-- Mathlib/MeasureTheory/Measure/Doubling.lean
theorem exists_eventually_forall_measure_closedBall_le_mul (K : ℝ) :
    ∃ C : ℝ≥0, ∀ᶠ ε in 𝓝[>] 0, ∀ x, ∀ t ≤ K, μ (closedBall x (t * ε)) ≤ C * μ (closedBall x ε)

Proof state:
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
n : ℕ
ih : ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * (2 * ε))) ≤ ↑(C ^ n) * μ (closedBall x (2 * ε))
ε : ℝ
hε :
  (∀ (x : α), μ (closedBall x (2 ^ n * (2 * ε))) ≤ ↑(C ^ n) * μ (closedBall x (2 * ε))) ∧
    ∀ (x : α), μ (closedBall x (2 * ε)) ≤ ↑(doublingConstant μ) * μ (closedBall x ε)
x : α
⊢ ↑(C ^ n) * (↑C * μ (closedBall x ε)) = ↑(C ^ (n + 1)) * μ (closedBall x ε)

Tactic:
rw [← mul_assoc, pow_succ, ENNReal.coe_mul]

Result:
no goals

Proof state:
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
n : ℕ
⊢ ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑(C ^ n) * μ (closedBall x ε)

Tactic:
induction' n with n ih

Result:
case zero
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
⊢ ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ 0 * ε)) ≤ ↑(C ^ 0) * μ (closedBall x ε)

case succ
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
n : ℕ
ih : ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑(C ^ n) * μ (closedBall x ε)
⊢ ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ (n + 1) * ε)) ≤ ↑(C ^ (n + 1)) * μ (closedBall x ε)

Proof state:
case h.h.h.h
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
hμ : ∀ (n : ℕ), ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑(C ^ n) * μ (closedBall x ε)
hK : 1 ≤ K
ε : ℝ
hε : ∀ (x : α), μ (closedBall x (2 ^ ⌈Real.logb 2 K⌉₊ * ε)) ≤ ↑(C ^ ⌈Real.logb 2 K⌉₊) * μ (closedBall x ε)
hε₀ : ε ∈ Ioi 0
x : α
t : ℝ
ht : t ≤ K
⊢ Real.logb 2 K ≤ ↑⌈Real.logb 2 K⌉₊

case h.h.h.h.hb
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
hμ : ∀ (n : ℕ), ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑(C ^ n) * μ (closedBall x ε)
hK : 1 ≤ K
ε : ℝ
hε : ∀ (x : α), μ (closedBall x (2 ^ ⌈Real.logb 2 K⌉₊ * ε)) ≤ ↑(C ^ ⌈Real.logb 2 K⌉₊) * μ (closedBall x ε)
hε₀ : ε ∈ Ioi 0
x : α
t : ℝ
ht : t ≤ K
⊢ 1 < 2

case h.h.h.h.hx
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
hμ : ∀ (n : ℕ), ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑(C ^ n) * μ (closedBall x ε)
hK : 1 ≤ K
ε : ℝ
hε : ∀ (x : α), μ (closedBall x (2 ^ ⌈Real.logb 2 K⌉₊ * ε)) ≤ ↑(C ^ ⌈Real.logb 2 K⌉₊) * μ (closedBall x ε)
hε₀ : ε ∈ Ioi 0
x : α
t : ℝ
ht : t ≤ K
⊢ 0 < K

Tactic:
exacts [Nat.le_ceil _, by norm_num, by linarith]

Result:
no goals

Proof state:
case succ
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
n : ℕ
ih : ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑(C ^ n) * μ (closedBall x ε)
⊢ ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ (n + 1) * ε)) ≤ ↑(C ^ (n + 1)) * μ (closedBall x ε)

Tactic:
replace ih := eventually_nhdsGT_zero_mul_left (two_pos : 0 < (2 : ℝ)) ih

Result:
case succ
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
n : ℕ
ih : ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * (2 * ε))) ≤ ↑(C ^ n) * μ (closedBall x (2 * ε))
⊢ ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ (n + 1) * ε)) ≤ ↑(C ^ (n + 1)) * μ (closedBall x ε)

Proof state:
case inl.h.h
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
hμ : ∀ (n : ℕ), ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑(C ^ n) * μ (closedBall x ε)
hK : K < 1
ε : ℝ
hε : ε ∈ Ioi 0
x : α
t : ℝ
ht : t ≤ K
⊢ t * ε ≤ ε

Tactic:
nlinarith [mem_Ioi.mp hε]

Result:
no goals

Proof state:
case succ
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
n : ℕ
ih : ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * (2 * ε))) ≤ ↑(C ^ n) * μ (closedBall x (2 * ε))
ε : ℝ
hε :
  (∀ (x : α), μ (closedBall x (2 ^ n * (2 * ε))) ≤ ↑(C ^ n) * μ (closedBall x (2 * ε))) ∧
    ∀ (x : α), μ (closedBall x (2 * ε)) ≤ ↑(doublingConstant μ) * μ (closedBall x ε)
x : α
⊢ μ (closedBall x (2 ^ (n + 1) * ε)) ≤ ↑(C ^ (n + 1)) * μ (closedBall x ε)

Tactic:
calc
  μ (closedBall x ((2 : ℝ) ^ (n + 1) * ε)) = μ (closedBall x ((2 : ℝ) ^ n * (2 * ε))) := by
    rw [pow_succ, mul_assoc]
  _ ≤ ↑(C ^ n) * μ (closedBall x (2 * ε)) := hε.1 x
  _ ≤ ↑(C ^ n) * (C * μ (closedBall x ε)) := by gcongr; exact hε.2 x
  _ = ↑(C ^ (n + 1)) * μ (closedBall x ε) := by rw [← mul_assoc, pow_succ, ENNReal.coe_mul]

Result:
no goals

Proof state:
case h
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
hμ : ∀ (n : ℕ), ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑(C ^ n) * μ (closedBall x ε)
hK : 1 ≤ K
⊢ ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), ∀ t ≤ K, μ (closedBall x (t * ε)) ≤ ↑(C ^ ⌈Real.logb 2 K⌉₊) * μ (closedBall x ε)

Tactic:
filter_upwards [hμ ⌈Real.logb 2 K⌉₊, eventually_mem_nhdsWithin] with ε hε hε₀ x t ht

Result:
case h
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
hμ : ∀ (n : ℕ), ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑(C ^ n) * μ (closedBall x ε)
hK : 1 ≤ K
ε : ℝ
hε : ∀ (x : α), μ (closedBall x (2 ^ ⌈Real.logb 2 K⌉₊ * ε)) ≤ ↑(C ^ ⌈Real.logb 2 K⌉₊) * μ (closedBall x ε)
hε₀ : ε ∈ Ioi 0
x : α
t : ℝ
ht : t ≤ K
⊢ μ (closedBall x (t * ε)) ≤ ↑(C ^ ⌈Real.logb 2 K⌉₊) * μ (closedBall x ε)

Proof state:
case inl
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
hμ : ∀ (n : ℕ), ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑(C ^ n) * μ (closedBall x ε)
hK : K < 1
⊢ ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), ∀ t ≤ K, μ (closedBall x (t * ε)) ≤ μ (closedBall x ε)

Tactic:
refine eventually_mem_nhdsWithin.mono fun ε hε x t ht ↦ ?_

Result:
case inl
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
hμ : ∀ (n : ℕ), ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * ε)) ≤ ↑(C ^ n) * μ (closedBall x ε)
hK : K < 1
ε : ℝ
hε : ε ∈ Ioi 0
x : α
t : ℝ
ht : t ≤ K
⊢ μ (closedBall x (t * ε)) ≤ μ (closedBall x ε)

Proof state:
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
⊢ ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), ∀ t ≤ K, μ (closedBall x (t * ε)) ≤ ↑C * μ (closedBall x ε)

Tactic:
let C := doublingConstant μ

Result:
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
⊢ ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), ∀ t ≤ K, μ (closedBall x (t * ε)) ≤ ↑C * μ (closedBall x ε)