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

Goal state before:
pp: case nil
α : Type u_1
inst✝¹ : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst✝ : CompleteSpace α
u : ℕ → ℝ≥0∞
upos' : ∀ (n : ℕ), u n ∈ Ioo 0 1
hu : Tendsto u atTop (nhds 0)
upos : ∀ (n : ℕ), 0 < u n
P : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }
C0 C1 : {C : Set α} → Perfect C → C.Nonempty → {ε : ℝ≥0∞} → 0 < ε → Set α
h0 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C0 hC hnonempty hε) ∧
      (C0 hC hnonempty hε).Nonempty ∧ C0 hC hnonempty hε ⊆ C ∧ ediam (C0 hC hnonempty hε) ≤ ε
h1 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C1 hC hnonempty hε) ∧
      (C1 hC hnonempty hε).Nonempty ∧ C1 hC hnonempty hε ⊆ C ∧ ediam (C1 hC hnonempty hε) ≤ ε
hdisj :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)
⊢ P

Tactic:
| nil =>

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
h : StandardBorelSpace α
⊢ UpgradedStandardBorel α

Tactic:
choose τ hb hp using h.polish

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
h : StandardBorelSpace α
τ : TopologicalSpace α
hb : BorelSpace α
hp : PolishSpace α
⊢ UpgradedStandardBorel α
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: α : Type u_1
inst✝² : TopologicalSpace α
β : Type u_3
inst✝¹ : TopologicalSpace β
inst✝ : PolishSpace β
f : β → α
f_cont : Continuous[inst✝¹, inst✝²] f
⊢ AnalyticSet (range f)

Tactic:
by
  cases isEmpty_or_nonempty β
  · rw [range_eq_empty]
    exact analyticSet_empty
  · rw [AnalyticSet]
    obtain ⟨g, g_cont, hg⟩ : ∃ g : (ℕ → ℕ) → β, Continuous g ∧ Surjective g :=
      exists_nat_nat_continuous_surjective β
    refine Or.inr ⟨f ∘ g, f_cont.comp g_cont, ?_⟩
    rw [hg.range_comp]

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

Goal state before:
pp: case mp.inl
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
h : s = ∅
⊢ ∃ β h, ∃ (_ : PolishSpace β), ∃ f, Continuous[_, inst✝] f ∧ range f = s

Tactic:
refine ⟨Empty, inferInstance, inferInstance, Empty.elim, continuous_bot, ?_⟩

Goal state after:
case mp.inl
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
h : s = ∅
⊢ range Empty.elim = s
Module: Mathlib.MeasureTheory.Measure.Count

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : Set α
s_fin : s.Finite
s_mble : MeasurableSet s
⊢ count s = ↑(#s_fin.toFinset)

Tactic:
simp [←
    @count_apply_finset' _ _ s_fin.toFinset (by simpa only [Finite.coe_toFinset] using s_mble)]

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

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : Countable α
inst✝ : MeasurableSingletonClass α
μ : Measure α
s : Set α
hs : MeasurableSet s
⊢ ∑' (x : α), s.indicator (fun x ↦ μ {x}) x = μ s

Tactic:
classical
  calc
    (∑' x : α, s.indicator (fun x => μ {x}) x) =
      Measure.sum (fun a => μ {a} • Measure.dirac a) s := by
      simp only [Measure.sum_apply _ hs, Measure.smul_apply, smul_eq_mul, Measure.dirac_apply,
        Set.indicator_apply, mul_ite, Pi.one_apply, mul_one, mul_zero]
    _ = μ s := by rw [μ.sum_smul_dirac]

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

Goal state before:
pp: α : Type u_4
mα : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
μ : Measure α
s : Finset α
hμ : ∀ᵐ (a : α) ∂μ, a ∈ s
t : Set α
ht : MeasurableSet t
⊢ μ (t ∩ ⋃ x ∈ s, {x}) = (∑ a ∈ s, μ {a} • dirac a) t

Tactic:
Set.inter_iUnion

Goal state after:
α : Type u_4
mα : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
μ : Measure α
s : Finset α
hμ : ∀ᵐ (a : α) ∂μ, a ∈ s
t : Set α
ht : MeasurableSet t
⊢ μ (⋃ i ∈ s, t ∩ {i}) = (∑ a ∈ s, μ {a} • dirac a) t
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
m n : ℕ
hmn : m + 1 ≤ n
⊢ (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m → (↑(p (n + 1))).2.1 m = (↑(p (m + 1))).2.1 m

