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 : Group G
inst_2 : IsTopologicalGroup G
h3 :
  ∀ (g : G) {K : Compacts G},
    Eq (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (Compacts.map (fun (x : G) ↦ g * x) ⋯ K) : ℝ≥0∞)
      ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
K : Compacts G
hK : Ne (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (0 : ℝ≥0∞)
U : Opens G
hU : (↑U : Set G).Nonempty
this : (interior (↑U : Set G)).Nonempty
⊢ 0 < μ.innerContent U
have rcases : ∀ (s : Finset G) (hs : K.carrier ⊆ ⋃ g ∈ s, (fun (x : G) ↦ g * x) ⁻¹' (↑U : Set G)), 0 < μ.innerContent U
Tactic: rcases compact_covered_by_mul_left_translates K.2 this with ⟨s, hs⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : Group G
inst_2 : IsTopologicalGroup G
h3 :
  ∀ (g : G) {K : Compacts G},
    Eq (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (Compacts.map (fun (x : G) ↦ g * x) ⋯ K) : ℝ≥0∞)
      ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
K : Compacts G
hK : Ne (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (0 : ℝ≥0∞)
U : Opens G
hU : (↑U : Set G).Nonempty
this : (interior (↑U : Set G)).Nonempty
s : Finset G
hs : K.carrier ⊆ ⋃ g ∈ s, (fun (x : G) ↦ g * x) ⁻¹' (↑U : Set G)
⊢ 0 < μ.innerContent U
have suffices : (μ : (a : Compacts G) → ℝ≥0∞) K ≤ (↑s.card : ℝ≥0∞) * μ.innerContent U
Tactic: suffices μ K ≤ s.card * μ.innerContent U by
    exact (ENNReal.mul_pos_iff.mp <| hK.bot_lt.trans_le this).2
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (hypothesis)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : Group G
inst_2 : IsTopologicalGroup G
h3 :
  ∀ (g : G) {K : Compacts G},
    Eq (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (Compacts.map (fun (x : G) ↦ g * x) ⋯ K) : ℝ≥0∞)
      ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
K : Compacts G
hK : Ne (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (0 : ℝ≥0∞)
U : Opens G
hU : (↑U : Set G).Nonempty
this : (interior (↑U : Set G)).Nonempty
s : Finset G
hs : K.carrier ⊆ ⋃ g ∈ s, (fun (x : G) ↦ g * x) ⁻¹' (↑U : Set G)
⊢ (μ : (a : Compacts G) → ℝ≥0∞) K ≤ (↑s.card : ℝ≥0∞) * μ.innerContent U
have this₁ : (↑K : Set G) ⊆ (↑(⨆ g ∈ s, (Opens.comap (↑(Homeomorph.mulLeft g) : C(G, G)) : (a : Opens G) → Opens G) U) : Set G)
Tactic: have : (K : Set G) ⊆ ↑(⨆ g ∈ s, Opens.comap (Homeomorph.mulLeft g : C(G, G)) U) := by
    simpa only [Opens.iSup_def, Opens.coe_comap, Subtype.coe_mk]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : Group G
inst_2 : IsTopologicalGroup G
h3 :
  ∀ (g : G) {K : Compacts G},
    Eq (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (Compacts.map (fun (x : G) ↦ g * x) ⋯ K) : ℝ≥0∞)
      ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
K : Compacts G
hK : Ne (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (0 : ℝ≥0∞)
U : Opens G
hU : (↑U : Set G).Nonempty
this_1 : (interior (↑U : Set G)).Nonempty
s : Finset G
hs : K.carrier ⊆ ⋃ g ∈ s, (fun (x : G) ↦ g * x) ⁻¹' (↑U : Set G)
this :
  (↑K : Set G) ⊆ (↑(⨆ g ∈ s, (Opens.comap (↑(Homeomorph.mulLeft g) : C(G, G)) : (a : Opens G) → Opens G) U) : Set G)
⊢ (μ : (a : Compacts G) → ℝ≥0∞) K ≤ (↑s.card : ℝ≥0∞) * μ.innerContent U
have refine : μ.innerContent (⨆ g ∈ s, (Opens.comap (↑(Homeomorph.mulLeft g) : C(G, G)) : (a : Opens G) → Opens G) U) ≤
  (↑s.card : ℝ≥0∞) * μ.innerContent U
Tactic: refine (μ.le_innerContent _ _ this).trans ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : Group G
inst_2 : IsTopologicalGroup G
h3 :
  ∀ (g : G) {K : Compacts G},
    Eq (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (Compacts.map (fun (x : G) ↦ g * x) ⋯ K) : ℝ≥0∞)
      ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
K : Compacts G
hK : Ne (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (0 : ℝ≥0∞)
U : Opens G
hU : (↑U : Set G).Nonempty
this_1 : (interior (↑U : Set G)).Nonempty
s : Finset G
hs : K.carrier ⊆ ⋃ g ∈ s, (fun (x : G) ↦ g * x) ⁻¹' (↑U : Set G)
this :
  (↑K : Set G) ⊆ (↑(⨆ g ∈ s, (Opens.comap (↑(Homeomorph.mulLeft g) : C(G, G)) : (a : Opens G) → Opens G) U) : Set G)
⊢ μ.innerContent (⨆ g ∈ s, (Opens.comap (↑(Homeomorph.mulLeft g) : C(G, G)) : (a : Opens G) → Opens G) U) ≤
    (↑s.card : ℝ≥0∞) * μ.innerContent U
have _ : ∑ d ∈ s, μ.innerContent ((Opens.comap (↑(Homeomorph.mulLeft d) : C(G, G)) : (a : Opens G) → Opens G) U) ≤
  (↑s.card : ℝ≥0∞) * μ.innerContent U
Tactic: refine
    (rel_iSup_sum μ.innerContent μ.innerContent_bot (· ≤ ·) μ.innerContent_iSup_nat _ _).trans ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
K : Compacts G
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞)
have rw : LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
  (⨅ (t : Set G), ⨅ (ht : IsOpen t), ⨅ (_ : (↑K : Set G) ⊆ t), μ.innerContent { carrier := t, is_open' := ht } : ℝ≥0∞)
Tactic: rw [Content.outerMeasure, inducedOuterMeasure_eq_iInf]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
K : Compacts G
rw :
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
    (⨅ (t : Set G), ⨅ (ht : IsOpen t), ⨅ (_ : (↑K : Set G) ⊆ t), μ.innerContent { carrier := t, is_open' := ht } : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞)
have PU : ∀ ⦃f : ℕ → Set G⦄ (_hm : ∀ (i : ℕ), IsOpen (f i)), IsOpen.{w} (X := G) (⋃ (i : ℕ), f i)
Tactic: rw [Content.outerMeasure, inducedOuterMeasure_eq_iInf]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
K : Compacts G
rw :
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
    (⨅ (t : Set G), ⨅ (ht : IsOpen t), ⨅ (_ : (↑K : Set G) ⊆ t), μ.innerContent { carrier := t, is_open' := ht } : ℝ≥0∞)
PU : ∀ ⦃f : ℕ → Set G⦄ (_hm : ∀ (i : ℕ), IsOpen (f i)), IsOpen.{w} (X := G) (⋃ (i : ℕ), f i)
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞)
have msU : ∀ ⦃f : ℕ → Set G⦄ (hm : ∀ (i : ℕ), IsOpen (f i)),
  μ.innerContent { carrier := ⋃ (i : ℕ), f i, is_open' := sorry : Opens G } ≤
    ∑' (i : ℕ), μ.innerContent { carrier := f i, is_open' := sorry }
Tactic: rw [Content.outerMeasure, inducedOuterMeasure_eq_iInf]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
K : Compacts G
rw :
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
    (⨅ (t : Set G), ⨅ (ht : IsOpen t), ⨅ (_ : (↑K : Set G) ⊆ t), μ.innerContent { carrier := t, is_open' := ht } : ℝ≥0∞)
PU : ∀ ⦃f : ℕ → Set G⦄ (_hm : ∀ (i : ℕ), IsOpen (f i)), IsOpen.{w} (X := G) (⋃ (i : ℕ), f i)
msU :
  ∀ ⦃f : ℕ → Set G⦄ (hm : ∀ (i : ℕ), IsOpen (f i)),
    μ.innerContent { carrier := ⋃ (i : ℕ), f i, is_open' := ⋯ : Opens G } ≤
      ∑' (i : ℕ), μ.innerContent { carrier := f i, is_open' := ⋯ }
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞)
have m_mono : ∀ ⦃s₁ s₂ : Set G⦄ (hs₁ : IsOpen s₁) (hs₂ : IsOpen s₂),
  s₁ ⊆ s₂ →
    μ.innerContent { carrier := s₁, is_open' := hs₁ : Opens G } ≤ μ.innerContent { carrier := s₂, is_open' := hs₂ }
Tactic: rw [Content.outerMeasure, inducedOuterMeasure_eq_iInf]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : Opens G
hU : Ne (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U : Set G) : ℝ≥0∞) (∞ : ℝ≥0∞)
ε : ℝ≥0
hε : ε ≠ 0
⊢ ∃ (K : Compacts G),
    (↑K : Set G) ⊆ (↑U : Set G) ∧
      (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U : Set G) ≤
        HAdd.hAdd (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞) (↑ε : ℝ≥0∞)
have rw : ∀ (hU : μ.innerContent U ≠ ∞) {ε : ℝ≥0} (hε : ε ≠ 0),
  ∃ (K : Compacts G),
    (↑K : Set G) ⊆ (↑U : Set G) ∧
      μ.innerContent U ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞) (↑ε : ℝ≥0∞)
Tactic: rw [μ.outerMeasure_opens] at hU ⊢
Cleared hypotheses: hU, ε, hε
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : Opens G
hU : μ.innerContent U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
⊢ ∃ (K : Compacts G),
    (↑K : Set G) ⊆ (↑U : Set G) ∧
      μ.innerContent U ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞) (↑ε : ℝ≥0∞)
