mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: 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)
K' : (x : α) → x ∈ s → ℝ≥0
hK' : ∀ (x : α) (hx : x ∈ s), ∀ y ∈ s.diff (ball x (ε x hx)), edist (f x) (f y) ≤ ↑(K' x hx) * edist x y
t : Finset ↑s
ht : s ⊆ ⋃ x ∈ t, s ∩ ball (↑x) (ε ↑x ⋯ / 2)
x : α
hx : x ∈ s
y : α
hy : y ∈ s
z : ↑s
hz : z ∈ t
hx' : x ∈ s ∩ ball (↑z) (ε ↑z ⋯ / 2)
hy' : y ∉ ball (↑z) (ε ↑z ⋯)
⊢ ↑(K ↑z ⋯) * edist (↑z) x + ↑(K' ↑z ⋯) * (edist x ↑z + edist x y) ≤
    ↑(K ↑z ⋯) * edist x y + ↑(K' ↑z ⋯) * (edist x y + edist x y)

Tactic:
simp_rw [edist_dist, dist_comm _ x]
        gcongr <;> linarith [mem_ball.1 hx'.2, (not_lt (α := ℝ)).1 hy', dist_triangle_left y z x]

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: i j : ↑I
h : i ≠ 0 ∧ j < 1
⊢ j * i < i

Tactic:
← coe_lt_one,

Goal state after:
i j : ↑I
h : i ≠ 0 ∧ ↑j < 1
⊢ j * i < i
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: a b : ℝ
x y z : ↑(Icc a b)
hxy : x ≤ y
hyz : y ≤ z
⊢ 0 ≤ (↑y - ↑x) / (↑z - ↑x)

Tactic:
by_cases h : (z - x : ℝ) = 0

Goal state after:
case pos
a b : ℝ
x y z : ↑(Icc a b)
hxy : x ≤ y
hyz : y ≤ z
h : ↑z - ↑x = 0
⊢ 0 ≤ (↑y - ↑x) / (↑z - ↑x)

case neg
a b : ℝ
x y z : ↑(Icc a b)
hxy : x ≤ y
hyz : y ≤ z
h : ¬↑z - ↑x = 0
⊢ 0 ≤ (↑y - ↑x) / (↑z - ↑x)
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: a b : ℝ
x y z : ↑(Icc a b)
hxy : x ≤ y
hyz : y ≤ z
⊢ (↑y - ↑x) / (↑z - ↑x) ≤ 1

Tactic:
by_cases h : (z - x : ℝ) = 0

Goal state after:
case pos
a b : ℝ
x y z : ↑(Icc a b)
hxy : x ≤ y
hyz : y ≤ z
h : ↑z - ↑x = 0
⊢ (↑y - ↑x) / (↑z - ↑x) ≤ 1

case neg
a b : ℝ
x y z : ↑(Icc a b)
hxy : x ≤ y
hyz : y ≤ z
h : ¬↑z - ↑x = 0
⊢ (↑y - ↑x) / (↑z - ↑x) ≤ 1
Module: Mathlib.Topology.Metrizable.CompletelyMetrizable

Goal state before:
pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun x y ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
u : ℕ → X
hu : CauchySeq u
⊢ ∃ a, Tendsto u atTop (𝓝 a)

Tactic:
Metric.cauchySeq_iff'

Goal state after:
case refine_2
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun x y ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
u : ℕ → X
hu : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε
⊢ ∃ a, Tendsto u atTop (𝓝 a)
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: a b : ℝ
x y z : ↑(Icc a b)
hxy : x ≤ y
hyz : y ≤ z
⊢ ↑y = (1 - (↑y - ↑x) / (↑z - ↑x)) * ↑x + (↑y - ↑x) / (↑z - ↑x) * ↑z

