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

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
s t : Set ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
hs : s ∈ measurableCylinders α
ht : t ∈ measurableCylinders α
hst : Disjoint s t
I : Finset ι
S : Set ((i : ↥I) → α ↑i)
hS : MeasurableSet S
hs_eq : s = cylinder I S
J : Finset ι
T : Set ((i : ↥J) → α ↑i)
hT : MeasurableSet T
ht_eq : t = cylinder J T
S' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' S
T' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' T
hS' : MeasurableSet S'
hT' : MeasurableSet T'
⊢ cylinder I S = cylinder (I ∪ J) S'

Tactic:
]

Goal state after:
ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
s t : Set ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
hs : s ∈ measurableCylinders α
ht : t ∈ measurableCylinders α
hst : Disjoint s t
I : Finset ι
S : Set ((i : ↥I) → α ↑i)
hS : MeasurableSet S
hs_eq : s = cylinder I S
J : Finset ι
T : Set ((i : ↥J) → α ↑i)
hT : MeasurableSet T
ht_eq : t = cylinder J T
S' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' S
T' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' T
hS' : MeasurableSet S'
hT' : MeasurableSet T'
⊢ cylinder I S = cylinder (I ∪ J) S'
Module: Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
s t : Set ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
hs : s ∈ measurableCylinders α
ht : t ∈ measurableCylinders α
hst : Disjoint s t
I : Finset ι
S : Set ((i : ↥I) → α ↑i)
hS : MeasurableSet S
hs_eq : s = cylinder I S
J : Finset ι
T : Set ((i : ↥J) → α ↑i)
hT : MeasurableSet T
ht_eq : t = cylinder J T
S' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' S
T' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' T
hS' : MeasurableSet S'
hT' : MeasurableSet T'
h_eq1 : s = cylinder (I ∪ J) S'
⊢ cylinder J T = cylinder (I ∪ J) T'

Tactic:
]

Goal state after:
ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
s t : Set ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
hs : s ∈ measurableCylinders α
ht : t ∈ measurableCylinders α
hst : Disjoint s t
I : Finset ι
S : Set ((i : ↥I) → α ↑i)
hS : MeasurableSet S
hs_eq : s = cylinder I S
J : Finset ι
T : Set ((i : ↥J) → α ↑i)
hT : MeasurableSet T
ht_eq : t = cylinder J T
S' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' S
T' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' T
hS' : MeasurableSet S'
hT' : MeasurableSet T'
h_eq1 : s = cylinder (I ∪ J) S'
⊢ cylinder J T = cylinder (I ∪ J) T'
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
⊢ t ∈ C

Tactic:
]

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.MeasureTheory.Constructions.ProjectiveFamilyContent

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
s t : Set ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
hs : s ∈ measurableCylinders α
ht : t ∈ measurableCylinders α
hst : Disjoint s t
I : Finset ι
S : Set ((i : ↥I) → α ↑i)
hS : MeasurableSet S
hs_eq : s = cylinder I S
J : Finset ι
T : Set ((i : ↥J) → α ↑i)
hT : MeasurableSet T
ht_eq : t = cylinder J T
S' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' S
T' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' T
hS' : MeasurableSet S'
hT' : MeasurableSet T'
h_eq1 : s = cylinder (I ∪ J) S'
h_eq2 : t = cylinder (I ∪ J) T'
⊢ cylinder I S ∪ cylinder J T = cylinder (I ∪ J) (S' ∪ T')

Tactic:
]

Goal state after:
ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
s t : Set ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
hs : s ∈ measurableCylinders α
ht : t ∈ measurableCylinders α
hst : Disjoint s t
I : Finset ι
S : Set ((i : ↥I) → α ↑i)
hS : MeasurableSet S
hs_eq : s = cylinder I S
J : Finset ι
T : Set ((i : ↥J) → α ↑i)
hT : MeasurableSet T
ht_eq : t = cylinder J T
S' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' S
T' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' T
hS' : MeasurableSet S'
hT' : MeasurableSet T'
h_eq1 : s = cylinder (I ∪ J) S'
h_eq2 : t = cylinder (I ∪ J) T'
⊢ cylinder I S ∪ cylinder J T = cylinder (I ∪ J) (S' ∪ T')
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
hC : IsSetSemiring C
m : AddContent G C
s : Set α
J : Finset (Set α)
hJ : ↑J ⊆ C
h'J : (↑J).PairwiseDisjoint id
hs : s = ⋃₀ ↑J
h : ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s = ⋃₀ ↑J
⊢ ⋃₀ ↑h.choose = s