have rcases : ∀ (K : Compacts G) (h1K : (↑K : Set G) ⊆ (↑U : Set G))
  (h2K : μ.innerContent U ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)),
  ∃ (K : Compacts G),
    (↑K : Set G) ⊆ (↑U : Set G) ∧
      μ.innerContent U ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞) (↑ε : ℝ≥0∞)
Tactic: rcases μ.innerContent_exists_compact hU hε with ⟨K, h1K, h2K⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : Opens G
hU : μ.innerContent U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
K : Compacts G
h1K : (↑K : Set G) ⊆ (↑U : Set G)
h2K : μ.innerContent U ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)
⊢ μ.innerContent U ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞) (↑ε : ℝ≥0∞)
have grw : HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞) ≤
  HAdd.hAdd (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞) (↑ε : ℝ≥0∞)
Tactic: grw [h2K, μ.le_outerMeasure_compacts K]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : Opens G
hU : μ.innerContent U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
K : Compacts G
h1K : (↑K : Set G) ⊆ (↑U : Set G)
h2K : μ.innerContent U ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)
grw :
  HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞) ≤
    HAdd.hAdd (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞) (↑ε : ℝ≥0∞)
⊢ μ.innerContent U ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞) (↑ε : ℝ≥0∞)
have grw₁ : HAdd.hAdd (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞) (↑ε : ℝ≥0∞) ≤
  HAdd.hAdd (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑K : Set G) : ℝ≥0∞) (↑ε : ℝ≥0∞)
Tactic: grw [h2K, μ.le_outerMeasure_compacts K]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
A : Set G
hA : Ne (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) A : ℝ≥0∞) (∞ : ℝ≥0∞)
ε : ℝ≥0
hε : ε ≠ 0
⊢ ∃ (U : Opens G),
    A ⊆ (↑U : Set G) ∧
      (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U : Set G) ≤
        HAdd.hAdd (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) A : ℝ≥0∞) (↑ε : ℝ≥0∞)
have rcases : ∀ (U : Set G) (hU : IsOpen U) (h2U : A ⊆ U)
  (h3U :
    ((inducedOuterMeasure (P := IsOpen)
              (fun (s₁ : Set G) (hs₁ : IsOpen s₁) ↦ μ.innerContent { carrier := s₁, is_open' := hs₁ }) sorry sorry :
            OuterMeasure G) :
          (a : Set G) → ℝ≥0∞)
        U ≤
      HAdd.hAdd (α := ℝ≥0∞)
        (((inducedOuterMeasure (P := IsOpen)
                  (fun (s₁ : Set G) (hs₁ : IsOpen s₁) ↦ μ.innerContent { carrier := s₁, is_open' := hs₁ : Opens G }) sorry
                  sorry :
                OuterMeasure G) :
              (a : Set G) → ℝ≥0∞)
            A :
          ℝ≥0∞)
        (↑ε : ℝ≥0∞)),
  ∃ (U : Opens G),
    A ⊆ (↑U : Set G) ∧
      (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U : Set G) ≤
        HAdd.hAdd (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) A : ℝ≥0∞) (↑ε : ℝ≥0∞)
Tactic: rcases inducedOuterMeasure_exists_set _ μ.innerContent_iUnion_nat μ.innerContent_mono hA
      (ENNReal.coe_ne_zero.2 hε) with
    ⟨U, hU, h2U, h3U⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space 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∞)
A : Set G
⊢ Eq (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((⇑f : (a : G) → G) ⁻¹' A) : ℝ≥0∞)
    ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) A : ℝ≥0∞)
have refine : ∀ (s : Set G) (hs : IsOpen s),
  μ.innerContent { carrier := (⇑f.toEquiv : (a : G) → G) ⁻¹' s, is_open' := sorry : Opens G } =
    μ.innerContent { carrier := s, is_open' := hs }
Tactic: refine inducedOuterMeasure_preimage _ μ.innerContent_iUnion_nat μ.innerContent_mono _
    (fun _ => f.isOpen_preimage) ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
inst_2 : WeaklyLocallyCompactSpace G
K : Set G
hK : IsCompact K
⊢ LT.lt (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) K : ℝ≥0∞) (∞ : ℝ≥0∞)
