Module: Mathlib.Probability.Distributions.Fernique

Goal state before:
pp: case a
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ μ {x | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
C : ℝ := logRatio c' * a⁻¹ ^ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    μ {x | ‖x‖ ≤ a} * ENNReal.ofReal (rexp (logRatio c'))
ha : ¬μ {x | ‖x‖ ≤ a} = 1
ha_lt : μ {x | ‖x‖ ≤ a} < 1
hc'_lt : c' < 1
h_iUnion : Set.univ = closedBall 0 a ∪ ⋃ n, closedBall 0 (t (n + 1)) \ closedBall 0 (t n)
this :
  ∫⁻ (x : E) in closedBall 0 (t 0) ∪ ⋃ n, closedBall 0 (t (n + 1)) \ closedBall 0 (t n),
      ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    ∫⁻ (x : E) in closedBall 0 (t 0), ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ +
      ∑' (i : ℕ), ∫⁻ (x : E) in closedBall 0 (t (i + 1)) \ closedBall 0 (t i), ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ
n : ℕ
⊢ 2⁻¹ * Real.log (c'.toReal / (1 - c').toReal) * 2 ^ n ≤
    2⁻¹ * Real.log ((μ {x | ‖x‖ ≤ a}).toReal / (1 - μ {x | ‖x‖ ≤ a}).toReal) * 2 ^ n

Tactic:
gcongr

Goal state after:
case hx
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ μ {x | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := ⋯
c : ℝ≥0∞ := ⋯
C : ℝ := ⋯
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    μ {x | ‖x‖ ≤ a} * ENNReal.ofReal (rexp (logRatio c'))
ha : ¬μ {x | ‖x‖ ≤ a} = 1
ha_lt : μ {x | ‖x‖ ≤ a} < 1
hc'_lt : c' < 1
h_iUnion : Set.univ = closedBall 0 a ∪ ⋃ n, closedBall 0 (t (n + 1)) \ closedBall 0 (t n)
this :
  ∫⁻ (x : E) in closedBall 0 (t 0) ∪ ⋃ n, closedBall 0 (t (n + 1)) \ closedBall 0 (t n),
      ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    ∫⁻ (x : E) in closedBall 0 (t 0), ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ +
      ∑' (i : ℕ), ∫⁻ (x : E) in closedBall 0 (t (i + 1)) \ closedBall 0 (t i), ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ
n : ℕ
⊢ 0 < c'.toReal / (1 - c').toReal

case hd
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ μ {x | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := ⋯
c : ℝ≥0∞ := ⋯
C : ℝ := ⋯
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    μ {x | ‖x‖ ≤ a} * ENNReal.ofReal (rexp (logRatio c'))
ha : ¬μ {x | ‖x‖ ≤ a} = 1
ha_lt : μ {x | ‖x‖ ≤ a} < 1
hc'_lt : c' < 1
h_iUnion : Set.univ = closedBall 0 a ∪ ⋃ n, closedBall 0 (t (n + 1)) \ closedBall 0 (t n)
this :
  ∫⁻ (x : E) in closedBall 0 (t 0) ∪ ⋃ n, closedBall 0 (t (n + 1)) \ closedBall 0 (t n),
      ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    ∫⁻ (x : E) in closedBall 0 (t 0), ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ +
      ∑' (i : ℕ), ∫⁻ (x : E) in closedBall 0 (t (i + 1)) \ closedBall 0 (t i), ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ
n : ℕ
⊢ 0 < (1 - μ {x | ‖x‖ ≤ a}).toReal

case hb
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ μ {x | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := ⋯
c : ℝ≥0∞ := ⋯
C : ℝ := ⋯
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    μ {x | ‖x‖ ≤ a} * ENNReal.ofReal (rexp (logRatio c'))
ha : ¬μ {x | ‖x‖ ≤ a} = 1
ha_lt : μ {x | ‖x‖ ≤ a} < 1
hc'_lt : c' < 1
h_iUnion : Set.univ = closedBall 0 a ∪ ⋃ n, closedBall 0 (t (n + 1)) \ closedBall 0 (t n)
this :
  ∫⁻ (x : E) in closedBall 0 (t 0) ∪ ⋃ n, closedBall 0 (t (n + 1)) \ closedBall 0 (t n),
      ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    ∫⁻ (x : E) in closedBall 0 (t 0), ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ +
      ∑' (i : ℕ), ∫⁻ (x : E) in closedBall 0 (t (i + 1)) \ closedBall 0 (t i), ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ
n : ℕ
⊢ μ {x | ‖x‖ ≤ a} ≠ ∞

case hb
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ μ {x | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := ⋯
c : ℝ≥0∞ := ⋯
C : ℝ := ⋯
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    μ {x | ‖x‖ ≤ a} * ENNReal.ofReal (rexp (logRatio c'))
ha : ¬μ {x | ‖x‖ ≤ a} = 1
ha_lt : μ {x | ‖x‖ ≤ a} < 1
hc'_lt : c' < 1
h_iUnion : Set.univ = closedBall 0 a ∪ ⋃ n, closedBall 0 (t (n + 1)) \ closedBall 0 (t n)
this :
  ∫⁻ (x : E) in closedBall 0 (t 0) ∪ ⋃ n, closedBall 0 (t (n + 1)) \ closedBall 0 (t n),
      ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    ∫⁻ (x : E) in closedBall 0 (t 0), ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ +
      ∑' (i : ℕ), ∫⁻ (x : E) in closedBall 0 (t (i + 1)) \ closedBall 0 (t i), ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ
n : ℕ
⊢ 1 - c' ≠ ∞