awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
refine :
  ∀ (i : { i : Compacts G // (↑i : Set G) ⊆ (↑U' : Set G) ∩ U }),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (↑i : Compacts G) : ℝ≥0∞)
        ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑U' : Set G) \ U) : ℝ≥0∞) ≤
      (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)
⊢ LE.le (α := ℝ≥0∞)
    (⨆ (i : { i : Compacts G // (↑i : Set G) ⊆ (↑U' : Set G) ∩ U }),
        HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (↑i : Compacts G) : ℝ≥0∞)
          ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑U' : Set G) \ U) : ℝ≥0∞) :
      ℝ≥0∞)
    ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G) : ℝ≥0∞)

Tactic:
refine iSup_le ?_

Premises used: iSup_le
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∩ U
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
dsimp :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L : ℝ≥0∞)
      ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑U' : Set G) \ U) : ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
      ((μ : (a : Compacts G) → ℝ≥0∞)
          (↑(Subtype.mk (p := fun (i : Compacts G) ↦ (↑i : Set G) ⊆ (↑U' : Set G) ∩ U) L hL) : Compacts G) :
        ℝ≥0∞)
      ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑U' : Set G) \ U) : ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)

Tactic:
dsimp
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∩ U
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
grw :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L' : ℝ≥0∞)
      ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑U' : Set G) \ U) : ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L : ℝ≥0∞)
      ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑U' : Set G) \ U) : ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)

Tactic:
grw [show μ L ≤ μ L' from μ.mono _ _ subset_closure]

Premises used: subset_closure, MeasureTheory.Content.mono
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∩ U
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
⊢ (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U

Tactic:
simp only [subset_inter_iff] at hL

Premises used: Set.subset_inter_iff
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
⊢ (↑L' : Set G) ⊆ U

Tactic:
have hL'U : (L' : Set G) ⊆ U := IsCompact.closure_subset_of_isOpen L.2 hU hL.2
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
hL'U : (↑L' : Set G) ⊆ U
⊢ (↑L' : Set G) ⊆ (↑U' : Set G)

Tactic:
have hL'U' : (L' : Set G) ⊆ (U' : Set G) := IsCompact.closure_subset_of_isOpen L.2 U'.2 hL.1
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
hL'U : (↑L' : Set G) ⊆ U
hL'U' : (↑L' : Set G) ⊆ (↑U' : Set G)
⊢ (↑U' : Set G) \ U ⊆ (↑U' : Set G) \ (↑L' : Set G)

Tactic:
have : ↑U' \ U ⊆ U' \ L' := diff_subset_diff_right hL'U
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this_1 : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
hL'U : (↑L' : Set G) ⊆ U
hL'U' : (↑L' : Set G) ⊆ (↑U' : Set G)
this : (↑U' : Set G) \ U ⊆ (↑U' : Set G) \ (↑L' : Set G)
grw :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L' : ℝ≥0∞)
      ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑U' : Set G) \ (↑L' : Set G)) : ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L' : ℝ≥0∞)
      ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑U' : Set G) \ U) : ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)

Tactic:
grw [this]
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this_1 : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
hL'U : (↑L' : Set G) ⊆ U
hL'U' : (↑L' : Set G) ⊆ (↑U' : Set G)
this : (↑U' : Set G) \ U ⊆ (↑U' : Set G) \ (↑L' : Set G)
rw :
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L' : ℝ≥0∞)
      (μ.innerContent { carrier := (↑U' : Set G) \ (↑L' : Set G), is_open' := ⋯ : Opens G }) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L' : ℝ≥0∞)
      ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑U' : Set G) \ (↑L' : Set G)) : ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)

Tactic:
rw [μ.outerMeasure_of_isOpen (↑U' \ L') (IsOpen.sdiff U'.2 isClosed_closure)]

