awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (hypothesis)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
U : Opens G
hU : μ.innerContent U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : (↑ε : ℝ≥0∞) ≠ 0
h : (↑ε : ℝ≥0∞) < μ.innerContent U
⊢ ∃ (K : Compacts G),
    (↑K : Set G) ⊆ (↑U : Set G) ∧
      μ.innerContent U ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)
have h₂ : μ.innerContent U - (↑ε : ℝ≥0∞) < μ.innerContent U
Tactic: have h₂ := ENNReal.sub_lt_self hU h.ne_bot h'ε
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
U : Opens G
hU : μ.innerContent U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : (↑ε : ℝ≥0∞) ≠ 0
h : (↑ε : ℝ≥0∞) < μ.innerContent U
h₂ : μ.innerContent U - (↑ε : ℝ≥0∞) < μ.innerContent U
| μ.innerContent U - (↑ε : ℝ≥0∞) < μ.innerContent U
have rhs : μ.innerContent U
Tactic: rhs
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
U : Opens G
hU : μ.innerContent U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : (↑ε : ℝ≥0∞) ≠ 0
h : (↑ε : ℝ≥0∞) < μ.innerContent U
h₂ : μ.innerContent U - (↑ε : ℝ≥0∞) < μ.innerContent U
| μ.innerContent U - (↑ε : ℝ≥0∞) < μ.innerContent U
have rw : ⨆ (K : Compacts G), ⨆ (_ : (↑K : Set G) ⊆ (↑U : Set G)), (μ : (a : Compacts G) → ℝ≥0∞) K
Tactic: rhs; rw [innerContent]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (hypothesis)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
U : Opens G
hU : μ.innerContent U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : (↑ε : ℝ≥0∞) ≠ 0
h : (↑ε : ℝ≥0∞) < μ.innerContent U
h₂ : μ.innerContent U - (↑ε : ℝ≥0∞) < μ.innerContent U
⊢ ∃ (K : Compacts G),
    (↑K : Set G) ⊆ (↑U : Set G) ∧
      μ.innerContent U ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)
have h₃ : μ.innerContent U - (↑ε : ℝ≥0∞) <
  ⨆ (K : Compacts G), ⨆ (_ : (↑K : Set G) ⊆ (↑U : Set G)), (μ : (a : Compacts G) → ℝ≥0∞) K
Tactic: conv at h₂ => rhs; rw [innerContent]
Cleared hypotheses: h₂
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (hypothesis)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
U : Opens G
hU : μ.innerContent U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : (↑ε : ℝ≥0∞) ≠ 0
h : (↑ε : ℝ≥0∞) < μ.innerContent U
h₂ :
  μ.innerContent U - (↑ε : ℝ≥0∞) <
    ⨆ (K : Compacts G), ⨆ (_ : (↑K : Set G) ⊆ (↑U : Set G)), (μ : (a : Compacts G) → ℝ≥0∞) K
⊢ ∃ (K : Compacts G),
    (↑K : Set G) ⊆ (↑U : Set G) ∧
      μ.innerContent U ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)
have h₃ : ∃ (i : Compacts G) (_ : (↑i : Set G) ⊆ (↑U : Set G)), μ.innerContent U - (↑ε : ℝ≥0∞) < (μ : (a : Compacts G) → ℝ≥0∞) i
Tactic: simp only [lt_iSup_iff] at h₂
Cleared hypotheses: h₂
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
U : Opens G
hU : μ.innerContent U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : (↑ε : ℝ≥0∞) ≠ 0
h : (↑ε : ℝ≥0∞) < μ.innerContent U
h₂ :
  ∃ (i : Compacts G) (_ : (↑i : Set G) ⊆ (↑U : Set G)), μ.innerContent U - (↑ε : ℝ≥0∞) < (μ : (a : Compacts G) → ℝ≥0∞) i
⊢ ∃ (K : Compacts G),
    (↑K : Set G) ⊆ (↑U : Set G) ∧
      μ.innerContent U ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)
have inr : ∀ (U_1 : Compacts G) (h1U : (↑U_1 : Set G) ⊆ (↑U : Set G))
  (h2U : μ.innerContent U - (↑ε : ℝ≥0∞) < (μ : (a : Compacts G) → ℝ≥0∞) U_1),
  ∃ (K : Compacts G),
    (↑K : Set G) ⊆ (↑U : Set G) ∧
      μ.innerContent U ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)
Tactic: rcases h₂ with ⟨U, h1U, h2U⟩
Cleared hypotheses: h₂
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
U_1 : Opens G
hU : μ.innerContent U_1 ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : (↑ε : ℝ≥0∞) ≠ 0
h : (↑ε : ℝ≥0∞) < μ.innerContent U_1
U : Compacts G
h1U : (↑U : Set G) ⊆ (↑U_1 : Set G)
h2U : μ.innerContent U_1 - (↑ε : ℝ≥0∞) < (μ : (a : Compacts G) → ℝ≥0∞) U
⊢ ∃ (K : Compacts G),
    (↑K : Set G) ⊆ (↑U_1 : Set G) ∧
      μ.innerContent U_1 ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (↑ε : ℝ≥0∞)
have inr : μ.innerContent U_1 ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) U : ℝ≥0∞) (↑ε : ℝ≥0∞)
Tactic: refine ⟨U, h1U, ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
U_1 : Opens G
hU : μ.innerContent U_1 ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : (↑ε : ℝ≥0∞) ≠ 0
h : (↑ε : ℝ≥0∞) < μ.innerContent U_1
U : Compacts G
h1U : (↑U : Set G) ⊆ (↑U_1 : Set G)
h2U : μ.innerContent U_1 - (↑ε : ℝ≥0∞) < (μ : (a : Compacts G) → ℝ≥0∞) U
⊢ μ.innerContent U_1 ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) U : ℝ≥0∞) (↑ε : ℝ≥0∞)
have inr : μ.innerContent U_1 - (↑ε : ℝ≥0∞) ≤ (μ : (a : Compacts G) → ℝ≥0∞) U
Tactic: rw [← tsub_le_iff_right]
-- 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 : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
    (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
have refine_1 : LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (Finset.sup ∅ K) : ℝ≥0∞)
  (∑ i ∈ ∅, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
Tactic: refine Finset.induction_on t ?_ ?_
-- 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 : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
refine_1 :
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (Finset.sup ∅ K) : ℝ≥0∞)
    (∑ i ∈ ∅, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
    (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
have refine_2 : ∀ (a : ℕ) (s : Finset ℕ),
  a ∉ s →
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (s.sup K) : ℝ≥0∞)
        (∑ i ∈ s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞) →
      LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) ((insert a s).sup K) : ℝ≥0∞)
        (∑ i ∈ insert a s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
Tactic: refine Finset.induction_on t ?_ ?_
-- 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 : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
n : ℕ
s : Finset ℕ
hn : n ∉ s
ih :
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (s.sup K) : ℝ≥0∞)
    (∑ i ∈ s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) ((insert n s).sup K) : ℝ≥0∞)
    (∑ i ∈ insert n s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
have refine_2 : LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n ⊔ s.sup K) : ℝ≥0∞)
  (∑ i ∈ insert n s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
Tactic: grw [Finset.sup_insert, Finset.sum_insert hn, μ.sup_le, ih]
-- 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 : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
n : ℕ
s : Finset ℕ
hn : n ∉ s
ih :
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (s.sup K) : ℝ≥0∞)
    (∑ i ∈ s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
refine_2 :
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n ⊔ s.sup K) : ℝ≥0∞)
    (∑ i ∈ insert n s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) ((insert n s).sup K) : ℝ≥0∞)
    (∑ i ∈ insert n s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
have refine_2₁ : (μ : (a : Compacts G) → ℝ≥0∞) (K n ⊔ s.sup K) ≤
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n) : ℝ≥0∞)
    (∑ x ∈ s, (μ : (a : Compacts G) → ℝ≥0∞) (K x) : ℝ≥0∞)
Tactic: grw [Finset.sup_insert, Finset.sum_insert hn, μ.sup_le, ih]
-- 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 : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
n : ℕ
s : Finset ℕ
hn : n ∉ s
ih :
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (s.sup K) : ℝ≥0∞)
    (∑ i ∈ s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
refine_2 :
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n ⊔ s.sup K) : ℝ≥0∞)
    (∑ i ∈ insert n s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
refine_2₁ :
  (μ : (a : Compacts G) → ℝ≥0∞) (K n ⊔ s.sup K) ≤
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n) : ℝ≥0∞)
      (∑ x ∈ s, (μ : (a : Compacts G) → ℝ≥0∞) (K x) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) ((insert n s).sup K) : ℝ≥0∞)
    (∑ i ∈ insert n s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
have refine_2₂ : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n) : ℝ≥0∞)
    ((μ : (a : Compacts G) → ℝ≥0∞) (s.sup K) : ℝ≥0∞) ≤
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n) : ℝ≥0∞)
    (∑ x ∈ s, (μ : (a : Compacts G) → ℝ≥0∞) (K x) : ℝ≥0∞)
Tactic: grw [Finset.sup_insert, Finset.sum_insert hn, μ.sup_le, ih]
-- 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 : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
n : ℕ
s : Finset ℕ
hn : n ∉ s
ih :
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (s.sup K) : ℝ≥0∞)
    (∑ i ∈ s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
refine_2 :
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n ⊔ s.sup K) : ℝ≥0∞)
    (∑ i ∈ insert n s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
refine_2₁ :
  (μ : (a : Compacts G) → ℝ≥0∞) (K n ⊔ s.sup K) ≤
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n) : ℝ≥0∞)
      (∑ x ∈ s, (μ : (a : Compacts G) → ℝ≥0∞) (K x) : ℝ≥0∞)
refine_2₂ :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n) : ℝ≥0∞)
      ((μ : (a : Compacts G) → ℝ≥0∞) (s.sup K) : ℝ≥0∞) ≤
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n) : ℝ≥0∞)
      (∑ x ∈ s, (μ : (a : Compacts G) → ℝ≥0∞) (K x) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) ((insert n s).sup K) : ℝ≥0∞)
    (∑ i ∈ insert n s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
have refine_2₃ : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n) : ℝ≥0∞)
    (∑ i ∈ s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞) ≤
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (K n) : ℝ≥0∞)
    (∑ i ∈ s, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
Tactic: grw [Finset.sup_insert, Finset.sum_insert hn, μ.sup_le, ih]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (hypothesis)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Opens G
⊢ μ.innerContent (⨆ (i : ℕ), U i) ≤ ∑' (i : ℕ), μ.innerContent (U i)
have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
    (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
Tactic: have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    refine Finset.induction_on t ?_ ?_
    · simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
    · intro n s hn ih
      grw [Finset.sup_insert, Finset.sum_insert hn, μ.sup_le, ih]
-- 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 : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
⊢ μ.innerContent (⨆ (i : ℕ), U i) ≤ ∑' (i : ℕ), μ.innerContent (U i)
have refine : ∀ (K : Compacts G) (hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)),
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (∑' (i : ℕ), μ.innerContent (U i) : ℝ≥0∞)
Tactic: refine iSup₂_le 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
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (∑' (i : ℕ), μ.innerContent (U i) : ℝ≥0∞)
have obtain : ∀ (t : Finset ℕ) (ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)),
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (∑' (i : ℕ), μ.innerContent (U i) : ℝ≥0∞)
Tactic: obtain ⟨t, ht⟩ :=
    K.isCompact.elim_finite_subcover _ (fun i => (U i).isOpen) (by rwa [← Opens.coe_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
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (∑' (i : ℕ), μ.innerContent (U i) : ℝ≥0∞)
have rcases : ∀ (K' : ℕ → Set G) (h1K' : ∀ (i : ℕ), IsCompact (K' i)) (h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i)
  (h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i),
  LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (∑' (i : ℕ), μ.innerContent (U i) : ℝ≥0∞)
Tactic: rcases K.isCompact.finite_compact_cover t (SetLike.coe ∘ U) (fun i _ => (U i).isOpen) ht with
    ⟨K', h1K', h2K', h3K'⟩
-- 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 : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (∑' (i : ℕ), μ.innerContent (U i) : ℝ≥0∞)
have h : K = t.sup L
Tactic: convert! le_trans (h3 t L) _
-- 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 : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
h : K = t.sup L
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞) (∑' (i : ℕ), μ.innerContent (U i) : ℝ≥0∞)
have convert_2 : LE.le (α := ℝ≥0∞) (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (L i)) (∑' (i : ℕ), μ.innerContent (U i))
Tactic: convert! le_trans (h3 t L) _
-- 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 : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
⊢ K = t.sup L
have h : (↑K : Set G) = (↑(t.sup L) : Set G)
Tactic: ext1
-- 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 : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
⊢ (↑K : Set G) = (↑(t.sup L) : Set G)
have h : (↑K : Set G) = ⨆ a ∈ t, (↑(L a) : Set G)
Tactic: rw [Compacts.coe_finset_sup, Finset.sup_eq_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
U : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
⊢ LE.le (α := ℝ≥0∞) (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (L i)) (∑' (i : ℕ), μ.innerContent (U i))
have convert_2 : ∀ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (L i) ≤ μ.innerContent (U i)
Tactic: refine le_trans (Finset.sum_le_sum ?_) (ENNReal.sum_le_tsum t)
-- 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 : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
i : ℕ
h : i ∈ t
⊢ (μ : (a : Compacts G) → ℝ≥0∞) (L i) ≤ μ.innerContent (U i)
have convert_2 : LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (L i) : ℝ≥0∞)
  (⨆ (_ : (↑(L i) : Set G) ⊆ (↑(U i) : Set G)), (μ : (a : Compacts G) → ℝ≥0∞) (L i) : ℝ≥0∞)
Tactic: refine le_trans ?_ (le_iSup _ (L i))
-- 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 : ℕ → Opens G
h3 :
  ∀ (t : Finset ℕ) (K : ℕ → Compacts G),
    LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (t.sup K) : ℝ≥0∞)
      (∑ i ∈ t, (μ : (a : Compacts G) → ℝ≥0∞) (K i) : ℝ≥0∞)
K : Compacts G
hK : (↑K : Set G) ⊆ (↑(⨆ (i : ℕ), U i) : Set G)
t : Finset ℕ
ht : (↑K : Set G) ⊆ ⋃ i ∈ t, (↑(U i) : Set G)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : (↑K : Set G) = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun (n : ℕ) ↦ { carrier := K' n, isCompact' := ⋯ }
i : ℕ
h : i ∈ t
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (L i) : ℝ≥0∞)
    (⨆ (_ : (↑(L i) : Set G) ⊆ (↑(U i) : Set G)), (μ : (a : Compacts G) → ℝ≥0∞) (L i) : ℝ≥0∞)
have convert_2 : LE.le (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (L i) : ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (L i) : ℝ≥0∞)
Tactic: refine le_trans ?_ (le_iSup _ (h2K' i))
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (hypothesis)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
inst_1 : R1Space G
U : ℕ → Set G
hU : ∀ (i : ℕ), IsOpen (U i)
⊢ μ.innerContent { carrier := ⋃ (i : ℕ), U i, is_open' := ⋯ : Opens G } ≤
    ∑' (i : ℕ), μ.innerContent { carrier := U i, is_open' := ⋯ }
have this : μ.innerContent (⨆ (i : ℕ), { carrier := U i, is_open' := sorry : Opens G }) ≤
  ∑' (i : ℕ), μ.innerContent { carrier := U i, is_open' := sorry }
Tactic: have := μ.innerContent_iSup_nat fun i => ⟨U i, hU i⟩
-- 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 : ℕ → Set G
hU : ∀ (i : ℕ), IsOpen (U i)
this :
  μ.innerContent (⨆ (i : ℕ), { carrier := U i, is_open' := ⋯ : Opens G }) ≤
    ∑' (i : ℕ), μ.innerContent { carrier := U i, is_open' := ⋯ }
⊢ μ.innerContent { carrier := ⋃ (i : ℕ), U i, is_open' := ⋯ : Opens G } ≤
    ∑' (i : ℕ), μ.innerContent { carrier := U i, is_open' := ⋯ }
have rwa : ∀
  (this :
    μ.innerContent
        { carrier := ⋃ (i : ℕ), (↑{ carrier := U i, is_open' := sorry : Opens G } : Set G), is_open' := sorry : Opens G } ≤
      ∑' (i : ℕ), μ.innerContent { carrier := U i, is_open' := sorry }),
  μ.innerContent { carrier := ⋃ (i : ℕ), U i, is_open' := sorry : Opens G } ≤
    ∑' (i : ℕ), μ.innerContent { carrier := U i, is_open' := sorry }
Tactic: rwa [Opens.iSup_def] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
f : G ≃ₜ G
h :
  ∀ ⦃K : Compacts G⦄,
    Eq (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (Compacts.map (⇑f : (a : G) → G) ⋯ K) : ℝ≥0∞)
      ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
U : Opens G
⊢ μ.innerContent ((Opens.comap (↑f : C(G, G)) : (a : Opens G) → Opens G) U) = μ.innerContent U
have refine : ∀ (K : Compacts G) (x : (↑K : Set G) ⊆ (↑((Opens.comap (↑f : C(G, G)) : (a : Opens G) → Opens G) U : Opens G) : Set G)),
  Eq (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) ((Compacts.equiv f : (a : Compacts G) → Compacts G) K) : ℝ≥0∞)
    ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
Tactic: refine (Compacts.equiv f).surjective.iSup_congr _ fun K => iSup_congr_Prop image_subset_iff ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Content (goal)
Goal:
G : Type w
inst : TopologicalSpace G
μ : Content G
f : G ≃ₜ G
h :
  ∀ ⦃K : Compacts G⦄,
    Eq (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (Compacts.map (⇑f : (a : G) → G) ⋯ K) : ℝ≥0∞)
      ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
U : Opens G
K : Compacts G
hK : (↑K : Set G) ⊆ (↑((Opens.comap (↑f : C(G, G)) : (a : Opens G) → Opens G) U : Opens G) : Set G)
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) ((Compacts.equiv f : (a : Compacts G) → Compacts G) K) : ℝ≥0∞)
    ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
have simp : Eq (α := ℝ≥0∞) ((μ : (a : Compacts G) → ℝ≥0∞) (Compacts.map (⇑f : (a : G) → G) sorry K) : ℝ≥0∞)
  ((μ : (a : Compacts G) → ℝ≥0∞) K : ℝ≥0∞)
Tactic: simp only [Equiv.coe_fn_mk, Compacts.equiv]
-- 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
⊢ 0 < μ.innerContent U
have this : (interior (↑U : Set G)).Nonempty
Tactic: have : (interior (U : Set G)).Nonempty := by rwa [U.isOpen.interior_eq]