awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
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 }
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
      (⨆ (x : { i : Compacts G // (↑i : Set G) ⊆ (↑U' : Set G) ∩ U }), (μ : (a : Compacts G) → ℝ≥0∞) (↑x : 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)
have rw : 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: rw [ENNReal.iSup_add]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
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 }
⊢ 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∞)
have 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)
Tactic: refine iSup_le ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
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' := ⋯ }
⊢ 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)
have 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)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
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' := ⋯ }
⊢ 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)
have 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)
Tactic: grw [show μ L ≤ μ L' from μ.mono _ _ subset_closure]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (hypothesis)
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' := ⋯ }
⊢ 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)
have hL₁ : (↑L : Set G) ⊆ (↑U' : Set G) ∧ (↑L : Set G) ⊆ U
Tactic: simp only [subset_inter_iff] at hL
Cleared hypotheses: hL
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (hypothesis)
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
⊢ 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)
have hL'U : (↑L' : Set G) ⊆ U
Tactic: have hL'U : (L' : Set G) ⊆ U := IsCompact.closure_subset_of_isOpen L.2 hU hL.2
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (hypothesis)
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
⊢ 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)
have hL'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
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (hypothesis)
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)
⊢ 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)
have this₁ : (↑U' : Set G) \ U ⊆ (↑U' : Set G) \ (↑L' : Set G)
Tactic: have : ↑U' \ U ⊆ U' \ L' := diff_subset_diff_right hL'U
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
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)
⊢ 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)
have 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)
Tactic: grw [this]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
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)
⊢ 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)
have 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' := sorry : Opens G }) ≤
  (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)
Tactic: rw [μ.outerMeasure_of_isOpen (↑U' \ L') (IsOpen.sdiff U'.2 isClosed_closure)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
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)
⊢ 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)
have 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' := sorry : 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: simp only [innerContent, iSup_subtype']
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
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)
⊢ 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)
have 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)
Tactic: rw [Opens.coe_mk]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (hypothesis)
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)
⊢ 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)
have this₁ : 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 _⟩⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
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) }
⊢ 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)
have 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∞)
Tactic: rw [ENNReal.add_iSup]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
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∞)
have 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)
Tactic: refine iSup_le ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
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' := ⋯ }
⊢ 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)
have 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)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
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' := ⋯ }
⊢ 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)
have 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)
Tactic: grw [show μ M ≤ μ M' from μ.mono _ _ subset_closure]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (hypothesis)
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' := ⋯ }
⊢ 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)
have hM' : (↑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
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
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)
have simp : (↑M' : Set G) ⊆ (↑U' : Set G)
Tactic: simp only [Compacts.coe_sup, union_subset_iff, hL'U', true_and]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (hypothesis)
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)
⊢ 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)
have this₁ : (↑(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
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
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_3 : 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)
this : (↑(L' ⊔ M') : Set G) ⊆ (↑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)
have rw : 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.innerContent { carrier := (↑U' : Set G), is_open' := sorry }
Tactic: rw [μ.outerMeasure_of_isOpen (↑U') U'.2]