Tactic:
by_cases h : (z - x : ℝ) = 0

Goal state after:
case pos
a b : ℝ
x y z : ↑(Icc a b)
hxy : x ≤ y
hyz : y ≤ z
h : ↑z - ↑x = 0
⊢ ↑y = (1 - (↑y - ↑x) / (↑z - ↑x)) * ↑x + (↑y - ↑x) / (↑z - ↑x) * ↑z

case neg
a b : ℝ
x y z : ↑(Icc a b)
hxy : x ≤ y
hyz : y ≤ z
h : ¬↑z - ↑x = 0
⊢ ↑y = (1 - (↑y - ↑x) / (↑z - ↑x)) * ↑x + (↑y - ↑x) / (↑z - ↑x) * ↑z
Module: Mathlib.Topology.Metrizable.CompletelyMetrizable

Goal state before:
pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun x y ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
⊢ CompleteSpace X

Tactic:
refine Metric.complete_of_cauchySeq_tendsto fun u hu ↦ ?_
    rw [Metric.cauchySeq_iff'] at hu
    obtain ⟨N, hN⟩ := hu 1 (by simp)
    refine ⟨u N, @tendsto_atTop_of_eventually_const X UniformSpace.toTopologicalSpace (u N) _ _ _ N
      fun n hn ↦ ?_⟩
    specialize hN n hn
    by_contra
    change (if u n = u N then 0 else 1) < 1 at hN
    simp_all

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

Goal state before:
pp: case inr
E : ℕ → Type u_1
x y : (n : ℕ) → E n
h : x ≠ y
⊢ dist x y ≤ 1

Tactic:
· simp only [dist, ne_eq, h, not_false_eq_true, ↓reduceIte, one_div, inv_pow]
    bound

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

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝³ : Nonempty Z
inst✝² : MetricSpace Z
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
ε : ℝ
hΦ : Isometry Φ
hΨ : Isometry Ψ
p q : Z
⊢ |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * 0

Tactic:
by rw [hΦ.dist_eq, hΨ.dist_eq]; simp

Goal state after:
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)
x y : (n : ℕ) → E n
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
H : longestPrefix x s < firstDiff x y
xs : x ∉ s
ys : y ∉ s
l_eq : longestPrefix y s = longestPrefix x s
⊢ x ∈ cylinder y (longestPrefix x s)

Tactic:
exact cylinder_anti y H.le (mem_cylinder_firstDiff x y)

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ≥0∞
⊢ ∫⁻ (x : ↑s), f ↑x ∂Measure.comap Subtype.val μ = ∫⁻ (x : α) in s, f x ∂μ

Tactic:
rw [← (MeasurableEmbedding.subtype_coe hs).lintegral_map, map_comap_subtype_coe hs]

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

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
c : β
s : Set β
hs : MeasurableSet s
hsc : c ∉ s
⊢ μ ∅ = μ univ * 0

Tactic:
measure_empty,

Goal state after:
case neg
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
c : β
s : Set β
hs : MeasurableSet s
hsc : c ∉ s
⊢ 0 = μ univ * 0
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : Countable α
μ1 μ2 : Measure α
inst✝¹ : SigmaFinite μ1
inst✝ : SigmaFinite μ2
⊢ μ1 = μ2 ↔ ∀ (x : α), μ1.real {x} = μ2.real {x}

Tactic:
rw [Measure.ext_iff_singleton]
  congr! with x
  rw [measureReal_def, measureReal_def, ENNReal.toReal_eq_toReal_iff]
  simp [measure_singleton_lt_top, ne_of_lt]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : Disjoint (ae μ) (ae ν)
⊢ Disjoint μ ν

Tactic:
by
  simp_rw [Filter.disjoint_iff, mem_ae_iff] at h
  obtain ⟨s, hs, t, ht, hst⟩ := h
  rw [disjoint_iff_inf_le]
  have : (⊥ : Measure α) = 0 := rfl
  refine Measure.le_intro fun u hu _ ↦ ?_
  simp only [Measure.inf_apply hu, this, coe_zero, Pi.zero_apply, nonpos_iff_eq_zero]
  refine csInf_eq_bot_of_bot_mem ⟨t, ?_⟩
  simp [measure_mono_null (inter_subset_left.trans hst.subset_compl_left) hs,
    measure_mono_null inter_subset_left ht]

