mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Constructions.Projective

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
I J : Finset ι
hP : IsProjectiveMeasureFamily P
S : Set ((i : ↥I) → α ↑i)
T : Set ((i : ↥J) → α ↑i)
hS : MeasurableSet S
hT : MeasurableSet T
h_eq : cylinder I S = cylinder J T
U : Set ((i : ↥(I ∪ J)) → α ↑i) := Finset.restrict₂ ⋯ ⁻¹' S ∩ Finset.restrict₂ ⋯ ⁻¹' T
⊢ cylinder J T = cylinder J T ∩ cylinder J T

Tactic:
inter_self

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
I J : Finset ι
hP : IsProjectiveMeasureFamily P
S : Set ((i : ↥I) → α ↑i)
T : Set ((i : ↥J) → α ↑i)
hS : MeasurableSet S
hT : MeasurableSet T
h_eq : cylinder I S = cylinder J T
U : Set ((i : ↥(I ∪ J)) → α ↑i) := Finset.restrict₂ ⋯ ⁻¹' S ∩ Finset.restrict₂ ⋯ ⁻¹' T
⊢ cylinder J T = cylinder J T
Module: Mathlib.MeasureTheory.Constructions.Cylinders

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
h_nonempty : Nonempty ((i : ι) → α i)
s : Finset ι
S : Set ((i : ↥s) → α ↑i)
h : cylinder s S = ∅
hS : ¬S = ∅
⊢ False

Tactic:
← Ne,

Goal state after:
ι : Type u_1
α : ι → Type u_2
h_nonempty : Nonempty ((i : ι) → α i)
s : Finset ι
S : Set ((i : ↥s) → α ↑i)
h : cylinder s S = ∅
hS : S ≠ ∅
⊢ False
Module: Mathlib.MeasureTheory.Constructions.Cylinders

Goal state before:
pp: ι : Type u_2
α : ι → Type u_1
inst✝ : (i : ι) → MeasurableSpace (α i)
⊢ ∅ ∈ ⋃ s, ⋃ S, ⋃ (_ : MeasurableSet S), {cylinder s S}

Tactic:
mem_iUnion,

Goal state after:
ι : Type u_2
α : ι → Type u_1
inst✝ : (i : ι) → MeasurableSpace (α i)
⊢ ∃ i i_1, ∃ (_ : MeasurableSet i_1), ∅ ∈ {cylinder i i_1}
Module: Mathlib.MeasureTheory.Constructions.Cylinders

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (α i)
t : Set ((i : ι) → α i)
⊢ t ∈ ⋃ s, ⋃ S, ⋃ (_ : MeasurableSet S), {cylinder s S} ↔ ∃ s S, MeasurableSet S ∧ t = cylinder s S

Tactic:
mem_iUnion,

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (α i)
t : Set ((i : ι) → α i)
⊢ (∃ i i_1, ∃ (_ : MeasurableSet i_1), t ∈ {cylinder i i_1}) ↔ ∃ s S, MeasurableSet S ∧ t = cylinder s S
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: case empty
α : Type u_1
C : Set (Set α)
hC : IsSetSemiring C
s : Set α
hs : s ∈ supClosure C
hTC : ↑∅ ⊆ C
⊢ s \ ∅.sup id ∈ supClosure C

Tactic:
simpa

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: K : Type u_1
inst✝² : DivisionRing K
V : Type u_2
inst✝¹ : AddCommGroup V
inst✝ : Module K V
e : V ≃ₗ[K] V
he : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x
h : 1 < finrank K (V ⧸ (↑e).fixedSubmodule)
u : V
hu : LinearIndependent K ![(↑e).fixedSubmodule.mkQ u, e.fixedReduce ((↑e).fixedSubmodule.mkQ u)]
f : Dual K V
hfu : f u ≠ 0
v : V := (f u)⁻¹ • u
hf : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker f
v_def : v = (f u)⁻¹ • u
hspan : K ∙ (e v - v) = K ∙ (e u - u)
⊢ f v = 1

Tactic:
simp only [_root_.map_smul, smul_eq_mul, v_def, inv_mul_cancel₀ hfu]

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

Goal state before:
pp: case refine_3
ι : Type u_2
α : ι → Type u_1
inst✝ : (i : ι) → MeasurableSpace (α i)
s : Set ((i : ι) → α i)
hs : MeasurableSet s
f : ℕ → Set ((a : ι) → α a)
df : Pairwise (Disjoint on f)
mf : ∀ (i : ℕ), MeasurableSet (f i)
I : ℕ → Set ι
t : (i : ℕ) → Set ((a : ↑(I i)) → α ↑a)
hI : ∀ (i : ℕ), (I i).Countable
hf : ∀ (i : ℕ), f i = (I i).domRestrict ⁻¹' t i
x : (a : ι) → α a
x✝ : ∃ i x_1, (I i).domRestrict x_1 ∈ t i ∧ (⋃ n, I n).domRestrict x_1 = (⋃ n, I n).domRestrict x
n : ℕ
x' : (a : ι) → α a
hn : (I n).domRestrict x' ∈ t n
hx : (⋃ n, I n).domRestrict x' = (⋃ n, I n).domRestrict x
this : ∀ (x : (i : ι) → α i), (I n).domRestrict x = (fun x i ↦ x ⟨↑i, ⋯⟩) ((⋃ k, I k).domRestrict x)
⊢ (fun x i ↦ x ⟨↑i, ⋯⟩) ((⋃ n, I n).domRestrict x') ∈ t n

Tactic:
← this

Goal state after:
case refine_3
ι : Type u_2
α : ι → Type u_1
inst✝ : (i : ι) → MeasurableSpace (α i)
s : Set ((i : ι) → α i)
hs : MeasurableSet s
f : ℕ → Set ((a : ι) → α a)
df : Pairwise (Disjoint on f)
mf : ∀ (i : ℕ), MeasurableSet (f i)
I : ℕ → Set ι
t : (i : ℕ) → Set ((a : ↑(I i)) → α ↑a)
hI : ∀ (i : ℕ), (I i).Countable
hf : ∀ (i : ℕ), f i = (I i).domRestrict ⁻¹' t i
x : (a : ι) → α a
x✝ : ∃ i x_1, (I i).domRestrict x_1 ∈ t i ∧ (⋃ n, I n).domRestrict x_1 = (⋃ n, I n).domRestrict x
n : ℕ
x' : (a : ι) → α a
hn : (I n).domRestrict x' ∈ t n
hx : (⋃ n, I n).domRestrict x' = (⋃ n, I n).domRestrict x
this : ∀ (x : (i : ι) → α i), (I n).domRestrict x = (fun x i ↦ x ⟨↑i, ⋯⟩) ((⋃ k, I k).domRestrict x)
⊢ (I n).domRestrict x' ∈ t n
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case convert_1
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝¹ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
inst✝ : Fintype ι
f : ι → Set α
hf : ∀ (i : ι), f i ∈ C
h_dis : Pairwise (Disjoint on f)
h_mem : ⋃ i, f i ∈ C
⊢ ∀ i ∈ Finset.univ, f i ∈ C

Tactic:
simpa

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

Goal state before:
pp: case convert_3
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝¹ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
inst✝ : Fintype ι
f : ι → Set α
hf : ∀ (i : ι), f i ∈ C
h_dis : Pairwise (Disjoint on f)
h_mem : ⋃ i, f i ∈ C
⊢ ⋃ i ∈ Finset.univ, f i ∈ C

Tactic:
simpa

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

Goal state before:
pp: case convert_1.«0»
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hs : s ∈ C
ht : t ∈ C
hst : s ∪ t ∈ C
h_dis : Disjoint s t
A : s ∪ t = ⋃ i, ![s, t] i
⊢ ![s, t] ((fun i ↦ i) ⟨0, ⋯⟩) ∈ C

Tactic:
simpa

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

Goal state before:
pp: case convert_1.«1»
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hs : s ∈ C
ht : t ∈ C
hst : s ∪ t ∈ C
h_dis : Disjoint s t
A : s ∪ t = ⋃ i, ![s, t] i
⊢ ![s, t] ((fun i ↦ i) ⟨1, ⋯⟩) ∈ C

Tactic:
simpa

Goal state after:
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inl.coe.coe.refine_1
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a b : ι
hb✝ : a < b
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Ioi a
⊢ (⋂ z ∈ f, Ioi ↑z) ∩ ⋂ z ∈ g, Iio ↑z ⊆ Ioi ↑a

pp: case a.inl.coe.coe.refine_2
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a b : ι
hb✝ : a < b
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Ioi a
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] ((⋂ z ∈ f, Ioi ↑z) ∩ ⋂ z ∈ g, Iio ↑z)

