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

tactic: refine (eq_of_boxes_subset_iUnion_superset (fun J₁ h₁ => ?_) ?_).symm
state_before: ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x : ι → ℝ
πi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J
hJ : J ∈ π
⊢ (π.biUnion πi).restrict J = πi J
state_after: case refine_1
ι : Type u_1
I J J₁✝ J₂ : Box ι
π π₁ π₂ : Prepartition I
x : ι → ℝ
πi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J
hJ : J ∈ π
J₁ : Box ι
h₁ : J₁ ∈ (πi J).boxes
⊢ J₁ ∈ ((π.biUnion πi).restrict J).boxes

case refine_2
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x : ι → ℝ
πi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J
hJ : J ∈ π
⊢ ((π.biUnion πi).restrict J).iUnion ⊆ (πi J).iUnion

tactic: refine (mem_restrict _).2 ⟨J₁, π.mem_biUnion.2 ⟨J, hJ, h₁⟩, (inf_of_le_right ?_).symm⟩
state_before: case refine_1
ι : Type u_1
I J J₁✝ J₂ : Box ι
π π₁ π₂ : Prepartition I
x : ι → ℝ
πi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J
hJ : J ∈ π
J₁ : Box ι
h₁ : J₁ ∈ (πi J).boxes
⊢ J₁ ∈ ((π.biUnion πi).restrict J).boxes
state_after: case refine_1
ι : Type u_1
I J J₁✝ J₂ : Box ι
π π₁ π₂ : Prepartition I
x : ι → ℝ
πi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J
hJ : J ∈ π
J₁ : Box ι
h₁ : J₁ ∈ (πi J).boxes
⊢ ↑J₁ ≤ ↑J

tactic: exact WithBot.coe_le_coe.2 (le_of_mem _ h₁)
state_before: case refine_1
ι : Type u_1
I J J₁✝ J₂ : Box ι
π π₁ π₂ : Prepartition I
x : ι → ℝ
πi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J
hJ : J ∈ π
J₁ : Box ι
h₁ : J₁ ∈ (πi J).boxes
⊢ ↑J₁ ≤ ↑J
state_after: no goals

tactic: simp only [iUnion_restrict, iUnion_biUnion, Set.subset_def, Set.mem_inter_iff, Set.mem_iUnion]
state_before: case refine_2
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x : ι → ℝ
πi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J
hJ : J ∈ π
⊢ ((π.biUnion πi).restrict J).iUnion ⊆ (πi J).iUnion
state_after: case refine_2
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x : ι → ℝ
πi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J
hJ : J ∈ π
⊢ ∀ (x : ι → ℝ), (x ∈ ↑J ∧ ∃ i, ∃ (_ : i ∈ π), x ∈ (πi i).iUnion) → x ∈ (πi J).iUnion

tactic: rintro x ⟨hxJ, J₁, h₁, hx⟩
state_before: case refine_2
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x : ι → ℝ
πi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J
hJ : J ∈ π
⊢ ∀ (x : ι → ℝ), (x ∈ ↑J ∧ ∃ i, ∃ (_ : i ∈ π), x ∈ (πi i).iUnion) → x ∈ (πi J).iUnion
state_after: case refine_2.intro.intro.intro
ι : Type u_1
I J J₁✝ J₂ : Box ι
π π₁ π₂ : Prepartition I
x✝ : ι → ℝ
πi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J
hJ : J ∈ π
x : ι → ℝ
hxJ : x ∈ ↑J
J₁ : Box ι
h₁ : J₁ ∈ π
hx : x ∈ (πi J₁).iUnion
⊢ x ∈ (πi J).iUnion

tactic: obtain rfl : J = J₁ := π.eq_of_mem_of_mem hJ h₁ hxJ (iUnion_subset _ hx)
state_before: case refine_2.intro.intro.intro
ι : Type u_1
I J J₁✝ J₂ : Box ι
π π₁ π₂ : Prepartition I
x✝ : ι → ℝ
πi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J
hJ : J ∈ π
x : ι → ℝ
hxJ : x ∈ ↑J
J₁ : Box ι
h₁ : J₁ ∈ π
hx : x ∈ (πi J₁).iUnion
⊢ x ∈ (πi J).iUnion
state_after: case refine_2.intro.intro.intro
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x✝ : ι → ℝ
πi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J
hJ : J ∈ π
x : ι → ℝ
hxJ : x ∈ ↑J
h₁ : J ∈ π
hx : x ∈ (πi J).iUnion
⊢ x ∈ (πi J).iUnion

tactic: exact hx
state_before: case refine_2.intro.intro.intro
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x✝ : ι → ℝ
πi✝ πi₁ πi₂ πi : (J : Box ι) → Prepartition J
hJ : J ∈ π
x : ι → ℝ
hxJ : x ∈ ↑J
h₁ : J ∈ π
hx : x ∈ (πi J).iUnion
⊢ x ∈ (πi J).iUnion
state_after: no goals