Tactic:
]

Goal state after:
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
hC : IsSetSemiring C
m : AddContent G C
s : Set α
J : Finset (Set α)
hJ : ↑J ⊆ C
h'J : (↑J).PairwiseDisjoint id
hs : s = ⋃₀ ↑J
h : ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s = ⋃₀ ↑J
⊢ ⋃₀ ↑h.choose = s
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m✝ m' m : AddContent G C
hC : IsSetSemiring C
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
s t : Set ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
hs : s ∈ measurableCylinders α
ht : t ∈ measurableCylinders α
hst : Disjoint s t
I : Finset ι
S : Set ((i : ↥I) → α ↑i)
hS : MeasurableSet S
hs_eq : s = cylinder I S
J : Finset ι
T : Set ((i : ↥J) → α ↑i)
hT : MeasurableSet T
ht_eq : t = cylinder J T
S' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' S
T' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' T
hS' : MeasurableSet S'
hT' : MeasurableSet T'
h_eq1 : s = cylinder (I ∪ J) S'
h_eq2 : t = cylinder (I ∪ J) T'
h_eq3 : s ∪ t = cylinder (I ∪ J) (S' ∪ T')
⊢ (P (I ∪ J)) (S' ∪ T') = (P (I ∪ J)) S' + (P (I ∪ J)) T'

Tactic:
]

Goal state after:
ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
s t : Set ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
hs : s ∈ measurableCylinders α
ht : t ∈ measurableCylinders α
hst : Disjoint s t
I : Finset ι
S : Set ((i : ↥I) → α ↑i)
hS : MeasurableSet S
hs_eq : s = cylinder I S
J : Finset ι
T : Set ((i : ↥J) → α ↑i)
hT : MeasurableSet T
ht_eq : t = cylinder J T
S' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' S
T' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' T
hS' : MeasurableSet S'
hT' : MeasurableSet T'
h_eq1 : s = cylinder (I ∪ J) S'
h_eq2 : t = cylinder (I ∪ J) T'
h_eq3 : s ∪ t = cylinder (I ∪ J) (S' ∪ T')
⊢ (P (I ∪ J)) (S' ∪ T') = (P (I ∪ J)) S' + (P (I ∪ J)) T'
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: case inr
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
key : ∀ (x : V), g (e x) • (v - e v) - c • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hc0 : c ≠ 0
⊢ ⊤ ≤ (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ⊔ K ∙ v

Tactic:
]

Goal state after:
case inr
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
key : ∀ (x : V), g (e x) • (v - e v) - c • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hc0 : c ≠ 0
⊢ ⊤ ≤ (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ⊔ K ∙ v
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m✝ m' m : AddContent G C
hC : IsSetSemiring C
I : Finset (Set α)
hI : ↑I ⊆ _root_.supClosure C
h'I : (↑I).PairwiseDisjoint id
hh'I : ⋃₀ ↑I ∈ _root_.supClosure C
J : (s : Set α) → Finpartition s
hJC : ∀ s ∈ I, ↑(J s).parts ⊆ C
K : Finpartition (I.sup id) := Finpartition.combine J ⋯
⊢ m.supClosureFun (⋃₀ ↑K.parts) = ∑ u ∈ I, m.supClosureFun u

Tactic:
m.supClosureFun_apply hC (by simpa [K] using hJC) K.supIndep.pairwiseDisjoint rfl,

