mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Field.Lemmas

Goal state before:
pp: case mpr
K : Type u_4
inst✝ : NormedField K
h : IsComplete (closedBall 0 1)
⊢ CompleteSpace K

Tactic:
rcases NormedField.discreteTopology_or_nontriviallyNormedField K with _ | ⟨_, rfl⟩

Goal state after:
case mpr.inl
K : Type u_4
inst✝ : NormedField K
h : IsComplete (closedBall 0 1)
h✝ : DiscreteTopology K
⊢ CompleteSpace K

case mpr.inr
K : Type u_4
val✝ : NontriviallyNormedField K
h : IsComplete (closedBall 0 1)
⊢ CompleteSpace K
Module: Mathlib.MeasureTheory.MeasurableSpace.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
b : β
s : Set β
x✝ : MeasurableSet s
⊢ MeasurableSet s

Tactic:
rw [map_def]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
b : β
s : Set β
x✝ : MeasurableSet s
⊢ MeasurableSet ((fun _a ↦ b) ⁻¹' s)
Module: Mathlib.MeasureTheory.MeasurableSpace.Constructions

Goal state before:
pp: δ : Type u_4
X : δ → Type u_6
inst✝¹ : (a : δ) → MeasurableSpace (X a)
a : δ
inst✝ : DecidableEq δ
⊢ Measurable fun p ↦ update p.1 a p.2

Tactic:
measurable_pi_iff

Goal state after:
δ : Type u_4
X : δ → Type u_6
inst✝¹ : (a : δ) → MeasurableSpace (X a)
a : δ
inst✝ : DecidableEq δ
⊢ ∀ (a_1 : δ), Measurable fun x ↦ update x.1 a x.2 a_1
Module: Mathlib.Topology.Algebra.InfiniteSum.Real

Goal state before:
pp: α : Type u_1
inst✝ : PseudoMetricSpace α
f : ℕ → α
d : ℕ → ℝ
hf : ∀ (n : ℕ), dist (f n) (f n.succ) ≤ d n
hd : Summable d
⊢ CauchySeq f

Tactic:
by
  lift d to ℕ → ℝ≥0 using fun n ↦ dist_nonneg.trans (hf n)
  apply cauchySeq_of_edist_le_of_summable d (α := α) (f := f)
  · exact_mod_cast hf
  · exact_mod_cast hd

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.Constructions

Goal state before:
pp: δ : Type u_4
δ' : Type u_5
X : δ → Type u_6
inst✝ : (a : δ) → MeasurableSpace (X a)
f : δ' ≃ δ
⊢ Measurable ⇑(Equiv.piCongrLeft X f)

Tactic:
measurable_pi_iff

Goal state after:
δ : Type u_4
δ' : Type u_5
X : δ → Type u_6
inst✝ : (a : δ) → MeasurableSpace (X a)
f : δ' ≃ δ
⊢ ∀ (a : δ), Measurable fun x ↦ (Equiv.piCongrLeft X f) x a
Module: Mathlib.Data.Sign.Basic

Goal state before:
pp: case neg
α : Type u
inst✝² : AddCommGroup α
inst✝¹ : LinearOrder α
inst✝ : IsOrderedAddMonoid α
ι : Type u_1
s : Finset ι
f : ι → α
hs : s.Nonempty
h : ∀ i ∈ s, sign (f i) = -1
⊢ sign (∑ i ∈ s, f i) = -1

Tactic:
sign_eq_neg_one_iff

Goal state after:
case neg
α : Type u
inst✝² : AddCommGroup α
inst✝¹ : LinearOrder α
inst✝ : IsOrderedAddMonoid α
ι : Type u_1
s : Finset ι
f : ι → α
hs : s.Nonempty
h : ∀ i ∈ s, f i < 0
⊢ ∑ i ∈ s, f i < 0
Module: Mathlib.Data.Sign.Defs

Goal state before:
pp: α : Type u_1
inst✝¹ : Zero α
inst✝ : LinearOrder α
a : α
⊢ sign a ≤ 0 ↔ a ≤ 0

Tactic:
rcases lt_trichotomy 0 a with (h | h | h)
  · simp [h, h.not_ge]
  · simp [← h]
  · simp [h, h.le]

Goal state after:
Module: Mathlib.Data.EReal.Inv

Goal state before:
pp: a : EReal
a_0 : a ∈ Ioi 0
⊢ ∀ ⦃b : EReal⦄, b ∈ Ioi 0 → a < b → (fun x ↦ x⁻¹) b < (fun x ↦ x⁻¹) a

Tactic:
b

Goal state after:
a : EReal
a_0 : a ∈ Ioi 0
b : EReal
⊢ b ∈ Ioi 0 → a < b → (fun x ↦ x⁻¹) b < (fun x ↦ x⁻¹) a
Module: Mathlib.Topology.Semicontinuity.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → Set β
h : HasOpenCGraph f
x : α
⊢ SemicontinuousAt (fun x b ↦ b ∈ f x) x

Tactic:
b

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → Set β
h : HasOpenCGraph f
x : α
b : β
⊢ (fun x b ↦ b ∈ f x) x b → ∀ᶠ (x' : α) in 𝓝 x, (fun x b ↦ b ∈ f x) x' b
Module: Mathlib.Topology.Instances.EReal.Lemmas

Goal state before:
pp: ⊢ (𝓝 ⊤).HasBasis (fun x ↦ True) fun x ↦ Ioi ↑x

Tactic:
refine (nhds_top_basis (α := EReal)).to_hasBasis (fun x hx => ?_)
    fun _ _ ↦ ⟨_, coe_lt_top _, Subset.rfl⟩

Goal state after:
x : EReal
hx : x < ⊤
⊢ ∃ i', True ∧ Ioi ↑i' ⊆ Ioi x
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
s : Set α
ι : Type u_4
f : ι → α → β
inst✝ : Preorder β
I : Set ι
hfy : ∀ i ∈ I, LowerSemicontinuousOn (f i) s
M : α → ι
M_mem : ∀ x ∈ s, M x ∈ I
M_max : ∀ x ∈ s, IsMaxOn (fun y ↦ f y x) I (M x)
x : α
hx : x ∈ s
⊢ SemicontinuousWithinAt (fun x1 x2 ↦ (fun x ↦ f (M x) x) x1 > x2) s x

Tactic:
b

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
s : Set α
ι : Type u_4
f : ι → α → β
inst✝ : Preorder β
I : Set ι
hfy : ∀ i ∈ I, LowerSemicontinuousOn (f i) s
M : α → ι
M_mem : ∀ x ∈ s, M x ∈ I
M_max : ∀ x ∈ s, IsMaxOn (fun y ↦ f y x) I (M x)
x : α
hx : x ∈ s
b : β
⊢ (fun x1 x2 ↦ (fun x ↦ f (M x) x) x1 > x2) x b → ∀ᶠ (x' : α) in 𝓝[s] x, (fun x1 x2 ↦ (fun x ↦ f (M x) x) x1 > x2) x' b
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
γ : Type u_4
inst✝² : LinearOrder γ
inst✝¹ : TopologicalSpace γ
inst✝ : ClosedIciTopology γ
f : α → γ
s : Set α
hs : IsClosed[inst✝³] s
⊢ LowerSemicontinuousOn f s ↔ IsClosed[instTopologicalSpaceProd] {p | p.1 ∈ s ∧ f p.1 ≤ p.2}

Tactic:
simp_rw [lowerSemicontinuousOn_iff, lowerSemicontinuousWithinAt_iff,
    eventually_nhdsWithin_iff, ← isOpen_compl_iff, compl_ofPred, isOpen_iff_eventually, mem_ofPred,
    not_and, not_le]
  constructor
  · intro hf ⟨x, y⟩ h
    by_cases hx : x ∈ s
    · have ⟨y', hy', z, hz, h₁⟩ := (h hx).exists_disjoint_Iio_Ioi
      filter_upwards [(hf x hx z hz).prodMk_nhds (eventually_lt_nhds hy')]
        with _ ⟨h₂, h₃⟩ h₄ using h₁ _ h₃ _ <| h₂ h₄
    · filter_upwards [(continuous_fst.tendsto _).eventually (hs.isOpen_compl.eventually_mem hx)]
        with _ h₁ h₂ using (h₁ h₂).elim
  · intro hf x _ y hy
    exact ((Continuous.prodMk_left y).tendsto x).eventually (hf (x, y) (fun _ => hy))

Goal state after:
Module: Mathlib.Topology.Instances.EReal.Lemmas

Goal state before:
pp: a : ℝ
⊢ ContinuousAt (fun p ↦ p.1 + p.2) (⊤, ↑a)

Tactic:
simp only [ContinuousAt, tendsto_nhds_top_iff_real, top_add_coe]
  refine fun r ↦ ((lt_mem_nhds (coe_lt_top (r - (a - 1)))).prod_nhds
    (lt_mem_nhds <| EReal.coe_lt_coe_iff.2 <| sub_one_lt _)).mono fun _ h ↦ ?_
  simpa only [← coe_add, _root_.sub_add_cancel] using add_lt_add h.1 h.2

Goal state after:
Module: Mathlib.Topology.Instances.EReal.Lemmas

Goal state before:
pp: ⊢ ContinuousAt (fun p ↦ p.1 + p.2) (⊥, ⊥)

Tactic:
by
  simp only [ContinuousAt, tendsto_nhds_bot_iff_real, bot_add]
  refine fun r ↦ ((gt_mem_nhds (bot_lt_coe 0)).prod_nhds
    (gt_mem_nhds <| bot_lt_coe r)).mono fun _ h ↦ ?_
  simpa only [coe_zero, zero_add] using add_lt_add h.1 h.2

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.OfFunction

Goal state before:
pp: α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
s : Set α
f : ℕ → Set α := fun i ↦ Nat.casesOn i s fun x ↦ ∅
⊢ ∀ (b' : ℕ), b' ≠ 0 → m (f b') = 0

Tactic:
rintro (_ | i)

Goal state after:
case zero
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
s : Set α
f : ℕ → Set α := fun i ↦ Nat.casesOn i s fun x ↦ ∅
⊢ 0 ≠ 0 → m (f 0) = 0

case succ
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
s : Set α
f : ℕ → Set α := fun i ↦ Nat.casesOn i s fun x ↦ ∅
i : ℕ
⊢ i + 1 ≠ 0 → m (f (i + 1)) = 0
Module: Mathlib.Analysis.SpecificLimits.Basic

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
r C : ℝ≥0∞
hr : r < 1
hC : C ≠ ∞
f : ℕ → α
hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * r ^ n
⊢ C * ∑' (i : ℕ), r ^ i ≠ ∞

Tactic:
ENNReal.tsum_geometric

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
r C : ℝ≥0∞
hr : r < 1
hC : C ≠ ∞
f : ℕ → α
hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * r ^ n
⊢ C * (1 - r)⁻¹ ≠ ∞
Module: Mathlib.Analysis.SpecificLimits.Basic

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
ι✝ : Type u_3
ε : ℝ
hε : 0 < ε
ι : Type ?u.11
inst✝ : Encodable ι
f : ℕ → ℝ := fun n ↦ ε / 2 / 2 ^ n
hf : HasSum f ε
f0 : ∀ (n : ℕ), 0 < f n
c : ℝ
hg : HasSum (f ∘ Encodable.encode) c
⊢ ∀ (i : ι), (f ∘ Encodable.encode) i ≤ f (Encodable.encode i)

Tactic:
intro n
    exact le_rfl

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
s : Set α
x : α
γ : Type u_4
inst✝² : LinearOrder γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f : α → γ
⊢ ContinuousWithinAt f s x ↔ LowerSemicontinuousWithinAt f s x ∧ UpperSemicontinuousWithinAt f s x

Tactic:
by
  refine ⟨fun h => ⟨h.lowerSemicontinuousWithinAt, h.upperSemicontinuousWithinAt⟩, ?_⟩
  rintro ⟨h₁, h₂⟩
  intro v hv
  simp only [Filter.mem_map]
  by_cases! Hl : ∃ l, l < f x
  · rcases exists_Ioc_subset_of_mem_nhds hv Hl with ⟨l, lfx, hl⟩
    by_cases! Hu : ∃ u, f x < u
    · rcases exists_Ico_subset_of_mem_nhds hv Hu with ⟨u, fxu, hu⟩
      filter_upwards [h₁ l lfx, h₂ u fxu] with a lfa fau
      rcases le_or_gt (f a) (f x) with h | h
      · exact hl ⟨lfa, h⟩
      · exact hu ⟨le_of_lt h, fau⟩
    · filter_upwards [h₁ l lfx] with a lfa using hl ⟨lfa, Hu (f a)⟩
  · by_cases! Hu : ∃ u, f x < u
    · rcases exists_Ico_subset_of_mem_nhds hv Hu with ⟨u, fxu, hu⟩
      filter_upwards [h₂ u fxu] with a lfa
      apply hu
      exact ⟨Hl (f a), lfa⟩
    · apply Filter.Eventually.of_forall
      intro a
      have : f a = f x := le_antisymm (Hu _) (Hl _)
      rw [this]
      exact mem_of_mem_nhds hv

Goal state after:
Module: Mathlib.Topology.Instances.EReal.Lemmas

Goal state before:
pp: case top.bot
h₁ : (⊤, ⊥).1 ≠ 0 ∨ (⊤, ⊥).2 ≠ ⊥
h₂ : (⊤, ⊥).1 ≠ 0 ∨ (⊤, ⊥).2 ≠ ⊤
h₃ : (⊤, ⊥).1 ≠ ⊥ ∨ (⊤, ⊥).2 ≠ 0
h₄ : (⊤, ⊥).1 ≠ ⊤ ∨ (⊤, ⊥).2 ≠ 0
⊢ ContinuousAt (fun p ↦ p.1 * p.2) (⊤, ⊥)

pp: case top.coe
a✝ : ℝ
h₁ : (⊤, ↑a✝).1 ≠ 0 ∨ (⊤, ↑a✝).2 ≠ ⊥
h₂ : (⊤, ↑a✝).1 ≠ 0 ∨ (⊤, ↑a✝).2 ≠ ⊤
h₃ : (⊤, ↑a✝).1 ≠ ⊥ ∨ (⊤, ↑a✝).2 ≠ 0
h₄ : (⊤, ↑a✝).1 ≠ ⊤ ∨ (⊤, ↑a✝).2 ≠ 0
⊢ ContinuousAt (fun p ↦ p.1 * p.2) (⊤, ↑a✝)

pp: case top.top
h₁ : (⊤, ⊤).1 ≠ 0 ∨ (⊤, ⊤).2 ≠ ⊥
h₂ : (⊤, ⊤).1 ≠ 0 ∨ (⊤, ⊤).2 ≠ ⊤
h₃ : (⊤, ⊤).1 ≠ ⊥ ∨ (⊤, ⊤).2 ≠ 0
h₄ : (⊤, ⊤).1 ≠ ⊤ ∨ (⊤, ⊤).2 ≠ 0
⊢ ContinuousAt (fun p ↦ p.1 * p.2) (⊤, ⊤)

Tactic:
· exact continuousAt_mul_symm2 continuousAt_mul_top_top

Goal state after:
case top.coe
a✝ : ℝ
h₁ : (⊤, ↑a✝).1 ≠ 0 ∨ (⊤, ↑a✝).2 ≠ ⊥
h₂ : (⊤, ↑a✝).1 ≠ 0 ∨ (⊤, ↑a✝).2 ≠ ⊤
h₃ : (⊤, ↑a✝).1 ≠ ⊥ ∨ (⊤, ↑a✝).2 ≠ 0
h₄ : (⊤, ↑a✝).1 ≠ ⊤ ∨ (⊤, ↑a✝).2 ≠ 0
⊢ ContinuousAt (fun p ↦ p.1 * p.2) (⊤, ↑a✝)

case top.top
h₁ : (⊤, ⊤).1 ≠ 0 ∨ (⊤, ⊤).2 ≠ ⊥
h₂ : (⊤, ⊤).1 ≠ 0 ∨ (⊤, ⊤).2 ≠ ⊤
h₃ : (⊤, ⊤).1 ≠ ⊥ ∨ (⊤, ⊤).2 ≠ 0
h₄ : (⊤, ⊤).1 ≠ ⊤ ∨ (⊤, ⊤).2 ≠ 0
⊢ ContinuousAt (fun p ↦ p.1 * p.2) (⊤, ⊤)
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: case h.left
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
⊢ a ∈ (⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b ↔
    a ∈ ⋂ b ∈ T_s ∪ T_t', if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅

Tactic:
simp_rw [Set.mem_inter_iff, Set.mem_iInter, Finset.mem_union]

Goal state after:
case h.left
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
⊢ ((∀ i ∈ T_s, a ∈ f_s i) ∧ ∀ i ∈ T_t', a ∈ f_t' i) ↔
    ∀ (i : β),
      i ∈ T_s ∨ i ∈ T_t' →
        a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: case refine_1.mp
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset ↑s
f : ↑s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
a : α
h1 : ∀ (a_1 : β) (b : a_1 ∈ s), ⟨a_1, b⟩ ∈ T → a ∈ f ⟨a_1, b⟩
⊢ ∀ (a_1 : β) (b : a_1 ∈ s), ⟨a_1, b⟩ ∈ T → a ∈ Function.extend (fun x ↦ ↑x) f (fun x ↦ ∅) ↑⟨a_1, b⟩

Tactic:
b

Goal state after:
case refine_1.mp
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset ↑s
f : ↑s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
a : α
h1 : ∀ (a_1 : β) (b : a_1 ∈ s), ⟨a_1, b⟩ ∈ T → a ∈ f ⟨a_1, b⟩
b : β
⊢ ∀ (b_1 : b ∈ s), ⟨b, b_1⟩ ∈ T → a ∈ Function.extend (fun x ↦ ↑x) f (fun x ↦ ∅) ↑⟨b, b_1⟩
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: case refine_1.mpr
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset ↑s
f : ↑s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
a : α
h1 : ∀ (a_1 : β) (b : a_1 ∈ s), ⟨a_1, b⟩ ∈ T → a ∈ Function.extend (fun x ↦ ↑x) f (fun x ↦ ∅) ↑⟨a_1, b⟩
⊢ ∀ (a_1 : β) (b : a_1 ∈ s), ⟨a_1, b⟩ ∈ T → a ∈ f ⟨a_1, b⟩

Tactic:
b

Goal state after:
case refine_1.mpr
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset ↑s
f : ↑s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
a : α
h1 : ∀ (a_1 : β) (b : a_1 ∈ s), ⟨a_1, b⟩ ∈ T → a ∈ Function.extend (fun x ↦ ↑x) f (fun x ↦ ∅) ↑⟨a_1, b⟩
b : β
⊢ ∀ (b_1 : b ∈ s), ⟨b, b_1⟩ ∈ T → a ∈ f ⟨b, b_1⟩
Module: Mathlib.MeasureTheory.Measure.MeasureSpaceDef

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
⊢ μ s = ⨅ t, μ ↑t

Tactic:
iInf_subtype,

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
⊢ μ s = ⨅ i, ⨅ (_ : s ⊆ i ∧ MeasurableSet i), μ i
Module: Mathlib.MeasureTheory.Function.AEMeasurableSequence

Goal state before:
pp: ι : Sort u_1
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : ι → α → β
μ : Measure α
p : α → (ι → β) → Prop
hf : ∀ (i : ι), AEMeasurable (f i) μ
x : α
hx : x ∈ aeSeqSet hf p
i : ι
⊢ aeSeq hf p i x = f i x

Tactic:
simp only [aeSeq_eq_mk_of_mem_aeSeqSet hf hx i, mk_eq_fun_of_mem_aeSeqSet hf hx i]

Goal state after:
Module: Mathlib.MeasureTheory.Function.AEMeasurableSequence

Goal state before:
pp: ι : Sort u_1
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : ι → α → β
μ : Measure α
p : α → (ι → β) → Prop
hf : ∀ (i : ι), AEMeasurable (f i) μ
x : α
hx : x ∈ aeSeqSet hf p
h_ss : aeSeqSet hf p ⊆ {x | p x fun n ↦ f n x}
hx' : x ∈ {x | p x fun n ↦ f n x}
⊢ p x fun n ↦ f n x

Tactic:
exact hx'

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.Induced

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
m : (s : Set α) → MeasurableSet s → ℝ≥0∞
m0 : m ∅ ⋯ = 0
mU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),
    Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯
f : ℕ → Set α
h : ∀ (i : ℕ), MeasurableSet (f i)
⊢ m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯

Tactic:
simp +singlePass only [iUnion_disjointed.symm]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
m : (s : Set α) → MeasurableSet s → ℝ≥0∞
m0 : m ∅ ⋯ = 0
mU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),
    Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯
f : ℕ → Set α
h : ∀ (i : ℕ), MeasurableSet (f i)
⊢ m (⋃ i, disjointed f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
π : δ' → Type u_6
π' : δ' → Type u_7
inst✝¹ : (x : δ') → MeasurableSpace (π x)
inst✝ : (x : δ') → MeasurableSpace (π' x)
f : δ ≃ δ'
⊢ Measurable ⇑(Equiv.piCongrLeft π f).symm

Tactic:
measurable_pi_iff

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
π : δ' → Type u_6
π' : δ' → Type u_7
inst✝¹ : (x : δ') → MeasurableSpace (π x)
inst✝ : (x : δ') → MeasurableSpace (π' x)
f : δ ≃ δ'
⊢ ∀ (a : δ), Measurable fun x ↦ (Equiv.piCongrLeft π f).symm x a
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α✝
inst✝⁵ : MeasurableSpace α✝
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
π : δ' → Type u_6
π' : δ' → Type u_7
inst✝² : (x : δ') → MeasurableSpace (π x)
inst✝¹ : (x : δ') → MeasurableSpace (π' x)
α : δ ⊕ δ' → Type u_8
inst✝ : (i : δ ⊕ δ') → MeasurableSpace (α i)
⊢ Measurable ⇑(Equiv.sumPiEquivProdPi α).symm

Tactic:
measurable_pi_iff

Goal state after:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α✝
inst✝⁵ : MeasurableSpace α✝
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
π : δ' → Type u_6
π' : δ' → Type u_7
inst✝² : (x : δ') → MeasurableSpace (π x)
inst✝¹ : (x : δ') → MeasurableSpace (π' x)
α : δ ⊕ δ' → Type u_8
inst✝ : (i : δ ⊕ δ') → MeasurableSpace (α i)
⊢ ∀ (a : δ ⊕ δ'), Measurable fun x ↦ (Equiv.sumPiEquivProdPi α).symm x a
Module: Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μa : Measure α
μb : Measure β
f : α → β
hf : QuasiMeasurePreserving f μa μb
f' : α → β
hf' : Measurable f'
h : f =ᵐ[μa] f'
⊢ map f' μa ≪ μb

Tactic:
Measure.map_congr h.symm

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μa : Measure α
μb : Measure β
f : α → β
hf : QuasiMeasurePreserving f μa μb
f' : α → β
hf' : Measurable f'
h : f =ᵐ[μa] f'
⊢ map f μa ≪ μb
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
⊢ (μ.restrict s) univ = μ s

Tactic:
rw [restrict_apply MeasurableSet.univ, Set.univ_inter]

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

Goal state before:
pp: α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
⊢ (μ.restrict t).restrict s = (μ.restrict s).restrict t

Tactic:
restrict_restrict hs,

Goal state after:
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
⊢ μ.restrict (s ∩ t) = (μ.restrict s).restrict t
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
δ : Type u_3
m0 : MeasurableSpace α
μ : Measure α
inst✝ : PseudoMetricSpace δ
f : α → δ
x : δ
hμ : μ ≠ 0
⊢ f ⁻¹' ⋃ i, Metric.ball x ↑i = univ

Tactic:
Metric.iUnion_ball_nat,

Goal state after:
α : Type u_1
δ : Type u_3
m0 : MeasurableSpace α
μ : Measure α
inst✝ : PseudoMetricSpace δ
f : α → δ
x : δ
hμ : μ ≠ 0
⊢ f ⁻¹' univ = univ
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : Filter α
t : Set α
ht : t ∈ f
u : Set α
hu : u ∈ ae μ
hμ : μ (t ∩ u) < ∞
⊢ μ.FiniteAtFilter f

Tactic:
suffices μ t ≤ μ (t ∩ u) from ⟨t, ht, this.trans_lt hμ⟩

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : Filter α
t : Set α
ht : t ∈ f
u : Set α
hu : u ∈ ae μ
hμ : μ (t ∩ u) < ∞
⊢ μ t ≤ μ (t ∩ u)
Module: Mathlib.MeasureTheory.Measure.Trim

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
⊢ μ.trim ⋯ = μ

Tactic:
by
  simp [Measure.trim]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
R : Type u_8
inst✝⁴ : Semiring R
inst✝³ : IsDomain R
inst✝² : Module R ℝ≥0∞
inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝ : Module.IsTorsionFree R ℝ≥0∞
c : R
hc : c ≠ 0
μ : Measure α
⊢ ae (c • μ) = ae μ

Tactic:
by
  ext; exact ae_smul_measure_iff hc

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
x✝ : μ (t ∩ s) + ν (tᶜ ∩ s) ∈ {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}
t' : ℕ → Set α := fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
ht' : t' = fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
n : ℕ
⊢ ¬n = 1 → ¬n = 0 → min (μ (t' n)) (ν (t' n)) = 0

Tactic:
hn₁

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
x✝ : μ (t ∩ s) + ν (tᶜ ∩ s) ∈ {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}
t' : ℕ → Set α := fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
ht' : t' = fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅
n : ℕ
hn₁ : ¬n = 1
⊢ ¬n = 0 → min (μ (t' n)) (ν (t' n)) = 0
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : CountablyGenerated α
⊢ Measurable (mapNatBool α)

Tactic:
measurable_pi_iff

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : CountablyGenerated α
⊢ ∀ (a : ℕ), Measurable fun x ↦ mapNatBool α x a
Module: Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
f : α → β
hf : MeasurableEmbedding f
inst✝ : SigmaFinite μ
⊢ range f ∪ (range f)ᶜ = univ

Tactic:
Set.union_compl_self

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
f : α → β
hf : MeasurableEmbedding f
inst✝ : SigmaFinite μ
⊢ univ = univ
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
f : α → β
h : NullMeasurable f μ
inst✝ : MeasurableSpace.CountablyGenerated ↑∅
hft : ∀ᵐ (x : α) ∂μ, f x ∈ ∅
⊢ μ = 0

Tactic:
simpa using hft

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

Goal state before:
pp: case refine_1
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ MeasurableSpace.generateFrom (range Ioi) ≤ MeasurableSpace.generateFrom (range Iic)

Tactic:
refine MeasurableSpace.generateFrom_le fun t ht => ?_

Goal state after:
case refine_1
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
t : Set α
ht : t ∈ range Ioi
⊢ MeasurableSet t
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case refine_2
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
⊢ MeasurableSpace.generateFrom (range Iic) ≤ MeasurableSpace.generateFrom (range Ioi)

Tactic:
refine MeasurableSpace.generateFrom_le fun t ht => ?_

Goal state after:
case refine_2
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
t : Set α
ht : t ∈ range Iic
⊢ MeasurableSet t