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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
K : ℝ≥0
f : α → β
hf : AntilipschitzWith K f
g : β → α
hg : Function.RightInverse g f
x y : β
this : edist (g x) (g y) ≤ ↑K * edist (f (g x)) (f (g y))
⊢ edist (g x) (g y) ≤ ↑K * edist x y

Tactic:
hg x,

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
K : ℝ≥0
f : α → β
hf : AntilipschitzWith K f
g : β → α
hg : Function.RightInverse g f
x y : β
this : edist (g x) (g y) ≤ ↑K * edist x (f (g y))
⊢ edist (g x) (g y) ≤ ↑K * edist x y
Module: Mathlib.Topology.Metrizable.Uniformity

Goal state before:
pp: X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
x y z : X
lxy lyz : List X
⊢ lxy.length + 1 = lxy.length + [y].length

Tactic:
length_singleton

Goal state after:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
x y z : X
lxy lyz : List X
⊢ lxy.length + 1 = lxy.length + 1
Module: Mathlib.Topology.Metrizable.Uniformity

Goal state before:
pp: X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
x y : X
l : List X
a b c : X
hab : d a b = 0
hbc : d b c = 0
⊢ d a c ≤ 0

Tactic:
simpa only [nonpos_iff_eq_zero, hab, hbc, dist_self c, max_self, mul_zero] using hd a b c c

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Continuity

Goal state before:
pp: α : Type u_1
ι : Type u_2
κ : Type u_3
E : Type u_4
F✝ : Type u_5
G : Type u_6
F : Type u_7
inst✝ : NormedGroup F
⊢ ∀ (x y : F), ‖x * y‖ₑ ≤ ‖x‖ₑ + ‖y‖ₑ

Tactic:
simp [enorm_eq_nnnorm, ← coe_add, nnnorm_mul_le']

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Continuity

Goal state before:
pp: ι : Type u_2
κ : Type u_3
G : Type u_6
inst✝ : SeminormedGroup G
f : ι → κ → G
s : Set κ
l : Filter ι
⊢ UniformCauchySeqOn f l s ↔ TendstoUniformlyOn (fun n z ↦ (f n.1 z)⁻¹ * f n.2 z) 1 (l ×ˢ l) s

Tactic:
by
  rw [tendstoUniformlyOn_iff_tendstoUniformlyOnFilter,
    uniformCauchySeqOn_iff_uniformCauchySeqOnFilter,
    SeminormedGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_one]

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

Goal state before:
pp: β : Type u_4
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → ℝ≥0∞
hf : Antitone f
⊢ Tendsto f atTop (𝓝 0) ↔ ∀ (ε : ℝ≥0∞), 0 < ε → ∃ n, f n ≤ ε

Tactic:
rw [ENNReal.tendsto_atTop_zero]
  refine ⟨fun h ↦ fun ε hε ↦ ?_, fun h ↦ fun ε hε ↦ ?_⟩
  · obtain ⟨n, hn⟩ := h ε hε
    exact ⟨n, hn n le_rfl⟩
  · obtain ⟨n, hn⟩ := h ε hε
    exact ⟨n, fun m hm ↦ (hf hm).trans hn⟩

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

Goal state before:
pp: b : ℝ≥0
x✝ : ∞ ≠ ∞ ∨ ↑b ≠ ∞
⊢ Tendsto (fun p ↦ p.1 - p.2) (𝓝 (∞, ↑b)) (𝓝 (∞ - ↑b))

Tactic:
rw [top_sub_coe, tendsto_nhds_top_iff_nnreal]
    refine fun x => ((lt_mem_nhds <| @coe_lt_top (b + 1 + x)).prod_nhds
      (ge_mem_nhds <| coe_lt_coe.2 <| lt_add_one b)).mono fun y hy => ?_
    grw [lt_tsub_iff_left, hy.2]
    exact hy.1

Goal state after:
Module: Mathlib.Topology.UniformSpace.Completion

Goal state before:
pp: case refine_2
α : Type u
inst✝ : UniformSpace α
f : CauchyFilter α
h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y, (f, pureCauchy y) ∈ s
s : Set (CauchyFilter α × CauchyFilter α)
hs : s ∈ 𝓤 (CauchyFilter α)
y : α
hy : (f, pureCauchy y) ∈ s
this : pureCauchy y ∈ range pureCauchy ∩ {y | (f, y) ∈ s}
⊢ (range pureCauchy ∩ UniformSpace.ball f s).Nonempty

Tactic:
exact ⟨_, this⟩

Goal state after:
Module: Mathlib.Order.Interval.Set.ProjIcc

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
a b : α
h : a ≤ b
x : ↑(Icc a b)
⊢ projIcc a b h ↑x = x

Tactic:
by
  cases x
  apply projIcc_of_mem

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

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
⊢ ∀ (x y : α × α), edist x.1 x.2 ≤ edist y.1 y.2 + 2 * edist x y

Tactic:
rintro ⟨x, y⟩ ⟨x', y'⟩

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
x y x' y' : α
⊢ edist (x, y).1 (x, y).2 ≤ edist (x', y').1 (x', y').2 + 2 * edist (x, y) (x', y')
Module: Mathlib.Topology.Metrizable.Uniformity

Goal state before:
pp: case refine_2
X : Type u_2
inst✝¹ : UniformSpace X
inst✝ : (𝓤 X).IsCountablyGenerated
U : ℕ → SetRel X X
hU_symm : ∀ (n : ℕ), (U n).IsSymm
hU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m
hB : (𝓤 X).HasAntitoneBasis U
d : X → X → ℝ≥0 := fun x y ↦ if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else 0
hd₀ : ∀ {x y : X}, d x y = 0 ↔ Inseparable x y
hd_symm : ∀ (x y : X), d x y = d y x
hr : ↑0 < ↑(1 / 2) ∧ ↑(1 / 2) < ↑1
I : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d ⋯ hd_symm
hdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)
hle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ (x, y) ∉ U n
hd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y
n : ℕ
x✝ : True
x : X × X
hx : x ∉ U n
⊢ ↑((1 / 2) ^ (n + 1)) ≤ ↑(d x.1 x.2 / 2)

Tactic:
NNReal.coe_le_coe,

Goal state after:
case refine_2
X : Type u_2
inst✝¹ : UniformSpace X
inst✝ : (𝓤 X).IsCountablyGenerated
U : ℕ → SetRel X X
hU_symm : ∀ (n : ℕ), (U n).IsSymm
hU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m
hB : (𝓤 X).HasAntitoneBasis U
d : X → X → ℝ≥0 := fun x y ↦ if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else 0
hd₀ : ∀ {x y : X}, d x y = 0 ↔ Inseparable x y
hd_symm : ∀ (x y : X), d x y = d y x
hr : ↑0 < ↑(1 / 2) ∧ ↑(1 / 2) < ↑1
I : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d ⋯ hd_symm
hdist_le : ∀ (x y : X), dist x y ≤ ↑(d x y)
hle_d : ∀ {x y : X} {n : ℕ}, (1 / 2) ^ n ≤ d x y ↔ (x, y) ∉ U n
hd_le : ∀ (x y : X), ↑(d x y) ≤ 2 * dist x y
n : ℕ
x✝ : True
x : X × X
hx : x ∉ U n
⊢ (1 / 2) ^ (n + 1) ≤ d x.1 x.2 / 2
Module: Mathlib.Topology.MetricSpace.IsometricSMul

Goal state before:
pp: M : Type u
G : Type v
X : Type w
Y : Type u_1
inst✝³ : PseudoEMetricSpace X
inst✝² : PseudoEMetricSpace Y
inst✝¹ : SMul M X
inst✝ : IsIsometricSMul M X
c : M
x y : Xᵐᵒᵖ
⊢ edist ((fun x ↦ c • x) x) ((fun x ↦ c • x) y) = edist x y

Tactic:
by simpa only using! edist_smul_left c x.unop y.unop

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

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
f : α → β
hf : Isometry f
h : ∃ x y, edist x y ≠ 0 ∧ edist x y ≠ ∞
⊢ ratio (Isometry.toDilation f hf) = 1

Tactic:
obtain ⟨x, y, h₁, h₂⟩ := h
    exact ratio_unique h₁ h₂ (by simp [hf x y]) |>.symm

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
g : β → α
hg : Function.Surjective g
⊢ MeasurableSpace.map g (MeasurableSpace.comap g m) = m

Tactic:
by
  refine le_antisymm (fun S hS => ?_) le_map_comap
  rw [map_def, measurableSet_comap] at hS
  aesop

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
E✝ : Type u_3
F : Type u_4
α : Type u_5
inst✝⁷ : NormedField 𝕜
inst✝⁶ : SeminormedAddCommGroup E✝
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 E✝
inst✝³ : NormedSpace 𝕜 F
ι : Type u_6
E : ι → Type u_7
inst✝² : Fintype ι
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
a : 𝕜
f : (i : ι) → E i
⊢ ↑‖a • f‖₊ ≤ ↑(‖a‖₊ * ‖f‖₊)

Tactic:
NNReal.coe_le_coe,

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
E✝ : Type u_3
F : Type u_4
α : Type u_5
inst✝⁷ : NormedField 𝕜
inst✝⁶ : SeminormedAddCommGroup E✝
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 E✝
inst✝³ : NormedSpace 𝕜 F
ι : Type u_6
E : ι → Type u_7
inst✝² : Fintype ι
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
a : 𝕜
f : (i : ι) → E i
⊢ ‖a • f‖₊ ≤ ‖a‖₊ * ‖f‖₊
Module: Mathlib.Analysis.Normed.Module.Basic

Goal state before:
pp: 𝕜✝ : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
α : Type u_5
inst✝⁵ : NormedField 𝕜✝
inst✝⁴ : SeminormedRing 𝕜'
inst✝³ : NormedAlgebra 𝕜✝ 𝕜'
𝕜 : Type ?u.15
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : CharZero 𝕜
inst✝ : NormedAlgebra ℝ 𝕜
q : ℚ
x : 𝕜
⊢ ‖q • x‖ ≤ ‖q‖ * ‖x‖

Tactic:
rw [← smul_one_smul ℝ q x, Rat.smul_one_eq_cast, norm_smul, Rat.norm_cast_real]

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

Goal state before:
pp: case inl
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), m (s (k + 1) \ s k) ≠ ∞
n : ℕ
h' : Monotone s
x : α
this : ∃ i, x ∈ s i
j : ℕ
hj : x ∈ s j
hlt : ∀ m < j, x ∉ s m
hjn : j ≤ n
⊢ x ∈ s n ∪ ⋃ i, s (i + n + 1) \ s (i + n)

Tactic:
exact Or.inl (h' hjn hj)

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

Goal state before:
pp: case h₁
β : Type u_2
inst✝¹ : MeasurableSpace β
inst✝ : Countable β
x : β
s : β → Set β
hs : ∀ y ∉ measurableAtom x, x ∈ s y ∧ MeasurableSet (s y) ∧ y ∉ s y
z : β
⊢ z ∈ measurableAtom x → z ∈ ⋂ y ∈ (measurableAtom x)ᶜ, s y

Tactic:
hz

Goal state after:
case h₁
β : Type u_2
inst✝¹ : MeasurableSpace β
inst✝ : Countable β
x : β
s : β → Set β
hs : ∀ y ∉ measurableAtom x, x ∈ s y ∧ MeasurableSet (s y) ∧ y ∉ s y
z : β
hz : z ∈ measurableAtom x
⊢ z ∈ ⋂ y ∈ (measurableAtom x)ᶜ, s y
Module: Mathlib.MeasureTheory.MeasurableSpace.Constructions

