mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: α : Type u_1
G : Type u_2
inst✝³ : MeasurableSpace G
inst✝² : DivInvMonoid G
inst✝¹ : MeasurableMul₂ G
inst✝ : MeasurableInv G
⊢ Measurable fun p ↦ p.1 / p.2

Tactic:
simp only [div_eq_mul_inv]
    exact measurable_fst.mul measurable_snd.inv

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

Goal state before:
pp: case succ
α✝ : Type u_1
M✝ : Type u_2
X : Type u_3
α : Type u_4
β : Type u_5
inst✝⁶ : MeasurableSpace X
inst✝⁵ : SMul M✝ X
m : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → M✝
g : α → X
inst✝⁴ : MeasurableSpace M✝
inst✝³ : MeasurableSMul M✝ X
M : Type u_6
inst✝² : AddMonoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableAdd₂ M
n : ℕ
ih : Measurable fun x ↦ (x, n).2 • (x, n).1
⊢ Measurable fun x ↦ (x, n + 1).2 • (x, n + 1).1

Tactic:
simp only [succ_nsmul]
      exact ih.add measurable_id

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

Goal state before:
pp: case H1
α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
bs : Bornology.IsBounded s
bt : Bornology.IsBounded t
cs : α
hcs : cs ∈ s
ct : α
hct : ct ∈ t
rs : ℝ
hrs : s ⊆ closedBall ct rs
rt : ℝ
hrt : t ⊆ closedBall cs rt
x : α
xs : x ∈ s
this : dist x ct ≤ max rs rt
⊢ 0 ≤ max rs rt

Tactic:
exact le_trans dist_nonneg this

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

Goal state before:
pp: case H2
α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
bs : Bornology.IsBounded s
bt : Bornology.IsBounded t
cs : α
hcs : cs ∈ s
ct : α
hct : ct ∈ t
rs : ℝ
hrs : s ⊆ closedBall ct rs
rt : ℝ
hrt : t ⊆ closedBall cs rt
x : α
xt : x ∈ t
this : dist x cs ≤ max rs rt
⊢ 0 ≤ max rs rt

Tactic:
exact le_trans dist_nonneg this

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : Zero β
inst✝ : Zero γ
g : β → γ
hg : g 0 = 0
f : α →ₛ β
s : Set α
⊢ map g (f.restrict s) = (map g f).restrict s

Tactic:
exact ext fun x =>
    if hs : MeasurableSet s then by simp [hs, Set.indicator_comp_of_zero hg]
    else by simp [restrict_of_not_measurable hs, hg]

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Canonical.Basic

Goal state before:
pp: α : Type u_1
inst✝⁴ : LinearOrder α
a b c : α
inst✝³ : AddCommMagma α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftReflectLT α
inst✝ : IsLeftCancelAdd α
⊢ a < b + c ↔ a < c ∨ ∃ d < b, a = d + c

Tactic:
add_comm,

Goal state after:
α : Type u_1
inst✝⁴ : LinearOrder α
a b c : α
inst✝³ : AddCommMagma α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftReflectLT α
inst✝ : IsLeftCancelAdd α
⊢ a < c + b ↔ a < c ∨ ∃ d < b, a = d + c
Module: Mathlib.Algebra.Order.Monoid.Canonical.Basic

Goal state before:
pp: α : Type u_1
inst✝⁴ : LinearOrder α
a b c : α
inst✝³ : AddCommMagma α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftMono α
inst✝ : IsLeftCancelAdd α
⊢ a ≤ b + c ↔ a < c ∨ ∃ d ≤ b, a = d + c

Tactic:
add_comm,

Goal state after:
α : Type u_1
inst✝⁴ : LinearOrder α
a b c : α
inst✝³ : AddCommMagma α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftMono α
inst✝ : IsLeftCancelAdd α
⊢ a ≤ c + b ↔ a < c ∨ ∃ d ≤ b, a = d + c
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:

