ i ∈ s, f i b) =ᵐ[μ] fun b ↦ ⨆ i ∈ s, g i b

Tactic:
have : ∀ i ∈ s, ∀ᵐ b ∂μ, f i b = g i b :=
    fun i hi ↦ by simpa [g, hi] using! (hf i hi).ae_eq_mk

Goal state after:
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_5
μ : Measure δ
s : Set ι
f : ι → δ → α
hs : s.Countable
hf : ∀ i ∈ s, AEMeasurable (f i) μ
g : ι → δ → α := fun i ↦ if hi : i ∈ s then mk (f i) ⋯ else fun _b ↦ sSup ∅
this✝ : ∀ i ∈ s, Measurable (g i)
this : ∀ i ∈ s, ∀ᵐ (b : δ) ∂μ, f i b = g i b
⊢ (fun b ↦ ⨆ i ∈ s, f i b) =ᵐ[μ] fun b ↦ ⨆ i ∈ s, g i b
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α →ₛ ℝ≥0∞
h : f.lintegral μ ≠ ∞
⊢ f.FinMeasSupp μ

Tactic:
refine finMeasSupp_iff.2 fun b hb => ?_
  rw [f.lintegral_eq_of_subset' (Finset.subset_insert b _)] at h
  refine ENNReal.lt_top_of_mul_ne_top_right ?_ hb
  exact (lt_top_of_sum_ne_top h (Finset.mem_insert_self _ _)).ne

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case hbc
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
hf : StronglyMeasurable f
c : ℝ
x : α
hfx : ‖f x‖ ≤ c
h_tendsto : Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 0)
hfx0 : f x = 0
h_tendsto_norm : Tendsto (fun n ↦ ‖(hf.approx n) x‖) atTop (𝓝 0)
n : ℕ
⊢ ‖min 1 (c / ‖(hf.approx n) x‖)‖ ≤ ‖1‖

Tactic:
rw [norm_one, Real.norm_of_nonneg]

Goal state after:
case hbc
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
hf : StronglyMeasurable f
c : ℝ
x : α
hfx : ‖f x‖ ≤ c
h_tendsto : Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 0)
hfx0 : f x = 0
h_tendsto_norm : Tendsto (fun n ↦ ‖(hf.approx n) x‖) atTop (𝓝 0)
n : ℕ
⊢ min 1 (c / ‖(hf.approx n) x‖) ≤ 1

case hbc
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
hf : StronglyMeasurable f
c : ℝ
x : α
hfx : ‖f x‖ ≤ c
h_tendsto : Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 0)
hfx0 : f x = 0
h_tendsto_norm : Tendsto (fun n ↦ ‖(hf.approx n) x‖) atTop (𝓝 0)
n : ℕ
⊢ 0 ≤ min 1 (c / ‖(hf.approx n) x‖)
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
hf : StronglyMeasurable f
c : ℝ
x : α
hfx : ‖f x‖ ≤ c
⊢ Tendsto (fun n ↦ (hf.approxBounded c n) x) atTop (𝓝 (f x))

Tactic:
have h_tendsto := hf.tendsto_approx x
  simp only [StronglyMeasurable.approxBounded, SimpleFunc.coe_map, Function.comp_apply]
  by_cases hfx0 : ‖f x‖ = 0
  · rw [norm_eq_zero] at hfx0
    rw [hfx0] at h_tendsto ⊢
    have h_tendsto_norm : Tendsto (fun n => ‖hf.approx n x‖) atTop (𝓝 0) := by
      convert! h_tendsto.norm
      rw [norm_zero]
    refine squeeze_zero_norm (fun n => ?_) h_tendsto_norm
    calc
      ‖min 1 (c / ‖hf.approx n x‖) • hf.approx n x‖ =
          ‖min 1 (c / ‖hf.approx n x‖)‖ * ‖hf.approx n x‖ :=
        norm_smul _ _
      _ ≤ ‖(1 : ℝ)‖ * ‖hf.approx n x‖ := by
        gcongr
        rw [norm_one, Real.norm_of_nonneg]
        · exact min_le_left _ _
        · exact le_min zero_le_one (div_nonneg ((norm_nonneg _).trans hfx) (norm_nonneg _))
      _ = ‖hf.approx n x‖ := by rw [norm_one, one_mul]
  rw [← one_smul ℝ (f x)]
  refine Tendsto.smul ?_ h_tendsto
  have : min 1 (c / ‖f x‖) = 1 := by
    rw [min_eq_left_iff, one_le_div (lt_of_le_of_ne (norm_nonneg _) (Ne.symm hfx0))]
    exact hfx
  nth_rw 2 [this.symm]
  refine Tendsto.min tendsto_const_nhds ?_
  exact Tendsto.div tendsto_const_nhds h_tendsto.norm hfx0

