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_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.innerContent { carrier := (↑U' : Set G), is_open' := ⋯ }
have refine : (μ : (a : Compacts G) → ℝ≥0∞) (L' ⊔ M') =
  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∞)
Tactic: refine le_trans (ge_of_eq ?_) (μ.le_innerContent _ _ 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
⊢ Measure.OuterRegular μ.measure
have refine : ∀ (A : Set G) (hA : MeasurableSet A) (r : ℝ≥0∞) (hr : ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) A < r),
  ∃ U ⊇ A, IsOpen U ∧ ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) U < r
Tactic: refine ⟨fun A hA r (hr : _ < _) ↦ ?_⟩
-- 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
A : Set G
hA : MeasurableSet A
r : ℝ≥0∞
hr : ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) A < r
⊢ ∃ U ⊇ A, IsOpen U ∧ ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) U < r
have hr₁ : ⨅ (U : Set G), ⨅ (hU : IsOpen U), ⨅ (_ : A ⊆ U), μ.innerContent { carrier := U, is_open' := hU } < r
Tactic: rw [μ.measure_apply hA, outerMeasure_eq_iInf] at hr
Cleared hypotheses: hr
-- 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
A : Set G
hA : MeasurableSet A
r : ℝ≥0∞
hr : ⨅ (U : Set G), ⨅ (hU : IsOpen U), ⨅ (_ : A ⊆ U), μ.innerContent { carrier := U, is_open' := hU } < r
⊢ ∃ U ⊇ A, IsOpen U ∧ ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) U < r
have hr₁ : ∃ (i : Set G) (h : IsOpen i) (_ : A ⊆ i), μ.innerContent { carrier := i, is_open' := sorry : Opens G } < r
Tactic: simp only [iInf_lt_iff] at hr
Cleared hypotheses: hr
-- 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
A : Set G
hA : MeasurableSet A
r : ℝ≥0∞
hr : ∃ (i : Set G) (h : IsOpen i) (_ : A ⊆ i), μ.innerContent { carrier := i, is_open' := ⋯ : Opens G } < r
⊢ ∃ U ⊇ A, IsOpen U ∧ ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) U < r
have rcases : ∀ (U : Set G) (hUo : IsOpen U) (hAU : A ⊆ U) (hr : μ.innerContent { carrier := U, is_open' := sorry : Opens G } < r),
  ∃ U ⊇ A, IsOpen U ∧ ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) U < r
Tactic: rcases hr with ⟨U, hUo, hAU, hr⟩
Cleared hypotheses: hr
-- 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
A : Set G
hA : MeasurableSet A
r : ℝ≥0∞
U : Set G
hUo : IsOpen U
hAU : A ⊆ U
hr : μ.innerContent { carrier := U, is_open' := ⋯ : Opens G } < r
⊢ ∃ U ⊇ A, IsOpen U ∧ ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) U < r
have hr₁ : ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) U < r
Tactic: rw [← μ.outerMeasure_of_isOpen U hUo, ← μ.measure_apply hUo.measurableSet] at hr
Cleared hypotheses: hr
-- 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
inst_3 : WeaklyLocallyCompactSpace G
⊢ IsFiniteMeasureOnCompacts (m0 := S) μ.measure
have refine : ∀ (K : Set G) (hK : IsCompact K), LT.lt (α := ℝ≥0∞) (((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) K : ℝ≥0∞) (∞ : ℝ≥0∞)
Tactic: refine ⟨fun K hK => ?_⟩
-- 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
inst_3 : WeaklyLocallyCompactSpace G
K : Set G
hK : IsCompact K
⊢ LT.lt (α := ℝ≥0∞) (((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) K : ℝ≥0∞) (∞ : ℝ≥0∞)
have apply : LT.lt (α := ℝ≥0∞) (((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) (closure K) : ℝ≥0∞) (∞ : ℝ≥0∞)
Tactic: apply (measure_mono subset_closure).trans_lt _
-- 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
inst_3 : WeaklyLocallyCompactSpace G
K : Set G
hK : IsCompact K
⊢ LT.lt (α := ℝ≥0∞) (((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) (closure K) : ℝ≥0∞) (∞ : ℝ≥0∞)
have rw : LT.lt (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (closure K) : ℝ≥0∞) (∞ : ℝ≥0∞)
Tactic: rw [measure_apply _ isClosed_closure.measurableSet]
-- 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
inst_3 : WeaklyLocallyCompactSpace G