Tactic:
thickening (δ / 2) (f '' V)
      _ ⊆ thickening (δ / 2) (thickening (δ / 2) {f x}) :=
        thickening_subset_of_subset _ (image_preimage_subset f _)
      _ ⊆ thickening ((δ / 2) + (δ / 2)) ({f x}) :=
        thickening_thickening_subset (δ / 2) (δ / 2) {f x}
      _ ⊆ U := by simp [hthick]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ι : Type u_5
inst✝ : Countable ι
f : α →ₛ ℝ≥0∞
s : ι → Set α
hd : Directed (fun x1 x2 ↦ x1 ⊆ x2) s
μ : Measure α
⊢ f.lintegral (μ.restrict (⋃ i, s i)) = ⨆ i, f.lintegral (μ.restrict (s i))

Tactic:
simp only [lintegral, Measure.restrict_iUnion_apply_eq_iSup hd (measurableSet_preimage ..),
    ENNReal.mul_iSup]

Goal state after:
α : Type u_1
m : MeasurableSpace α
ι : Type u_5
inst✝ : Countable ι
f : α →ₛ ℝ≥0∞
s : ι → Set α
hd : Directed (fun x1 x2 ↦ x1 ⊆ x2) s
μ : Measure α
⊢ ∑ x ∈ f.range, ⨆ i, x * (μ.restrict (s i)) (⇑f ⁻¹' {x}) = ⨆ i, ∑ x ∈ f.range, x * (μ.restrict (s i)) (⇑f ⁻¹' {x})
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: case mpr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : Zero β
μ : Measure α
f : α →ₛ β
⊢ (∀ (y : β), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞) → f.FinMeasSupp μ

Tactic:
intro H

Goal state after:
case mpr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : Zero β
μ : Measure α
f : α →ₛ β
H : ∀ (y : β), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
⊢ f.FinMeasSupp μ
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: a : ℝ
this : IsLUB (range Rat.cast ∩ Iio a) a
⊢ MeasurableSet (Iio a)

Tactic:
rw [← this.biUnion_Iio_eq, ← image_univ, ← image_inter_preimage, univ_inter, biUnion_image]

Goal state after:
a : ℝ
this : IsLUB (range Rat.cast ∩ Iio a) a
⊢ MeasurableSet (⋃ y ∈ Rat.cast ⁻¹' Iio a, Iio ↑y)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ι : Type u_5
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), Measurable (f i)
⊢ Measurable fun x ↦ ∑' (i : ι), f i x

Tactic:
simp_rw [ENNReal.tsum_eq_iSup_sum]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
ι : Type u_5
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), Measurable (f i)
⊢ Measurable fun x ↦ ⨆ s, ∑ i ∈ s, f i x
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ι : Type u_5
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
μ : Measure α
h : ∀ (i : ι), AEMeasurable (f i) μ
⊢ AEMeasurable (fun x ↦ ∑' (i : ι), f i x) μ

Tactic:
simp_rw [ENNReal.tsum_eq_iSup_sum]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
ι : Type u_5
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
μ : Measure α
h : ∀ (i : ι), AEMeasurable (f i) μ
⊢ AEMeasurable (fun x ↦ ⨆ s, ∑ i ∈ s, f i x) μ
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case neg
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_5
ι' : Type u_6
f : ι → δ → α
v : Filter ι
hf : ∀ (i : ι), Measurable (f i)
p : ι' → Prop
s : ι' → Set ι
hv : v.HasCountableBasis p s
hs : ∀ (j : ι'), (s j).Countable
this✝ : Countable (Subtype p)
hp : Nonempty (Subtype p)
H : ¬∃ j, s ↑j = ∅
this : ∀ (i : ι'), Countable ↑(s i)
m : Subtype p → Set δ := fun j ↦ {x | BddBelow (range fun i ↦ f (↑i) x)}
m_meas : ∀ (j : Subtype p), MeasurableSet (m j)
mc_meas : MeasurableSet {x | ∀ (j : Subtype p), x ∉ m j}
j : Subtype p
reparam : δ → Subtype p → Subtype p := fun x ↦ liminfReparam (fun i ↦ f i x) s p
F0 : Subtype p → δ → α := fun j x ↦ ⨅ i, f (↑i) x
F0_meas : ∀ (j : Subtype p), Measurable (F0 j)
F1 : δ → α := fun x ↦ F0 (reparam x j) x
hF1 : F1 = fun x ↦ F0 (reparam x j) x
g : ℕ → Subtype p := Classical.choose ⋯
Z : ∀ (x : δ), ∃ n, x ∈ m (g n) ∨ ∀ (k : Subtype p), x ∉ m k
x : δ
A : reparam x j = if x ∈ m j then j else g (Nat.find ⋯)
hjx : x ∉ m j
⊢ F1 x = F0 (g (Nat.find ⋯)) x