Tactic:
· intro w hw
          induction w with
          | top => simp
          | coe w =>
            simp only [mem_Ioi, WithTop.coe_lt_coe]
            apply hfg
            simpa using hw

Goal state after:
case a.inl.coe.coe.refine_2
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a b : ι
hb✝ : a < b
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Ioi a
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] ((⋂ z ∈ f, Ioi ↑z) ∩ ⋂ z ∈ g, Iio ↑z)
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inl.coe.coe.refine_2.hs.h
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := ⋯
d : Set (WithTop ι) := ⋯
basis : Set (Set ι) := ⋯
h_basis : IsTopologicalBasis basis
a b : ι
hb✝ : a < b
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Ioi a
⊢ ∀ i ∈ f, IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Ioi ↑i)

Tactic:
intro i hi

Goal state after:
case a.inl.coe.coe.refine_2.hs.h
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a b : ι
hb✝ : a < b
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Ioi a
i : ι
hi : i ∈ f
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Ioi ↑i)
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case h_mem
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hC : IsSetSemiring C
J J' : Finset (Set α)
hJ : ↑J ⊆ C
hJdisj : (↑J).PairwiseDisjoint id
hJ' : ↑J' ⊆ C
hJ'disj : (↑J').PairwiseDisjoint id
h : ⋃₀ ↑J = ⋃₀ ↑J'
s : Set α
hs : s ∈ J
this : s = ⋃ t ∈ J', s ∩ t
⊢ ⋃ i ∈ J', s ∩ i ∈ C

Tactic:
← this

Goal state after:
case h_mem
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hC : IsSetSemiring C
J J' : Finset (Set α)
hJ : ↑J ⊆ C
hJdisj : (↑J).PairwiseDisjoint id
hJ' : ↑J' ⊆ C
hJ'disj : (↑J').PairwiseDisjoint id
h : ⋃₀ ↑J = ⋃₀ ↑J'
s : Set α
hs : s ∈ J
this : s = ⋃ t ∈ J', s ∩ t
⊢ s ∈ C
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inl.coe.coe.refine_2.ht.h
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := ⋯
d : Set (WithTop ι) := ⋯
basis : Set (Set ι) := ⋯
h_basis : IsTopologicalBasis basis
a b : ι
hb✝ : a < b
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Ioi a
⊢ ∀ i ∈ g, IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Iio ↑i)

Tactic:
intro i hi

Goal state after:
case a.inl.coe.coe.refine_2.ht.h
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a b : ι
hb✝ : a < b
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Ioi a
i : ι
hi : i ∈ g
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Iio ↑i)
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hC : IsSetSemiring C
J J' : Finset (Set α)
hJ : ↑J ⊆ C
hJdisj : (↑J).PairwiseDisjoint id
hJ' : ↑J' ⊆ C
hJ'disj : (↑J').PairwiseDisjoint id
h : ⋃₀ ↑J = ⋃₀ ↑J'
t : Set α
ht : t ∈ J'
this : t = ⋃ s ∈ J, s ∩ t
⊢ ⋃ i ∈ J, i ∩ t ∈ C

Tactic:
← this

Goal state after:
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hC : IsSetSemiring C
J J' : Finset (Set α)
hJ : ↑J ⊆ C
hJdisj : (↑J).PairwiseDisjoint id
hJ' : ↑J' ⊆ C
hJ'disj : (↑J').PairwiseDisjoint id
h : ⋃₀ ↑J = ⋃₀ ↑J'
t : Set α
ht : t ∈ J'
this : t = ⋃ s ∈ J, s ∩ t
⊢ t ∈ C
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inr.coe.coe.refine_2.hs.hs.h
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := ⋯
d : Set (WithTop ι) := ⋯
basis : Set (Set ι) := ⋯
h_basis : IsTopologicalBasis basis
a b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ ∀ i ∈ f, IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Ioi ↑i)