⊢ Measure.Regular μ.measure
have this : IsFiniteMeasureOnCompacts (m0 := S) μ.measure
Tactic: have : IsFiniteMeasureOnCompacts μ.measure := by
    refine ⟨fun K hK => ?_⟩
    apply (measure_mono subset_closure).trans_lt _
    rw [measure_apply _ isClosed_closure.measurableSet]
    exact μ.outerMeasure_lt_top_of_isCompact hK.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
inst_3 : WeaklyLocallyCompactSpace G
this : IsFiniteMeasureOnCompacts (m0 := S) μ.measure
⊢ Measure.Regular μ.measure
have refine : ∀ (U : Set G) (hU : IsOpen U) (r : ℝ≥0∞) (hr : r < ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) U),
  ∃ K ⊆ U, IsCompact K ∧ r < ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) K
Tactic: refine ⟨fun U hU r hr => ?_⟩
-- 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
inst_3 : WeaklyLocallyCompactSpace G
this : IsFiniteMeasureOnCompacts (m0 := S) μ.measure
U : Set G
hU : IsOpen U
r : ℝ≥0∞
hr : r < ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) U
⊢ ∃ K ⊆ U, IsCompact K ∧ r < ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) K
have hr₁ : r < μ.innerContent { carrier := U, is_open' := hU }
Tactic: rw [measure_apply _ hU.measurableSet, μ.outerMeasure_of_isOpen U hU] at hr
Cleared hypotheses: hr
-- 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
inst_3 : WeaklyLocallyCompactSpace G
this : IsFiniteMeasureOnCompacts (m0 := S) μ.measure
U : Set G
hU : IsOpen U
r : ℝ≥0∞
hr : r < μ.innerContent { carrier := U, is_open' := hU }
⊢ ∃ K ⊆ U, IsCompact K ∧ r < ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) K
have hr₁ : ∃ (i : Compacts G) (_ : (↑i : Set G) ⊆ (↑{ carrier := U, is_open' := hU : Opens G } : Set G)),
  r < (μ : (a : Compacts G) → ℝ≥0∞) i
Tactic: simp only [innerContent, lt_iSup_iff] at hr
Cleared hypotheses: hr
-- 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
inst_3 : WeaklyLocallyCompactSpace G
this : IsFiniteMeasureOnCompacts (m0 := S) μ.measure
U : Set G
hU : IsOpen U
r : ℝ≥0∞
hr :
  ∃ (i : Compacts G) (_ : (↑i : Set G) ⊆ (↑{ carrier := U, is_open' := hU : Opens G } : Set G)),
    r < (μ : (a : Compacts G) → ℝ≥0∞) i
⊢ ∃ K ⊆ U, IsCompact K ∧ r < ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) K
have rcases : ∀ (K : Compacts G) (hKU : (↑K : Set G) ⊆ (↑{ carrier := U, is_open' := hU : Opens G } : Set G))
  (hr : r < (μ : (a : Compacts G) → ℝ≥0∞) K),
  ∃ K ⊆ U, IsCompact K ∧ r < ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) K
Tactic: rcases hr with ⟨K, hKU, hr⟩
Cleared hypotheses: hr
-- 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
inst_3 : WeaklyLocallyCompactSpace G
this : IsFiniteMeasureOnCompacts (m0 := S) μ.measure
U : Set G
hU : IsOpen U
r : ℝ≥0∞
K : Compacts G
hKU : (↑K : Set G) ⊆ (↑{ carrier := U, is_open' := hU : Opens G } : Set G)
hr : r < (μ : (a : Compacts G) → ℝ≥0∞) K
⊢ ∃ K ⊆ U, IsCompact K ∧ r < ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) K
have refine : LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
  (((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞)
Tactic: refine ⟨K, hKU, K.2, hr.trans_le ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
H : μ.ContentRegular
K : Compacts G
ε : ℝ≥0
hε : ε ≠ 0
⊢ ∃ (K' : Compacts G),
    K.carrier ⊆ interior K'.carrier ∧
      (μ : (a : Compacts G) → ℝ≥0∞) K' ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)
have by_contra : ∀
  (hc :
    ¬∃ (K' : Compacts G),
        K.carrier ⊆ interior K'.carrier ∧
          (μ : (a : Compacts G) → ℝ≥0∞) K' ≤
            HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)),
  False
Tactic: by_contra hc
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (hypothesis)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
H : μ.ContentRegular
K : Compacts G
ε : ℝ≥0
hε : ε ≠ 0
hc :
  ¬∃ (K' : Compacts G),
      K.carrier ⊆ interior K'.carrier ∧
        (μ : (a : Compacts G) → ℝ≥0∞) K' ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)
⊢ False
have hc₁ : ∀ (x : Compacts G),
  K.carrier ⊆ interior x.carrier →
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞) < (μ : (a : Compacts G) → ℝ≥0∞) x
Tactic: simp only [not_exists, not_and, not_le] at hc
Cleared hypotheses: hc
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (hypothesis)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
H : μ.ContentRegular
K : Compacts G
ε : ℝ≥0
hε : ε ≠ 0
hc :
  ∀ (x : Compacts G),
    K.carrier ⊆ interior x.carrier →
      HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞) < (μ : (a : Compacts G) → ℝ≥0∞) x
⊢ False
have lower_bound_iInf : HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞) ≤
  ⨅ (K' : Compacts G), ⨅ (_ : (↑K : Set G) ⊆ interior (↑K' : Set G)), (μ : (a : Compacts G) → ℝ≥0∞) K'
Tactic: have lower_bound_iInf : μ K + ε ≤
      ⨅ (K' : TopologicalSpace.Compacts G) (_ : (K : Set G) ⊆ interior (K' : Set G)), μ K' :=
    le_iInf fun K' => le_iInf fun K'_hyp => le_of_lt (hc K' K'_hyp)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (hypothesis)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
H : μ.ContentRegular
K : Compacts G
ε : ℝ≥0
hε : ε ≠ 0
hc :
  ∀ (x : Compacts G),
    K.carrier ⊆ interior x.carrier →
      HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞) < (μ : (a : Compacts G) → ℝ≥0∞) x
lower_bound_iInf :
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞) ≤
    ⨅ (K' : Compacts G), ⨅ (_ : (↑K : Set G) ⊆ interior (↑K' : Set G)), (μ : (a : Compacts G) → ℝ≥0∞) K'
⊢ False
have lower_bound_iInf₁ : HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞) ≤ (μ : (a : Compacts G) → ℝ≥0∞) K
Tactic: rw [← H] at lower_bound_iInf
Cleared hypotheses: lower_bound_iInf
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : MeasurableSpace G
inst_2 : R1Space G
inst_3 : BorelSpace G
H : μ.ContentRegular
K : Compacts G
⊢ Eq (α := ℝ≥0∞) (((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞)
    ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
have refine_1 : LE.le (α := ℝ≥0∞) (((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞)
  ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
Tactic: refine le_antisymm ?_ ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : MeasurableSpace G
inst_2 : R1Space G
inst_3 : BorelSpace G
H : μ.ContentRegular
K : Compacts G
refine_1 :
  LE.le (α := ℝ≥0∞) (((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞)
    ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) (((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞)
    ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
have refine_2 : LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
  (((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞)
Tactic: refine le_antisymm ?_ ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : MeasurableSpace G
inst_2 : R1Space G
inst_3 : BorelSpace G
H : μ.ContentRegular
K : Compacts G
⊢ LE.le (α := ℝ≥0∞) (((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞)
    ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
have refine_1 : ∀ (ε : ℝ≥0),
  0 < ε →
    LT.lt (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (∞ : ℝ≥0∞) →
      ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) (↑K : Set G) ≤
        HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)
Tactic: apply ENNReal.le_of_forall_pos_le_add
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : MeasurableSpace G
inst_2 : R1Space G
inst_3 : BorelSpace G
H : μ.ContentRegular
K : Compacts G
ε : ℝ≥0
εpos : 0 < ε
h : LT.lt (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) (↑K : Set G) ≤
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)
have refine_1 : ∀ (K' : Compacts G)
  (K'_hyp :
    K.carrier ⊆ interior K'.carrier ∧
      (μ : (a : Compacts G) → ℝ≥0∞) K' ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)),
  ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) (↑K : Set G) ≤
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)
Tactic: obtain ⟨K', K'_hyp⟩ := contentRegular_exists_compact μ H K (ne_bot_of_gt εpos)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : MeasurableSpace G
inst_2 : R1Space G
inst_3 : BorelSpace G
H : μ.ContentRegular
K : Compacts G
ε : ℝ≥0
εpos : 0 < ε
h : LT.lt (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (∞ : ℝ≥0∞)
K' : Compacts G
K'_hyp :
  K.carrier ⊆ interior K'.carrier ∧
    (μ : (a : Compacts G) → ℝ≥0∞) K' ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) (((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) (interior (↑K' : Set G)) : ℝ≥0∞)
    ((μ : (a : Compacts G) → ℝ≥0∞) K' : ℝ≥0∞)
have rw : LE.le (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (interior (↑K' : Set G)) : ℝ≥0∞)
  ((μ : (a : Compacts G) → ℝ≥0∞) K' : ℝ≥0∞)
Tactic: rw [μ.measure_apply (IsOpen.measurableSet isOpen_interior)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : MeasurableSpace G
inst_2 : R1Space G
inst_3 : BorelSpace G
H : μ.ContentRegular
K : Compacts G
ε : ℝ≥0
εpos : 0 < ε
h : LT.lt (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (∞ : ℝ≥0∞)
K' : Compacts G
K'_hyp :
  K.carrier ⊆ interior K'.carrier ∧
    (μ : (a : Compacts G) → ℝ≥0∞) K' ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)
⊢ ((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) (↑K : Set G) ≤
    HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)
have measure_mono : LE.le (α := ℝ≥0∞) (((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) (interior (↑K' : Set G)) : ℝ≥0∞)
  ((μ : (a : Compacts G) → ℝ≥0∞) K' : ℝ≥0∞)
Tactic: calc
      μ.measure ↑K ≤ μ.measure (interior ↑K') := measure_mono K'_hyp.1
      _ ≤ μ K' := by
        rw [μ.measure_apply (IsOpen.measurableSet isOpen_interior)]
        exact μ.outerMeasure_interior_compacts K'
      _ ≤ μ K + ε := K'_hyp.right
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : MeasurableSpace G
inst_2 : R1Space G
inst_3 : BorelSpace G
H : μ.ContentRegular
K : Compacts G
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) { carrier := closure (↑K : Set G), isCompact' := ⋯ } : ℝ≥0∞)
    (((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) (closure (↑K : Set G)) : ℝ≥0∞)
have rw : LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) { carrier := closure (↑K : Set G), isCompact' := sorry } : ℝ≥0∞)
  ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (closure (↑K : Set G)) : ℝ≥0∞)
Tactic: rw [μ.measure_apply (isClosed_closure.measurableSet)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : MeasurableSpace G
inst_2 : R1Space G
inst_3 : BorelSpace G
H : μ.ContentRegular
K : Compacts G
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
    (((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞)
have μ₁ : LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) { carrier := closure (↑K : Set G), isCompact' := sorry } : ℝ≥0∞)
  (((μ.measure : Measure G) : (a : Set G) → ℝ≥0∞) (closure (↑K : Set G)) : ℝ≥0∞)
Tactic: calc
    μ K ≤ μ ⟨closure K, K.2.closure⟩ := μ.mono _ _ subset_closure
    _ ≤ μ.measure (closure K) := by
      rw [μ.measure_apply (isClosed_closure.measurableSet)]
      exact μ.le_outerMeasure_compacts _
    _ = μ.measure K := K.2.measure_closure _
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.ContinuousPreimage (goal)
Goal:
α : Type u_1
X : Type u_2
Y : Type u_3
inst : TopologicalSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
inst_3 : R1Space X
inst_4 : TopologicalSpace Y
inst_5 : MeasurableSpace Y
inst_6 : BorelSpace Y
inst_7 : R1Space Y
μ : Measure X
ν : Measure Y
inst_8 : μ.InnerRegularCompactLTTop
inst_9 : IsLocallyFiniteMeasure ν
l : Filter α
f : α → C(X, Y)
g : C(X, Y)
s : Set Y
hfg : Tendsto f l (𝓝 g)
hf : ∀ᶠ (a : α) in l, MeasurePreserving (⇑(f a) : (a : X) → Y) μ ν
hg : MeasurePreserving (⇑g : (a : X) → Y) μ ν
hs : NullMeasurableSet s ν
hνs : Ne (α := ℝ≥0∞) ((ν : (a : Set Y) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ ν.InnerRegularCompactLTTop
have rw : Measure.InnerRegularCompactLTTop.{u_3} (α := Y) (Measure.map (⇑g : (a : X) → Y) μ)
Tactic: rw [← hg.map_eq]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.ContinuousPreimage (hypothesis)
Goal:
α : Type u_1
X : Type u_2
Y : Type u_3
inst : TopologicalSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
inst_3 : R1Space X
inst_4 : TopologicalSpace Y
inst_5 : MeasurableSpace Y
inst_6 : BorelSpace Y
inst_7 : R1Space Y
μ : Measure X
ν : Measure Y
inst_8 : μ.InnerRegularCompactLTTop
inst_9 : IsLocallyFiniteMeasure ν
l : Filter α
f : α → C(X, Y)
g : C(X, Y)
s : Set Y
hfg : Tendsto f l (𝓝 g)
hf : ∀ᶠ (a : α) in l, MeasurePreserving (⇑(f a) : (a : X) → Y) μ ν
hg : MeasurePreserving (⇑g : (a : X) → Y) μ ν
hs : NullMeasurableSet s ν
hνs : Ne (α := ℝ≥0∞) ((ν : (a : Set Y) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ Tendsto (β := ℝ≥0∞)
    (fun (a : α) ↦ (μ : (a : Set X) → ℝ≥0∞) (((⇑(f a) : (a : X) → Y) ⁻¹' s) ∆ ((⇑g : (a : X) → Y) ⁻¹' s))) l (𝓝 0)
have this : ν.InnerRegularCompactLTTop
Tactic: have : ν.InnerRegularCompactLTTop := by
    rw [← hg.map_eq]
    exact .map_of_continuous (map_continuous _)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.ContinuousPreimage (goal)
Goal:
α : Type u_1
X : Type u_2
Y : Type u_3
inst : TopologicalSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
inst_3 : R1Space X
inst_4 : TopologicalSpace Y
inst_5 : MeasurableSpace Y
inst_6 : BorelSpace Y
inst_7 : R1Space Y
μ : Measure X
ν : Measure Y
inst_8 : μ.InnerRegularCompactLTTop
inst_9 : IsLocallyFiniteMeasure ν
l : Filter α
f : α → C(X, Y)
g : C(X, Y)
s : Set Y
hfg : Tendsto f l (𝓝 g)
hf : ∀ᶠ (a : α) in l, MeasurePreserving (⇑(f a) : (a : X) → Y) μ ν
hg : MeasurePreserving (⇑g : (a : X) → Y) μ ν
hs : NullMeasurableSet s ν
hνs : Ne (α := ℝ≥0∞) ((ν : (a : Set Y) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
this : ν.InnerRegularCompactLTTop
⊢ Tendsto (β := ℝ≥0∞)
    (fun (a : α) ↦ (μ : (a : Set X) → ℝ≥0∞) (((⇑(f a) : (a : X) → Y) ⁻¹' s) ∆ ((⇑g : (a : X) → Y) ⁻¹' s))) l (𝓝 0)
have rw : ∀ ε > 0, ∀ᶠ (x : α) in l, (μ : (a : Set X) → ℝ≥0∞) (((⇑(f x) : (a : X) → Y) ⁻¹' s) ∆ ((⇑g : (a : X) → Y) ⁻¹' s)) ≤ ε
Tactic: rw [ENNReal.tendsto_nhds_zero]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.ContinuousPreimage (goal)
Goal:
α : Type u_1
X : Type u_2
Y : Type u_3
inst : TopologicalSpace X
inst_1 : MeasurableSpace X
inst_2 : BorelSpace X
inst_3 : R1Space X
inst_4 : TopologicalSpace Y
inst_5 : MeasurableSpace Y
inst_6 : BorelSpace Y
inst_7 : R1Space Y
μ : Measure X
ν : Measure Y
inst_8 : μ.InnerRegularCompactLTTop
inst_9 : IsLocallyFiniteMeasure ν
l : Filter α
f : α → C(X, Y)
g : C(X, Y)
s : Set Y
hfg : Tendsto f l (𝓝 g)
hf : ∀ᶠ (a : α) in l, MeasurePreserving (⇑(f a) : (a : X) → Y) μ ν
hg : MeasurePreserving (⇑g : (a : X) → Y) μ ν
hs : NullMeasurableSet s ν
hνs : Ne (α := ℝ≥0∞) ((ν : (a : Set Y) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
this : ν.InnerRegularCompactLTTop
ε : ℝ≥0∞
hε : ε > 0
⊢ ∀ᶠ (x : α) in l, (μ : (a : Set X) → ℝ≥0∞) (((⇑(f x) : (a : X) → Y) ⁻¹' s) ∆ ((⇑g : (a : X) → Y) ⁻¹' s)) ≤ ε
have inr : ∀
  (this :
    ∀ {s : Set Y} (hs : NullMeasurableSet s ν) (hνs : Ne (α := ℝ≥0∞) ((ν : (a : Set Y) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞))
      (ε : ℝ≥0∞) (hε : ε > 0) (hso : IsOpen[inst_4] s),
      ∀ᶠ (x : α) in l, (μ : (a : Set X) → ℝ≥0∞) (((⇑(f x) : (a : X) → Y) ⁻¹' s) ∆ ((⇑g : (a : X) → Y) ⁻¹' s)) ≤ ε)
  (hso : ¬IsOpen[inst_4] s),
  ∀ᶠ (x : α) in l, (μ : (a : Set X) → ℝ≥0∞) (((⇑(f x) : (a : X) → Y) ⁻¹' s) ∆ ((⇑g : (a : X) → Y) ⁻¹' s)) ≤ ε
Tactic: wlog hso : IsOpen s generalizing s ε