Tactic:
have : reparam x j = g (Nat.find (Z x)) := by rw [A, if_neg hjx]

Goal state after:
case neg
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_5
ι' : Type u_6
f : ι → δ → α
v : Filter ι
hf : ∀ (i : ι), Measurable (f i)
p : ι' → Prop
s : ι' → Set ι
hv : v.HasCountableBasis p s
hs : ∀ (j : ι'), (s j).Countable
this✝¹ : Countable (Subtype p)
hp : Nonempty (Subtype p)
H : ¬∃ j, s ↑j = ∅
this✝ : ∀ (i : ι'), Countable ↑(s i)
m : Subtype p → Set δ := fun j ↦ {x | BddBelow (range fun i ↦ f (↑i) x)}
m_meas : ∀ (j : Subtype p), MeasurableSet (m j)
mc_meas : MeasurableSet {x | ∀ (j : Subtype p), x ∉ m j}
j : Subtype p
reparam : δ → Subtype p → Subtype p := fun x ↦ liminfReparam (fun i ↦ f i x) s p
F0 : Subtype p → δ → α := fun j x ↦ ⨅ i, f (↑i) x
F0_meas : ∀ (j : Subtype p), Measurable (F0 j)
F1 : δ → α := fun x ↦ F0 (reparam x j) x
hF1 : F1 = fun x ↦ F0 (reparam x j) x
g : ℕ → Subtype p := Classical.choose ⋯
Z : ∀ (x : δ), ∃ n, x ∈ m (g n) ∨ ∀ (k : Subtype p), x ∉ m k
x : δ
A : reparam x j = if x ∈ m j then j else g (Nat.find ⋯)
hjx : x ∉ m j
this : reparam x j = g (Nat.find ⋯)
⊢ F1 x = F0 (g (Nat.find ⋯)) x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

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

Tactic:
gcongr; exact iInf_le _ _

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : Monotone f
c : ℝ≥0 → ℝ≥0∞ := ofNNReal
F : α → ℝ≥0∞ := fun a ↦ ⨆ n, f n a
s : α →ₛ ℝ≥0
hsf : ∀ (x : α), ↑(s x) ≤ ⨆ n, f n x
r : ℝ≥0
right✝ ha✝ : ↑r < 1
ha : r < 1
rs : α →ₛ ℝ≥0 := SimpleFunc.map (fun a ↦ r * a) s
eq_rs : SimpleFunc.map c rs = const α ↑r * SimpleFunc.map c s
x : α
p_eq : ¬c (rs x) = 0
this✝ : r * s x ≠ 0
this : s x ≠ 0
⊢ (SimpleFunc.map c rs) x < ⨆ n, f n x

Tactic:
refine lt_of_lt_of_le (ENNReal.coe_lt_coe.2 ?_) (hsf x)

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : Monotone f
c : ℝ≥0 → ℝ≥0∞ := ofNNReal
F : α → ℝ≥0∞ := fun a ↦ ⨆ n, f n a
s : α →ₛ ℝ≥0
hsf : ∀ (x : α), ↑(s x) ≤ ⨆ n, f n x
r : ℝ≥0
right✝ ha✝ : ↑r < 1
ha : r < 1
rs : α →ₛ ℝ≥0 := SimpleFunc.map (fun a ↦ r * a) s
eq_rs : SimpleFunc.map c rs = const α ↑r * SimpleFunc.map c s
x : α
p_eq : ¬c (rs x) = 0
this✝ : r * s x ≠ 0
this : s x ≠ 0
⊢ rs x < s x
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
m₀ : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoMetrizableSpace β
inst✝ : PseudoMetrizableSpace γ
g : β → γ
f : α → β
hg : IsEmbedding g
this : PseudoMetricSpace γ := pseudoMetrizableSpacePseudoMetric γ
this✝³ : MeasurableSpace β := borel β
this✝² : BorelSpace β
this✝¹ : MeasurableSpace γ := borel γ
this✝ : BorelSpace γ
H : AEStronglyMeasurable (fun x ↦ g (f x)) μ
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂μ, f x ∈ t