Goal state after:
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
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = Ax.some
Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty
fy : f y = Ay.some
H : firstDiff x y ≤ longestPrefix x s ∧ firstDiff x y ≤ longestPrefix y s
⊢ Ax.some ∈ cylinder x (firstDiff x y)

Tactic:
exact cylinder_anti x H.1 Ax.some_mem.2

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

Goal state before:
pp: case neg
α : Type u_1
inst✝¹ : MeasurableSpace α
s : Set α
a : α
hs : MeasurableSet s
inst✝ : Decidable (a ∈ s)
has : a ∉ s
⊢ (dirac a) s = 0

Tactic:
dirac_apply' _ hs,

Goal state after:
case neg
α : Type u_1
inst✝¹ : MeasurableSpace α
s : Set α
a : α
hs : MeasurableSet s
inst✝ : Decidable (a ∈ s)
has : a ∉ s
⊢ s.indicator 1 a = 0
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: case mpr
α : Type u_1
inst✝ : MeasurableSpace α
x y : α
⊢ (∀ (A : Set α), MeasurableSet A → (x ∈ A ↔ y ∈ A)) → dirac x = dirac y

Tactic:
· intro h
    ext A A_mble
    by_cases x_in_A : x ∈ A
    · simp only [Measure.dirac_apply' _ A_mble, x_in_A, indicator_of_mem, Pi.one_apply,
                 (h A A_mble).mp x_in_A]
    · have y_notin_A : y ∉ A := by simp_all only [not_false_eq_true]
      simp only [Measure.dirac_apply' _ A_mble, x_in_A, y_notin_A,
                 not_false_eq_true, indicator_of_notMem]

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

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : Countable α
inst✝ : MeasurableSingletonClass α
f : α → ℝ≥0∞
| ∫⁻ (a : α), f a ∂μ

Tactic:
← sum_smul_dirac μ,

Goal state after:
α : Type u_1
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : Countable α
inst✝ : MeasurableSingletonClass α
f : α → ℝ≥0∞
| ∫⁻ (a : α), f a ∂sum fun a ↦ μ {a} • dirac a
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case succ
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst✝³] f
hg : Continuous[_, inst✝³] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x ↦ 0, fun x ↦ 0), ⋯⟩
p : ℕ → A := fun n ↦ F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
m n : ℕ
hmn : m + 1 ≤ n
IH : (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
⊢ (↑(p (n + 1))).2.2 m = (↑(p (m + 1))).2.2 m

Tactic:
have I : (F (p n)).val.snd.snd m = (p n).val.snd.snd m := by
      apply hFy (p n) m
      rw [pn_fst]
      exact hmn

Goal state after:
case succ
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst✝³] f
hg : Continuous[_, inst✝³] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I✝ :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x ↦ 0, fun x ↦ 0), ⋯⟩
p : ℕ → A := fun n ↦ F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
m n : ℕ
hmn : m + 1 ≤ n
IH : (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
I : (↑(F (p n))).2.2 m = (↑(p n)).2.2 m
⊢ (↑(p (n + 1))).2.2 m = (↑(p (m + 1))).2.2 m
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Countable

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : Set α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ∞ → f s ≤ C
h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)
S : ℕ → Set α := spanningSets (μ.trim hm)
hS_mono : Monotone S
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
⊢ f univ ≤ C

Tactic:
rw [← @iUnion_spanningSets _ m (μ.trim hm)]

Goal state after:
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : Set α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ∞ → f s ≤ C
h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)
S : ℕ → Set α := spanningSets (μ.trim hm)
hS_mono : Monotone S
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
⊢ f (⋃ i, spanningSets (μ.trim hm) i) ≤ C
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
m : ∀ {α : Type ?u.42} {β : Type ?u.41} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)
f g : α × β → ℝ≥0∞
hf : Measurable f
h2f : Measurable fun x ↦ ∫⁻ (y : β), f (x, y) ∂ν
h2g : Measurable fun x ↦ ∫⁻ (y : β), g (x, y) ∂ν
⊢ Measurable fun x ↦ ∫⁻ (y : β), (f + g) (x, y) ∂ν

Tactic:
simp only [Pi.add_apply]

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
m : ∀ {α : Type ?u.42} {β : Type ?u.41} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)
f g : α × β → ℝ≥0∞
hf : Measurable f
h2f : Measurable fun x ↦ ∫⁻ (y : β), f (x, y) ∂ν
h2g : Measurable fun x ↦ ∫⁻ (y : β), g (x, y) ∂ν
⊢ Measurable fun x ↦ ∫⁻ (y : β), f (x, y) + g (x, y) ∂ν
Module: Mathlib.MeasureTheory.Group.Convolution

Goal state before:
pp: M : Type u_2
M' : Type u_3
mM : MeasurableSpace M
inst✝⁵ : Monoid M
inst✝⁴ : MeasurableMul₂ M
mM' : MeasurableSpace M'
inst✝³ : Monoid M'
inst✝² : MeasurableMul₂ M'
μ ν : Measure M
inst✝¹ : SFinite μ
inst✝ : SFinite ν
L : M →* M'
hL : Measurable ⇑L
this : (⇑L ∘ fun p ↦ p.1 * p.2) = (fun p ↦ p.1 * p.2) ∘ Prod.map ⇑L ⇑L
⊢ map (⇑L ∘ fun x ↦ x.1 * x.2) (μ.prod ν) = map (fun x ↦ x.1 * x.2) ((map (⇑L) μ).prod (map (⇑L) ν))

Tactic:
this,

Goal state after:
M : Type u_2
M' : Type u_3
mM : MeasurableSpace M
inst✝⁵ : Monoid M
inst✝⁴ : MeasurableMul₂ M
mM' : MeasurableSpace M'
inst✝³ : Monoid M'
inst✝² : MeasurableMul₂ M'
μ ν : Measure M
inst✝¹ : SFinite μ
inst✝ : SFinite ν
L : M →* M'
hL : Measurable ⇑L
this : (⇑L ∘ fun p ↦ p.1 * p.2) = (fun p ↦ p.1 * p.2) ∘ Prod.map ⇑L ⇑L
⊢ map ((fun p ↦ p.1 * p.2) ∘ Prod.map ⇑L ⇑L) (μ.prod ν) = map (fun x ↦ x.1 * x.2) ((map (⇑L) μ).prod (map (⇑L) ν))
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: case inl
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p q : Set α → Prop
U : Set α
ε : ℝ≥0∞
H : μ.InnerRegularWRT p q
h0 : p ∅
hU : q U
hμU : μ U ≠ ∞
hε : ε ≠ 0
h₀ : μ U = 0
⊢ μ U < μ ∅ + ε

Tactic:
measure_empty,

Goal state after:
case inl
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p q : Set α → Prop
U : Set α
ε : ℝ≥0∞
H : μ.InnerRegularWRT p q
h0 : p ∅
hU : q U
hμU : μ U ≠ ∞
hε : ε ≠ 0
h₀ : μ U = 0
⊢ μ U < 0 + ε
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
pa qa : Set α → Prop
H : μ.InnerRegularWRT pa qa
f : α ≃ᵐ β
pb qb : Set β → Prop
hAB : ∀ (U : Set β), qb U → qa (⇑f ⁻¹' U)
hAB' : ∀ (K : Set α), pa K → pb (⇑f '' K)
⊢ (map (⇑f) μ).InnerRegularWRT pb qb

Tactic:
by
  intro U hU r hr
  rw [f.map_apply U] at hr
  rcases H (hAB U hU) r hr with ⟨K, hKU, hKc, hK⟩
  refine ⟨f '' K, image_subset_iff.2 hKU, hAB' _ hKc, ?_⟩
  rwa [f.map_apply, f.preimage_image]

