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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
⊢ s \ ⋃₀ ↑I ⊆ s

Tactic:
]

Goal state after:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
⊢ s \ ⋃₀ ↑I ⊆ s
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
⊢ ⋃₀ ↑(hC.disjointOfDiffUnion hs hI) ⊆ ⋃₀ ↑(hC.disjointOfDiffUnion hs hI)

Tactic:
]

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

Goal state before:
pp: ι : Type u_2
X : ι → Type u_3
m : (i : ι) → MeasurableSpace (X i)
Δ : Set ι
⊢ ∀ (a : ↑Δ), Measurable fun x ↦ Δ.domRestrict x a

Tactic:
]

Goal state after:
ι : Type u_2
X : ι → Type u_3
m : (i : ι) → MeasurableSpace (X i)
Δ : Set ι
⊢ ∀ (a : ↑Δ), Measurable fun x ↦ Δ.domRestrict x a
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
f : Dual K V
hu : LinearIndependent K ![(f u)⁻¹ • (↑e).fixedSubmodule.mkQ u, (f u)⁻¹ • e.fixedReduce ((↑e).fixedSubmodule.mkQ u)]
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)
⊢ LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]

Tactic:
]

Goal state after:
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
f : Dual K V
hu : LinearIndependent K ![(f u)⁻¹ • (↑e).fixedSubmodule.mkQ u, (f u)⁻¹ • e.fixedReduce ((↑e).fixedSubmodule.mkQ u)]
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)
⊢ LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
⊢ Disjoint (⋃₀ ↑I) (s \ ⋃₀ ↑I)

Tactic:
]

Goal state after:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
⊢ Disjoint (⋃₀ ↑I) (s \ ⋃₀ ↑I)
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
h : ∃ a ∈ I, a ∈ hC.disjointOfDiffUnion hs hI
⊢ False

Tactic:
]

Goal state after:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
h : ∃ a ∈ I, a ∈ hC.disjointOfDiffUnion hs hI
⊢ False
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
u : Set α
huI : u ∈ I
hu_disjointOfDiffUnion : ∅ ∈ hC.disjointOfDiffUnion hs hI
h_disj : u = ∅
⊢ ∅ ∈ hC.disjointOfDiffUnion hs hI

Tactic:
]

Goal state after:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
u : Set α
huI : u ∈ I
hu_disjointOfDiffUnion : ∅ ∈ hC.disjointOfDiffUnion hs hI
h_disj : u = ∅
⊢ ∅ ∈ hC.disjointOfDiffUnion hs hI
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
h_dis : (↑I).PairwiseDisjoint id
⊢ (↑I).PairwiseDisjoint id ∧
    (↑(hC.disjointOfDiffUnion hs hI)).PairwiseDisjoint id ∧
      ∀ ⦃i : Set α⦄, i ∈ ↑I → ∀ ⦃j : Set α⦄, j ∈ ↑(hC.disjointOfDiffUnion hs hI) → i ≠ j → Disjoint (id i) (id j)

Tactic:
]

Goal state after:
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
h_dis : (↑I).PairwiseDisjoint id
⊢ (↑I).PairwiseDisjoint id ∧
    (↑(hC.disjointOfDiffUnion hs hI)).PairwiseDisjoint id ∧
      ∀ ⦃i : Set α⦄, i ∈ ↑I → ∀ ⦃j : Set α⦄, j ∈ ↑(hC.disjointOfDiffUnion hs hI) → i ≠ j → Disjoint (id i) (id j)
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: 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}
H : instWithTopOfOrderTopology = generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}
⊢ instWithTopOfOrderTopology = instWithTopOfOrderTopology

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
hI_ss : ∀ t ∈ I, t ⊆ s
⊢ ⋃₀ ↑I ∪ ⋃₀ ↑(hC.disjointOfDiffUnion hs hI) = ⋃₀ ↑I ∪ ⋃₀ ↑(hC.disjointOfDiffUnion hs hI)

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
j : ↥J
⊢ ⋯.choose.parts = ⋯.choose.parts

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
j : ↥J
⊢ ↑⋯.choose.parts ⊆ C

Tactic:
]

Goal state after:
α : Type u_1
C : Set (Set α)
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
j : ↥J
⊢ ↑⋯.choose.parts ⊆ C
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
j : ↥J
⊢ ⋯.choose.parts.SupIndep id

Tactic:
]

Goal state after:
α : Type u_1
C : Set (Set α)
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
j : ↥J
⊢ ⋯.choose.parts.SupIndep id
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: 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}
H : instWithTopOfOrderTopology = generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}
d_count : d.Countable
this : {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a} = Ioi '' d ∪ Iio '' d
⊢ (Ioi '' d ∪ Iio '' d).Countable

Tactic:
]

Goal state after:
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}
H : instWithTopOfOrderTopology = generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}
d_count : d.Countable
this : {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a} = Ioi '' d ∪ Iio '' d
⊢ (Ioi '' d ∪ Iio '' d).Countable
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
j : ↥J
⊢ ↑j ⊆ ↑j

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
j : Set α
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
hj : j ∈ J
⊢ ⋃₀ ↑(hC.disjointOfUnion hJ j) ⊆ j

Tactic:
lift j to J using hj
  grw [disjointOfUnion_coe, ← Finset.sup_id_set_eq_sUnion, Finpartition.sup_parts,
    disjointed_subset, Equiv.symm_apply_apply]

Goal state after:
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
C : Set (Set α)
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
j : ↥J
⊢ ∅ ∉ ⋯.choose.parts

Tactic:
]

Goal state after:
α : Type u_1
C : Set (Set α)
J : Finset (Set α)
hC : IsSetSemiring C
hJ : ↑J ⊆ C
j : ↥J
⊢ ∅ ∉ ⋯.choose.parts
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : Nonempty α
u v : α
huv : u ≤ v
u' v' : α
hu'v' : u' ≤ v'
⊢ Set.Ioc (max u u') (min v v') ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : Nonempty α
u v : α
huv : u ≤ v
u' v' : α
hu'v' : u' ≤ v'
⊢ Set.Ioc (max u u') (min v v') ∈ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v}
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
f g : Dual K V
v : V
a b : K
hv : LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]
hf : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker f
hfv : f v = 1
hg : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker (f + g)
hfgv : (f + g) v = 1
hg0 : g ≠ 0
hne_top : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ⊔ K ∙ v < ⊤
ha✝ : ∀ (v_1 : V), (auxTransvection hf * e) v_1 - a • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hb✝ : ∀ (v_1 : V), (auxTransvection hg * e) v_1 - b • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
c : K := b - a
hc : c = b - a
x : V
ha : e x + f (e x) • (v - e v) - a • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hb : e x + f (e x) • (v - e v) + g (e x) • (v - e v) - b • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
⊢ g (e x) • (v - e v) - c • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v

Tactic:
]

Goal state after:
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
f g : Dual K V
v : V
a b : K
hv : LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]
hf : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker f
hfv : f v = 1
hg : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker (f + g)
hfgv : (f + g) v = 1
hg0 : g ≠ 0
hne_top : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ⊔ K ∙ v < ⊤
ha✝ : ∀ (v_1 : V), (auxTransvection hf * e) v_1 - a • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hb✝ : ∀ (v_1 : V), (auxTransvection hg * e) v_1 - b • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
c : K := b - a
hc : c = b - a
x : V
ha : e x + f (e x) • (v - e v) - a • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hb : e x + f (e x) • (v - e v) + g (e x) • (v - e v) - b • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
⊢ g (e x) • (v - e v) - c • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
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)
⊢ (I n).domRestrict x' ∈ t n

Tactic:
]

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.SetSemiring

Goal state before:
pp: case inr.inr
α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : Nonempty α
u v : α
huv : u ≤ v
u' v' : α
hu'v' : u' ≤ v'
hu : u < u'
hv : v' < v
⊢ ∃ I, ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} ∧ (↑I).PairwiseDisjoint id ∧ Set.Ioc u u' ∪ Set.Ioc v' v = ⋃₀ ↑I

Tactic:
]

Goal state after:
case inr.inr
α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : Nonempty α
u v : α
huv : u ≤ v
u' v' : α
hu'v' : u' ≤ v'
hu : u < u'
hv : v' < v
⊢ ∃ I, ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} ∧ (↑I).PairwiseDisjoint id ∧ Set.Ioc u u' ∪ Set.Ioc v' v = ⋃₀ ↑I
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a
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
⊢ generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a} ≤ Preorder.topology (WithTop ι)

Tactic:
]

Goal state after:
case a
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
⊢ generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a} ≤ Preorder.topology (WithTop ι)
Module: Mathlib.MeasureTheory.SetAlgebra

Goal state before:
pp: case pos
α : Type u_1
𝒜 : Set (Set α)
ι : Type u_2
h𝒜 : IsSetAlgebra 𝒜
s : ι → Set α
S : Finset ι
hs : ∀ i ∈ S, s i ∈ 𝒜
h : S = ∅
⊢ univ ∈ 𝒜

Tactic:
]

Goal state after:
case pos
α : Type u_1
𝒜 : Set (Set α)
ι : Type u_2
h𝒜 : IsSetAlgebra 𝒜
s : ι → Set α
S : Finset ι
hs : ∀ i ∈ S, s i ∈ 𝒜
h : S = ∅
⊢ univ ∈ 𝒜
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
ι : Type u_3
a : Finset ι
f : ι → Set α
hf : ∀ i ∈ a, f i ∈ C
h_dis : (↑a).PairwiseDisjoint f
h_mem : ⋃ i ∈ a, f i ∈ C
A : ⋃ i ∈ a, f i = ⋃₀ ↑(Finset.image f a)
⊢ ∑ u ∈ Finset.image f a, m u = ∑ i ∈ a, m (f i)

pp: case h_ss
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
a : Finset ι
f : ι → Set α
hf : ∀ i ∈ a, f i ∈ C
h_dis : (↑a).PairwiseDisjoint f
h_mem : ⋃ i ∈ a, f i ∈ C
A : ⋃ i ∈ a, f i = ⋃₀ ↑(Finset.image f a)
⊢ ↑(Finset.image f a) ⊆ C

pp: case h_dis
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
a : Finset ι
f : ι → Set α
hf : ∀ i ∈ a, f i ∈ C
h_dis : (↑a).PairwiseDisjoint f
h_mem : ⋃ i ∈ a, f i ∈ C
A : ⋃ i ∈ a, f i = ⋃₀ ↑(Finset.image f a)
⊢ (↑(Finset.image f a)).PairwiseDisjoint id

pp: case h_mem
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
a : Finset ι
f : ι → Set α
hf : ∀ i ∈ a, f i ∈ C
h_dis : (↑a).PairwiseDisjoint f
h_mem : ⋃ i ∈ a, f i ∈ C
A : ⋃ i ∈ a, f i = ⋃₀ ↑(Finset.image f a)
⊢ ⋃₀ ↑(Finset.image f a) ∈ C

Tactic:
]

Goal state after:
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
a : Finset ι
f : ι → Set α
hf : ∀ i ∈ a, f i ∈ C
h_dis : (↑a).PairwiseDisjoint f
h_mem : ⋃ i ∈ a, f i ∈ C
A : ⋃ i ∈ a, f i = ⋃₀ ↑(Finset.image f a)
⊢ ∑ u ∈ Finset.image f a, m u = ∑ i ∈ a, m (f i)

case h_ss
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
a : Finset ι
f : ι → Set α
hf : ∀ i ∈ a, f i ∈ C
h_dis : (↑a).PairwiseDisjoint f
h_mem : ⋃ i ∈ a, f i ∈ C
A : ⋃ i ∈ a, f i = ⋃₀ ↑(Finset.image f a)
⊢ ↑(Finset.image f a) ⊆ C

case h_dis
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
a : Finset ι
f : ι → Set α
hf : ∀ i ∈ a, f i ∈ C
h_dis : (↑a).PairwiseDisjoint f
h_mem : ⋃ i ∈ a, f i ∈ C
A : ⋃ i ∈ a, f i = ⋃₀ ↑(Finset.image f a)
⊢ (↑(Finset.image f a)).PairwiseDisjoint id

case h_mem
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
a : Finset ι
f : ι → Set α
hf : ∀ i ∈ a, f i ∈ C
h_dis : (↑a).PairwiseDisjoint f
h_mem : ⋃ i ∈ a, f i ∈ C
A : ⋃ i ∈ a, f i = ⋃₀ ↑(Finset.image f a)
⊢ ⋃₀ ↑(Finset.image f a) ∈ C
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case a.inl.coe
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 : ι
⊢ ∀ x ∈ Ioi ↑a, ∃ t ⊆ Ioi ↑a, IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] t ∧ x ∈ t

Tactic:
]

Goal state after:
case a.inl.coe
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 : ι
⊢ ∀ x ∈ Ioi ↑a, ∃ t ⊆ Ioi ↑a, IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] t ∧ x ∈ t
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
f g : Dual K V
v : V
a b : K
hv : LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]
hf : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker f
hfv : f v = 1
hg : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker (f + g)
hfgv : (f + g) v = 1
hg0 : g ≠ 0
hne_top : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ⊔ K ∙ v < ⊤
ha✝ : ∀ (v_1 : V), (auxTransvection hf * e) v_1 - a • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hb✝ : ∀ (v_1 : V), (auxTransvection hg * e) v_1 - b • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
c : K := b - a
hc : c = b - a
x : V
ha : e x + f (e x) • (v - e v) - a • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hb : e x + f (e x) • (v - e v) + g (e x) • (v - e v) - b • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
⊢ g (e x) • (v - e v) + -(c • x) =
    g (e x) • (v - e v) + e x + f (e x) • (v - e v) - b • x - (e x + f (e x) • (v - e v) - a • x)

Tactic:
]

Goal state after:
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
f g : Dual K V
v : V
a b : K
hv : LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]
hf : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker f
hfv : f v = 1
hg : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker (f + g)
hfgv : (f + g) v = 1
hg0 : g ≠ 0
hne_top : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ⊔ K ∙ v < ⊤
ha✝ : ∀ (v_1 : V), (auxTransvection hf * e) v_1 - a • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hb✝ : ∀ (v_1 : V), (auxTransvection hg * e) v_1 - b • v_1 ∈ (↑e).fixedSubmodule ⊔ K ∙ v
c : K := b - a
hc : c = b - a
x : V
ha : e x + f (e x) • (v - e v) - a • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hb : e x + f (e x) • (v - e v) + g (e x) • (v - e v) - b • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
⊢ g (e x) • (v - e v) + -(c • x) =
    g (e x) • (v - e v) + e x + f (e x) • (v - e v) - b • x - (e x + f (e x) • (v - e v) - a • x)
Module: Mathlib.MeasureTheory.SetAlgebra

Goal state before:
pp: case base
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
u_mem : u ∈ 𝒜
a : Set (Set α)
ha : {u} ∈ {{u}}
t : Set α
ht : u ∈ {u}
⊢ u ∈ 𝒜 ∨ uᶜ ∈ 𝒜

Tactic:
]

Goal state after:
case base
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
u_mem : u ∈ 𝒜
a : Set (Set α)
ha : {u} ∈ {{u}}
t : Set α
ht : u ∈ {u}
⊢ u ∈ 𝒜 ∨ uᶜ ∈ 𝒜
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
ι : Type u_3
a : Finset ι
f : ι → Set α
hf : ∀ i ∈ a, f i ∈ C
h_dis : (↑a).PairwiseDisjoint f
h_mem : ⋃ i ∈ a, f i ∈ C
A : ⋃ i ∈ a, f i = ⋃₀ ↑(Finset.image f a)
⊢ ⋃ i ∈ a, f i ∈ C

Tactic:
]

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
ι : Type u_3
a : Finset ι
f : ι → Set α
hf : ∀ i ∈ a, f i ∈ C
h_dis : (↑a).PairwiseDisjoint f
h_mem : ⋃ i ∈ a, f i ∈ C
A : ⋃ i ∈ a, f i = ⋃₀ ↑(Finset.image f a)
⊢ ⋃ i ∈ a, f i ∈ C
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
ι : Type u_3
a : Finset ι
f : ι → Set α
hf : ∀ i ∈ a, f i ∈ C
h_dis : (↑a).PairwiseDisjoint f
h_mem : ⋃ i ∈ a, f i ∈ C
A : ⋃ i ∈ a, f i = ⋃₀ ↑(Finset.image f a)
⊢ ∑ i ∈ a, m (f i) = ∑ i ∈ a, m (f i)

pp: α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
a : Finset ι
f : ι → Set α
hf : ∀ i ∈ a, f i ∈ C
h_dis : (↑a).PairwiseDisjoint f
h_mem : ⋃ i ∈ a, f i ∈ C
A : ⋃ i ∈ a, f i = ⋃₀ ↑(Finset.image f a)
⊢ (↑a).Pairwise fun i j ↦ f i = f j → m (f i) = 0

Tactic:
]

Goal state after:
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
ι : Type u_3
a : Finset ι
f : ι → Set α
hf : ∀ i ∈ a, f i ∈ C
h_dis : (↑a).PairwiseDisjoint f
h_mem : ⋃ i ∈ a, f i ∈ C
A : ⋃ i ∈ a, f i = ⋃₀ ↑(Finset.image f a)
⊢ (↑a).Pairwise fun i j ↦ f i = f j → m (f i) = 0
Module: Mathlib.MeasureTheory.SetAlgebra

Goal state before:
pp: case compl.refine_1
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs✝ : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this✝ : Finite ↑A
this : ∀ (a : ↑A), Finite ↑↑a
f : (a : ↑A) → ↑↑a
a : ↑A
⊢ ↑(f a) ∈ 𝒜 ∨ (↑(f a))ᶜ ∈ 𝒜

Tactic:
]

Goal state after:
case compl.refine_1
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs✝ : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this✝ : Finite ↑A
this : ∀ (a : ↑A), Finite ↑↑a
f : (a : ↑A) → ↑↑a
a : ↑A
⊢ ↑(f a) ∈ 𝒜 ∨ (↑(f a))ᶜ ∈ 𝒜