Goal state after:
α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m✝ m' m : AddContent G C
hC : IsSetSemiring C
I : Finset (Set α)
hI : ↑I ⊆ _root_.supClosure C
h'I : (↑I).PairwiseDisjoint id
hh'I : ⋃₀ ↑I ∈ _root_.supClosure C
J : (s : Set α) → Finpartition s
hJC : ∀ s ∈ I, ↑(J s).parts ⊆ C
K : Finpartition (I.sup id) := Finpartition.combine J ⋯
⊢ ∑ s ∈ K.parts, m s = ∑ u ∈ I, m.supClosureFun u
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m✝ m' m : AddContent G C
hC : IsSetSemiring C
I : Finset (Set α)
hI : ↑I ⊆ _root_.supClosure C
h'I : (↑I).PairwiseDisjoint id
hh'I : ⋃₀ ↑I ∈ _root_.supClosure C
J : (s : Set α) → Finpartition s
hJC : ∀ s ∈ I, ↑(J s).parts ⊆ C
K : Finpartition (I.sup id) := Finpartition.combine J ⋯
⊢ ∑ i ∈ I, ∑ p ∈ (J i).parts, m p = ∑ u ∈ I, m.supClosureFun u

Tactic:
]

Goal state after:
α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m✝ m' m : AddContent G C
hC : IsSetSemiring C
I : Finset (Set α)
hI : ↑I ⊆ _root_.supClosure C
h'I : (↑I).PairwiseDisjoint id
hh'I : ⋃₀ ↑I ∈ _root_.supClosure C
J : (s : Set α) → Finpartition s
hJC : ∀ s ∈ I, ↑(J s).parts ⊆ C
K : Finpartition (I.sup id) := Finpartition.combine J ⋯
⊢ ∑ i ∈ I, ∑ p ∈ (J i).parts, m p = ∑ u ∈ I, m.supClosureFun u
Module: Mathlib.Topology.Order.WithTop

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

Tactic:
]

Goal state after:
case a.inr.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 ∈ Iio ↑a, ∃ t ⊆ Iio ↑a, IsOpen[generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}] t ∧ x ∈ t
Module: Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent

Goal state before:
pp: case inr
ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
s t : Set ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
hs : s ∈ measurableCylinders α
ht : t ∈ measurableCylinders α
hst : Disjoint s t
I : Finset ι
S : Set ((i : ↥I) → α ↑i)
hS : MeasurableSet S
hs_eq : s = cylinder I S
J : Finset ι
T : Set ((i : ↥J) → α ↑i)
hT : MeasurableSet T
ht_eq : t = cylinder J T
S' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' S
T' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' T
hS' : MeasurableSet S'
hT' : MeasurableSet T'
h_eq1 : s = cylinder (I ∪ J) S'
h_eq2 : t = cylinder (I ∪ J) T'
h_eq3 : s ∪ t = cylinder (I ∪ J) (S' ∪ T')
h : Nonempty ((i : ι) → α i)
⊢ (P (I ∪ J)) S' + (P (I ∪ J)) T' = (P (I ∪ J)) S' + (P (I ∪ J)) T'

pp: ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
s t : Set ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
hs : s ∈ measurableCylinders α
ht : t ∈ measurableCylinders α
hst : Disjoint s t
I : Finset ι
S : Set ((i : ↥I) → α ↑i)
hS : MeasurableSet S
hs_eq : s = cylinder I S
J : Finset ι
T : Set ((i : ↥J) → α ↑i)
hT : MeasurableSet T
ht_eq : t = cylinder J T
S' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' S
T' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' T
hS' : MeasurableSet S'
hT' : MeasurableSet T'
h_eq1 : s = cylinder (I ∪ J) S'
h_eq2 : t = cylinder (I ∪ J) T'
h_eq3 : s ∪ t = cylinder (I ∪ J) (S' ∪ T')
h : Nonempty ((i : ι) → α i)
⊢ Disjoint S' T'

Tactic:
]