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case neg.inr
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : c ≤ ‖(hf.approx n) x‖
⊢ ‖c‖ / ‖(hf.approx n) x‖ * ‖(hf.approx n) x‖ ≤ c

Tactic:
mul_comm,

Goal state after:
case neg.inr
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : c ≤ ‖(hf.approx n) x‖
⊢ ‖(hf.approx n) x‖ * (‖c‖ / ‖(hf.approx n) x‖) ≤ c
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case neg.inr
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : c ≤ ‖(hf.approx n) x‖
⊢ ‖(hf.approx n) x‖ * ‖c‖ * ‖(hf.approx n) x‖⁻¹ ≤ c

Tactic:
mul_comm,

Goal state after:
case neg.inr
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : c ≤ ‖(hf.approx n) x‖
⊢ ‖(hf.approx n) x‖⁻¹ * (‖(hf.approx n) x‖ * ‖c‖) ≤ c
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case neg.inr
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : c ≤ ‖(hf.approx n) x‖
⊢ 1 * ‖c‖ ≤ c

Tactic:
one_mul,

Goal state after:
case neg.inr
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : c ≤ ‖(hf.approx n) x‖
⊢ ‖c‖ ≤ c
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f g : α → β
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
hf : AEStronglyMeasurable f (μ.restrict s)
hg : AEStronglyMeasurable g (μ.restrict sᶜ)
⊢ ∀ᵐ (x : α) ∂μ.restrict sᶜ,
    s.piecewise f g x = s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg) x

Tactic:
have h := hg.ae_eq_mk
    rw [Filter.EventuallyEq, ae_restrict_iff' hs.compl] at h
    rw [ae_restrict_iff' hs.compl]
    filter_upwards [h] with x hx
    intro hx_mem
    rw [Set.mem_compl_iff] at hx_mem
    simp only [hx_mem, not_false_eq_true, Set.piecewise_eq_of_notMem, hx hx_mem]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case neg
α : Type u_1
m✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Monotone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
n m : ℕ
hnm : n ≤ m
x : α
hx : x ∉ aeSeqSet hf p
⊢ aeSeq hf p n x ≤ aeSeq hf p m x

Tactic:
simp only [aeSeq, hx, if_false, le_rfl]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Markov

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hle : f ≤ᵐ[μ] g
hg : AEMeasurable g μ
ε : ℝ≥0∞
φ : α → ℝ≥0∞
hφm : Measurable φ
hφ_le : φ ≤ f
hφ_eq : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), φ a ∂μ
⊢ ∫⁻ (x : α), f x ∂μ + ε * μ {x | f x + ε ≤ g x} = ∫⁻ (x : α), φ x ∂μ + ε * μ {x | f x + ε ≤ g x}

Tactic:
rw [hφ_eq]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Markov

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hle : f ≤ᵐ[μ] g
hg : AEMeasurable g μ
ε : ℝ≥0∞
φ : α → ℝ≥0∞
hφm : Measurable φ
hφ_le : φ ≤ f
hφ_eq : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), φ a ∂μ
⊢ ∫⁻ (x : α), φ x ∂μ + ε * μ {x | φ x + ε ≤ g x} = ∫⁻ (x : α), φ x + {x | φ x + ε ≤ g x}.indicator (fun x ↦ ε) x ∂μ

Tactic:
by
      rw [lintegral_add_left hφm, lintegral_indicator₀, setLIntegral_const]
      exact measurableSet_le (hφm.nullMeasurable.measurable'.add_const _) hg.nullMeasurable

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Markov

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
ε : ℝ≥0∞
⊢ ε * μ {x | ε ≤ f x} ≤ ∫⁻ (a : α), f a ∂μ

Tactic:
simpa only [lintegral_zero, zero_add] using
    lintegral_add_mul_meas_add_le_le_lintegral (ae_of_all _ fun x => zero_le) hf ε

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∈ aeSeqSet hf p
⊢ f b₁ x ≤ f z x

Tactic:
apply_rules [hz₁, hz₂]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Sub

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
h_ae_seq_mono : Antitone (aeSeq h_meas p)
⊢ ∫⁻ (a : α), ⨅ i, aeSeq h_meas p i a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ

Tactic:
lintegral_iInf (aeSeq.measurable h_meas p) h_ae_seq_mono

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
h_ae_seq_mono : Antitone (aeSeq h_meas p)
⊢ ⨅ n, ∫⁻ (a : α), aeSeq h_meas p n a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ

α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
h_ae_seq_mono : Antitone (aeSeq h_meas p)
⊢ ∫⁻ (a : α), aeSeq h_meas p 0 a ∂μ ≠ ∞
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∈ aeSeqSet hf p
⊢ f b₂ x ≤ f z x

Tactic:
apply_rules [hz₁, hz₂]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : Measurable f
⊢ ∫⁻ (a : α), f a * r ∂μ = (∫⁻ (a : α), f a ∂μ) * r

Tactic:
mul_comm,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : Measurable f
⊢ ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : AEMeasurable f μ
⊢ ∫⁻ (a : α), f a * r ∂μ = (∫⁻ (a : α), f a ∂μ) * r

Tactic:
mul_comm,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : AEMeasurable f μ
⊢ ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
⊢ (∫⁻ (a : α), f a ∂μ) * r ≤ ∫⁻ (a : α), f a * r ∂μ

Tactic:
mul_comm,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
⊢ r * ∫⁻ (a : α), f a ∂μ ≤ ∫⁻ (a : α), r * f a ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ∞
⊢ ∫⁻ (a : α), f a * r ∂μ = (∫⁻ (a : α), f a ∂μ) * r

Tactic:
mul_comm,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ∞
⊢ ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case add
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : AddZeroClass β
inst✝ : TopologicalSpace β
P : (f : α → β) → StronglyMeasurable f → Prop
ind : ∀ (c : β) ⦃s : Set α⦄ (hs : MeasurableSet s), P (s.indicator fun x ↦ c) ⋯
add :
  ∀ ⦃f g : α → β⦄ (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) (hfg : StronglyMeasurable (f + g)),
    Disjoint (support f) (support g) → P f hf → P g hg → P (f + g) hfg
lim :
  ∀ ⦃f : ℕ → α → β⦄ ⦃g : α → β⦄ (hf : ∀ (n : ℕ), StronglyMeasurable (f n)) (hg : StronglyMeasurable g),
    (∀ (n : ℕ), P (f n) ⋯) → (∀ (x : α), Tendsto (fun x_1 ↦ f x_1 x) atTop (𝓝 (g x))) → P g hg
f✝ : α → β
hf✝ : StronglyMeasurable f✝
s : ℕ → α →ₛ β := hf✝.approx
n : ℕ
f g : α →ₛ β
h_supp : Disjoint (support ⇑f) (support ⇑g)
hf : P ⇑f ⋯
hg : P ⇑g ⋯
⊢ P ⇑(f + g) ⋯

Tactic:
| @add f g h_supp hf hg =>

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
⊢ StronglyMeasurable f

Tactic:
have hs_m₂ : MeasurableSet[m₂] s := by
    have : MeasurableSet (s ∩ univ) := hs univ (by simpa)
    simpa

Goal state after:
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
⊢ StronglyMeasurable f
Module: Mathlib.Analysis.Normed.Group.InfiniteSum

Goal state before:
pp: ι : Type u_1
E : Type u_3
inst✝ : SeminormedAddCommGroup E
f : ι → E
g : ι → ℝ
a : E
b : ℝ
hf : HasSum f a
hg : HasSum g b
h : ∀ (i : ι), ‖f i‖ ≤ g i
⊢ ‖a‖ ≤ b

Tactic:
exact le_of_tendsto_of_tendsto' hf.norm hg fun _s ↦ norm_sum_le_of_le _ fun i _hi ↦ h i

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case refine_1
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous[inst✝⁷, inst✝²] fun i ↦ u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n ↦ u ((t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p ↦ u ((t_sf n) p.1) p.2
h_tendsto : Tendsto U atTop (𝓝 fun p ↦ u p.1 p.2)
n : ℕ
⊢ Measurable fun p ↦ u (↑p.1) p.2

Tactic:
have :
        (fun p : ↥(t_sf n).range × α => u (↑p.fst) p.snd) =
          (fun p : α × (t_sf n).range => u (↑p.snd) p.fst) ∘ Prod.swap :=
        rfl

