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

Goal state before:
pp: case inr.inr
α : Type u
inst✝ : PseudoEMetricSpace α
ε δ : ℝ
s : Set α
hε : 0 ≤ ε
hδ : 0 ≤ δ
x : α
⊢ x ∈ thickening ε (thickening δ s) → x ∈ thickening (ε + δ) s

Tactic:
[

Goal state after:
case inr.inr
α : Type u
inst✝ : PseudoEMetricSpace α
ε δ : ℝ
s : Set α
hε : 0 ≤ ε
hδ : 0 ≤ δ
x : α
⊢ x ∈ thickening ε (thickening δ s) → x ∈ thickening (ε + δ) s
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: β : Type u_2
α : Type u_5
f : α →ₛ β
inst✝ : Nonempty β
hf_meas : ∀ (x : β), MeasurableSet (⇑f ⁻¹' {x})
⊢ ∃ c, ∀ (x : α), f x = c

Tactic:
[

Goal state after:
β : Type u_2
α : Type u_5
f : α →ₛ β
inst✝ : Nonempty β
hf_meas : ∀ (x : β), MeasurableSet (⇑f ⁻¹' {x})
⊢ ∃ c, ∀ (x : α), f x = c
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
r : α → β → Prop
f : α →ₛ β
h : ∀ (b : β), MeasurableSet {a | r a b}
this : {a | r a (f a)} = ⋃ b ∈ range ⇑f, {a | r a b} ∩ ⇑f ⁻¹' {b}
⊢ MeasurableSet {a | r a (f a)}

Tactic:
this

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
r : α → β → Prop
f : α →ₛ β
h : ∀ (b : β), MeasurableSet {a | r a b}
this : {a | r a (f a)} = ⋃ b ∈ range ⇑f, {a | r a b} ∩ ⇑f ⁻¹' {b}
⊢ MeasurableSet (⋃ b ∈ range ⇑f, {a | r a b} ∩ ⇑f ⁻¹' {b})
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: case inr
α : Type u
inst✝ : PseudoEMetricSpace α
δ ε : ℝ
hδ : 0 ≤ δ
s : Set α
hε : 0 ≤ ε
x : α
⊢ x ∈ thickening ε (cthickening δ s) → x ∈ thickening (ε + δ) s

Tactic:
[

Goal state after:
case inr
α : Type u
inst✝ : PseudoEMetricSpace α
δ ε : ℝ
hδ : 0 ≤ δ
s : Set α
hε : 0 ≤ ε
x : α
⊢ x ∈ thickening ε (cthickening δ s) → x ∈ thickening (ε + δ) s
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case refine_3
α : Type u_5
inst✝⁴ : TopologicalSpace α
m : MeasurableSpace α
inst✝³ : SecondCountableTopology α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : BorelSpace α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a ≤ b → μ (Icc a b) ≠ ∞
h : ∀ ⦃a b : α⦄, a ≤ b → μ (Icc a b) = ν (Icc a b)
s : Set α
hsc : s.Countable
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
hst : ∀ (x : α), IsTop x → x ∈ s
this : {x | ∃ i, ∃ (_ : i ∈ s), ∃ i_2, ∃ (_ : i_2 ∈ s) (_ : i ≤ i_2), x = Icc i i_2}.Countable
l u : α
hlt : l ≤ u
⊢ μ (Icc l u) ≠ ∞

Tactic:
exact hμ hlt

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

Goal state before:
pp: case inr
α : Type u
inst✝ : PseudoEMetricSpace α
ε : ℝ
hε : 0 ≤ ε
δ : ℝ
s : Set α
hδ : 0 ≤ δ
x : α
⊢ x ∈ cthickening ε (thickening δ s) → x ∈ cthickening (ε + δ) s

Tactic:
[

Goal state after:
case inr
α : Type u
inst✝ : PseudoEMetricSpace α
ε : ℝ
hε : 0 ≤ ε
δ : ℝ
s : Set α
hδ : 0 ≤ δ
x : α
⊢ x ∈ cthickening ε (thickening δ s) → x ∈ cthickening (ε + δ) s
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
δ ε : ℝ
hε : 0 ≤ ε
hδ : 0 ≤ δ
s : Set α
x : α
⊢ x ∈ cthickening ε (cthickening δ s) → x ∈ cthickening (ε + δ) s

Tactic:
[

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
δ ε : ℝ
hε : 0 ≤ ε
hδ : 0 ≤ δ
s : Set α
x : α
⊢ x ∈ cthickening ε (cthickening δ s) → x ∈ cthickening (ε + δ) s
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: β : Type u_2
δ : Type u_4
inst✝⁶ : TopologicalSpace β
mβ : MeasurableSpace β
inst✝⁵ : BorelSpace β
mδ : MeasurableSpace δ
inst✝⁴ : CompleteLinearOrder β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
ι : Type u_5
inst✝¹ : TopologicalSpace ι
inst✝ : SeparableSpace ι
f : ι → δ → β
mf : ∀ (t : ι), Measurable (f t)
cf : ∀ (x : δ), LowerSemicontinuous fun x_1 ↦ f x_1 x
c : β
J : Set ι
cJ : J.Countable
dJ : Dense J
this : (⨆ i, f i) ⁻¹' Ioi c = ⋃ j ∈ J, {x | c < f j x}
⊢ MeasurableSet ((⨆ i, f i) ⁻¹' Ioi c)

Tactic:
this

Goal state after:
β : Type u_2
δ : Type u_4
inst✝⁶ : TopologicalSpace β
mβ : MeasurableSpace β
inst✝⁵ : BorelSpace β
mδ : MeasurableSpace δ
inst✝⁴ : CompleteLinearOrder β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
ι : Type u_5
inst✝¹ : TopologicalSpace ι
inst✝ : SeparableSpace ι
f : ι → δ → β
mf : ∀ (t : ι), Measurable (f t)
cf : ∀ (x : δ), LowerSemicontinuous fun x_1 ↦ f x_1 x
c : β
J : Set ι
cJ : J.Countable
dJ : Dense J
this : (⨆ i, f i) ⁻¹' Ioi c = ⋃ j ∈ J, {x | c < f j x}
⊢ MeasurableSet (⋃ j ∈ J, {x | c < f j x})
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: β : Type u_2
δ : Type u_4
inst✝⁶ : TopologicalSpace β
mβ : MeasurableSpace β
inst✝⁵ : BorelSpace β
mδ : MeasurableSpace δ
inst✝⁴ : CompleteLinearOrder β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
ι : Type u_5
inst✝¹ : TopologicalSpace ι
inst✝ : SeparableSpace ι
f : ι → δ → β
mf : ∀ (t : ι), Measurable (f t)
cf : ∀ (x : δ), LowerSemicontinuous fun x_1 ↦ f x_1 x
c : β
J : Set ι
cJ : J.Countable
dJ : Dense J
x : δ
⊢ c < (⨆ i, f i) x ↔ ∃ i, ∃ (_ : i ∈ J), c < f i x

Tactic:
[

Goal state after:
β : Type u_2
δ : Type u_4
inst✝⁶ : TopologicalSpace β
mβ : MeasurableSpace β
inst✝⁵ : BorelSpace β
mδ : MeasurableSpace δ
inst✝⁴ : CompleteLinearOrder β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
ι : Type u_5
inst✝¹ : TopologicalSpace ι
inst✝ : SeparableSpace ι
f : ι → δ → β
mf : ∀ (t : ι), Measurable (f t)
cf : ∀ (x : δ), LowerSemicontinuous fun x_1 ↦ f x_1 x
c : β
J : Set ι
cJ : J.Countable
dJ : Dense J
x : δ
⊢ c < (⨆ i, f i) x ↔ ∃ i, ∃ (_ : i ∈ J), c < f i x
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 : SemilatticeSup β
inst✝ : OrderBot β
f : γ → α →ₛ β
s : Finset γ
a : α
⊢ (s.sup f) a = s.sup fun c ↦ (f c) a

Tactic:
refine Finset.induction_on s rfl ?_
  intro a s _ ih
  rw [Finset.sup_insert, Finset.sup_insert, sup_apply, ih]

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

Goal state before:
pp: α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iic x)
⊢ ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x)

Tactic:
[

Goal state after:
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iic x)
⊢ ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ici x)
⊢ ∀ (x : α), MeasurableSet (f ⁻¹' Iio x)

Tactic:
[

Goal state after:
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ici x)
⊢ ∀ (x : α), MeasurableSet (f ⁻¹' Iio x)
Module: Mathlib.MeasureTheory.MeasurableSpace.Prod

Goal state before:
pp: case a.refine_2
α : Type u_3
β : Type u_4
C : Set (Set α)
D : Set (Set β)
hD : IsCountablySpanning D
s : Set β
hs : s ∈ D
t : ℕ → Set α
h1t : ∀ (n : ℕ), t n ∈ C
h2t : ⋃ n, t n = univ
n : ℕ
⊢ t n ×ˢ s ∈ image2 (fun x1 x2 ↦ x1 ×ˢ x2) C D

Tactic:
exact mem_image2_of_mem (h1t n) hs

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

Goal state before:
pp: α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsLUB (range fun i ↦ f i b) (g b)
a : α
⊢ MeasurableSet (g ⁻¹' Ioi a)

Tactic:
[

Goal state after:
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsLUB (range fun i ↦ f i b) (g b)
a : α
⊢ MeasurableSet (g ⁻¹' Ioi a)
Module: Mathlib.MeasureTheory.MeasurableSpace.Prod

Goal state before:
pp: α : Type u_3
β : Type u_4
γ : Type u_5
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
X : α → β
Y : α → γ
⊢ MeasurableSpace.comap (fun ω ↦ (X ω, Y ω)) (mβ.prod mγ) = MeasurableSpace.comap X mβ ⊔ MeasurableSpace.comap Y mγ

Tactic:
[

Goal state after:
α : Type u_3
β : Type u_4
γ : Type u_5
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
X : α → β
Y : α → γ
⊢ MeasurableSpace.comap (fun ω ↦ (X ω, Y ω)) (mβ.prod mγ) = MeasurableSpace.comap X mβ ⊔ MeasurableSpace.comap Y mγ
Module: Mathlib.MeasureTheory.MeasurableSpace.Prod

Goal state before:
pp: α : Type u_3
β : Type u_4
γ : Type u_5
δ : Type u_6
mα : MeasurableSpace α
mβ : MeasurableSpace β
X : γ → α
Y : δ → β
⊢ MeasurableSpace.comap (Prod.map X Y) (mα.prod mβ) = (MeasurableSpace.comap X mα).prod (MeasurableSpace.comap Y mβ)

Tactic:
[

Goal state after:
α : Type u_3
β : Type u_4
γ : Type u_5
δ : Type u_6
mα : MeasurableSpace α
mβ : MeasurableSpace β
X : γ → α
Y : δ → β
⊢ MeasurableSpace.comap (Prod.map X Y) (mα.prod mβ) = (MeasurableSpace.comap X mα).prod (MeasurableSpace.comap Y mβ)
Module: Mathlib.Algebra.Order.Monoid.Canonical.Basic

Goal state before:
pp: α : Type u_1
inst✝⁴ : LinearOrder α
P : α → Prop
b c : α
inst✝³ : Add α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftReflectLT α
inst✝ : IsLeftCancelAdd α
⊢ (∀ a < b + c, P a) ↔ (∀ a < b, P a) ∧ ∀ d < c, P (b + d)

Tactic:
[

Goal state after:
α : Type u_1
inst✝⁴ : LinearOrder α
P : α → Prop
b c : α
inst✝³ : Add α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftReflectLT α
inst✝ : IsLeftCancelAdd α
⊢ (∀ a < b + c, P a) ↔ (∀ a < b, P a) ∧ ∀ d < c, P (b + d)
Module: Mathlib.Algebra.Order.Monoid.Canonical.Basic

Goal state before:
pp: α : Type u_1
inst✝⁴ : LinearOrder α
P : α → Prop
b c : α
inst✝³ : Add α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftReflectLT α
inst✝ : IsLeftCancelAdd α
⊢ (∃ a < b + c, P a) ↔ (∃ a < b, P a) ∨ ∃ d < c, P (b + d)

Tactic:
[

Goal state after:
α : Type u_1
inst✝⁴ : LinearOrder α
P : α → Prop
b c : α
inst✝³ : Add α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftReflectLT α
inst✝ : IsLeftCancelAdd α
⊢ (∃ a < b + c, P a) ↔ (∃ a < b, P a) ∨ ∃ d < c, P (b + d)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case inl.inr
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
g g' : δ → α
hf : ∀ (i : ι), Measurable (f i)
s : Set δ
hs : MeasurableSet s
hg : ∀ b ∈ s, IsLUB {a | ∃ i, f i b = a} (g b)
hg' : EqOn g g' sᶜ
g'_meas : Measurable g'
hι : IsEmpty ι
x : δ
hx : x ∈ s
A : ∀ b ∈ s, IsBot (g b)
B : ∀ b ∈ s, g b = g x
this : g = s.piecewise (fun _y ↦ g x) g'
⊢ Measurable g

Tactic:
this

Goal state after:
case inl.inr
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
g g' : δ → α
hf : ∀ (i : ι), Measurable (f i)
s : Set δ
hs : MeasurableSet s
hg : ∀ b ∈ s, IsLUB {a | ∃ i, f i b = a} (g b)
hg' : EqOn g g' sᶜ
g'_meas : Measurable g'
hι : IsEmpty ι
x : δ
hx : x ∈ s
A : ∀ b ∈ s, IsBot (g b)
B : ∀ b ∈ s, g b = g x
this : g = s.piecewise (fun _y ↦ g x) g'
⊢ Measurable (s.piecewise (fun _y ↦ g x) g')
Module: Mathlib.Algebra.Order.Monoid.Canonical.Basic

Goal state before:
pp: α : Type u_1
inst✝⁴ : LinearOrder α
P : α → Prop
b c : α
inst✝³ : Add α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftMono α
inst✝ : IsLeftCancelAdd α
⊢ (∀ a ≤ b + c, P a) ↔ (∀ a < b, P a) ∧ ∀ d ≤ c, P (b + d)

Tactic:
[

Goal state after:
α : Type u_1
inst✝⁴ : LinearOrder α
P : α → Prop
b c : α
inst✝³ : Add α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftMono α
inst✝ : IsLeftCancelAdd α
⊢ (∀ a ≤ b + c, P a) ↔ (∀ a < b, P a) ∧ ∀ d ≤ c, P (b + d)
Module: Mathlib.Algebra.Order.Monoid.Canonical.Basic

Goal state before:
pp: α : Type u_1
inst✝⁴ : LinearOrder α
P : α → Prop
b c : α
inst✝³ : Add α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftMono α
inst✝ : IsLeftCancelAdd α
⊢ (∃ a ≤ b + c, P a) ↔ (∃ a < b, P a) ∨ ∃ d ≤ c, P (b + d)

Tactic:
[

Goal state after:
α : Type u_1
inst✝⁴ : LinearOrder α
P : α → Prop
b c : α
inst✝³ : Add α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftMono α
inst✝ : IsLeftCancelAdd α
⊢ (∃ a ≤ b + c, P a) ↔ (∃ a < b, P a) ∨ ∃ d ≤ c, P (b + d)
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 < c ∨ ∃ d < b, a = c + d) ↔ a < c ∨ ∃ d < b, a = d + c

Tactic:
[

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 ∨ ∃ d < b, a = c + d) ↔ 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
P : α → Prop
b c : α
inst✝³ : AddCommMagma α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftReflectLT α
inst✝ : IsLeftCancelAdd α
⊢ (∀ a < b + c, P a) ↔ (∀ a < c, P a) ∧ ∀ d < b, P (d + c)

Tactic:
[

Goal state after:
α : Type u_1
inst✝⁴ : LinearOrder α
P : α → Prop
b c : α
inst✝³ : AddCommMagma α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftReflectLT α
inst✝ : IsLeftCancelAdd α
⊢ (∀ a < b + c, P a) ↔ (∀ a < c, P a) ∧ ∀ d < b, P (d + c)
Module: Mathlib.Algebra.Order.Monoid.Canonical.Basic

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

Tactic:
[

Goal state after:
α : Type u_1
inst✝⁴ : LinearOrder α
P : α → Prop
b c : α
inst✝³ : AddCommMagma α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftReflectLT α
inst✝ : IsLeftCancelAdd α
⊢ (∃ a < b + c, P a) ↔ (∃ a < c, P a) ∨ ∃ d < b, P (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 < c ∨ ∃ d ≤ b, a = c + d) ↔ a < c ∨ ∃ d ≤ b, a = d + c

Tactic:
[

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 ∨ ∃ d ≤ b, a = c + d) ↔ a < c ∨ ∃ d ≤ b, a = d + c
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ≥0∞
n : ℕ
a : α
⊢ (eapprox f n) a < ∞

Tactic:
by
  simp only [eapprox, approx, finset_sup_apply, restrict]
  rw [Finset.sup_lt_iff (α := ℝ≥0∞) bot_lt_top]
  intro b _
  split_ifs
  · simp only [coe_zero, coe_piecewise, piecewise_eq_indicator, coe_const]
    calc
      { a : α | ennrealRatEmbed b ≤ f a }.indicator (fun _ => ennrealRatEmbed b) a ≤
          ennrealRatEmbed b :=
        indicator_le_self _ _ a
      _ < ⊤ := ENNReal.coe_lt_top
  · exact WithTop.top_pos

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : LinearOrder α
P : α → Prop
b c : α
inst✝³ : AddCommMagma α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftMono α
inst✝ : IsLeftCancelAdd α
⊢ (∀ a ≤ b + c, P a) ↔ (∀ a < c, P a) ∧ ∀ d ≤ b, P (d + c)

Tactic:
[

Goal state after:
α : Type u_1
inst✝⁴ : LinearOrder α
P : α → Prop
b c : α
inst✝³ : AddCommMagma α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftMono α
inst✝ : IsLeftCancelAdd α
⊢ (∀ a ≤ b + c, P a) ↔ (∀ a < c, P a) ∧ ∀ d ≤ b, P (d + c)
Module: Mathlib.Algebra.Order.Monoid.Canonical.Basic

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

Tactic:
[

Goal state after:
α : Type u_1
inst✝⁴ : LinearOrder α
P : α → Prop
b c : α
inst✝³ : AddCommMagma α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftMono α
inst✝ : IsLeftCancelAdd α
⊢ (∃ a ≤ b + c, P a) ↔ (∃ a < c, P a) ∨ ∃ d ≤ b, P (d + c)
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ≥0∞
hf : Measurable f
a : α
⊢ ∑' (n : ℕ), ↑((eapproxDiff f n) a) = f a

Tactic:
[

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ≥0∞
hf : Measurable f
a : α
⊢ ∑' (n : ℕ), ↑((eapproxDiff f n) a) = f a
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i) μ
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
a✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
p : δ → (ι → α) → Prop := fun x f' ↦ IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := (aeSeqSet hf p).piecewise g fun x ↦ hα.some
b : δ
h : b ∈ aeSeqSet hf p
x : α
⊢ x ∈ {a | ∃ i, mk (f i) ⋯ b = a} ↔ x ∈ {a | ∃ i, f i b = a}

Tactic:
[

Goal state after:
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i) μ
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
a✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
p : δ → (ι → α) → Prop := fun x f' ↦ IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := (aeSeqSet hf p).piecewise g fun x ↦ hα.some
b : δ
h : b ∈ aeSeqSet hf p
x : α
⊢ x ∈ {a | ∃ i, mk (f i) ⋯ b = a} ↔ x ∈ {a | ∃ i, f i b = a}
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
M : α → α
hM : ∀ x ∈ s, x < M x
H : (s \ interior s).Countable
this : s = interior s ∪ s \ interior s
⊢ MeasurableSet s