Tactic:
IH

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
m n : ℕ
hmn : m + 1 ≤ n
IH : (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
⊢ (↑(p (n + 1))).2.1 m = (↑(p (m + 1))).2.1 m
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
m n : ℕ
hmn : m + 1 ≤ n
IH : (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
I : (↑(F (p n))).2.1 m = (↑(p n)).2.1 m
⊢ (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m

Tactic:
IH

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
m n : ℕ
hmn : m + 1 ≤ n
IH : (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
I : (↑(F (p n))).2.1 m = (↑(p n)).2.1 m
⊢ (↑(p (m + 1))).2.1 m = (↑(p (m + 1))).2.1 m
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
⊢ (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m → (↑(p (n + 1))).2.2 m = (↑(p (m + 1))).2.2 m

Tactic:
IH

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
⊢ (↑(p (n + 1))).2.2 m = (↑(p (m + 1))).2.2 m
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
I : (↑(F (p n))).2.2 m = (↑(p n)).2.2 m
⊢ (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m

Tactic:
IH

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 (m + 1))).2.2 m = (↑(p (m + 1))).2.2 m
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: X : Type u_3
inst✝¹ : MetricSpace X
inst✝ : SeparableSpace X
x y : X
hxy : x ≠ y
n : ℕ
hn : {y}ᶜ ∈ comap (distDenseSeq X n) (𝓝 (distDenseSeq X n x))
e : distDenseSeq X n x = distDenseSeq X n y
⊢ False

Tactic:
simp +contextual [e, ← exists_prop, mem_of_mem_nhds] at hn

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

Goal state before:
pp: M : Type u_1
inst✝³ : Monoid M
inst✝² : MeasurableSpace M
inst✝¹ : MeasurableMul₂ M
μ : Measure M
inst✝ : SFinite μ
x : M
⊢ map (fun x ↦ x.1 * x.2) (μ.prod (dirac x)) = map (fun y ↦ y * x) μ

Tactic:
prod_dirac,

Goal state after:
M : Type u_1
inst✝³ : Monoid M
inst✝² : MeasurableSpace M
inst✝¹ : MeasurableMul₂ M
μ : Measure M
inst✝ : SFinite μ
x : M
⊢ map (fun x ↦ x.1 * x.2) (map (fun x_1 ↦ (x_1, x)) μ) = map (fun y ↦ y * x) μ
Module: Mathlib.MeasureTheory.Group.Convolution

Goal state before:
pp: M : Type u_1
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
x y : M
⊢ map (fun y_1 ↦ y_1 * y) (dirac x) = dirac (x * y)

Tactic:
map_dirac' (by fun_prop)

Goal state after:
M : Type u_1
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
x y : M
⊢ dirac (x * y) = dirac (x * y)
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
S : Set G
⊢ IsComplement S ↑H ↔ ∀ (q : G ⧸ H), ∃! s, ↑↑s = q

Tactic:
by
  simp_rw [isComplement_iff_existsUnique_inv_mul_mem, SetLike.mem_coe, ← QuotientGroup.eq,
    QuotientGroup.forall_mk]

Goal state after:
Module: Mathlib.GroupTheory.Complement

Goal state before:
pp: case refine_1
G : Type u_1
inst✝³ : Group G
H K : Subgroup G
S T✝ : Set G
F : Type u_2
inst✝² : Group F
inst✝¹ : MulAction F G
inst✝ : QuotientAction F H
f : F
T : H.LeftTransversal
g : G
t : ↑↑T
ht1 : (↑t)⁻¹ * f⁻¹ • g ∈ ↑H
ht2 : ∀ (y : ↑↑T), (fun s ↦ (↑s)⁻¹ * f⁻¹ • g ∈ ↑H) y → y = t
⊢ (fun s ↦ (↑s)⁻¹ * g ∈ ↑H) ⟨f • ↑t, ⋯⟩

Tactic:
exact smul_inv_smul f g ▸ QuotientAction.inv_mul_mem f ht1

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

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst✝¹] (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ∞
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r
n : ℕ
H₁ : ∀ (t : Set α), (μ.restrict (s n)) t = μ (t ∩ s n)
Ht : (μ.restrict (s n)) (A n) ≠ ∞
U : Set α
hAU : U ⊇ A n
hUo : IsOpen[inst✝¹] U
hU : (μ.restrict (s n)) U < (μ.restrict (s n)) (A n) + δ n
⊢ ∃ U ⊇ A n, IsOpen[inst✝¹] U ∧ μ U < μ (A n) + δ n