Tactic:
intro i hi

Goal state after:
case a.inr.coe.coe.refine_2.hs.hs.h
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
i : ι
hi : i ∈ f
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Ioi ↑i)
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case succ.inr
α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G✝ : Type u_2
inst✝² : AddCommMonoid G✝
m m' : AddContent G✝ C
inst✝¹ : LinearOrder α
G : Type u_3
inst✝ : AddCommGroup G
f : α → G
n : ℕ
ih :
  ∀ (I : Finset (Set α)),
    ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} →
      (↑I).PairwiseDisjoint id →
        ⋃₀ ↑I ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} → #I = n → onIocAux f (⋃₀ ↑I) = ∑ u ∈ I, onIocAux f u
I : Finset (Set α)
hI : ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}
h'I : (↑I).PairwiseDisjoint id
hn : #I = n + 1
u v : α
huv✝ : u ≤ v
h'uv : ⋃₀ ↑I = Set.Ioc u v
huv : u < v
this : v ∈ ⋃₀ ↑I
u' v' : α
hu'v' : u' ≤ v'
tI : Set.Ioc u' v' ∈ I
ht : v ∈ Set.Ioc u' v'
⊢ onIocAux f (⋃₀ ↑I) = ∑ u ∈ I, onIocAux f u

Tactic:
have ⟨_, uu'⟩ : v' ≤ v ∧ u ≤ u' := (Ioc_subset_Ioc_iff (by grind)).1 (by grind)

Goal state after:
case succ.inr
α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G✝ : Type u_2
inst✝² : AddCommMonoid G✝
m m' : AddContent G✝ C
inst✝¹ : LinearOrder α
G : Type u_3
inst✝ : AddCommGroup G
f : α → G
n : ℕ
ih :
  ∀ (I : Finset (Set α)),
    ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} →
      (↑I).PairwiseDisjoint id →
        ⋃₀ ↑I ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} → #I = n → onIocAux f (⋃₀ ↑I) = ∑ u ∈ I, onIocAux f u
I : Finset (Set α)
hI : ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}
h'I : (↑I).PairwiseDisjoint id
hn : #I = n + 1
u v : α
huv✝ : u ≤ v
h'uv : ⋃₀ ↑I = Set.Ioc u v
huv : u < v
this : v ∈ ⋃₀ ↑I
u' v' : α
hu'v' : u' ≤ v'
tI : Set.Ioc u' v' ∈ I
ht : v ∈ Set.Ioc u' v'
left✝ : v' ≤ v
uu' : u ≤ u'
⊢ onIocAux f (⋃₀ ↑I) = ∑ u ∈ I, onIocAux f u
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inr.coe.coe.refine_2.hs.ht.h
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := ⋯
d : Set (WithTop ι) := ⋯
basis : Set (Set ι) := ⋯
h_basis : IsTopologicalBasis basis
a b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
⊢ ∀ i ∈ g, IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Iio ↑i)

Tactic:
intro i hi

Goal state after:
case a.inr.coe.coe.refine_2.hs.ht.h
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a b : ι
hb✝ : b < a
f g : Set ι
hfc : f ⊆ c
hgc : g ⊆ c
f_fin : f.Finite
g_fin : g.Finite
hb : b ∈ (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a
hfg : (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a ⊆ Iio a
i : ι
hi : i ∈ g
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Iio ↑i)
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inr
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : WithTop ι
⊢ IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] (Iio a)

Tactic:
induction a with
    | top =>
      -- for `a = ⊤`, this is trivial as `Iio ⊤` is in our set by design
      apply isOpen_generateFrom_of_mem
      exact ⟨⊤, by simp [d]⟩
    | coe a =>
      -- for `a : ι`, we consider an element `b ∈ Iio a`, and seek a finite intersection of our
      -- sets containing it and contained in `Iio a`.
      rw [@isOpen_iff_forall_mem_open]
      intro b hb
      induction b with
      | top => simp at hb
      | coe b =>
        -- `b` can not be equal to `⊤`, so it comes from `ι`. In `ι` we can find a finite
        -- intersection of our sets containing `b` and contained in `Iio a`. We lift it
        -- to `WithTop ι`, and intersect with `Iio ⊤` (which is also in our set) to exclude `⊤`.
        simp only [mem_Iio, WithTop.coe_lt_coe] at hb
        obtain ⟨t, ⟨f, g, hfc, hgc, f_fin, g_fin, rfl⟩, hb, hfg⟩ :
          ∃ t ∈ basis, b ∈ t ∧ t ⊆ Iio a := h_basis.isOpen_iff.1 (isOpen_Iio' a) b hb
        refine ⟨(⋂ z ∈ f, Ioi z) ∩ (⋂ z ∈ g, Iio z) ∩ Iio ⊤, ?_, ?_, by simpa using hb⟩
        · intro w hw
          induction w with
          | top => simp at hw
          | coe w =>
            simp only [mem_Iio, WithTop.coe_lt_coe]
            apply hfg
            simpa using hw
        · apply @IsOpen.inter _ (generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}); swap
          · apply isOpen_generateFrom_of_mem
            simp [d]
          apply @IsOpen.inter _ (generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a})
          · apply @Finite.isOpen_biInter _ _ (generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a})
            · exact f_fin
            · intro i hi
              apply isOpen_generateFrom_of_mem
              grind
          · apply @Finite.isOpen_biInter _ _ (generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a})
            · exact g_fin
            · intro i hi
              apply isOpen_generateFrom_of_mem
              grind