Tactic:
this

Goal state after:
α : Type u_1
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
M : α → α
hM : ∀ x ∈ s, x < M x
H : (s \ interior s).Countable
this : s = interior s ∪ s \ interior s
⊢ MeasurableSet (interior s ∪ s \ interior s)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case pos
α : Type u_1
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
this : s = {x₀} ∪ s \ {x₀}
⊢ MeasurableSet s

Tactic:
this

Goal state after:
case pos
α : Type u_1
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
this : s = {x₀} ∪ s \ {x₀}
⊢ MeasurableSet ({x₀} ∪ s \ {x₀})
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
⊢ MeasurableSet s

Tactic:
by_cases H : ∃ x ∈ s, IsTop x
  · rcases H with ⟨x₀, x₀s, h₀⟩
    have : s = { x₀ } ∪ s \ { x₀ } := by rw [union_sdiff_cancel (singleton_subset_iff.2 x₀s)]
    rw [this]
    refine (measurableSet_singleton _).union ?_
    have A : ∀ x ∈ s \ { x₀ }, x < x₀ := fun x hx => lt_of_le_of_ne (h₀ _) (by simpa using hx.2)
    refine .of_mem_nhdsGT_aux (fun x hx => ?_) fun x hx => ⟨x₀, A x hx⟩
    obtain ⟨u, hu, us⟩ : ∃ (u : α), u ∈ Ioi x ∧ Ioo x u ⊆ s :=
      (mem_nhdsGT_iff_exists_Ioo_subset' (A x hx)).1 (h x hx.1)
    refine (mem_nhdsGT_iff_exists_Ioo_subset' (A x hx)).2 ⟨u, hu, fun y hy => ⟨us hy, ?_⟩⟩
    exact ne_of_lt (hy.2.trans_le (h₀ _))
  · refine .of_mem_nhdsGT_aux h ?_
    simp only [IsTop] at H
    push Not at H
    exact H

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α →ₛ ℝ≥0∞
x : ℝ≥0∞
⊢ ∑ r ∈ f.range, x * r * μ (⇑f ⁻¹' {r}) = x * ∑ r ∈ f.range, r * μ (⇑f ⁻¹' {r})

Tactic:
[

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α →ₛ ℝ≥0∞
x : ℝ≥0∞
⊢ ∑ r ∈ f.range, x * r * μ (⇑f ⁻¹' {r}) = x * ∑ r ∈ f.range, r * μ (⇑f ⁻¹' {r})
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: δ : Type u_4
mδ : MeasurableSpace δ
α : Type u_5
mα : MeasurableSpace α
inst✝ : ConditionallyCompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
⊢ Measurable fun b ↦ ⨆ (_ : p), f b

Tactic:
[

Goal state after:
δ : Type u_4
mδ : MeasurableSpace δ
α : Type u_5
mα : MeasurableSpace α
inst✝ : ConditionallyCompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
⊢ Measurable fun b ↦ ⨆ (_ : p), f b
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: δ : Type u_4
mδ : MeasurableSpace δ
α : Type u_5
mα : MeasurableSpace α
inst✝ : ConditionallyCompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
⊢ Measurable fun b ↦ ⨅ (_ : p), f b

Tactic:
[

Goal state after:
δ : Type u_4
mδ : MeasurableSpace δ
α : Type u_5
mα : MeasurableSpace α
inst✝ : ConditionallyCompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
⊢ Measurable fun b ↦ ⨅ (_ : p), f b
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: 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
f : ι → δ → α
s : Set ι
hs : s.Countable
hf : ∀ i ∈ s, Measurable (f i)
⊢ Measurable fun x ↦ sSup ((fun i ↦ f i x) '' s)

Tactic:
[

Goal state after:
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
f : ι → δ → α
s : Set ι
hs : s.Countable
hf : ∀ i ∈ s, Measurable (f i)
⊢ Measurable fun x ↦ sSup ((fun i ↦ f i x) '' s)