Goal state before:
pp: case h₂
β : Type u_2
inst✝¹ : MeasurableSpace β
inst✝ : Countable β
x : β
s : β → Set β
hs : ∀ y ∉ measurableAtom x, x ∈ s y ∧ MeasurableSet (s y) ∧ y ∉ s y
z : β
⊢ z ∈ (measurableAtom x)ᶜ → z ∈ (⋂ y ∈ (measurableAtom x)ᶜ, s y)ᶜ

Tactic:
hz

Goal state after:
case h₂
β : Type u_2
inst✝¹ : MeasurableSpace β
inst✝ : Countable β
x : β
s : β → Set β
hs : ∀ y ∉ measurableAtom x, x ∈ s y ∧ MeasurableSet (s y) ∧ y ∉ s y
z : β
hz : z ∈ (measurableAtom x)ᶜ
⊢ z ∈ (⋂ y ∈ (measurableAtom x)ᶜ, s y)ᶜ
Module: Mathlib.MeasureTheory.MeasurableSpace.Constructions

Goal state before:
pp: β : Type u_2
inst✝ : MeasurableSpace β
x y : β
hx : x ∈ measurableAtom y
z : β
⊢ z ∈ measurableAtom x → z ∈ measurableAtom y

Tactic:
hz

Goal state after:
β : Type u_2
inst✝ : MeasurableSpace β
x y : β
hx : x ∈ measurableAtom y
z : β
hz : z ∈ measurableAtom x
⊢ z ∈ measurableAtom y
Module: Mathlib.Topology.Algebra.Algebra

Goal state before:
pp: R : Type u_1
inst✝⁷ : CommSemiring R
A : Type u_2
inst✝⁶ : Semiring A
inst✝⁵ : TopologicalSpace A
B : Type u_3
inst✝⁴ : Semiring B
inst✝³ : TopologicalSpace B
inst✝² : Algebra R A
inst✝¹ : Algebra R B
inst✝ : IsSemitopologicalSemiring A
s : Subalgebra R A
r : R
⊢ (algebraMap R A) r ∈ s.topologicalClosure.carrier

Tactic:
simp only [Subsemiring.coe_carrier_toSubmonoid, Subsemiring.topologicalClosure_coe,
      Subalgebra.coe_toSubsemiring]

Goal state after:
R : Type u_1
inst✝⁷ : CommSemiring R
A : Type u_2
inst✝⁶ : Semiring A
inst✝⁵ : TopologicalSpace A
B : Type u_3
inst✝⁴ : Semiring B
inst✝³ : TopologicalSpace B
inst✝² : Algebra R A
inst✝¹ : Algebra R B
inst✝ : IsSemitopologicalSemiring A
s : Subalgebra R A
r : R
⊢ (algebraMap R A) r ∈ closure[inst✝⁵] ↑s
Module: Mathlib.Data.EReal.Inv

Goal state before:
pp: b✝ : EReal
a : ℝ
a_0 : 0 < ↑a
b : ℝ
b_0 : 0 < ↑b
a_b : ↑a < ↑b
⊢ ↑b⁻¹ < ↑a⁻¹

Tactic:
EReal.coe_lt_coe_iff

Goal state after:
b✝ : EReal
a : ℝ
a_0 : 0 < ↑a
b : ℝ
b_0 : 0 < ↑b
a_b : ↑a < ↑b
⊢ b⁻¹ < a⁻¹
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
f : α → β
inst✝ : LinearOrder β
s : Set α
ne_s : s.Nonempty
hs : IsCompact s
hf : LowerSemicontinuousOn f s
x✝¹ : Nonempty α
x✝ : Nonempty ↑s
φ : β → Filter α := fun b ↦ 𝓟 (s ∩ f ⁻¹' Iic b)
ℱ : Filter α := ⨅ a, φ (f ↑a)
x : ↑s
⊢ (φ (f ↑x)).NeBot

Tactic:
have : (pure x : Filter α) ≤ φ (f x) := le_principal_iff.mpr ⟨x.2, le_refl (f x)⟩

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
f : α → β
inst✝ : LinearOrder β
s : Set α
ne_s : s.Nonempty
hs : IsCompact s
hf : LowerSemicontinuousOn f s
x✝¹ : Nonempty α
x✝ : Nonempty ↑s
φ : β → Filter α := fun b ↦ 𝓟 (s ∩ f ⁻¹' Iic b)
ℱ : Filter α := ⨅ a, φ (f ↑a)
x : ↑s
this : pure ↑x ≤ φ (f ↑x)
⊢ (φ (f ↑x)).NeBot
Module: Mathlib.Data.EReal.Inv

Goal state before:
pp: a b c : EReal
h : b > 0
h' : b ≠ ⊤
⊢ a ≤ c / b ↔ a * b ≤ c

Tactic:
by
  nth_rw 1 [← @mul_div_cancel a b (ne_bot_of_gt h) h' h.ne']
  rw [mul_div b a b, mul_comm a b]
  exact StrictMono.le_iff_le (strictMono_div_right_of_pos h h')

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
s : Set α
y : β
inst✝¹ : Zero β
inst✝ : Preorder β
hs : IsOpen[inst✝²] s
hy : 0 ≤ y
x : α
z : β
⊢ (fun x1 x2 ↦ s.indicator (fun _x ↦ y) x1 > x2) x z →
    ∀ᶠ (x' : α) in 𝓝 x, (fun x1 x2 ↦ s.indicator (fun _x ↦ y) x1 > x2) x' z

Tactic:
hz

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
s : Set α
y : β
inst✝¹ : Zero β
inst✝ : Preorder β
hs : IsOpen[inst✝²] s
hy : 0 ≤ y
x : α
z : β
hz : s.indicator (fun _x ↦ y) x > z
⊢ ∀ᶠ (x' : α) in 𝓝 x, (fun x1 x2 ↦ s.indicator (fun _x ↦ y) x1 > x2) x' z
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
s : Set α
y : β
inst✝¹ : Zero β
inst✝ : Preorder β
hs : IsClosed[inst✝²] s
hy : y ≤ 0
x : α
z : β
⊢ (fun x1 x2 ↦ s.indicator (fun _x ↦ y) x1 > x2) x z →
    ∀ᶠ (x' : α) in 𝓝 x, (fun x1 x2 ↦ s.indicator (fun _x ↦ y) x1 > x2) x' z

Tactic:
hz

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
s : Set α
y : β
inst✝¹ : Zero β
inst✝ : Preorder β
hs : IsClosed[inst✝²] s
hy : y ≤ 0
x : α
z : β
hz : s.indicator (fun _x ↦ y) x > z
⊢ ∀ᶠ (x' : α) in 𝓝 x, (fun x1 x2 ↦ s.indicator (fun _x ↦ y) x1 > x2) x' z
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
b : β
hb : (fun x ↦ f (M x) x) x > b
z : α
⊢ z ∈ s → (fun x1 x2 ↦ f (M x) x1 > x2) z b → (fun x1 x2 ↦ (fun x ↦ f (M x) x) x1 > x2) z b

Tactic:
hz

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
hb : (fun x ↦ f (M x) x) x > b
z : α
hz : z ∈ s
⊢ (fun x1 x2 ↦ f (M x) x1 > x2) z b → (fun x1 x2 ↦ (fun x ↦ f (M x) x) x1 > x2) z b
Module: Mathlib.Analysis.SpecificLimits.Basic

Goal state before:
pp: C : EReal
h : C ≠ ⊥
h' : C ≠ ⊤
⊢ Tendsto (fun n ↦ C / ↑n) atTop (𝓝 0)

Tactic:
have : (fun n : ℕ ↦ C / n) = fun n : ℕ ↦ ((C.toReal / n : ℝ) : EReal) := by
    ext n
    nth_rw 1 [← coe_toReal h' h, ← coe_coe_eq_natCast n, ← coe_div C.toReal n]

Goal state after:
C : EReal
h : C ≠ ⊥
h' : C ≠ ⊤
this : (fun n ↦ C / ↑n) = fun n ↦ ↑(C.toReal / ↑n)
⊢ Tendsto (fun n ↦ C / ↑n) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.OuterMeasure.OfFunction

Goal state before:
pp: case a
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
P : Set α → Prop
m_top : ∀ (s : Set α), ¬P s → m s = ∞
s : Set α
⊢ ⨅ t, ⨅ (_ : ∀ (i : ℕ), P (t i)), ⨅ (_ : s ⊆ ⋃ i, t i), ∑' (i : ℕ), m (t i) ≤
    ⨅ t, ⨅ (_ : s ⊆ iUnion t), ∑' (n : ℕ), m (t n)

Tactic:
le_iInf_iff

Goal state after:
case a
α : Type u_1
m : Set α → ℝ≥0∞
m_empty : m ∅ = 0
P : Set α → Prop
m_top : ∀ (s : Set α), ¬P s → m s = ∞
s : Set α
⊢ ∀ (i : ℕ → Set α),
    s ⊆ iUnion i → ⨅ t, ⨅ (_ : ∀ (i : ℕ), P (t i)), ⨅ (_ : s ⊆ ⋃ i, t i), ∑' (i : ℕ), m (t i) ≤ ∑' (n : ℕ), m (i n)
Module: Mathlib.Analysis.SpecificLimits.Basic

Goal state before:
pp: r : ℝ≥0
hr : r < 1
⊢ HasSum (fun a ↦ ↑r ^ a) (↑(1 - r))⁻¹

Tactic:
rw [NNReal.coe_sub (le_of_lt hr)]

Goal state after:
r : ℝ≥0
hr : r < 1
⊢ HasSum (fun a ↦ ↑r ^ a) (↑1 - ↑r)⁻¹
Module: Mathlib.Analysis.SpecificLimits.Basic

Goal state before:
pp: ⊢ 2⁻¹ * (1 - 2⁻¹)⁻¹ = 1

Tactic:
one_sub_inv_two,

Goal state after:
⊢ 2⁻¹ * 2⁻¹⁻¹ = 1
Module: Mathlib.MeasureTheory.OuterMeasure.OfFunction

Goal state before:
pp: case refine_2
α : Type u_1
m : Set α → ℝ≥0∞
β : Type u_2
f : β → α
h : (Monotone fun s ↦ m ↑s) ∨ Surjective f
⊢ OuterMeasure.ofFunction (fun s ↦ ⨆ (_ : (f '' s).Nonempty), m (f '' s)) ⋯ =
    OuterMeasure.ofFunction (fun s ↦ ⨆ (_ : s.Nonempty), m (f '' s)) ⋯

Tactic:
congr with s : 1

Goal state after:
case refine_2
α : Type u_1
m : Set α → ℝ≥0∞
β : Type u_2
f : β → α
h : (Monotone fun s ↦ m ↑s) ∨ Surjective f
s : Set β
⊢ ⨆ (_ : (f '' s).Nonempty), m (f '' s) = ⨆ (_ : s.Nonempty), m (f '' s)
Module: Mathlib.Topology.Instances.EReal.Lemmas

Goal state before:
pp: case inr
a : ℝ
h : a ≠ 0
a_pos : 0 < a
⊢ ContinuousAt (fun p ↦ p.1 * p.2) (⊤, ↑a)

Tactic:
exact continuousAt_mul_top_pos a_pos

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.Basic

Goal state before:
pp: case refine_2
k : ℕ
hn : 0 < k.succ
x : ℕ
hx : x < k
⊢ (↑x + 1 + 1) * (↑k + 1)⁻¹ ≤ 1

Tactic:
· refine (div_le_one <| mod_cast hn).mpr ?_
        norm_cast
        lia

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

Goal state before:
pp: α : Type u
m : OuterMeasure α
s₁ s₂ : Set α
h₁ : m.IsCaratheodory s₁
h₂ : m.IsCaratheodory s₂
t : Set α
⊢ m t = m (t ∩ (s₁ ∪ s₂)) + m (t \ (s₁ ∪ s₂))

Tactic:
rw [h₁ t, h₂ (t ∩ s₁), h₂ (t \ s₁), h₁ (t ∩ (s₁ ∪ s₂)), inter_sdiff_assoc _ _ s₁,
    Set.inter_assoc _ _ s₁, inter_eq_self_of_subset_right Set.subset_union_left,
    union_sdiff_left, h₂ (t ∩ s₁)]

Goal state after:
α : Type u
m : OuterMeasure α
s₁ s₂ : Set α
h₁ : m.IsCaratheodory s₁
h₂ : m.IsCaratheodory s₂
t : Set α
⊢ m (t ∩ s₁ ∩ s₂) + m ((t ∩ s₁) \ s₂) + (m (t \ s₁ ∩ s₂) + m ((t \ s₁) \ s₂)) =
    m (t ∩ s₁ ∩ s₂) + m ((t ∩ s₁) \ s₂) + m (t ∩ (s₂ \ s₁)) + m (t \ (s₁ ∪ s₂))
Module: Mathlib.MeasureTheory.OuterMeasure.Induced

Goal state before:
pp: case pos
α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
s : ℕ → Set α
h : ∀ (i : ℕ), P (s i)
⊢ m (⋃ i, s i) ⋯ ≤ ∑' (i : ℕ), extend m (s i)

Tactic:
congr_arg tsum _

Goal state after:
case pos
α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
s : ℕ → Set α
h : ∀ (i : ℕ), P (s i)
⊢ m (⋃ i, s i) ⋯ ≤ tsum ?m.89

α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
s : ℕ → Set α
h : ∀ (i : ℕ), P (s i)
⊢ ℕ → ℝ≥0∞

α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
s : ℕ → Set α
h : ∀ (i : ℕ), P (s i)
⊢ (fun i ↦ extend m (s i)) = ?m.89
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: α : Type u_3
ι : Type u_4
π : ι → Set (Set α)
hpi : ∀ (x : ι), IsPiSystem (π x)
S : Set ι
t1 : Set α
p1 : Finset ι
hp1S : ↑p1 ⊆ S
f1 : ι → Set α
hf1m : ∀ x ∈ p1, f1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
t2 : Set α
p2 : Finset ι
hp2S : ↑p2 ⊆ S
f2 : ι → Set α
hf2m : ∀ x ∈ p2, f2 x ∈ π x
ht2_eq : t2 = ⋂ x ∈ p2, f2 x
h_nonempty : (t1 ∩ t2).Nonempty
g : ι → Set α := fun n ↦ (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ
⊢ ∃ t, ∃ (_ : ↑t ⊆ S), ∃ f, ∃ (_ : ∀ x ∈ t, f x ∈ π x), t1 ∩ t2 = ⋂ x ∈ t, f x

Tactic:
have hp_union_ss : ↑(p1 ∪ p2) ⊆ S := by
    simp only [hp1S, hp2S, Finset.coe_union, union_subset_iff, and_self_iff]

Goal state after:
α : Type u_3
ι : Type u_4
π : ι → Set (Set α)
hpi : ∀ (x : ι), IsPiSystem (π x)
S : Set ι
t1 : Set α
p1 : Finset ι
hp1S : ↑p1 ⊆ S
f1 : ι → Set α
hf1m : ∀ x ∈ p1, f1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
t2 : Set α
p2 : Finset ι
hp2S : ↑p2 ⊆ S
f2 : ι → Set α
hf2m : ∀ x ∈ p2, f2 x ∈ π x
ht2_eq : t2 = ⋂ x ∈ p2, f2 x
h_nonempty : (t1 ∩ t2).Nonempty
g : ι → Set α := fun n ↦ (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ
hp_union_ss : ↑(p1 ∪ p2) ⊆ S
⊢ ∃ t, ∃ (_ : ↑t ⊆ S), ∃ f, ∃ (_ : ∀ x ∈ t, f x ∈ π x), t1 ∩ t2 = ⋂ x ∈ t, f x
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f : β → γ
e : α ≃ᵐ β
hfe : Measurable (f ∘ ⇑e)
⊢ Measurable f

Tactic:
by
      have : Measurable (f ∘ (e.symm.trans e).toEquiv) := hfe.comp e.symm.measurable
      rwa [coe_toEquiv, symm_trans_self] at this

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

Goal state before:
pp: case refine_2
ι : Type u_1
α : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
s : ι → Set α
hd : Pairwise (AEDisjoint μ on s)
i j : ι
hne : i ≠ j
x : α
hi : x ∈ s i
hU : x ∉ toMeasurable μ (s i ∩ ⋃ j ∈ {i}ᶜ, s j)
hj : x ∈ s j
⊢ x ∈ toMeasurable μ (s j ∩ ⋃ j_1 ∈ {j}ᶜ, s j_1)

Tactic:
replace hU : x ∉ s i ∩ iUnion fun j ↦ iUnion fun _ ↦ s j :=
      fun h ↦ hU (subset_toMeasurable _ _ h)

Goal state after:
case refine_2
ι : Type u_1
α : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
s : ι → Set α
hd : Pairwise (AEDisjoint μ on s)
i j : ι
hne : i ≠ j
x : α
hi : x ∈ s i
hj : x ∈ s j
hU : x ∉ s i ∩ ⋃ j ∈ {i}ᶜ, s j
⊢ x ∈ toMeasurable μ (s j ∩ ⋃ j_1 ∈ {j}ᶜ, s j_1)
Module: Mathlib.MeasureTheory.Measure.NullMeasurable

Goal state before:
pp: α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝² : MeasurableSingletonClass α
mβ : MeasurableSpace β
inst✝¹ : MeasurableSingletonClass β
inst✝ : Countable β
μ : Measure (α × β)
x : α
⊢ Pairwise (Disjoint on fun x_1 ↦ {(x, x_1)})

Tactic:
by simp [Pairwise]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : AEMeasurable f μ
⊢ map f μ = 0 ↔ μ = 0

Tactic:
simp_rw [← measure_univ_eq_zero, map_apply_of_aemeasurable hf .univ, preimage_univ]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hfi : Injective f
hf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s t : Set α
hst : (comap f μ) {a | ¬s a = t a} = 0
h_eq_α : {a | ¬s a = t a} = s \ t ∪ t \ s
h_eq_β : {a | ¬(f '' s) a = (f '' t) a} = f '' (s \ t) ∪ f '' (t \ s)
⊢ μ {a | ¬(f '' s) a = (f '' t) a} = 0

Tactic:
← Set.image_union

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hfi : Injective f
hf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s t : Set α
hst : (comap f μ) {a | ¬s a = t a} = 0
h_eq_α : {a | ¬s a = t a} = s \ t ∪ t \ s
h_eq_β : {a | ¬(f '' s) a = (f '' t) a} = f '' (s \ t ∪ t \ s)
⊢ μ {a | ¬(f '' s) a = (f '' t) a} = 0
Module: Mathlib.MeasureTheory.Measure.Comap

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hfi : Injective f
hf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s t : Set α
hst : (comap f μ) {a | ¬s a = t a} = 0
h_eq_α : {a | ¬s a = t a} = s \ t ∪ t \ s
h_eq_β : {a | ¬(f '' s) a = (f '' t) a} = f '' (s \ t ∪ t \ s)
⊢ μ {a | ¬(f '' s) a = (f '' t) a} = 0

Tactic:
rw [h_eq_β]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
μ : Measure β
hfi : Injective f
hf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ
s t : Set α
hst : (comap f μ) {a | ¬s a = t a} = 0
h_eq_α : {a | ¬s a = t a} = s \ t ∪ t \ s
h_eq_β : {a | ¬(f '' s) a = (f '' t) a} = f '' (s \ t ∪ t \ s)
⊢ μ (f '' (s \ t ∪ t \ s)) = 0