Tactic:
· rcases (aestronglyMeasurable_iff_aemeasurable_separable.1 H).2 with ⟨t, ht, h't⟩
    exact ⟨g ⁻¹' t, hg.isSeparable_preimage ht, h't⟩

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

Goal state before:
pp: case inr.refine_1
α : Type u_1
β : Type u_2
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : Nonempty β
inst✝ : T2Space β
hα : Nonempty α
hf : StronglyMeasurable f
fs : ℕ → α →ₛ β := hf.approx
h_fs_tendsto : ∀ (x : α), Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))
this : ∀ (n : ℕ), ∃ c, ∀ (x : α), (fs n) x = c
cs : ℕ → β := fun n ↦ ⋯.choose
h_cs_eq : ∀ (n : ℕ), ⇑(fs n) = fun x ↦ cs n
⊢ f = fun x ↦ f hα.some

Tactic:
conv at h_fs_tendsto => enter [x, 1, n]; rw [h_cs_eq]

Goal state after:
case inr.refine_1
α : Type u_1
β : Type u_2
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : Nonempty β
inst✝ : T2Space β
hα : Nonempty α
hf : StronglyMeasurable f
fs : ℕ → α →ₛ β := hf.approx
this : ∀ (n : ℕ), ∃ c, ∀ (x : α), (fs n) x = c
cs : ℕ → β := fun n ↦ ⋯.choose
h_fs_tendsto : ∀ (x : α), Tendsto (fun n ↦ (fun x ↦ cs n) x) atTop (𝓝 (f x))
h_cs_eq : ∀ (n : ℕ), ⇑(fs n) = fun x ↦ cs n
⊢ f = fun x ↦ f hα.some
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∫⁻ (a : α), f a ∂μ = 0
meas_levels_0 : ∀ ε > 0, μ {x | ε ≤ f x} = 0
u : ℕ → ℝ≥0∞
bu : ∀ (n : ℕ), u n ∈ Ioo 0 1
tu : Tendsto u atTop (𝓝 0)
⊢ f =ᵐ[μ] 0

Tactic:
have u_union : {x | f x ≠ 0} = ⋃ n, {x | u n ≤ f x} := by
    ext x
    rw [mem_iUnion, mem_ofPred_eq, ← pos_iff_ne_zero]
    rw [ENNReal.tendsto_atTop_zero] at tu
    constructor <;> intro h'
    · obtain ⟨n, hn⟩ := tu _ h'; use n, hn _ le_rfl
    · obtain ⟨n, hn⟩ := h'; exact (bu n).1.trans_le hn

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∫⁻ (a : α), f a ∂μ = 0
meas_levels_0 : ∀ ε > 0, μ {x | ε ≤ f x} = 0
u : ℕ → ℝ≥0∞
bu : ∀ (n : ℕ), u n ∈ Ioo 0 1
tu : Tendsto u atTop (𝓝 0)
u_union : {x | f x ≠ 0} = ⋃ n, {x | u n ≤ f x}
⊢ f =ᵐ[μ] 0
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

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

Tactic:
rw [← ENNReal.mul_iSup, lintegral_eq_iSup_eapprox_lintegral hf]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
μ : Measure α
f : α → β
inst✝² : Zero β
inst✝¹ : TopologicalSpace β
inst✝ : T2Space β
fs : ℕ → α →ₛ β
hT_lt_top : ∀ (n : ℕ), μ (support ⇑(fs n)) < ∞
h_approx : ∀ (x : α), Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))
T : ℕ → Set α := fun n ↦ support ⇑(fs n)
hT_meas : ∀ (n : ℕ), MeasurableSet (T n)
t : Set α := ⋃ n, T n
n : ℕ
x : α
hxt : x ∈ tᶜ
⊢ (fs n) x = 0