Premises used: SDiff.sdiff, IsOpen.sdiff, SetLike.coe, TopologicalSpace.Opens.is_open', isClosed_closure, MeasureTheory.Content.outerMeasure_of_isOpen
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this_1 : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
hL'U : (↑L' : Set G) ⊆ U
hL'U' : (↑L' : Set G) ⊆ (↑U' : Set G)
this : (↑U' : Set G) \ U ⊆ (↑U' : Set G) \ (↑L' : Set G)
simp :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L' : ℝ≥0∞)
      (⨆ (x :
          { i : Compacts G //
            (↑i : Set G) ⊆ (↑{ carrier := (↑U' : Set G) \ (↑L' : Set G), is_open' := ⋯ : Opens G } : Set G) }),
          (μ : (a : Compacts G) → ℝ≥0∞) (↑x : Compacts G) :
        ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)
⊢ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L' : ℝ≥0∞)
      (μ.innerContent { carrier := (↑U' : Set G) \ (↑L' : Set G), is_open' := ⋯ : Opens G }) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)

Tactic:
simp only [innerContent, iSup_subtype']

Premises used: iSup_subtype', MeasureTheory.Content.innerContent
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this_1 : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
hL'U : (↑L' : Set G) ⊆ U
hL'U' : (↑L' : Set G) ⊆ (↑U' : Set G)
this : (↑U' : Set G) \ U ⊆ (↑U' : Set G) \ (↑L' : Set G)
rw :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L' : ℝ≥0∞)
      (⨆ (x : { i : Compacts G // (↑i : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G) }),
          (μ : (a : Compacts G) → ℝ≥0∞) (↑x : Compacts G) :
        ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L' : ℝ≥0∞)
      (⨆ (x :
          { i : Compacts G //
            (↑i : Set G) ⊆ (↑{ carrier := (↑U' : Set G) \ (↑L' : Set G), is_open' := ⋯ : Opens G } : Set G) }),
          (μ : (a : Compacts G) → ℝ≥0∞) (↑x : Compacts G) :
        ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)

Tactic:
rw [Opens.coe_mk]

Premises used: TopologicalSpace.Opens.coe_mk
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this_1 : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
hL'U : (↑L' : Set G) ⊆ U
hL'U' : (↑L' : Set G) ⊆ (↑U' : Set G)
this : (↑U' : Set G) \ U ⊆ (↑U' : Set G) \ (↑L' : Set G)
⊢ Nonempty.{max 1 (w + 1)} { M : Compacts G // (↑M : Set G) ⊆ (↑U' : Set G) \ closure (↑L : Set G) }

Tactic:
haveI : Nonempty { M : Compacts G // (M : Set G) ⊆ ↑U' \ closure L } := ⟨⟨⊥, empty_subset _⟩⟩
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this_1 : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
hL'U : (↑L' : Set G) ⊆ U
hL'U' : (↑L' : Set G) ⊆ (↑U' : Set G)
this_2 : (↑U' : Set G) \ U ⊆ (↑U' : Set G) \ (↑L' : Set G)
this : Nonempty.{max 1 (w + 1)} { M : Compacts G // (↑M : Set G) ⊆ (↑U' : Set G) \ closure (↑L : Set G) }
rw :
  LE.le (α := ℝ≥0∞)
    (⨆ (i : { i : Compacts G // (↑i : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G) }),
        HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L' : ℝ≥0∞)
          ((μ : (a : Compacts G) → ℝ≥0∞) (↑i : Compacts G) : ℝ≥0∞) :
      ℝ≥0∞)
    ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G) : ℝ≥0∞)
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L' : ℝ≥0∞)
      (⨆ (x : { i : Compacts G // (↑i : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G) }),
          (μ : (a : Compacts G) → ℝ≥0∞) (↑x : Compacts G) :
        ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)

Tactic:
rw [ENNReal.add_iSup]

Premises used: ENNReal.add_iSup
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this_1 : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
hL'U : (↑L' : Set G) ⊆ U
hL'U' : (↑L' : Set G) ⊆ (↑U' : Set G)
this_2 : (↑U' : Set G) \ U ⊆ (↑U' : Set G) \ (↑L' : Set G)
this : Nonempty.{max 1 (w + 1)} { M : Compacts G // (↑M : Set G) ⊆ (↑U' : Set G) \ closure (↑L : Set G) }
⊢ LE.le (α := ℝ≥0∞)
    (⨆ (i : { i : Compacts G // (↑i : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G) }),
        HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L' : ℝ≥0∞)
          ((μ : (a : Compacts G) → ℝ≥0∞) (↑i : Compacts G) : ℝ≥0∞) :
      ℝ≥0∞)
    ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G) : ℝ≥0∞)

Tactic:
iSup_le ?_

Premises used: iSup_le
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this_1 : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
hL'U : (↑L' : Set G) ⊆ U
hL'U' : (↑L' : Set G) ⊆ (↑U' : Set G)
this_2 : (↑U' : Set G) \ U ⊆ (↑U' : Set G) \ (↑L' : Set G)
this : Nonempty.{max 1 (w + 1)} { M : Compacts G // (↑M : Set G) ⊆ (↑U' : Set G) \ closure (↑L : Set G) }
refine :
  ∀ (i : { i : Compacts G // (↑i : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G) }),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L' : ℝ≥0∞)
        ((μ : (a : Compacts G) → ℝ≥0∞) (↑i : Compacts G) : ℝ≥0∞) ≤
      (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)
⊢ LE.le (α := ℝ≥0∞)
    (⨆ (i : { i : Compacts G // (↑i : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G) }),
        HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L' : ℝ≥0∞)
          ((μ : (a : Compacts G) → ℝ≥0∞) (↑i : Compacts G) : ℝ≥0∞) :
      ℝ≥0∞)
    ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G) : ℝ≥0∞)

Tactic:
refine iSup_le ?_

Premises used: iSup_le
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this_1 : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
hL'U : (↑L' : Set G) ⊆ U
hL'U' : (↑L' : Set G) ⊆ (↑U' : Set G)
this_2 : (↑U' : Set G) \ U ⊆ (↑U' : Set G) \ (↑L' : Set G)
this : Nonempty.{max 1 (w + 1)} { M : Compacts G // (↑M : Set G) ⊆ (↑U' : Set G) \ closure (↑L : Set G) }
M : Compacts G
hM : (↑M : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G)
M' : Compacts G := { carrier := closure (↑M : Set G), isCompact' := ⋯ }
dsimp :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L' : ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) M : ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L' : ℝ≥0∞)
      ((μ : (a : Compacts G) → ℝ≥0∞)
          (↑(Subtype.mk (p := fun (i : Compacts G) ↦ (↑i : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G)) M hM) : Compacts G) :
        ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)

Tactic:
dsimp
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this_1 : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
hL'U : (↑L' : Set G) ⊆ U
hL'U' : (↑L' : Set G) ⊆ (↑U' : Set G)
this_2 : (↑U' : Set G) \ U ⊆ (↑U' : Set G) \ (↑L' : Set G)
this : Nonempty.{max 1 (w + 1)} { M : Compacts G // (↑M : Set G) ⊆ (↑U' : Set G) \ closure (↑L : Set G) }
M : Compacts G
hM : (↑M : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G)
M' : Compacts G := { carrier := closure (↑M : Set G), isCompact' := ⋯ }
grw :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L' : ℝ≥0∞)
      ((μ : (a : Compacts G) → ℝ≥0∞) M' : ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) L' : ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) M : ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)

Tactic:
grw [show μ M ≤ μ M' from μ.mono _ _ subset_closure]

Premises used: subset_closure, MeasureTheory.Content.mono
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this_1 : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
hL'U : (↑L' : Set G) ⊆ U
hL'U' : (↑L' : Set G) ⊆ (↑U' : Set G)
this_2 : (↑U' : Set G) \ U ⊆ (↑U' : Set G) \ (↑L' : Set G)
this : Nonempty.{max 1 (w + 1)} { M : Compacts G // (↑M : Set G) ⊆ (↑U' : Set G) \ closure (↑L : Set G) }
M : Compacts G
hM : (↑M : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G)
M' : Compacts G := { carrier := closure (↑M : Set G), isCompact' := ⋯ }
⊢ (↑M' : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G)

Tactic:
have hM' : (M' : Set G) ⊆ U' \ L' :=
    IsCompact.closure_subset_of_isOpen M.2 (IsOpen.sdiff U'.2 isClosed_closure) hM
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this_1 : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
hL'U : (↑L' : Set G) ⊆ U
hL'U' : (↑L' : Set G) ⊆ (↑U' : Set G)
this_2 : (↑U' : Set G) \ U ⊆ (↑U' : Set G) \ (↑L' : Set G)
this : Nonempty.{max 1 (w + 1)} { M : Compacts G // (↑M : Set G) ⊆ (↑U' : Set G) \ closure (↑L : Set G) }
M : Compacts G
hM : (↑M : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G)
M' : Compacts G := { carrier := closure (↑M : Set G), isCompact' := ⋯ }
hM' : (↑M' : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G)
simp : (↑M' : Set G) ⊆ (↑U' : Set G)
⊢ (↑(L' ⊔ M') : Set G) ⊆ (↑U' : Set G)

Tactic:
simp only [Compacts.coe_sup, union_subset_iff, hL'U', true_and]

Premises used: Set.union_subset_iff, TopologicalSpace.Compacts.coe_sup, true_and
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this_1 : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
hL'U : (↑L' : Set G) ⊆ U
hL'U' : (↑L' : Set G) ⊆ (↑U' : Set G)
this_2 : (↑U' : Set G) \ U ⊆ (↑U' : Set G) \ (↑L' : Set G)
this : Nonempty.{max 1 (w + 1)} { M : Compacts G // (↑M : Set G) ⊆ (↑U' : Set G) \ closure (↑L : Set G) }
M : Compacts G
hM : (↑M : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G)
M' : Compacts G := { carrier := closure (↑M : Set G), isCompact' := ⋯ }
hM' : (↑M' : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G)
⊢ (↑M' : Set G) ⊆ (↑U' : Set G)

Tactic:
hM'.trans diff_subset

Premises used: Set.diff_subset, HasSubset.Subset.trans
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this_1 : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
hL'U : (↑L' : Set G) ⊆ U
hL'U' : (↑L' : Set G) ⊆ (↑U' : Set G)
this_2 : (↑U' : Set G) \ U ⊆ (↑U' : Set G) \ (↑L' : Set G)
this : Nonempty.{max 1 (w + 1)} { M : Compacts G // (↑M : Set G) ⊆ (↑U' : Set G) \ closure (↑L : Set G) }
M : Compacts G
hM : (↑M : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G)
M' : Compacts G := { carrier := closure (↑M : Set G), isCompact' := ⋯ }
hM' : (↑M' : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G)
⊢ (↑M' : Set G) ⊆ (↑U' : Set G)

Tactic:
exact hM'.trans diff_subset

Premises used: Set.diff_subset, HasSubset.Subset.trans
-- Module: Mathlib.MeasureTheory.Measure.Content
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
S : MeasurableSpace G
inst_2 : BorelSpace G
U : Set G
hU : Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} U
U' : Opens G
this_1 : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure (↑L : Set G), isCompact' := ⋯ }
hL : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
hL'U : (↑L' : Set G) ⊆ U
hL'U' : (↑L' : Set G) ⊆ (↑U' : Set G)
this_2 : (↑U' : Set G) \ U ⊆ (↑U' : Set G) \ (↑L' : Set G)
this : Nonempty.{max 1 (w + 1)} { M : Compacts G // (↑M : Set G) ⊆ (↑U' : Set G) \ closure (↑L : Set G) }
M : Compacts G
hM : (↑M : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G)
M' : Compacts G := { carrier := closure (↑M : Set G), isCompact' := ⋯ }
hM' : (↑M' : Set G) ⊆ (↑U' : Set G) \ (↑L' : Set G)
⊢ (↑(L' ⊔ M') : Set G) ⊆ (↑U' : Set G)

Tactic:
have : (↑(L' ⊔ M') : Set G) ⊆ U' := by
    simp only [Compacts.coe_sup, union_subset_iff, hL'U', true_and]
    exact hM'.trans diff_subset