Goal state after:
ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
s t : Set ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
hs : s ∈ measurableCylinders α
ht : t ∈ measurableCylinders α
hst : Disjoint s t
I : Finset ι
S : Set ((i : ↥I) → α ↑i)
hS : MeasurableSet S
hs_eq : s = cylinder I S
J : Finset ι
T : Set ((i : ↥J) → α ↑i)
hT : MeasurableSet T
ht_eq : t = cylinder J T
S' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' S
T' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' T
hS' : MeasurableSet S'
hT' : MeasurableSet T'
h_eq1 : s = cylinder (I ∪ J) S'
h_eq2 : t = cylinder (I ∪ J) T'
h_eq3 : s ∪ t = cylinder (I ∪ J) (S' ∪ T')
h : Nonempty ((i : ι) → α i)
⊢ Disjoint S' T'
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s t : Set α
I✝ : Finset (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m✝ m' m : AddContent G C
hC : IsSetSemiring C
I : Finset (Set α)
hI : ↑I ⊆ _root_.supClosure C
h'I : (↑I).PairwiseDisjoint id
hh'I : ⋃₀ ↑I ∈ _root_.supClosure C
J : (s : Set α) → Finpartition s
hJC : ∀ s ∈ I, ↑(J s).parts ⊆ C
K : Finpartition (I.sup id) := Finpartition.combine J ⋯
i : Set α
hi : i ∈ I
⊢ i = i

Tactic:
]

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
hC : IsSetSemiring C
m : AddContent G C
s : Set α
J : Finpartition s
hJ : ↑J.parts ⊆ C
⊢ ∑ s ∈ J.parts, m s = ∑ s ∈ J.parts, m s

pp: α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
hC : IsSetSemiring C
m : AddContent G C
s : Set α
J : Finpartition s
hJ : ↑J.parts ⊆ C
⊢ s = ⋃₀ ↑J.parts

Tactic:
]

Goal state after:
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
hC : IsSetSemiring C
m : AddContent G C
s : Set α
J : Finpartition s
hJ : ↑J.parts ⊆ C
⊢ s = ⋃₀ ↑J.parts
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
t : Set α
I : Finset (Set α)
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
h_ss : ↑I ⊆ C
h_dis : (↑I).PairwiseDisjoint id
ht : t ∈ C
hJt : ∀ s ∈ I, s ⊆ t
⊢ ∑ u ∈ I, m u ≤ ∑ i ∈ I, m i + ∑ i ∈ hC.disjointOfDiffUnion ht h_ss, m i

Tactic:
]

Goal state after:
α : Type u_1
C : Set (Set α)
t : Set α
I : Finset (Set α)
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
h_ss : ↑I ⊆ C
h_dis : (↑I).PairwiseDisjoint id
ht : t ∈ C
hJt : ∀ s ∈ I, s ⊆ t
⊢ ∑ u ∈ I, m u ≤ ∑ i ∈ I, m i + ∑ i ∈ hC.disjointOfDiffUnion ht h_ss, m i
Module: Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
s t : Set ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
hs : s ∈ measurableCylinders α
ht : t ∈ measurableCylinders α
I : Finset ι
S : Set ((i : ↥I) → α ↑i)
hS : MeasurableSet S
hs_eq : s = cylinder I S
J : Finset ι
T : Set ((i : ↥J) → α ↑i)
hst : Disjoint (restrict₂ ⋯ ⁻¹' S) (restrict₂ ⋯ ⁻¹' T)
hT : MeasurableSet T
ht_eq : t = cylinder J T
S' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' S
T' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' T
hS' : MeasurableSet S'
hT' : MeasurableSet T'
h_eq1 : s = cylinder (I ∪ J) S'
h_eq2 : t = cylinder (I ∪ J) T'
h_eq3 : s ∪ t = cylinder (I ∪ J) (S' ∪ T')
h : Nonempty ((i : ι) → α i)
⊢ Disjoint S' T'

Tactic:
]

Goal state after:
ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
s t : Set ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
hs : s ∈ measurableCylinders α
ht : t ∈ measurableCylinders α
I : Finset ι
S : Set ((i : ↥I) → α ↑i)
hS : MeasurableSet S
hs_eq : s = cylinder I S
J : Finset ι
T : Set ((i : ↥J) → α ↑i)
hst : Disjoint (restrict₂ ⋯ ⁻¹' S) (restrict₂ ⋯ ⁻¹' T)
hT : MeasurableSet T
ht_eq : t = cylinder J T
S' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' S
T' : Set ((i : ↥(I ∪ J)) → α ↑i) := restrict₂ ⋯ ⁻¹' T
hS' : MeasurableSet S'
hT' : MeasurableSet T'
h_eq1 : s = cylinder (I ∪ J) S'
h_eq2 : t = cylinder (I ∪ J) T'
h_eq3 : s ∪ t = cylinder (I ∪ J) (S' ∪ T')
h : Nonempty ((i : ι) → α i)
⊢ Disjoint S' T'
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
⊢ m s ≤ m t

Tactic:
have h := sum_addContent_le_of_subset (m := m) hC (I := {s}) ?_ ?_ ht ?_

Goal state after:
case refine_4
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
h : ∑ u ∈ {s}, m u ≤ m t
⊢ m s ≤ m t

case refine_1
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
⊢ ↑{s} ⊆ C

case refine_2
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
⊢ (↑{s}).PairwiseDisjoint id

case refine_3
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
⊢ ∀ s_1 ∈ {s}, s_1 ⊆ t
Module: Mathlib.MeasureTheory.Measure.AddContent

Goal state before:
pp: case refine_1
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
⊢ s ∈ C

Tactic:
]

Goal state after:
case refine_1
α : Type u_1
C : Set (Set α)
s t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
m : AddContent G C
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
hst : s ⊆ t
⊢ s ∈ C
Module: Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
I : Finset ι
S : Set ((i : ↥I) → α ↑i)
hP : IsProjectiveMeasureFamily P
s : Set ((i : ι) → α i)
hs_eq : s = cylinder I S
hS : MeasurableSet S
⊢ (P I) S = (P I) S

Tactic:
]

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

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
s : Set ((i : ι) → α i)
inst✝ : ∀ (J : Finset ι), IsFiniteMeasure (P J)
hP : IsProjectiveMeasureFamily P
⊢ (if hs : s ∈ measurableCylinders α then (P (measurableCylinders.finset hs)) (measurableCylinders.set hs) else 0) ≠ ∞

Tactic:
]

Goal state after:
ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)
s : Set ((i : ι) → α i)
inst✝ : ∀ (J : Finset ι), IsFiniteMeasure (P J)
hP : IsProjectiveMeasureFamily P
⊢ (if hs : s ∈ measurableCylinders α then (P (measurableCylinders.finset hs)) (measurableCylinders.set hs) else 0) ≠ ∞
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
key : ∀ (x : V), g (e x) • (v - e v) - c • x ∈ (↑e).fixedSubmodule ⊔ K ∙ v
hc0 : c ≠ 0
x : V
a✝ : x ∈ ⊤
⊢ g (e x) • v - g (e x) • e v = g (e x) • v - g (e x) • e v

Tactic:
]

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : ↑J ⊆ C
ht : t ∈ C
f : Fin #J → Set α := disjointed fun j ↦ ↑(J.equivFin.symm j)
htJ : t ⊆ ⋃ i, f i
h1 : ∀ (j : Fin #J), f j ∈ supClosure C
h2 : Pairwise (Disjoint on f)
h3 : ⋃ i, f i ∈ supClosure C
⊢ ∑ i, (m.supClosure hC) (f i) ≤ ∑ i, m ↑(J.equivFin.symm i)

Tactic:
]

Goal state after:
α : Type u_1
C : Set (Set α)
t : Set α
G : Type u_2
inst✝² : AddCommMonoid G
inst✝¹ : PartialOrder G
inst✝ : CanonicallyOrderedAdd G
m : AddContent G C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : ↑J ⊆ C
ht : t ∈ C
f : Fin #J → Set α := disjointed fun j ↦ ↑(J.equivFin.symm j)
htJ : t ⊆ ⋃ i, f i
h1 : ∀ (j : Fin #J), f j ∈ supClosure C
h2 : Pairwise (Disjoint on f)
h3 : ⋃ i, f i ∈ supClosure C
⊢ ∑ i, (m.supClosure hC) (f i) ≤ ∑ i, m ↑(J.equivFin.symm i)