Tactic:
Set.mem_compl_iff,

Goal state after:
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
μ : Measure α
f : α → β
inst✝² : Zero β
inst✝¹ : TopologicalSpace β
inst✝ : T2Space β
fs : ℕ → α →ₛ β
hT_lt_top : ∀ (n : ℕ), μ (support ⇑(fs n)) < ∞
h_approx : ∀ (x : α), Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))
T : ℕ → Set α := fun n ↦ support ⇑(fs n)
hT_meas : ∀ (n : ℕ), MeasurableSet (T n)
t : Set α := ⋃ n, T n
n : ℕ
x : α
hxt : x ∉ t
⊢ (fs n) x = 0
Module: Mathlib.Analysis.Normed.Group.InfiniteSum

Goal state before:
pp: E : Type u_3
inst✝ : SeminormedAddCommGroup E
f : ℕ → E
g : ℕ → ℝ
hg : CauchySeq fun n ↦ ∑ i ∈ range n, g i
hf : ∀ (i : ℕ), ‖f i‖ ≤ g i
⊢ CauchySeq fun n ↦ ∑ i ∈ range n, f i

Tactic:
refine Metric.cauchySeq_iff'.2 fun ε hε => ?_

Goal state after:
E : Type u_3
inst✝ : SeminormedAddCommGroup E
f : ℕ → E
g : ℕ → ℝ
hg : CauchySeq fun n ↦ ∑ i ∈ range n, g i
hf : ∀ (i : ℕ), ‖f i‖ ≤ g i
ε : ℝ
hε : ε > 0
⊢ ∃ N, ∀ n ≥ N, dist (∑ i ∈ range n, f i) (∑ i ∈ range N, f i) < ε
Module: Mathlib.Analysis.Normed.Group.InfiniteSum

Goal state before:
pp: case pos
ι : Type u_1
ε : Type u_5
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddCommMonoid ε
f : ι → ε
g : ι → ℝ≥0∞
a : ℝ≥0∞
hg : HasSum g a
h : ∀ (i : ι), ‖f i‖ₑ ≤ g i
hf : Summable f
⊢ ‖∑' (i : ι), f i‖ₑ ≤ a

Tactic:
exact hf.hasSum.enorm_le_of_bounded hg h

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: t : ↑I
⊢ 1 / 2 + ↑t ≤ 1 ↔ ↑t ≤ 1 / 2

Tactic:
add_comm,

Goal state after:
t : ↑I
⊢ ↑t + 1 / 2 ≤ 1 ↔ ↑t ≤ 1 / 2
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: α : Type u_1
inst✝³ : AddCommGroup α
inst✝² : LinearOrder α
inst✝¹ : IsOrderedAddMonoid α
a b : α
h : a ≤ b
δ : α
inst✝ : Archimedean α
hδ : 0 < δ
m : ℕ
hm : b - a ≤ m • δ
n : ℕ
hn : n ≥ m
⊢ ↑(projIcc a b h (a + n • δ)) = b

Tactic:
coe_projIcc,

Goal state after:
α : Type u_1
inst✝³ : AddCommGroup α
inst✝² : LinearOrder α
inst✝¹ : IsOrderedAddMonoid α
a b : α
h : a ≤ b
δ : α
inst✝ : Archimedean α
hδ : 0 < δ
m : ℕ
hm : b - a ≤ m • δ
n : ℕ
hn : n ≥ m
⊢ max a (min b (a + n • δ)) = b
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: α : Type u_1
inst✝³ : AddCommGroup α
inst✝² : LinearOrder α
inst✝¹ : IsOrderedAddMonoid α
a b : α
h : a ≤ b
δ : α
inst✝ : Archimedean α
hδ : 0 < δ
m : ℕ
hm : b - a ≤ m • δ
n : ℕ
hn : n ≥ m
⊢ max a (min b (a + n • δ)) = b

Tactic:
add_comm,

Goal state after:
α : Type u_1
inst✝³ : AddCommGroup α
inst✝² : LinearOrder α
inst✝¹ : IsOrderedAddMonoid α
a b : α
h : a ≤ b
δ : α
inst✝ : Archimedean α
hδ : 0 < δ
m : ℕ
hm : b - a ≤ m • δ
n : ℕ
hn : n ≥ m
⊢ max a (min b (n • δ + a)) = b
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : Nonempty Z
Φ : Z → X
Ψ : Z → Y
ε : ℝ
H : ∀ (p q : Z), |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε
x : Y
y : X
z : Y
⊢ glueDist Φ Ψ ε (Sum.inr x) (Sum.inr z) ≤
    glueDist Φ Ψ ε (Sum.inr x) (Sum.inl y) + glueDist Φ Ψ ε (Sum.inl y) (Sum.inr z)

Tactic:
by
    simp only [← glueDist_swap Φ]
    apply glueDist_triangle_inl_inr_inl
    simpa only [abs_sub_comm]

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

Goal state before:
pp: case mpr
E : ℕ → Type u_1
x : (n : ℕ) → E n
n : ℕ
y : (n : ℕ) → E n
⊢ (∀ i < n, y i = x i) → ∃ i, ∀ i_1 < n + 1, y i_1 = update x n i i_1

Tactic:
intro H

Goal state after:
case mpr
E : ℕ → Type u_1
x : (n : ℕ) → E n
n : ℕ
y : (n : ℕ) → E n
H : ∀ i < n, y i = x i
⊢ ∃ i, ∀ i_1 < n + 1, y i_1 = update x n i i_1
Module: Mathlib.Topology.MetricSpace.Gluing

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

Tactic:
by cases x <;> simp only [Sum.dist, dist_self]

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

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

Tactic:
intro H

Goal state after:
case mp
E : ℕ → Type u_1
α : Type u_2
inst✝ : PseudoMetricSpace α
f : ((n : ℕ) → E n) → α
H : ∀ (x y : (n : ℕ) → E n), dist (f x) (f y) ≤ dist x y
⊢ ∀ (x y : (n : ℕ) → E n) (n : ℕ), y ∈ cylinder x n → dist (f x) (f y) ≤ (1 / 2) ^ n
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: ι : Sort u_1
c : ι → Set (↑I × ↑I)
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : univ ⊆ ⋃ i, c i
δ : ℝ
δ_pos : δ > 0
ball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i
hδ : 0 < δ / 2
⊢ ∃ t,
    ↑(t 0) = ↑0 ∧
      Monotone t ∧ (∃ n, ∀ m ≥ n, ↑(t m) = ↑1) ∧ ∀ (n m : ℕ), ∃ i, Icc (t n) (t (n + 1)) ×ˢ Icc (t m) (t (m + 1)) ⊆ c i

Tactic:
have h : (0 : ℝ) ≤ 1 := zero_le_one

Goal state after:
ι : Sort u_1
c : ι → Set (↑I × ↑I)
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : univ ⊆ ⋃ i, c i
δ : ℝ
δ_pos : δ > 0
ball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i
hδ : 0 < δ / 2
h : 0 ≤ 1
⊢ ∃ t,
    ↑(t 0) = ↑0 ∧
      Monotone t ∧ (∃ n, ∀ m ≥ n, ↑(t m) = ↑1) ∧ ∀ (n m : ℕ), ∃ i, Icc (t n) (t (n + 1)) ×ˢ Icc (t m) (t (m + 1)) ⊆ c i
Module: Mathlib.Topology.MetricSpace.PiNat

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

Tactic:
intro H

Goal state after:
case mpr
E : ℕ → Type u_1
α : Type u_2
inst✝ : PseudoMetricSpace α
f : ((n : ℕ) → E n) → α
H : ∀ (x y : (n : ℕ) → E n) (n : ℕ), y ∈ cylinder x n → dist (f x) (f y) ≤ (1 / 2) ^ n
⊢ ∀ (x y : (n : ℕ) → E n), dist (f x) (f y) ≤ dist x y
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case h_nhds
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
x : (n : ℕ) → E n
u : Set ((n : ℕ) → E n)
hx : x ∈ u
u_open : IsOpen[Pi.topologicalSpace] u
U : (i : ℕ) → Set (E i)
F : Finset ℕ
xU : x ∈ (↑F).pi U
Uu : (↑F).pi U ⊆ u
n : ℕ
hn : n ∈ upperBounds ↑F
y : (n : ℕ) → E n
hy : y ∈ cylinder x (n + 1)
⊢ y ∈ (↑F).pi U

Tactic:
suffices ∀ i : ℕ, i ∈ F → y i ∈ U i by simpa

Goal state after:
case h_nhds
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
x : (n : ℕ) → E n
u : Set ((n : ℕ) → E n)
hx : x ∈ u
u_open : IsOpen[Pi.topologicalSpace] u
U : (i : ℕ) → Set (E i)
F : Finset ℕ
xU : x ∈ (↑F).pi U
Uu : (↑F).pi U ⊆ u
n : ℕ
hn : n ∈ upperBounds ↑F
y : (n : ℕ) → E n
hy : y ∈ cylinder x (n + 1)
⊢ ∀ i ∈ F, y i ∈ U i
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : ℕ → Type u
inst✝ : (n : ℕ) → MetricSpace (X n)
f : (n : ℕ) → X n → X (n + 1)
I : ∀ (n : ℕ), Isometry (f n)
x y : (n : ℕ) × X n
m : ℕ
hx : x.fst ≤ m + 1
hy : y.fst ≤ m + 1
⊢ inductiveLimitDist f x y = dist (leRecOn hx (fun {k} ↦ f k) x.snd) (leRecOn hy (fun {k} ↦ f k) y.snd)

Tactic:
by_cases h : max x.1 y.1 = (m + 1)

Goal state after:
case pos
X : ℕ → Type u
inst✝ : (n : ℕ) → MetricSpace (X n)
f : (n : ℕ) → X n → X (n + 1)
I : ∀ (n : ℕ), Isometry (f n)
x y : (n : ℕ) × X n
m : ℕ
hx : x.fst ≤ m + 1
hy : y.fst ≤ m + 1
h : max x.fst y.fst = m + 1
⊢ inductiveLimitDist f x y = dist (leRecOn hx (fun {k} ↦ f k) x.snd) (leRecOn hy (fun {k} ↦ f k) y.snd)

case neg
X : ℕ → Type u
inst✝ : (n : ℕ) → MetricSpace (X n)
f : (n : ℕ) → X n → X (n + 1)
I : ∀ (n : ℕ), Isometry (f n)
x y : (n : ℕ) × X n
m : ℕ
hx : x.fst ≤ m + 1
hy : y.fst ≤ m + 1
h : ¬max x.fst y.fst = m + 1
⊢ inductiveLimitDist f x y = dist (leRecOn hx (fun {k} ↦ f k) x.snd) (leRecOn hy (fun {k} ↦ f k) y.snd)
Module: Mathlib.Topology.MetricSpace.Perfect

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
this✝ : x ∈ eball x (ε / 2)
this : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty
⊢ closure (eball x (ε / 2) ∩ C) ⊆ C

Tactic:
rw [IsClosed.closure_subset_iff hC.closed]

Goal state after:
case refine_1
α : Type u_1
inst✝ : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
this✝ : x ∈ eball x (ε / 2)
this : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty
⊢ eball x (ε / 2) ∩ C ⊆ C
Module: Mathlib.Topology.MetricSpace.Perfect

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

Tactic:
rw [mul_comm, ENNReal.div_mul_cancel]

Goal state after:
case e'_4.ha₀
α : Type u_1
inst✝ : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
this✝ : x ∈ eball x (ε / 2)
this : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty
⊢ 2 ≠ 0

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

Goal state before:
pp: case cons
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)
a : Bool
l : List Bool
ih : P
⊢ P

Tactic:
| cons a l ih =>

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

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
g : α → β
hg : MeasurableEmbedding g
f : β → ℝ≥0∞
f₀ : SimpleFunc β ℝ≥0∞
hf₀ : ⇑f₀ ≤ fun a ↦ f a
⊢ f₀.lintegral (Measure.map g μ) ≤ ⨆ g_1, ⨆ (_ : ⇑g_1 ≤ fun a ↦ f (g a)), g_1.lintegral μ

Tactic:
rw [SimpleFunc.lintegral_map _ hg.measurable]
    have : (f₀.comp g hg.measurable : α → ℝ≥0∞) ≤ f ∘ g := fun x => hf₀ (g x)
    exact le_iSup_of_le (comp f₀ g hg.measurable) (by exact le_iSup (α := ℝ≥0∞) _ this)

Goal state after: