awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
U_1 : Opens G
hU : μ.innerContent U_1 ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : (↑ε : ℝ≥0∞) ≠ 0
h : (↑ε : ℝ≥0∞) < μ.innerContent U_1
U : Compacts G
h1U : (↑U : Set G) ⊆ (↑U_1 : Set G)
h2U : μ.innerContent U_1 - (↑ε : ℝ≥0∞) < (μ : (a : Compacts G) → ℝ≥0∞) U
⊢ μ.innerContent U_1 - (↑ε : ℝ≥0∞) ≤ (μ : (a : Compacts G) → ℝ≥0∞) U

Tactic:
le_of_lt h2U

Premises used: le_of_lt
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
U_1 : Opens G
hU : μ.innerContent U_1 ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : (↑ε : ℝ≥0∞) ≠ 0
h : (↑ε : ℝ≥0∞) < μ.innerContent U_1
U : Compacts G
h1U : (↑U : Set G) ⊆ (↑U_1 : Set G)
h2U : μ.innerContent U_1 - (↑ε : ℝ≥0∞) < (μ : (a : Compacts G) → ℝ≥0∞) U
⊢ μ.innerContent U_1 - (↑ε : ℝ≥0∞) ≤ (μ : (a : Compacts G) → ℝ≥0∞) U

Tactic:
exact le_of_lt h2U

Premises used: le_of_lt
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
    (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)

Tactic:
Finset.induction_on t ?_ ?_

Premises used: Finset.induction_on
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
refine_1 :
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (Finset.sup ∅ K) : ℝ≥0∞)
    (∑ i ∈ ∅, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
refine_2 :
  ∀ (a : ℕ) (s : Finset ℕ),
    a ∉ s →
      LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (s.sup K) : ℝ≥0∞)
          (∑ i ∈ s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞) →
        LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) ((insert a s).sup K) : ℝ≥0∞)
          (∑ i ∈ insert a s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
    (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)

Tactic:
refine Finset.induction_on t ?_ ?_
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (Finset.sup ∅ K) : ℝ≥0∞)
    (∑ i ∈ ∅, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)

Tactic:
simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]

Premises used: Finset.sup_empty, MeasureTheory.Content.empty, Finset.sum_empty, nonpos_iff_eq_zero
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
n : ℕ
s : Finset ℕ
hn : n ∉ s
ih :
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (s.sup K) : ℝ≥0∞)
    (∑ i ∈ s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
refine_2 :
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n ⊔ s.sup K) : ℝ≥0∞)
    (∑ i ∈ insert n s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
refine_2₁ :
  (μ : (a : Compacts G) → ℝ≥0∞) (K n ⊔ s.sup K) ≤
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n) : ℝ≥0∞)
      (∑ x ∈ s, (μ : (a : Compacts G) → ℝ≥0∞) (K x) : ℝ≥0∞)
refine_2₂ :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n) : ℝ≥0∞)
      ((μ : (a : Compacts G) → ℝ≥0∞) (s.sup K) : ℝ≥0∞) ≤
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n) : ℝ≥0∞)
      (∑ x ∈ s, (μ : (a : Compacts G) → ℝ≥0∞) (K x) : ℝ≥0∞)
refine_2₃ refine_2₄ :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n) : ℝ≥0∞)
      (∑ i ∈ s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞) ≤
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n) : ℝ≥0∞)
      (∑ i ∈ s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) ((insert n s).sup K) : ℝ≥0∞)
    (∑ i ∈ insert n s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)

Tactic:
grw [Finset.sup_insert, Finset.sum_insert hn, μ.sup_le, ih]

Premises used: MeasureTheory.Content.sup_le, Finset.sup_insert, Finset.sum_insert
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
⊢ ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)

Tactic:
have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    refine Finset.induction_on t ?_ ?_
    · simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
    · intro n s hn ih
      grw [Finset.sup_insert, Finset.sum_insert hn, μ.sup_le, ih]
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
⊢ μ.innerContent (⨆ (i : ℕ), U i) ≤ ∑' (i : ℕ), μ.innerContent (U i)

Tactic:
iSup₂_le fun K hK => ?_

Premises used: iSup₂_le, MeasureTheory.Content.innerContent
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
refine :
  ∀ (K : Compacts G) (hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (∑' (i : ℕ), μ.innerContent (U i) : ℝ≥0∞)
⊢ μ.innerContent (⨆ (i : ℕ), U i) ≤ ∑' (i : ℕ), μ.innerContent (U i)

Tactic:
refine iSup₂_le fun K hK => ?_

Premises used: iSup₂_le
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK rwa rwa₁ rwa₂ rwa₃ : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
⊢ (↑K : Set G) ⊆ ⋃ (i : ℕ), (↑(U i) : Set G)

Tactic:
rwa [← Opens.coe_iSup]

