Theorem: BoxIntegral.Prepartition.iUnion_filter_not
File: Mathlib/Analysis/BoxIntegral/Partition/Basic.lean

tactic: simp only [Prepartition.iUnion]
state_before: ι : Type u_1
I J J₁ J₂ : Box ι
π✝ π₁ π₂ : Prepartition I
x : ι → ℝ
πi πi₁ πi₂ : (J : Box ι) → Prepartition J
π : Prepartition I
p : Box ι → Prop
⊢ (π.filter fun J => ¬p J).iUnion = π.iUnion \ (π.filter p).iUnion
state_after: ι : Type u_1
I J J₁ J₂ : Box ι
π✝ π₁ π₂ : Prepartition I
x : ι → ℝ
πi πi₁ πi₂ : (J : Box ι) → Prepartition J
π : Prepartition I
p : Box ι → Prop
⊢ ⋃ J ∈ π.filter fun J => ¬p J, ↑J = (⋃ J ∈ π, ↑J) \ ⋃ J ∈ π.filter p, ↑J

tactic: convert (@Set.biUnion_diff_biUnion_eq (ι → ℝ) (Box ι) π.boxes (π.filter p).boxes (↑) _).symm
state_before: ι : Type u_1
I J J₁ J₂ : Box ι
π✝ π₁ π₂ : Prepartition I
x : ι → ℝ
πi πi₁ πi₂ : (J : Box ι) → Prepartition J
π : Prepartition I
p : Box ι → Prop
⊢ ⋃ J ∈ π.filter fun J => ¬p J, ↑J = (⋃ J ∈ π, ↑J) \ ⋃ J ∈ π.filter p, ↑J
state_after: case h.e'_2.h.e'_3.h.pq.a.a
ι : Type u_1
I J J₁ J₂ : Box ι
π✝ π₁ π₂ : Prepartition I
x : ι → ℝ
πi πi₁ πi₂ : (J : Box ι) → Prepartition J
π : Prepartition I
p : Box ι → Prop
x✝ : Box ι
⊢ (x✝ ∈ π.filter fun J => ¬p J) ↔ x✝ ∈ ↑π.boxes \ ↑(π.filter p).boxes

ι : Type u_1
I J J₁ J₂ : Box ι
π✝ π₁ π₂ : Prepartition I
x : ι → ℝ
πi πi₁ πi₂ : (J : Box ι) → Prepartition J
π : Prepartition I
p : Box ι → Prop
⊢ (↑π.boxes ∪ ↑(π.filter p).boxes).PairwiseDisjoint Box.toSet

tactic: simp (config := { contextual := true })
state_before: case h.e'_2.h.e'_3.h.pq.a.a
ι : Type u_1
I J J₁ J₂ : Box ι
π✝ π₁ π₂ : Prepartition I
x : ι → ℝ
πi πi₁ πi₂ : (J : Box ι) → Prepartition J
π : Prepartition I
p : Box ι → Prop
x✝ : Box ι
⊢ (x✝ ∈ π.filter fun J => ¬p J) ↔ x✝ ∈ ↑π.boxes \ ↑(π.filter p).boxes
state_after: no goals

tactic: rw [Set.PairwiseDisjoint]
state_before: ι : Type u_1
I J J₁ J₂ : Box ι
π✝ π₁ π₂ : Prepartition I
x : ι → ℝ
πi πi₁ πi₂ : (J : Box ι) → Prepartition J
π : Prepartition I
p : Box ι → Prop
⊢ (↑π.boxes ∪ ↑(π.filter p).boxes).PairwiseDisjoint Box.toSet
state_after: ι : Type u_1
I J J₁ J₂ : Box ι
π✝ π₁ π₂ : Prepartition I
x : ι → ℝ
πi πi₁ πi₂ : (J : Box ι) → Prepartition J
π : Prepartition I
p : Box ι → Prop
⊢ (↑π.boxes ∪ ↑(π.filter p).boxes).Pairwise (Disjoint on Box.toSet)

tactic: convert π.pairwiseDisjoint
state_before: ι : Type u_1
I J J₁ J₂ : Box ι
π✝ π₁ π₂ : Prepartition I
x : ι → ℝ
πi πi₁ πi₂ : (J : Box ι) → Prepartition J
π : Prepartition I
p : Box ι → Prop
⊢ (↑π.boxes ∪ ↑(π.filter p).boxes).Pairwise (Disjoint on Box.toSet)
state_after: case h.e'_2
ι : Type u_1
I J J₁ J₂ : Box ι
π✝ π₁ π₂ : Prepartition I
x : ι → ℝ
πi πi₁ πi₂ : (J : Box ι) → Prepartition J
π : Prepartition I
p : Box ι → Prop
⊢ ↑π.boxes ∪ ↑(π.filter p).boxes = ↑π.boxes

tactic: rw [Set.union_eq_left, filter_boxes, coe_filter]
state_before: case h.e'_2
ι : Type u_1
I J J₁ J₂ : Box ι
π✝ π₁ π₂ : Prepartition I
x : ι → ℝ
πi πi₁ πi₂ : (J : Box ι) → Prepartition J
π : Prepartition I
p : Box ι → Prop
⊢ ↑π.boxes ∪ ↑(π.filter p).boxes = ↑π.boxes
state_after: case h.e'_2
ι : Type u_1
I J J₁ J₂ : Box ι
π✝ π₁ π₂ : Prepartition I
x : ι → ℝ
πi πi₁ πi₂ : (J : Box ι) → Prepartition J
π : Prepartition I
p : Box ι → Prop
⊢ {x | x ∈ π.boxes ∧ p x} ⊆ ↑π.boxes

tactic: exact fun _ ⟨h, _⟩ => h
state_before: case h.e'_2
ι : Type u_1
I J J₁ J₂ : Box ι
π✝ π₁ π₂ : Prepartition I
x : ι → ℝ
πi πi₁ πi₂ : (J : Box ι) → Prepartition J
π : Prepartition I
p : Box ι → Prop
⊢ {x | x ∈ π.boxes ∧ p x} ⊆ ↑π.boxes
state_after: no goals