Tactic:
H₁,

Goal state after:
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst✝¹] (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ∞
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r
n : ℕ
H₁ : ∀ (t : Set α), (μ.restrict (s n)) t = μ (t ∩ s n)
Ht : (μ.restrict (s n)) (A n) ≠ ∞
U : Set α
hAU : U ⊇ A n
hUo : IsOpen[inst✝¹] U
hU : μ (U ∩ s n) < (μ.restrict (s n)) (A n) + δ n
⊢ ∃ U ⊇ A n, IsOpen[inst✝¹] U ∧ μ U < μ (A n) + δ n
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst✝¹] (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ∞
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r
n : ℕ
H₁ : ∀ (t : Set α), (μ.restrict (s n)) t = μ (t ∩ s n)
Ht : (μ.restrict (s n)) (A n) ≠ ∞
U : Set α
hAU : U ⊇ A n
hUo : IsOpen[inst✝¹] U
hU : μ (U ∩ s n) < (μ.restrict (s n)) (A n) + δ n
⊢ ∃ U ⊇ A n, IsOpen[inst✝¹] U ∧ μ U < μ (A n) + δ n

Tactic:
H₁,

Goal state after:
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst✝¹] (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet (⋃ n, A n)
hr : r > μ (⋃ n, A n)
HA : μ (⋃ n, A n) < ∞
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r
n : ℕ
H₁ : ∀ (t : Set α), (μ.restrict (s n)) t = μ (t ∩ s n)
Ht : (μ.restrict (s n)) (A n) ≠ ∞
U : Set α
hAU : U ⊇ A n
hUo : IsOpen[inst✝¹] U
hU : μ (U ∩ s n) < μ (A n ∩ s n) + δ n
⊢ ∃ U ⊇ A n, IsOpen[inst✝¹] U ∧ μ U < μ (A n) + δ n
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
p : Set α → Prop
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < ⨆ n, μ (F ∩ s n)
hBU : ⋃ n, F ∩ s n = F
this : μ F = ⨆ n, μ (F ∩ s n)
n : ℕ
hn : r < μ (F ∩ s n)
⊢ ∃ K ⊆ F, p K ∧ r < μ K

Tactic:
rw [← restrict_apply hF] at hn

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
p : Set α → Prop
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < ⨆ n, μ (F ∩ s n)
hBU : ⋃ n, F ∩ s n = F
this : μ F = ⨆ n, μ (F ∩ s n)
n : ℕ
hn : r < (μ.restrict (s n)) F
⊢ ∃ K ⊆ F, p K ∧ r < μ K
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
x : α
y : β
⊢ (dirac x).prod (dirac y) = dirac (x, y)

Tactic:
prod_dirac,

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
x : α
y : β
⊢ map (fun x ↦ (x, y)) (dirac x) = dirac (x, y)
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝⁴ : MeasurableSpace G
inst✝³ : Group G
inst✝² : MeasurableMul G
H : Subgroup G
inst✝¹ : H.FiniteIndex
hH : MeasurableSet ↑H
μ : Measure G
inst✝ : μ.IsMulLeftInvariant
s : Set G
hs : Subgroup.IsComplement s ↑H
⊢ ↑H.index * μ ↑H = μ univ

Tactic:
have hs' : Finite s := hs.finite_left_iff.mpr inferInstance

Goal state after:
G : Type u_1
inst✝⁴ : MeasurableSpace G
inst✝³ : Group G
inst✝² : MeasurableMul G
H : Subgroup G
inst✝¹ : H.FiniteIndex
hH : MeasurableSet ↑H
μ : Measure G
inst✝ : μ.IsMulLeftInvariant
s : Set G
hs : Subgroup.IsComplement s ↑H
hs' : Finite ↑s
⊢ ↑H.index * μ ↑H = μ univ
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: X : Type u_6
E✝ : Type u_7
ι : Type u_8
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : CommMonoid E✝
inst✝⁸ : TopologicalSpace E✝
inst✝⁷ : IsCompletelyPseudoMetrizableSpace E✝
inst✝⁶ : SecondCountableTopology E✝
inst✝⁵ : MeasurableSpace E✝
inst✝⁴ : BorelSpace E✝
inst✝³ : MeasurableMul₂ E✝
inst✝² : Countable ι
L : SummationFilter ι
inst✝¹ : L.NeBot
inst✝ : L.filter.IsCountablyGenerated
f : ι → X → E✝
h : ∀ (i : ι), Measurable (f i)
E : Set X := {x | Multipliable (fun x_1 ↦ f x_1 x) L}
hE : MeasurableSet E
h0 : (Eᶜ.domRestrict fun x ↦ ∏'[L] (i : ι), f i x) = fun x ↦ 1
⊢ Measurable fun x ↦ ∏'[L] (i : ι), f i x