Premises used: TopologicalSpace.Opens.coe_iSup
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
obtain :
  ∀ (t : Finset ℕ) (ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (∑' (i : ℕ), μ.innerContent (U i) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (∑' (i : ℕ), μ.innerContent (U i) : ℝ≥0∞)

Tactic:
obtain ⟨t, ht⟩ :=
    K.isCompact.elim_finite_subcover _ (fun i => (U i).isOpen) (by rwa [← Opens.coe_iSup])

Premises used: IsCompact.elim_finite_subcover, TopologicalSpace.Opens.isOpen, TopologicalSpace.Compacts.isCompact
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
rcases :
  ∀ (K' : ℕ → Set G) (h1K' : ∀ (i : ℕ), IsCompact (K' i)) (h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i)
    (h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (∑' (i : ℕ), μ.innerContent (U i) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (∑' (i : ℕ), μ.innerContent (U i) : ℝ≥0∞)

Tactic:
rcases K.isCompact.finite_compact_cover t (SetLike.coe ∘ U) (fun i _ => (U i).isOpen) ht with
    ⟨K', h1K', h2K', h3K'⟩

Premises used: Function.comp, IsCompact.finite_compact_cover, SetLike.coe, TopologicalSpace.Opens.isOpen, TopologicalSpace.Compacts.isCompact
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup L) : ℝ≥0∞) (∑' (i : ℕ), μ.innerContent (U i) : ℝ≥0∞)

Tactic:
le_trans (h3 t L) _

Premises used: le_trans
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
h : K = t.sup L
convert_2 : LE.le (α := ℝ≥0∞) (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (L i)) (∑' (i : ℕ), μ.innerContent (U i))
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (∑' (i : ℕ), μ.innerContent (U i) : ℝ≥0∞)

Tactic:
convert! le_trans (h3 t L) _

Premises used: le_trans
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
h : (↑K : Set G) = (↑(t.sup L) : Set G)
⊢ K = t.sup L

Tactic:
ext1
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
h : (↑K : Set G) = ⨆ a ∈ t, (↑(L a) : Set G)
⊢ (↑K : Set G) = (↑(t.sup L) : Set G)

Tactic:
rw [Compacts.coe_finset_sup, Finset.sup_eq_iSup]

Premises used: TopologicalSpace.Compacts.coe_finset_sup, Finset.sup_eq_iSup
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
⊢ (↑K : Set G) = ⨆ a ∈ t, (↑(L a) : Set G)

Tactic:
h3K'

Premises used: Set.iUnion
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
⊢ (↑K : Set G) = ⨆ a ∈ t, (↑(L a) : Set G)

Tactic:
exact h3K'
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
⊢ LE.le (α := ℝ≥0∞) (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (L i)) (∑' (i : ℕ), μ.innerContent (U i))

Tactic:
le_trans (Finset.sum_le_sum ?_) (ENNReal.sum_le_tsum t)

Premises used: ENNReal.sum_le_tsum, Finset.sum_le_sum, le_trans
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
convert_2 : ∀ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (L i) ≤ μ.innerContent (U i)
⊢ LE.le (α := ℝ≥0∞) (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (L i)) (∑' (i : ℕ), μ.innerContent (U i))

Tactic:
refine le_trans (Finset.sum_le_sum ?_) (ENNReal.sum_le_tsum t)

Premises used: ENNReal.sum_le_tsum, Finset.sum_le_sum, le_trans
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
i : ℕ
convert_2 : i ∈ t → (μ : (a : Compacts G) → ℝ≥0∞) (L i) ≤ μ.innerContent (U i)
⊢ i ∈ t → (μ : (a : Compacts G) → ℝ≥0∞) (L i) ≤ μ.innerContent (U i)

Tactic:
_
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
i : ℕ
h : i ∈ t
⊢ (μ : (a : Compacts G) → ℝ≥0∞) (L i) ≤ μ.innerContent (U i)

Tactic:
le_trans ?_ (le_iSup _ (L i))

Premises used: le_iSup, le_trans
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
i : ℕ
h : i ∈ t
convert_2 :
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (L i) : ℝ≥0∞)
    (⨆ (_ : (↑(L i) : Set G) ⊆ (↑(U i) : Set G)), (μ : (a : Compacts G) → ℝ≥0∞) (L i) : ℝ≥0∞)
⊢ (μ : (a : Compacts G) → ℝ≥0∞) (L i) ≤ μ.innerContent (U i)

Tactic:
refine le_trans ?_ (le_iSup _ (L i))

Premises used: le_iSup, le_trans
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
i : ℕ
h : i ∈ t
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (L i) : ℝ≥0∞)
    (⨆ (_ : (↑(L i) : Set G) ⊆ (↑(U i) : Set G)), (μ : (a : Compacts G) → ℝ≥0∞) (L i) : ℝ≥0∞)

Tactic:
le_trans ?_ (le_iSup _ (h2K' i))

Premises used: le_iSup, le_trans
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
i : ℕ
h : i ∈ t
convert_2 : LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (L i) : ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (L i) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (L i) : ℝ≥0∞)
    (⨆ (_ : (↑(L i) : Set G) ⊆ (↑(U i) : Set G)), (μ : (a : Compacts G) → ℝ≥0∞) (L i) : ℝ≥0∞)

Tactic:
refine le_trans ?_ (le_iSup _ (h2K' i))

Premises used: le_iSup, le_trans
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
i : ℕ
h : i ∈ t
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (L i) : ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (L i) : ℝ≥0∞)

Tactic:
rfl
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Set G
hU : ∀ (i : ℕ), IsOpen (U i)
⊢ μ.innerContent (⨆ (i : ℕ), { carrier := U i, is_open' := ⋯ : Opens G }) ≤
    ∑' (i : ℕ), μ.innerContent { carrier := U i, is_open' := ⋯ }

Tactic:
have := μ.innerContent_iSup_nat fun i => ⟨U i, hU i⟩
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Set G
hU : ∀ (i : ℕ), IsOpen (U i)
this :
  μ.innerContent (⨆ (i : ℕ), { carrier := U i, is_open' := ⋯ : Opens G }) ≤
    ∑' (i : ℕ), μ.innerContent { carrier := U i, is_open' := ⋯ }
rwa rwa₁ rwa₂ rwa₃ :
  ∀
    (this :
      μ.innerContent
          { carrier := ⋃ (i : ℕ), (↑{ carrier := U i, is_open' := ⋯ : Opens G } : Set G), is_open' := ⋯ : Opens G } ≤
        ∑' (i : ℕ), μ.innerContent { carrier := U i, is_open' := ⋯ }),
    μ.innerContent { carrier := ⋃ (i : ℕ), U i, is_open' := ⋯ : Opens G } ≤
      ∑' (i : ℕ), μ.innerContent { carrier := U i, is_open' := ⋯ }
⊢ μ.innerContent { carrier := ⋃ (i : ℕ), U i, is_open' := ⋯ : Opens G } ≤
    ∑' (i : ℕ), μ.innerContent { carrier := U i, is_open' := ⋯ }

Tactic:
rwa [Opens.iSup_def] at this

Premises used: TopologicalSpace.Opens.iSup_def
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
f : G ≃ₜ G
h :
  ∀ ⦃K : Compacts G⦄,
    Eq (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (Compacts.map (⇑f : (a : G) → G) ⋯ K) : ℝ≥0∞)
      ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
U : Opens G
⊢ μ.innerContent ((Opens.comap (↑f : C(G, G)) : (a : Opens G) → Opens G) U) = μ.innerContent U

Tactic:
(Compacts.equiv f).surjective.iSup_congr _ fun K => iSup_congr_Prop image_subset_iff ?_

Premises used: Function.Surjective.iSup_congr, TopologicalSpace.Compacts.equiv, iSup_congr_Prop, Set.image_subset_iff, Equiv.surjective, MeasureTheory.Content.innerContent
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
f : G ≃ₜ G
h :
  ∀ ⦃K : Compacts G⦄,
    Eq (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (Compacts.map (⇑f : (a : G) → G) ⋯ K) : ℝ≥0∞)
      ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
U : Opens G
refine :
  ∀ (K : Compacts G)
    (x : (↑K : Set G) ⊆ (↑((Opens.comap (↑f : C(G, G)) : (a : Opens G) → Opens G) U : Opens G) : Set G)),
    Eq (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) ((Compacts.equiv f : (a : Compacts G) → Compacts G) K) : ℝ≥0∞)
      ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
⊢ μ.innerContent ((Opens.comap (↑f : C(G, G)) : (a : Opens G) → Opens G) U) = μ.innerContent U

Tactic:
refine (Compacts.equiv f).surjective.iSup_congr _ fun K => iSup_congr_Prop image_subset_iff ?_

Premises used: Function.Surjective.iSup_congr, TopologicalSpace.Compacts.equiv, Equiv.surjective
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
f : G ≃ₜ G
h :
  ∀ ⦃K : Compacts G⦄,
    Eq (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (Compacts.map (⇑f : (a : G) → G) ⋯ K) : ℝ≥0∞)
      ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
U : Opens G
K : Compacts G
hK : (↑K : Set G) ⊆ (↑((Opens.comap (↑f : C(G, G)) : (a : Opens G) → Opens G) U : Opens G) : Set G)
simp :
  Eq (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (Compacts.map (⇑f : (a : G) → G) ⋯ K) : ℝ≥0∞)
    ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) ((Compacts.equiv f : (a : Compacts G) → Compacts G) K) : ℝ≥0∞)
    ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)

Tactic:
simp only [Equiv.coe_fn_mk, Compacts.equiv]

Premises used: TopologicalSpace.Compacts.equiv, Equiv.coe_fn_mk