Theorem: MeasureTheory.Content.innerContent_iSup_nat
File: Mathlib/MeasureTheory/Measure/Content.lean

tactic: refine iSup₂_le fun K hK => ?_
state_before: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
⊢ μ.innerContent (⨆ i, U i) ≤ ∑' (i : ℕ), μ.innerContent (U i)
state_after: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
⊢ (fun s => ↑(μ.toFun s)) K ≤ ∑' (i : ℕ), μ.innerContent (U i)

tactic: obtain ⟨t, ht⟩ :=
  K.isCompact.elim_finite_subcover _ (fun i => (U i).isOpen) (by rwa [← Opens.coe_iSup])
state_before: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
⊢ (fun s => ↑(μ.toFun s)) K ≤ ∑' (i : ℕ), μ.innerContent (U i)
state_after: case intro
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
⊢ (fun s => ↑(μ.toFun s)) K ≤ ∑' (i : ℕ), μ.innerContent (U i)

tactic: rcases K.isCompact.finite_compact_cover t (SetLike.coe ∘ U) (fun i _ => (U i).isOpen) ht with
  ⟨K', h1K', h2K', h3K'⟩
state_before: case intro
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
⊢ (fun s => ↑(μ.toFun s)) K ≤ ∑' (i : ℕ), μ.innerContent (U i)
state_after: case intro.intro.intro.intro
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
⊢ (fun s => ↑(μ.toFun s)) K ≤ ∑' (i : ℕ), μ.innerContent (U i)

tactic: let L : ℕ → Compacts G := fun n => ⟨K' n, h1K' n⟩
state_before: case intro.intro.intro.intro
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
⊢ (fun s => ↑(μ.toFun s)) K ≤ ∑' (i : ℕ), μ.innerContent (U i)
state_after: case intro.intro.intro.intro
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
⊢ (fun s => ↑(μ.toFun s)) K ≤ ∑' (i : ℕ), μ.innerContent (U i)

tactic: convert le_trans (h3 t L) _
state_before: case intro.intro.intro.intro
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
⊢ (fun s => ↑(μ.toFun s)) K ≤ ∑' (i : ℕ), μ.innerContent (U i)
state_after: case h.e'_3.h.e'_1
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
⊢ K = t.sup L

case intro.intro.intro.intro.convert_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
⊢ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (L i) ≤ ∑' (i : ℕ), μ.innerContent (U i)

tactic: refine le_trans (Finset.sum_le_sum ?_) (ENNReal.sum_le_tsum t)
state_before: case intro.intro.intro.intro.convert_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
⊢ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (L i) ≤ ∑' (i : ℕ), μ.innerContent (U i)
state_after: case intro.intro.intro.intro.convert_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
⊢ ∀ i ∈ t, (fun s => ↑(μ.toFun s)) (L i) ≤ μ.innerContent (U i)

tactic: intro i _
state_before: case intro.intro.intro.intro.convert_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
⊢ ∀ i ∈ t, (fun s => ↑(μ.toFun s)) (L i) ≤ μ.innerContent (U i)
state_after: case intro.intro.intro.intro.convert_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
i : ℕ
a✝ : i ∈ t
⊢ (fun s => ↑(μ.toFun s)) (L i) ≤ μ.innerContent (U i)

tactic: refine le_trans ?_ (le_iSup _ (L i))
state_before: case intro.intro.intro.intro.convert_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
i : ℕ
a✝ : i ∈ t
⊢ (fun s => ↑(μ.toFun s)) (L i) ≤ μ.innerContent (U i)
state_after: case intro.intro.intro.intro.convert_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
i : ℕ
a✝ : i ∈ t
⊢ (fun s => ↑(μ.toFun s)) (L i) ≤ ⨆ (_ : ↑(L i) ⊆ ↑(U i)), (fun s => ↑(μ.toFun s)) (L i)

tactic: refine le_trans ?_ (le_iSup _ (h2K' i))
state_before: case intro.intro.intro.intro.convert_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
i : ℕ
a✝ : i ∈ t
⊢ (fun s => ↑(μ.toFun s)) (L i) ≤ ⨆ (_ : ↑(L i) ⊆ ↑(U i)), (fun s => ↑(μ.toFun s)) (L i)
state_after: case intro.intro.intro.intro.convert_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
i : ℕ
a✝ : i ∈ t
⊢ (fun s => ↑(μ.toFun s)) (L i) ≤ (fun s => ↑(μ.toFun s)) (L i)

tactic: rfl
state_before: case intro.intro.intro.intro.convert_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
i : ℕ
a✝ : i ∈ t
⊢ (fun s => ↑(μ.toFun s)) (L i) ≤ (fun s => ↑(μ.toFun s)) (L i)
state_after: no goals