Goal state after:
case refine_1
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous[inst✝⁷, inst✝²] fun i ↦ u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n ↦ u ((t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p ↦ u ((t_sf n) p.1) p.2
h_tendsto : Tendsto U atTop (𝓝 fun p ↦ u p.1 p.2)
n : ℕ
this : (fun p ↦ u (↑p.1) p.2) = (fun p ↦ u (↑p.2) p.1) ∘ Prod.swap
⊢ Measurable fun p ↦ u (↑p.1) p.2
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
s : Set α
hs : IsCompact s
hf' : ContinuousOn f s
ε : (x : α) → x ∈ s → ℝ
hε : ∀ (x : α) (a : x ∈ s), ε x a > 0
K : (x : α) → x ∈ s → ℝ≥0
hf : ∀ (x : α) (a : x ∈ s), LipschitzOnWith (K x a) f (ball x (ε x a) ∩ s)
x : α
hx : x ∈ s
K' : ℝ
hK' : K' ∈ upperBounds ((fun y ↦ dist (f x) (f y) / dist x y) '' (s \ ball x (ε x hx)))
y : α
hy : y ∈ s.diff (ball x (ε x hx))
⊢ 0 < nndist x y

Tactic:
exact NNReal.coe_pos.1 <| coe_nndist x y ▸
        ((hε x hx).trans_le <| not_lt.1 <| dist_comm x y ▸ hy.2)

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

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
p q : X ⊕ Y
h : Sum.dist p q = 0
⊢ p = q

Tactic:
rcases p with p | p <;> rcases q with q | q
    · rw [eq_of_dist_eq_zero h]
    · exact eq_of_glueDist_eq_zero _ _ _ one_pos _ _ ((Sum.dist_eq_glueDist p q).symm.trans h)
    · exact eq_of_glueDist_eq_zero _ _ _ one_pos _ _ ((Sum.dist_eq_glueDist q p).symm.trans h)
    · rw [eq_of_dist_eq_zero h]

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

Goal state before:
pp: case mpr.inl
E : ℕ → Type u_1
α : Type u_2
inst✝ : PseudoMetricSpace α
f : ((n : ℕ) → E n) → α
H : ∀ (x y : (n : ℕ) → E n) (n : ℕ), y ∈ cylinder x n → dist (f x) (f y) ≤ (1 / 2) ^ n
x : (n : ℕ) → E n
⊢ dist (f x) (f x) ≤ dist x x

Tactic:
simp [PiNat.dist_nonneg]

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

Goal state before:
pp: E : ℕ → Type u_1
α : Type u_2
inst✝ : PseudoMetricSpace α
f : ((n : ℕ) → E n) → α
⊢ (∀ (x y : (n : ℕ) → E n), dist (f x) (f y) ≤ dist x y) ↔
    ∀ (x y : (n : ℕ) → E n) (n : ℕ), y ∈ cylinder x n → dist (f x) (f y) ≤ (1 / 2) ^ n

Tactic:
constructor
  · intro H x y n hxy
    apply (H x y).trans
    rw [PiNat.dist_comm]
    exact mem_cylinder_iff_dist_le.1 hxy
  · intro H x y
    rcases eq_or_ne x y with (rfl | hne)
    · simp [PiNat.dist_nonneg]
    rw [dist_eq_of_ne hne]
    apply H x y (firstDiff x y)
    rw [firstDiff_comm]
    exact mem_cylinder_firstDiff _ _

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

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝¹ : (i : ι) → MetricSpace (E i)
inst✝ : ∀ (i : ι), CompleteSpace (E i)
s : ι → Set ((i : ι) × E i) := fun i ↦ Sigma.fst ⁻¹' {i}
U : Set (((k : ι) × E k) × (k : ι) × E k) := {p | dist p.1 p.2 < 1}
i : ι
⊢ IsComplete (s i)

Tactic:
by
    simp only [s, ← range_sigmaMk]
    exact (isometry_mk i).isUniformInducing.isComplete_range

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

Goal state before:
pp: case e'_4
α : Type u_1
inst✝ : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
this✝ : x ∈ eball x (ε / 2)
this : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty
⊢ ε = 2 * (ε / 2)

Tactic:
mul_comm,

Goal state after:
case e'_4
α : Type u_1
inst✝ : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
this✝ : x ∈ eball x (ε / 2)
this : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty
⊢ ε = ε / 2 * 2
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
x y : (n : ℕ) → E n
hx : x ∉ s
hy : y ∈ s
⊢ firstDiff x y ≤ longestPrefix x s

Tactic:
rw [longestPrefix, le_tsub_iff_right]

Goal state after:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
x y : (n : ℕ) → E n
hx : x ∉ s
hy : y ∈ s
⊢ firstDiff x y + 1 ≤ shortestPrefixDiff x s

E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
x y : (n : ℕ) → E n
hx : x ∉ s
hy : y ∈ s
⊢ 1 ≤ shortestPrefixDiff x s