Theorem: MeasureTheory.toFiniteAux_univ_le_one
File: Mathlib/MeasureTheory/Measure/WithDensityFinite.lean

tactic: rw [toFiniteAux_apply]
state_before: α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
⊢ μ.toFiniteAux Set.univ ≤ 1
state_after: α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
⊢ ∑' (n : ℕ), (2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ)⁻¹ * (sFiniteSeq μ n) Set.univ ≤ 1

tactic: refine (tsum_le_tsum h_le_pow ENNReal.summable ENNReal.summable).trans ?_
state_before: α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
h_le_pow : ∀ (n : ℕ), (2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ)⁻¹ * (sFiniteSeq μ n) Set.univ ≤ (2 ^ (n + 1))⁻¹
⊢ ∑' (n : ℕ), (2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ)⁻¹ * (sFiniteSeq μ n) Set.univ ≤ 1
state_after: α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
h_le_pow : ∀ (n : ℕ), (2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ)⁻¹ * (sFiniteSeq μ n) Set.univ ≤ (2 ^ (n + 1))⁻¹
⊢ ∑' (i : ℕ), (2 ^ (i + 1))⁻¹ ≤ 1

tactic: simp [ENNReal.inv_pow, ENNReal.tsum_geometric_add_one, ENNReal.inv_mul_cancel]
state_before: α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
h_le_pow : ∀ (n : ℕ), (2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ)⁻¹ * (sFiniteSeq μ n) Set.univ ≤ (2 ^ (n + 1))⁻¹
⊢ ∑' (i : ℕ), (2 ^ (i + 1))⁻¹ ≤ 1
state_after: no goals

tactic: intro n
state_before: α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
⊢ ∀ (n : ℕ), (2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ)⁻¹ * (sFiniteSeq μ n) Set.univ ≤ (2 ^ (n + 1))⁻¹
state_after: α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
n : ℕ
⊢ (2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ)⁻¹ * (sFiniteSeq μ n) Set.univ ≤ (2 ^ (n + 1))⁻¹

tactic: by_cases h_zero : sFiniteSeq μ n = 0
state_before: α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
n : ℕ
⊢ (2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ)⁻¹ * (sFiniteSeq μ n) Set.univ ≤ (2 ^ (n + 1))⁻¹
state_after: case pos
α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
n : ℕ
h_zero : sFiniteSeq μ n = 0
⊢ (2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ)⁻¹ * (sFiniteSeq μ n) Set.univ ≤ (2 ^ (n + 1))⁻¹

case neg
α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
n : ℕ
h_zero : ¬sFiniteSeq μ n = 0
⊢ (2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ)⁻¹ * (sFiniteSeq μ n) Set.univ ≤ (2 ^ (n + 1))⁻¹

tactic: simp [h_zero]
state_before: case pos
α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
n : ℕ
h_zero : sFiniteSeq μ n = 0
⊢ (2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ)⁻¹ * (sFiniteSeq μ n) Set.univ ≤ (2 ^ (n + 1))⁻¹
state_after: no goals

tactic: rw [ENNReal.le_inv_iff_mul_le, mul_assoc, mul_comm (sFiniteSeq μ n Set.univ),
  ENNReal.inv_mul_cancel]
state_before: case neg
α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
n : ℕ
h_zero : ¬sFiniteSeq μ n = 0
⊢ (2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ)⁻¹ * (sFiniteSeq μ n) Set.univ ≤ (2 ^ (n + 1))⁻¹
state_after: case neg.h0
α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
n : ℕ
h_zero : ¬sFiniteSeq μ n = 0
⊢ 2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ ≠ 0

case neg.ht
α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
n : ℕ
h_zero : ¬sFiniteSeq μ n = 0
⊢ 2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ ≠ ⊤

tactic: simp [h_zero]
state_before: case neg.h0
α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
n : ℕ
h_zero : ¬sFiniteSeq μ n = 0
⊢ 2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ ≠ 0
state_after: no goals

tactic: exact ENNReal.mul_ne_top (by simp) (measure_ne_top _ _)
state_before: case neg.ht
α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
n : ℕ
h_zero : ¬sFiniteSeq μ n = 0
⊢ 2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ ≠ ⊤
state_after: no goals

tactic: simp
state_before: α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
n : ℕ
h_zero : ¬sFiniteSeq μ n = 0
⊢ 2 ^ (n + 1) ≠ ⊤
state_after: no goals