tactic: intro t K
state_before: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
⊢ ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
state_after: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
⊢ (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)

tactic: refine Finset.induction_on t ?_ ?_
state_before: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
⊢ (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
state_after: case refine_1
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
⊢ (fun s => ↑(μ.toFun s)) (∅.sup K) ≤ ∑ i ∈ ∅, (fun s => ↑(μ.toFun s)) (K i)

case refine_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
⊢ ∀ ⦃a : ℕ⦄ {s : Finset ℕ},
    a ∉ s →
      (fun s => ↑(μ.toFun s)) (s.sup K) ≤ ∑ i ∈ s, (fun s => ↑(μ.toFun s)) (K i) →
        (fun s => ↑(μ.toFun s)) ((insert a s).sup K) ≤ ∑ i ∈ insert a s, (fun s => ↑(μ.toFun s)) (K i)

tactic: simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
state_before: case refine_1
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
⊢ (fun s => ↑(μ.toFun s)) (∅.sup K) ≤ ∑ i ∈ ∅, (fun s => ↑(μ.toFun s)) (K i)
state_after: no goals

tactic: intro n s hn ih
state_before: case refine_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
⊢ ∀ ⦃a : ℕ⦄ {s : Finset ℕ},
    a ∉ s →
      (fun s => ↑(μ.toFun s)) (s.sup K) ≤ ∑ i ∈ s, (fun s => ↑(μ.toFun s)) (K i) →
        (fun s => ↑(μ.toFun s)) ((insert a s).sup K) ≤ ∑ i ∈ insert a s, (fun s => ↑(μ.toFun s)) (K i)
state_after: case refine_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
n : ℕ
s : Finset ℕ
hn : n ∉ s
ih : (fun s => ↑(μ.toFun s)) (s.sup K) ≤ ∑ i ∈ s, (fun s => ↑(μ.toFun s)) (K i)
⊢ (fun s => ↑(μ.toFun s)) ((insert n s).sup K) ≤ ∑ i ∈ insert n s, (fun s => ↑(μ.toFun s)) (K i)

tactic: rw [Finset.sup_insert, Finset.sum_insert hn]
state_before: case refine_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
n : ℕ
s : Finset ℕ
hn : n ∉ s
ih : (fun s => ↑(μ.toFun s)) (s.sup K) ≤ ∑ i ∈ s, (fun s => ↑(μ.toFun s)) (K i)
⊢ (fun s => ↑(μ.toFun s)) ((insert n s).sup K) ≤ ∑ i ∈ insert n s, (fun s => ↑(μ.toFun s)) (K i)
state_after: case refine_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
n : ℕ
s : Finset ℕ
hn : n ∉ s
ih : (fun s => ↑(μ.toFun s)) (s.sup K) ≤ ∑ i ∈ s, (fun s => ↑(μ.toFun s)) (K i)
⊢ (fun s => ↑(μ.toFun s)) (K n ⊔ s.sup K) ≤ (fun s => ↑(μ.toFun s)) (K n) + ∑ x ∈ s, (fun s => ↑(μ.toFun s)) (K x)

tactic: exact le_trans (μ.sup_le _ _) (add_le_add_left ih _)
state_before: case refine_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
n : ℕ
s : Finset ℕ
hn : n ∉ s
ih : (fun s => ↑(μ.toFun s)) (s.sup K) ≤ ∑ i ∈ s, (fun s => ↑(μ.toFun s)) (K i)
⊢ (fun s => ↑(μ.toFun s)) (K n ⊔ s.sup K) ≤ (fun s => ↑(μ.toFun s)) (K n) + ∑ x ∈ s, (fun s => ↑(μ.toFun s)) (K x)
state_after: no goals

tactic: rwa [← Opens.coe_iSup]
state_before: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
⊢ ↑K ⊆ ⋃ i, ↑(U i)
state_after: no goals

tactic: ext1
state_before: case h.e'_3.h.e'_1
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
⊢ K = t.sup L
state_after: case h.e'_3.h.e'_1.h
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
⊢ ↑K = ↑(t.sup L)

tactic: rw [Compacts.coe_finset_sup, Finset.sup_eq_iSup]
state_before: case h.e'_3.h.e'_1.h
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
⊢ ↑K = ↑(t.sup L)
state_after: case h.e'_3.h.e'_1.h
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
⊢ ↑K = ⨆ a ∈ t, ↑(L a)

tactic: exact h3K'
state_before: case h.e'_3.h.e'_1.h
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }
⊢ ↑K = ⨆ a ∈ t, ↑(L a)
state_after: no goals