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous[inst✝⁵, inst✝³] f
f_inj : Injective f
this✝ : UpgradedIsCompletelyMetrizableSpace γ := upgradeIsCompletelyMetrizable γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s ↦ closure[inst✝³] (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n ↦ ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
x : β
hx : x ∈ ⋂ n, F n
n : ℕ
⊢ ∃ s, ∃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), x ∈ E s

Tactic:
simpa only [F, mem_iUnion] using mem_iInter.1 hx n

Goal state after:
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case snd
G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g : G
k : ↥K
this : (hSK.equiv (g * ↑k)).1 = (hSK.equiv g).1
⊢ (↑(hSK.equiv (g * ↑k)).1)⁻¹ * (g * ↑k) = ↑(hSK.equiv g).2 * ↑k

Tactic:
this,

Goal state after:
case snd
G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g : G
k : ↥K
this : (hSK.equiv (g * ↑k)).1 = (hSK.equiv g).1
⊢ (↑(hSK.equiv g).1)⁻¹ * (g * ↑k) = ↑(hSK.equiv g).2 * ↑k
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case refine_2.fst
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
h : ↥H
g : G
this : (hHT.equiv (↑h * g)).2 = (hHT.equiv g).2
⊢ ↑h * g * (↑(hHT.equiv (↑h * g)).2)⁻¹ = ↑h * ↑(hHT.equiv g).1

Tactic:
this,

Goal state after:
case refine_2.fst
G : Type u_1
inst✝ : Group G
H : Subgroup G
T : Set G
hHT : IsComplement (↑H) T
h : ↥H
g : G
this : (hHT.equiv (↑h * g)).2 = (hHT.equiv g).2
⊢ ↑h * g * (↑(hHT.equiv g).2)⁻¹ = ↑h * ↑(hHT.equiv g).1
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
S T : Set G
hST : IsComplement S T
hs1 : 1 ∈ S
ht1 : 1 ∈ T
⊢ hST.equiv 1 = (⟨1, hs1⟩, ⟨1, ht1⟩)

Tactic:
by
  rw [← Equiv.eq_symm_apply]; simp [equiv]

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous[inst✝⁵, inst✝³] f
f_inj : Injective f
this✝¹ : UpgradedIsCompletelyMetrizableSpace γ := upgradeIsCompletelyMetrizable γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this✝ : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s ↦ closure[inst✝³] (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n ↦ ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
x : β
hx : x ∈ ⋂ n, F n
s : ℕ → ↑b
hs : ∀ (n : ℕ), Bornology.IsBounded ↑(s n) ∧ diam ↑(s n) ≤ u n
hxs : ∀ (n : ℕ), x ∈ E (s n)
y : ℕ → γ
hy : ∀ (n : ℕ), y n ∈ ↑(s n)
I : ∀ (m n : ℕ), (↑(s m) ∩ ↑(s n)).Nonempty
this : Tendsto (fun n ↦ 2 * u n) atTop (𝓝 0)
m n : ℕ
hmn : m ≤ n
z : γ
zsm : z ∈ ↑(s m)
zsn : z ∈ ↑(s n)
⊢ u m + u n ≤ 2 * u m

Tactic:
by linarith [u_anti.antitone hmn]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
inst✝ : SFinite μ
s : Set α
this : ν = ν.restrict univ
⊢ (μ.restrict s).prod ν = (μ.prod ν).restrict (s ×ˢ univ)

Tactic:
this,

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
inst✝ : SFinite μ
s : Set α
this : ν = ν.restrict univ
⊢ (μ.restrict s).prod (ν.restrict univ) = (μ.prod (ν.restrict univ)).restrict (s ×ˢ univ)
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
x : α
i : ℕ
s : Set α
t : Set β
hs : MeasurableSet s
ht : MeasurableSet t
⊢ s.indicator (fun x ↦ (sfiniteSeq ν i) t) x = (dirac x) s * (sfiniteSeq ν i) t