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s t : Set α
m : AddContent ℝ≥0∞ C
hC : IsSetRing C
hs : s ∈ C
ht : t ∈ C
⊢ m (s ∩ t) + m (s \ t) - m t ≤ m (s \ t)

Tactic:
add_comm

Goal state after:
α : Type u_1
C : Set (Set α)
s t : Set α
m : AddContent ℝ≥0∞ C
hC : IsSetRing C
hs : s ∈ C
ht : t ∈ C
⊢ m (s \ t) + m (s ∩ t) - m t ≤ m (s \ t)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut

Goal state before:
pp: case neg
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝¹ : CompleteSpace E
hm : m ≤ mΩ
inst✝ : IsFiniteMeasure μ
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (ω : Ω) ∂μ, ‖f ω‖ ≤ c
fs : ℕ → SimpleFunc Ω F := hf.approxBounded c
hfs_tendsto : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun x ↦ (fs x) ω) atTop (𝓝 (f ω))
hμ : ¬μ = 0
this✝ : (ae μ).NeBot
hc : 0 ≤ c
hfs_bound : ∀ (n : ℕ) (ω : Ω), ‖(fs n) ω‖ ≤ c
this : μ[fun ω ↦ (B (f ω)) (μ[g | m] ω) | m] = fun ω ↦ (B (f ω)) (μ[g | m] ω)
⊢ μ[fun ω ↦ (B (f ω)) (g ω) | m] =ᵐ[μ] fun ω ↦ (B (f ω)) (μ[g | m] ω)

Tactic:
← this

Goal state after:
case neg
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝¹ : CompleteSpace E
hm : m ≤ mΩ
inst✝ : IsFiniteMeasure μ
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (ω : Ω) ∂μ, ‖f ω‖ ≤ c
fs : ℕ → SimpleFunc Ω F := hf.approxBounded c
hfs_tendsto : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun x ↦ (fs x) ω) atTop (𝓝 (f ω))
hμ : ¬μ = 0
this✝ : (ae μ).NeBot
hc : 0 ≤ c
hfs_bound : ∀ (n : ℕ) (ω : Ω), ‖(fs n) ω‖ ≤ c
this : μ[fun ω ↦ (B (f ω)) (μ[g | m] ω) | m] = fun ω ↦ (B (f ω)) (μ[g | m] ω)
⊢ μ[fun ω ↦ (B (f ω)) (g ω) | m] =ᵐ[μ] μ[fun ω ↦ (B (f ω)) (μ[g | m] ω) | m]
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case mpr
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v w : VectorMeasure α M
⊢ (∀ (i : Set α), MeasurableSet i → v i = w i) → v = w

Tactic:
· rw [DFunLike.ext_iff]
    intro h i
    by_cases hi : MeasurableSet i
    · exact h i hi
    · simp_rw [not_measurable _ hi]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : MemLp f 1 μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hM : ∫⁻ (x : α), ↑‖{x | M ≤ ↑‖f x‖₊}.indicator f x‖₊ ∂μ ≤ ENNReal.ofReal ε
⊢ ∫⁻ (x : α), ‖{x | max M 0 ≤ ↑‖f x‖₊}.indicator f x‖ₑ ∂μ ≤ ENNReal.ofReal ε

Tactic:
simpa

Goal state after: