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

tactic: refine ⟨fun H => ⟨H.trans (π.biUnion_le πi), fun J hJ => ?_⟩, ?_⟩
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
π' : Prepartition I
⊢ π' ≤ π.biUnion πi ↔ π' ≤ π ∧ ∀ J ∈ π, π'.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
π' : Prepartition I
H : π' ≤ π.biUnion πi
J : Box ι
hJ : J ∈ π
⊢ π'.restrict J ≤ πi J

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
π' : Prepartition I
⊢ (π' ≤ π ∧ ∀ J ∈ π, π'.restrict J ≤ πi J) → π' ≤ π.biUnion πi

tactic: rw [← π.restrict_biUnion πi hJ]
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
π' : Prepartition I
H : π' ≤ π.biUnion πi
J : Box ι
hJ : J ∈ π
⊢ π'.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
π' : Prepartition I
H : π' ≤ π.biUnion πi
J : Box ι
hJ : J ∈ π
⊢ π'.restrict J ≤ (π.biUnion πi).restrict J

tactic: exact restrict_mono H
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
π' : Prepartition I
H : π' ≤ π.biUnion πi
J : Box ι
hJ : J ∈ π
⊢ π'.restrict J ≤ (π.biUnion πi).restrict J
state_after: no goals

tactic: rintro ⟨H, Hi⟩ J' hJ'
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
π' : Prepartition I
⊢ (π' ≤ π ∧ ∀ J ∈ π, π'.restrict J ≤ πi J) → π' ≤ π.biUnion πi
state_after: case refine_2.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
π' : Prepartition I
H : π' ≤ π
Hi : ∀ J ∈ π, π'.restrict J ≤ πi J
J' : Box ι
hJ' : J' ∈ π'
⊢ ∃ I' ∈ π.biUnion πi, J' ≤ I'

tactic: rcases H hJ' with ⟨J, hJ, hle⟩
state_before: case refine_2.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
π' : Prepartition I
H : π' ≤ π
Hi : ∀ J ∈ π, π'.restrict J ≤ πi J
J' : Box ι
hJ' : J' ∈ π'
⊢ ∃ I' ∈ π.biUnion πi, J' ≤ I'
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
π' : Prepartition I
H : π' ≤ π
Hi : ∀ J ∈ π, π'.restrict J ≤ πi J
J' : Box ι
hJ' : J' ∈ π'
J : Box ι
hJ : J ∈ π
hle : J' ≤ J
⊢ ∃ I' ∈ π.biUnion πi, J' ≤ I'

tactic: have : J' ∈ π'.restrict J :=
  π'.mem_restrict.2 ⟨J', hJ', (inf_of_le_right <| WithBot.coe_le_coe.2 hle).symm⟩
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
π' : Prepartition I
H : π' ≤ π
Hi : ∀ J ∈ π, π'.restrict J ≤ πi J
J' : Box ι
hJ' : J' ∈ π'
J : Box ι
hJ : J ∈ π
hle : J' ≤ J
⊢ ∃ I' ∈ π.biUnion πi, J' ≤ I'
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
π' : Prepartition I
H : π' ≤ π
Hi : ∀ J ∈ π, π'.restrict J ≤ πi J
J' : Box ι
hJ' : J' ∈ π'
J : Box ι
hJ : J ∈ π
hle : J' ≤ J
this : J' ∈ π'.restrict J
⊢ ∃ I' ∈ π.biUnion πi, J' ≤ I'

tactic: rcases Hi J hJ this with ⟨Ji, hJi, hlei⟩
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
π' : Prepartition I
H : π' ≤ π
Hi : ∀ J ∈ π, π'.restrict J ≤ πi J
J' : Box ι
hJ' : J' ∈ π'
J : Box ι
hJ : J ∈ π
hle : J' ≤ J
this : J' ∈ π'.restrict J
⊢ ∃ I' ∈ π.biUnion πi, J' ≤ I'
state_after: case refine_2.intro.intro.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
π' : Prepartition I
H : π' ≤ π
Hi : ∀ J ∈ π, π'.restrict J ≤ πi J
J' : Box ι
hJ' : J' ∈ π'
J : Box ι
hJ : J ∈ π
hle : J' ≤ J
this : J' ∈ π'.restrict J
Ji : Box ι
hJi : Ji ∈ πi J
hlei : J' ≤ Ji
⊢ ∃ I' ∈ π.biUnion πi, J' ≤ I'

tactic: exact ⟨Ji, π.mem_biUnion.2 ⟨J, hJ, hJi⟩, hlei⟩
state_before: case refine_2.intro.intro.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
π' : Prepartition I
H : π' ≤ π
Hi : ∀ J ∈ π, π'.restrict J ≤ πi J
J' : Box ι
hJ' : J' ∈ π'
J : Box ι
hJ : J ∈ π
hle : J' ≤ J
this : J' ∈ π'.restrict J
Ji : Box ι
hJi : Ji ∈ πi J
hlei : J' ≤ Ji
⊢ ∃ I' ∈ π.biUnion πi, J' ≤ I'
state_after: no goals