have rcases : ∀ (F : Set G) (h1F : IsCompact F) (h2F : K ⊆ interior F),
  LT.lt (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) K : ℝ≥0∞) (∞ : ℝ≥0∞)
Tactic: rcases exists_compact_superset hK with ⟨F, h1F, h2F⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
inst_2 : WeaklyLocallyCompactSpace G
K : Set G
hK : IsCompact K
F : Set G
h1F : IsCompact F
h2F : K ⊆ interior F
⊢ LT.lt (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) K : ℝ≥0∞) (∞ : ℝ≥0∞)
have K₁ : LE.le (α := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) (interior F) : ℝ≥0∞)
  ((μ : (a : Compacts G) → ℝ≥0∞) { carrier := F, isCompact' := h1F } : ℝ≥0∞)
Tactic: calc
    μ.outerMeasure K ≤ μ.outerMeasure (interior F) := measure_mono h2F
    _ ≤ μ ⟨F, h1F⟩ := by
      apply μ.outerMeasure_le ⟨interior F, isOpen_interior⟩ ⟨F, h1F⟩ interior_subset
    _ < ⊤ := μ.lt_top _
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
A : Set G
⊢ MeasurableSet A ↔
    ∀ (U : Opens G),
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑U : Set G) ∩ A) : ℝ≥0∞)
          ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑U : Set G) \ A) : ℝ≥0∞) ≤
        (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U : Set G)
have rw : MeasurableSet A ↔
  ∀ (U : Set G) (hU : IsOpen U),
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑{ carrier := U, is_open' := hU : Opens G } : Set G) ∩ A) : ℝ≥0∞)
        ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑{ carrier := U, is_open' := hU : Opens G } : Set G) \ A) : ℝ≥0∞) ≤
      (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑{ carrier := U, is_open' := hU : Opens G } : Set G)
Tactic: rw [Opens.forall]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
A : Set G
⊢ MeasurableSet A ↔
    ∀ (U : Set G) (hU : IsOpen U),
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
          ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑{ carrier := U, is_open' := hU : Opens G } : Set G) ∩ A) : ℝ≥0∞)
          ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑{ carrier := U, is_open' := hU : Opens G } : Set G) \ A) : ℝ≥0∞) ≤
        (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑{ carrier := U, is_open' := hU : Opens G } : Set G)
have msU : ∀ ⦃f : ℕ → Set G⦄ (hm : ∀ (i : ℕ), IsOpen (f i)),
  μ.innerContent { carrier := ⋃ (i : ℕ), f i, is_open' := sorry : Opens G } ≤
    ∑' (i : ℕ), μ.innerContent { carrier := f i, is_open' := sorry }
Tactic: apply inducedOuterMeasure_caratheodory
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
A : Set G
msU :
  ∀ ⦃f : ℕ → Set G⦄ (hm : ∀ (i : ℕ), IsOpen (f i)),
    μ.innerContent { carrier := ⋃ (i : ℕ), f i, is_open' := ⋯ : Opens G } ≤
      ∑' (i : ℕ), μ.innerContent { carrier := f i, is_open' := ⋯ }
⊢ MeasurableSet A ↔
    ∀ (U : Set G) (hU : IsOpen U),
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
          ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑{ carrier := U, is_open' := hU : Opens G } : Set G) ∩ A) : ℝ≥0∞)
          ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑{ carrier := U, is_open' := hU : Opens G } : Set G) \ A) : ℝ≥0∞) ≤
        (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑{ carrier := U, is_open' := hU : Opens G } : Set G)
have m_mono : ∀ ⦃s₁ s₂ : Set G⦄ (hs₁ : IsOpen s₁) (hs₂ : IsOpen s₂),
  s₁ ⊆ s₂ →
    μ.innerContent { carrier := s₁, is_open' := hs₁ : Opens G } ≤ μ.innerContent { carrier := s₂, is_open' := hs₂ }
Tactic: apply inducedOuterMeasure_caratheodory
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
A : Set G
msU :
  ∀ ⦃f : ℕ → Set G⦄ (hm : ∀ (i : ℕ), IsOpen (f i)),
    μ.innerContent { carrier := ⋃ (i : ℕ), f i, is_open' := ⋯ : Opens G } ≤
      ∑' (i : ℕ), μ.innerContent { carrier := f i, is_open' := ⋯ }
m_mono :
  ∀ ⦃s₁ s₂ : Set G⦄ (hs₁ : IsOpen s₁) (hs₂ : IsOpen s₂),
    s₁ ⊆ s₂ →
      μ.innerContent { carrier := s₁, is_open' := hs₁ : Opens G } ≤ μ.innerContent { carrier := s₂, is_open' := hs₂ }
⊢ MeasurableSet A ↔
    ∀ (U : Set G) (hU : IsOpen U),
      HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
          ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑{ carrier := U, is_open' := hU : Opens G } : Set G) ∩ A) : ℝ≥0∞)
          ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑{ carrier := U, is_open' := hU : Opens G } : Set G) \ A) : ℝ≥0∞) ≤
        (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑{ carrier := U, is_open' := hU : Opens G } : Set G)
have PU : ∀ ⦃f : ℕ → Set G⦄ (_hm : ∀ (i : ℕ), IsOpen (f i)), IsOpen.{w} (X := G) (⋃ (i : ℕ), f i)
Tactic: apply inducedOuterMeasure_caratheodory
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
inst_2 : Group G
inst_3 : IsTopologicalGroup G
h3 :
  ∀ (g : G) {K : Compacts G},
    Eq (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (Compacts.map (fun (x : G) ↦ g * x) ⋯ K) : ℝ≥0∞)
      ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
K : Compacts G
hK : Ne (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (0 : ℝ≥0∞)
U : Set G
h1U : IsOpen U
h2U : U.Nonempty
⊢ LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) U : ℝ≥0∞)
have h : (μ.outerMeasure : (a : Set G) → ℝ≥0∞) U = μ.innerContent { carrier := U, is_open' := h1U : Opens G }
Tactic: convert! μ.innerContent_pos_of_is_mul_left_invariant h3 K hK ⟨U, h1U⟩ h2U
-- 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
⊢ S ≤ μ.outerMeasure.caratheodory
have rw : borel G ≤ μ.outerMeasure.caratheodory
Tactic: rw [BorelSpace.measurable_eq (α := G)]
-- 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
⊢ borel G ≤ μ.outerMeasure.caratheodory
have refine : ∀ (t : Set G), Membership.mem (γ := Set (Set G)) {s : Set G | IsOpen s} t → MeasurableSet t
Tactic: refine MeasurableSpace.generateFrom_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
⊢ MeasurableSet U
have rw : ∀ (U_1 : Opens G),
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑U_1 : Set G) ∩ U) : ℝ≥0∞)
      ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑U_1 : Set G) \ U) : ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U_1 : Set G)
Tactic: rw [μ.outerMeasure_caratheodory]
-- 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
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := 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) \ U) : ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)
have rw : HAdd.hAdd (β := ℝ≥0∞) (μ.innerContent { carrier := (↑U' : Set G) ∩ U, is_open' := sorry : Opens G })
    ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑U' : Set G) \ U) : ℝ≥0∞) ≤
  (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)
Tactic: rw [μ.outerMeasure_of_isOpen ((U' : Set G) ∩ U) (U'.isOpen.inter hU)]
-- 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
⊢ HAdd.hAdd (β := ℝ≥0∞) (μ.innerContent { carrier := (↑U' : Set G) ∩ U, is_open' := ⋯ : Opens G })
      ((μ.outerMeasure : (a : Set G) → ℝ≥0∞) ((↑U' : Set G) \ U) : ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)
have simp : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
    (⨆ (x : { i : Compacts G // (↑i : Set G) ⊆ (↑{ carrier := (↑U' : Set G) ∩ U, 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) \ U) : 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
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞)
      (⨆ (x :
          { i : Compacts G // (↑i : Set G) ⊆ (↑{ carrier := (↑U' : Set G) ∩ U, 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) \ U) : ℝ≥0∞) ≤
    (μ.outerMeasure : (a : Set G) → ℝ≥0∞) (↑U' : Set G)
have rw : 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)
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
⊢ 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 this : Nonempty.{max 1 (w + 1)} { L : Compacts G // (↑L : Set G) ⊆ (↑U' : Set G) ∩ U }
Tactic: haveI : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