Tactic:
dirac_apply' _ hs,

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
x : α
i : ℕ
s : Set α
t : Set β
hs : MeasurableSet s
ht : MeasurableSet t
⊢ s.indicator (fun x ↦ (sfiniteSeq ν i) t) x = s.indicator 1 x * (sfiniteSeq ν i) t
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
H✝ : Type u_2
inst✝⁹ : MeasurableSpace G
inst✝⁸ : MeasurableSpace H✝
inst✝⁷ : Mul G
μ : Measure G
inst✝⁶ : MeasurableMul G
inst✝⁵ : μ.IsMulLeftInvariant
inst✝⁴ : SFinite μ
H : Type u_3
inst✝³ : Mul H
mH : MeasurableSpace H
ν : Measure H
inst✝² : MeasurableMul H
inst✝¹ : ν.IsMulLeftInvariant
inst✝ : SFinite ν
g : G
h : H
⊢ μ.prod (map (fun x ↦ h * x) ν) = μ.prod ν

Tactic:
map_mul_left_eq_self ν h

Goal state after:
G : Type u_1
H✝ : Type u_2
inst✝⁹ : MeasurableSpace G
inst✝⁸ : MeasurableSpace H✝
inst✝⁷ : Mul G
μ : Measure G
inst✝⁶ : MeasurableMul G
inst✝⁵ : μ.IsMulLeftInvariant
inst✝⁴ : SFinite μ
H : Type u_3
inst✝³ : Mul H
mH : MeasurableSpace H
ν : Measure H
inst✝² : MeasurableMul H
inst✝¹ : ν.IsMulLeftInvariant
inst✝ : SFinite ν
g : G
h : H
⊢ μ.prod ν = μ.prod ν
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulLeftInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
this : ⇑f '' (fun x ↦ g * x) ⁻¹' s = (fun x ↦ f g * x) ⁻¹' ⇑f '' s
⊢ μ (⇑f '' (fun x ↦ g * x) ⁻¹' s) = μ (⇑f '' s)

Tactic:
this,

Goal state after:
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulLeftInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
this : ⇑f '' (fun x ↦ g * x) ⁻¹' s = (fun x ↦ f g * x) ⁻¹' ⇑f '' s
⊢ μ ((fun x ↦ f g * x) ⁻¹' ⇑f '' s) = μ (⇑f '' s)
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulRightInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
this : ⇑f '' (fun x ↦ x * g) ⁻¹' s = (fun x ↦ x * f g) ⁻¹' ⇑f '' s
⊢ μ (⇑f '' (fun x ↦ x * g) ⁻¹' s) = μ (⇑f '' s)

Tactic:
this,

Goal state after:
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
H : Type u_3
inst✝² : Group H
mH : MeasurableSpace H
inst✝¹ : MeasurableMul H
μ : Measure H
inst✝ : μ.IsMulRightInvariant
f : G →* H
hf : MeasurableEmbedding ⇑f
g : G
s : Set G
hs : MeasurableSet s
this : ⇑f '' (fun x ↦ x * g) ⁻¹' s = (fun x ↦ x * f g) ⁻¹' ⇑f '' s
⊢ μ ((fun x ↦ x * f g) ⁻¹' ⇑f '' s) = μ (⇑f '' s)
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
X : α → β
Y : α → γ
μ : Measure α
hY : AEMeasurable Y μ
hX : ¬AEMeasurable X μ
this : ¬AEMeasurable (fun x ↦ (X x, Y x)) μ
⊢ (map (fun a ↦ (X a, Y a)) μ).fst = map X μ

Tactic:
simp [map_of_not_aemeasurable, hX, this]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in s, (f + g) a ∂μ = ∫⁻ (a : α) in s, f a + g a ∂μ

Tactic:
simp only [Pi.add_apply]

Goal state after: