awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Probability.Distributions.Fernique
Goal:
E : Type u_1
inst : SeminormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : SecondCountableTopology E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
a : ℝ
inst_5 : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4))) : (a : E × E) → E × E) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
C : ℝ := logRatio c' * HPow.hPow (β := ℕ) a⁻¹ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (ENNReal.ofReal (rexp (logRatio c')))
ha : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
⊢ LT.lt (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)

Tactic:
have ha_lt : μ {x | ‖x‖ ≤ a} < 1 := lt_of_le_of_ne prob_le_one ha
-- Module: Mathlib.Probability.Distributions.Fernique
Goal:
E : Type u_1
inst : SeminormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : SecondCountableTopology E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
a : ℝ
inst_5 : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4))) : (a : E × E) → E × E) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
C : ℝ := logRatio c' * HPow.hPow (β := ℕ) a⁻¹ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (ENNReal.ofReal (rexp (logRatio c')))
ha : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
ha_lt : LT.lt (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
⊢ c' < 1

Tactic:
have hc'_lt : c' < 1 := lt_of_le_of_lt hc' ha_lt
-- Module: Mathlib.Probability.Distributions.Fernique
Goal:
E : Type u_1
inst : SeminormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : SecondCountableTopology E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
a : ℝ
inst_5 : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4))) : (a : E × E) → E × E) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
C : ℝ := logRatio c' * HPow.hPow (β := ℕ) a⁻¹ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (ENNReal.ofReal (rexp (logRatio c')))
ha : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
ha_lt : LT.lt (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
hc'_lt : c' < 1
h :
  ∀ (x : E),
    Membership.mem (γ := Set E) Set.univ x ↔
      Membership.mem (γ := Set E) (closedBall 0 a ∪ ⋃ (n : ℕ), closedBall 0 (t (n + 1)) \ closedBall 0 (t n)) x
⊢ Eq.{u_1 + 1} (α := Set E) Set.univ (closedBall 0 a ∪ ⋃ (n : ℕ), closedBall 0 (t (n + 1)) \ closedBall 0 (t n))

Tactic:
ext x
-- Module: Mathlib.Probability.Distributions.Fernique
Goal:
E : Type u_1
inst : SeminormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : SecondCountableTopology E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
a : ℝ
inst_5 : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4))) : (a : E × E) → E × E) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
C : ℝ := logRatio c' * HPow.hPow (β := ℕ) a⁻¹ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (ENNReal.ofReal (rexp (logRatio c')))
ha : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
ha_lt : LT.lt (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
hc'_lt : c' < 1
x : E
h : ‖x‖ ≤ a ∨ ∃ (i : ℕ), ‖x‖ ≤ t (i + 1) ∧ t i < ‖x‖
⊢ Membership.mem (γ := Set E) Set.univ x ↔
    Membership.mem (γ := Set E) (closedBall 0 a ∪ ⋃ (n : ℕ), closedBall 0 (t (n + 1)) \ closedBall 0 (t n)) x

Tactic:
simp only [Set.mem_univ, Set.mem_union, Metric.mem_closedBall, dist_zero_right, Set.mem_iUnion,
      Set.mem_diff, not_le, true_iff]

Premises used: Set.mem_union, not_le, Set.mem_iUnion, true_iff, Set.mem_diff, Set.mem_univ, Metric.mem_closedBall, dist_zero_right
-- Module: Mathlib.Probability.Distributions.Fernique
Goal:
E : Type u_1
inst : SeminormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : SecondCountableTopology E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
a : ℝ
inst_5 : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4))) : (a : E × E) → E × E) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
C : ℝ := logRatio c' * HPow.hPow (β := ℕ) a⁻¹ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (ENNReal.ofReal (rexp (logRatio c')))
ha : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
ha_lt : LT.lt (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
hc'_lt : c' < 1
x : E
h : ‖x‖ ≤ a ∨ ∃ (i : ℕ), t i < ‖x‖ ∧ ‖x‖ ≤ t (i + 1)
⊢ ‖x‖ ≤ a ∨ ∃ (i : ℕ), ‖x‖ ≤ t (i + 1) ∧ t i < ‖x‖

Tactic:
simp_rw [and_comm (b := t _ < ‖x‖)]

Premises used: and_comm
-- Module: Mathlib.Probability.Distributions.Fernique
Goal:
E : Type u_1
inst : SeminormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : SecondCountableTopology E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
a : ℝ
inst_5 : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4))) : (a : E × E) → E × E) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
C : ℝ := logRatio c' * HPow.hPow (β := ℕ) a⁻¹ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (ENNReal.ofReal (rexp (logRatio c')))
ha : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
ha_lt : LT.lt (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
hc'_lt : c' < 1
x : E
h : ∀ (ha' : ‖x‖ ≤ a), ‖x‖ ≤ a ∨ ∃ (i : ℕ), t i < ‖x‖ ∧ ‖x‖ ≤ t (i + 1)
h₁ : ∀ (ha' : a < ‖x‖), ‖x‖ ≤ a ∨ ∃ (i : ℕ), t i < ‖x‖ ∧ ‖x‖ ≤ t (i + 1)
⊢ ‖x‖ ≤ a ∨ ∃ (i : ℕ), t i < ‖x‖ ∧ ‖x‖ ≤ t (i + 1)

Tactic:
rcases le_or_gt (‖x‖) a with ha' | ha'

Premises used: Norm.norm, le_or_gt
-- Module: Mathlib.Probability.Distributions.Fernique
Goal:
E : Type u_1
inst : SeminormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : SecondCountableTopology E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
a : ℝ
inst_5 : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4))) : (a : E × E) → E × E) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
C : ℝ := logRatio c' * HPow.hPow (β := ℕ) a⁻¹ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (ENNReal.ofReal (rexp (logRatio c')))
ha : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
ha_lt : LT.lt (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
hc'_lt : c' < 1
x : E
ha' : ‖x‖ ≤ a
⊢ ‖x‖ ≤ a ∨ ∃ (i : ℕ), t i < ‖x‖ ∧ ‖x‖ ≤ t (i + 1)

Tactic:
Or.inl ha'

Premises used: Or.inl
-- Module: Mathlib.Probability.Distributions.Fernique
Goal:
E : Type u_1
inst : SeminormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : SecondCountableTopology E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
a : ℝ
inst_5 : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4))) : (a : E × E) → E × E) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
C : ℝ := logRatio c' * HPow.hPow (β := ℕ) a⁻¹ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (ENNReal.ofReal (rexp (logRatio c')))
ha : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
ha_lt : LT.lt (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
hc'_lt : c' < 1
x : E
ha' : ‖x‖ ≤ a
⊢ ‖x‖ ≤ a ∨ ∃ (i : ℕ), t i < ‖x‖ ∧ ‖x‖ ≤ t (i + 1)

Tactic:
exact Or.inl ha'

Premises used: Or.inl
-- Module: Mathlib.Probability.Distributions.Fernique
Goal:
E : Type u_1
inst : SeminormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : SecondCountableTopology E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
a : ℝ
inst_5 : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4))) : (a : E × E) → E × E) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
C : ℝ := logRatio c' * HPow.hPow (β := ℕ) a⁻¹ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (ENNReal.ofReal (rexp (logRatio c')))
ha : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
ha_lt : LT.lt (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
hc'_lt : c' < 1
x : E
ha' : a < ‖x‖
⊢ ‖x‖ ≤ a ∨ ∃ (i : ℕ), t i < ‖x‖ ∧ ‖x‖ ≤ t (i + 1)

Tactic:
Or.inr <| (normThreshold_strictMono ha_pos).exists_between_of_tendsto_atTop
        (tendsto_normThreshold_atTop ha_pos) ha'

Premises used: StrictMono.exists_between_of_tendsto_atTop, ProbabilityTheory.Fernique.normThreshold_strictMono, ProbabilityTheory.Fernique.tendsto_normThreshold_atTop, Or.inr
-- Module: Mathlib.Probability.Distributions.Fernique
Goal:
E : Type u_1
inst : SeminormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : SecondCountableTopology E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
a : ℝ
inst_5 : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4))) : (a : E × E) → E × E) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
C : ℝ := logRatio c' * HPow.hPow (β := ℕ) a⁻¹ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (ENNReal.ofReal (rexp (logRatio c')))
ha : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
ha_lt : LT.lt (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
hc'_lt : c' < 1
x : E
ha' : a < ‖x‖
⊢ ‖x‖ ≤ a ∨ ∃ (i : ℕ), t i < ‖x‖ ∧ ‖x‖ ≤ t (i + 1)

Tactic:
exact Or.inr <| (normThreshold_strictMono ha_pos).exists_between_of_tendsto_atTop
        (tendsto_normThreshold_atTop ha_pos) ha'

Premises used: StrictMono.exists_between_of_tendsto_atTop, ProbabilityTheory.Fernique.normThreshold_strictMono, ProbabilityTheory.Fernique.tendsto_normThreshold_atTop, Or.inr
-- Module: Mathlib.Probability.Distributions.Fernique
Goal:
E : Type u_1
inst : SeminormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : SecondCountableTopology E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
a : ℝ
inst_5 : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4))) : (a : E × E) → E × E) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
C : ℝ := logRatio c' * HPow.hPow (β := ℕ) a⁻¹ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (ENNReal.ofReal (rexp (logRatio c')))
ha : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
ha_lt : LT.lt (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
hc'_lt : c' < 1
⊢ Eq.{u_1 + 1} (α := Set E) Set.univ (closedBall 0 a ∪ ⋃ (n : ℕ), closedBall 0 (t (n + 1)) \ closedBall 0 (t n))

Tactic:
have h_iUnion : (Set.univ : Set E)
      = closedBall 0 a ∪ ⋃ n, closedBall 0 (t (n + 1)) \ closedBall 0 (t n) := by
    ext x
    simp only [Set.mem_univ, Set.mem_union, Metric.mem_closedBall, dist_zero_right, Set.mem_iUnion,
      Set.mem_diff, not_le, true_iff]
    simp_rw [and_comm (b := t _ < ‖x‖)]
    rcases le_or_gt (‖x‖) a with ha' | ha'
    · exact Or.inl ha'
    · exact Or.inr <| (normThreshold_strictMono ha_pos).exists_between_of_tendsto_atTop
        (tendsto_normThreshold_atTop ha_pos) ha'
-- Module: Mathlib.Probability.Distributions.Fernique
Goal:
E : Type u_1
inst : SeminormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : SecondCountableTopology E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
a : ℝ
inst_5 : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4))) : (a : E × E) → E × E) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
C : ℝ := logRatio c' * HPow.hPow (β := ℕ) a⁻¹ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (ENNReal.ofReal (rexp (logRatio c')))
ha : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
ha_lt : LT.lt (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
hc'_lt : c' < 1
h_iUnion :
  Eq.{u_1 + 1} (α := Set E) Set.univ (closedBall 0 a ∪ ⋃ (n : ℕ), closedBall 0 (t (n + 1)) \ closedBall 0 (t n))
neg :
  ∫⁻ (x : E) in closedBall 0 a ∪ ⋃ (n : ℕ), closedBall 0 (t (n + 1)) \ closedBall 0 (t n),
      ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    c *
      (ENNReal.ofReal (rexp (logRatio c')) +
        ∑' (n : ℕ), ENNReal.ofReal (rexp (-2⁻¹ * Real.log (c' / (1 - c')).toReal * HPow.hPow (α := ℝ) 2 n)))
⊢ ∫⁻ (x : E), ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    c *
      (ENNReal.ofReal (rexp (logRatio c')) +
        ∑' (n : ℕ), ENNReal.ofReal (rexp (-2⁻¹ * Real.log (c' / (1 - c')).toReal * HPow.hPow (α := ℝ) 2 n)))

Tactic:
rw [← setLIntegral_univ, h_iUnion]

Premises used: MeasureTheory.setLIntegral_univ
-- Module: Mathlib.Probability.Distributions.Fernique
Goal:
E : Type u_1
inst : SeminormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : SecondCountableTopology E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
a : ℝ
inst_5 : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4))) : (a : E × E) → E × E) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
C : ℝ := logRatio c' * HPow.hPow (β := ℕ) a⁻¹ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (ENNReal.ofReal (rexp (logRatio c')))
ha : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
ha_lt : LT.lt (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
hc'_lt : c' < 1
h_iUnion :
  Eq.{u_1 + 1} (α := Set E) Set.univ (closedBall 0 a ∪ ⋃ (n : ℕ), closedBall 0 (t (n + 1)) \ closedBall 0 (t n))
⊢ ∫⁻ (x : E) in closedBall 0 (t 0) ∪ ⋃ (n : ℕ), closedBall 0 (t (n + 1)) \ closedBall 0 (t n),
      ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    ∫⁻ (x : E) in closedBall 0 (t 0), ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ +
      ∑' (i : ℕ),
        ∫⁻ (x : E) in closedBall 0 (t (i + 1)) \ closedBall 0 (t i),
          ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ

Tactic:
(lintegral_union_le _ _ _).trans ?_

Premises used: MeasureTheory.lintegral_union_le, LE.le.trans
-- Module: Mathlib.Probability.Distributions.Fernique
Goal:
E : Type u_1
inst : SeminormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : SecondCountableTopology E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
a : ℝ
inst_5 : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4))) : (a : E × E) → E × E) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
C : ℝ := logRatio c' * HPow.hPow (β := ℕ) a⁻¹ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (ENNReal.ofReal (rexp (logRatio c')))
ha : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
ha_lt : LT.lt (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
hc'_lt : c' < 1
h_iUnion :
  Eq.{u_1 + 1} (α := Set E) Set.univ (closedBall 0 a ∪ ⋃ (n : ℕ), closedBall 0 (t (n + 1)) \ closedBall 0 (t n))
refine :
  ∫⁻ (a : E) in closedBall 0 (t 0), ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖a‖ 2)) ∂μ +
      ∫⁻ (a : E) in ⋃ (n : ℕ), closedBall 0 (t (n + 1)) \ closedBall 0 (t n),
        ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖a‖ 2)) ∂μ ≤
    ∫⁻ (x : E) in closedBall 0 (t 0), ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ +
      ∑' (i : ℕ),
        ∫⁻ (x : E) in closedBall 0 (t (i + 1)) \ closedBall 0 (t i),
          ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ
⊢ ∫⁻ (x : E) in closedBall 0 (t 0) ∪ ⋃ (n : ℕ), closedBall 0 (t (n + 1)) \ closedBall 0 (t n),
      ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    ∫⁻ (x : E) in closedBall 0 (t 0), ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ +
      ∑' (i : ℕ),
        ∫⁻ (x : E) in closedBall 0 (t (i + 1)) \ closedBall 0 (t i),
          ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ

Tactic:
refine (lintegral_union_le _ _ _).trans ?_

Premises used: MeasureTheory.lintegral_union_le, LE.le.trans
-- Module: Mathlib.Probability.Distributions.Fernique
Goal:
E : Type u_1
inst : SeminormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : SecondCountableTopology E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
a : ℝ
inst_5 : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4))) : (a : E × E) → E × E) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
C : ℝ := logRatio c' * HPow.hPow (β := ℕ) a⁻¹ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (ENNReal.ofReal (rexp (logRatio c')))
ha : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
ha_lt : LT.lt (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
hc'_lt : c' < 1
h_iUnion :
  Eq.{u_1 + 1} (α := Set E) Set.univ (closedBall 0 a ∪ ⋃ (n : ℕ), closedBall 0 (t (n + 1)) \ closedBall 0 (t n))
bc :
  ∫⁻ (a : E) in ⋃ (n : ℕ), closedBall 0 (t (n + 1)) \ closedBall 0 (t n),
      ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖a‖ 2)) ∂μ ≤
    ∑' (i : ℕ),
      ∫⁻ (x : E) in closedBall 0 (t (i + 1)) \ closedBall 0 (t i),
        ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ
⊢ ∫⁻ (a : E) in closedBall 0 (t 0), ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖a‖ 2)) ∂μ +
      ∫⁻ (a : E) in ⋃ (n : ℕ), closedBall 0 (t (n + 1)) \ closedBall 0 (t n),
        ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖a‖ 2)) ∂μ ≤
    ∫⁻ (x : E) in closedBall 0 (t 0), ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ +
      ∑' (i : ℕ),
        ∫⁻ (x : E) in closedBall 0 (t (i + 1)) \ closedBall 0 (t i),
          ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ

Tactic:
gcongr
-- Module: Mathlib.Probability.Distributions.Fernique
Goal:
E : Type u_1
inst : SeminormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : SecondCountableTopology E
inst_3 : MeasurableSpace E
inst_4 : BorelSpace E
μ : Measure E
a : ℝ
inst_5 : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4))) : (a : E × E) → E × E) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := (μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a}
C : ℝ := logRatio c' * HPow.hPow (β := ℕ) a⁻¹ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ ≤
    HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (ENNReal.ofReal (rexp (logRatio c')))
ha : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
ha_lt : LT.lt (α := ℝ≥0∞) ((μ : (a : Set E) → ℝ≥0∞) {x : E | ‖x‖ ≤ a} : ℝ≥0∞) (1 : ℝ≥0∞)
hc'_lt : c' < 1
h_iUnion :
  Eq.{u_1 + 1} (α := Set E) Set.univ (closedBall 0 a ∪ ⋃ (n : ℕ), closedBall 0 (t (n + 1)) \ closedBall 0 (t n))
⊢ ∫⁻ (a : E) in ⋃ (n : ℕ), closedBall 0 (t (n + 1)) \ closedBall 0 (t n),
      ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖a‖ 2)) ∂μ ≤
    ∑' (i : ℕ),
      ∫⁻ (x : E) in closedBall 0 (t (i + 1)) \ closedBall 0 (t i),
        ENNReal.ofReal (rexp (C * HPow.hPow (β := ℕ) ‖x‖ 2)) ∂μ

Tactic:
lintegral_iUnion_le _ _

Premises used: MeasureTheory.lintegral_iUnion_le