Tactic:
refine measurable_of_restrict_of_restrict_compl hE ?_ (h0 ▸ measurable_const)

Goal state after:
X : Type u_6
E✝ : Type u_7
ι : Type u_8
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : CommMonoid E✝
inst✝⁸ : TopologicalSpace E✝
inst✝⁷ : IsCompletelyPseudoMetrizableSpace E✝
inst✝⁶ : SecondCountableTopology E✝
inst✝⁵ : MeasurableSpace E✝
inst✝⁴ : BorelSpace E✝
inst✝³ : MeasurableMul₂ E✝
inst✝² : Countable ι
L : SummationFilter ι
inst✝¹ : L.NeBot
inst✝ : L.filter.IsCountablyGenerated
f : ι → X → E✝
h : ∀ (i : ι), Measurable (f i)
E : Set X := {x | Multipliable (fun x_1 ↦ f x_1 x) L}
hE : MeasurableSet E
h0 : (Eᶜ.domRestrict fun x ↦ ∏'[L] (i : ι), f i x) = fun x ↦ 1
⊢ Measurable (E.domRestrict fun x ↦ ∏'[L] (i : ι), f i x)
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
⊢ snd 0 = 0

Tactic:
by simp [snd]

Goal state after:
Module: Mathlib.Analysis.LConvolution

Goal state before:
pp: G : Type u_1
mG : MeasurableSpace G
inst✝⁴ : CommGroup G
inst✝³ : MeasurableMul₂ G
inst✝² : MeasurableInv G
μ : Measure G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : μ.IsInvInvariant
f g : G → ℝ≥0∞
x : G
⊢ ∫⁻ (y : G), f y * g (y⁻¹ * x) ∂μ = ∫⁻ (y : G), g y * f (y⁻¹ * x) ∂μ

Tactic:
rw [← lintegral_mul_left_eq_self _ x, ← lintegral_inv_eq_self]

Goal state after:
G : Type u_1
mG : MeasurableSpace G
inst✝⁴ : CommGroup G
inst✝³ : MeasurableMul₂ G
inst✝² : MeasurableInv G
μ : Measure G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : μ.IsInvInvariant
f g : G → ℝ≥0∞
x : G
⊢ ∫⁻ (x_1 : G), f (x * x_1⁻¹) * g ((x * x_1⁻¹)⁻¹ * x) ∂μ = ∫⁻ (y : G), g y * f (y⁻¹ * x) ∂μ
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
μ✝ : Measure α
inst✝² : TopologicalSpace α
μ ν : Measure α
inst✝¹ : μ.InnerRegular
inst✝ : ν.InnerRegular
⊢ (μ + ν).InnerRegular

Tactic:
by
  constructor
  intro s hs r hr
  simp only [Measure.coe_add, Pi.add_apply] at hr
  rcases eq_or_ne (μ s) 0 with h | h
  · simp only [h, zero_add] at hr
    rcases MeasurableSet.exists_lt_isCompact hs hr with ⟨K, Ks, hK, h'K⟩
    exact ⟨K, Ks, hK, h'K.trans_le (by simp)⟩
  rcases eq_or_ne (ν s) 0 with h' | h'
  · simp only [h', add_zero] at hr
    rcases MeasurableSet.exists_lt_isCompact hs hr with ⟨K, Ks, hK, h'K⟩
    exact ⟨K, Ks, hK, h'K.trans_le (by simp)⟩
  rcases ENNReal.exists_lt_add_of_lt_add hr h h' with ⟨u, hu, v, hv, huv⟩
  rcases MeasurableSet.exists_lt_isCompact hs hu with ⟨K, Ks, hK, h'K⟩
  rcases MeasurableSet.exists_lt_isCompact hs hv with ⟨K', K's, hK', h'K'⟩
  refine ⟨K ∪ K', union_subset Ks K's, hK.union hK', huv.trans_le ?_⟩
  apply (add_le_add h'K.le h'K'.le).trans
  simp only [Measure.coe_add, Pi.add_apply]
  gcongr <;> simp

Goal state after: