Theorem: ProbabilityTheory.iIndepFun.integrable_exp_mul_sum
File: Mathlib/Probability/Moments.lean

tactic: induction' s using Finset.induction_on with i s hi_notin_s h_rec h_int
state_before: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
s : Finset ι
h_int : ∀ i ∈ s, Integrable (fun ω => rexp (t * X i ω)) μ
⊢ Integrable (fun ω => rexp (t * (∑ i ∈ s, X i) ω)) μ
state_after: case empty
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
h_int : ∀ i ∈ ∅, Integrable (fun ω => rexp (t * X i ω)) μ
⊢ Integrable (fun ω => rexp (t * (∑ i ∈ ∅, X i) ω)) μ

case insert
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
i : ι
s : Finset ι
hi_notin_s : i ∉ s
h_rec : (∀ i ∈ s, Integrable (fun ω => rexp (t * X i ω)) μ) → Integrable (fun ω => rexp (t * (∑ i ∈ s, X i) ω)) μ
h_int : ∀ i_1 ∈ insert i s, Integrable (fun ω => rexp (t * X i_1 ω)) μ
⊢ Integrable (fun ω => rexp (t * (∑ i ∈ insert i s, X i) ω)) μ

tactic: simp only [Pi.zero_apply, sum_apply, sum_empty, mul_zero, exp_zero]
state_before: case empty
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
h_int : ∀ i ∈ ∅, Integrable (fun ω => rexp (t * X i ω)) μ
⊢ Integrable (fun ω => rexp (t * (∑ i ∈ ∅, X i) ω)) μ
state_after: case empty
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
h_int : ∀ i ∈ ∅, Integrable (fun ω => rexp (t * X i ω)) μ
⊢ Integrable (fun ω => 1) μ

tactic: exact integrable_const _
state_before: case empty
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
h_int : ∀ i ∈ ∅, Integrable (fun ω => rexp (t * X i ω)) μ
⊢ Integrable (fun ω => 1) μ
state_after: no goals

tactic: have : ∀ i : ι, i ∈ s → Integrable (fun ω : Ω => exp (t * X i ω)) μ := fun i hi =>
  h_int i (mem_insert_of_mem hi)
state_before: case insert
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
i : ι
s : Finset ι
hi_notin_s : i ∉ s
h_rec : (∀ i ∈ s, Integrable (fun ω => rexp (t * X i ω)) μ) → Integrable (fun ω => rexp (t * (∑ i ∈ s, X i) ω)) μ
h_int : ∀ i_1 ∈ insert i s, Integrable (fun ω => rexp (t * X i_1 ω)) μ
⊢ Integrable (fun ω => rexp (t * (∑ i ∈ insert i s, X i) ω)) μ
state_after: case insert
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
i : ι
s : Finset ι
hi_notin_s : i ∉ s
h_rec : (∀ i ∈ s, Integrable (fun ω => rexp (t * X i ω)) μ) → Integrable (fun ω => rexp (t * (∑ i ∈ s, X i) ω)) μ
h_int : ∀ i_1 ∈ insert i s, Integrable (fun ω => rexp (t * X i_1 ω)) μ
this : ∀ i ∈ s, Integrable (fun ω => rexp (t * X i ω)) μ
⊢ Integrable (fun ω => rexp (t * (∑ i ∈ insert i s, X i) ω)) μ

tactic: specialize h_rec this
state_before: case insert
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
i : ι
s : Finset ι
hi_notin_s : i ∉ s
h_rec : (∀ i ∈ s, Integrable (fun ω => rexp (t * X i ω)) μ) → Integrable (fun ω => rexp (t * (∑ i ∈ s, X i) ω)) μ
h_int : ∀ i_1 ∈ insert i s, Integrable (fun ω => rexp (t * X i_1 ω)) μ
this : ∀ i ∈ s, Integrable (fun ω => rexp (t * X i ω)) μ
⊢ Integrable (fun ω => rexp (t * (∑ i ∈ insert i s, X i) ω)) μ
state_after: case insert
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
i : ι
s : Finset ι
hi_notin_s : i ∉ s
h_int : ∀ i_1 ∈ insert i s, Integrable (fun ω => rexp (t * X i_1 ω)) μ
this : ∀ i ∈ s, Integrable (fun ω => rexp (t * X i ω)) μ
h_rec : Integrable (fun ω => rexp (t * (∑ i ∈ s, X i) ω)) μ
⊢ Integrable (fun ω => rexp (t * (∑ i ∈ insert i s, X i) ω)) μ

tactic: rw [sum_insert hi_notin_s]
state_before: case insert
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
i : ι
s : Finset ι
hi_notin_s : i ∉ s
h_int : ∀ i_1 ∈ insert i s, Integrable (fun ω => rexp (t * X i_1 ω)) μ
this : ∀ i ∈ s, Integrable (fun ω => rexp (t * X i ω)) μ
h_rec : Integrable (fun ω => rexp (t * (∑ i ∈ s, X i) ω)) μ
⊢ Integrable (fun ω => rexp (t * (∑ i ∈ insert i s, X i) ω)) μ
state_after: case insert
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
i : ι
s : Finset ι
hi_notin_s : i ∉ s
h_int : ∀ i_1 ∈ insert i s, Integrable (fun ω => rexp (t * X i_1 ω)) μ
this : ∀ i ∈ s, Integrable (fun ω => rexp (t * X i ω)) μ
h_rec : Integrable (fun ω => rexp (t * (∑ i ∈ s, X i) ω)) μ
⊢ Integrable (fun ω => rexp (t * (X i + ∑ x ∈ s, X x) ω)) μ

tactic: refine IndepFun.integrable_exp_mul_add ?_ (h_int i (mem_insert_self _ _)) h_rec
state_before: case insert
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
i : ι
s : Finset ι
hi_notin_s : i ∉ s
h_int : ∀ i_1 ∈ insert i s, Integrable (fun ω => rexp (t * X i_1 ω)) μ
this : ∀ i ∈ s, Integrable (fun ω => rexp (t * X i ω)) μ
h_rec : Integrable (fun ω => rexp (t * (∑ i ∈ s, X i) ω)) μ
⊢ Integrable (fun ω => rexp (t * (X i + ∑ x ∈ s, X x) ω)) μ
state_after: case insert
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
i : ι
s : Finset ι
hi_notin_s : i ∉ s
h_int : ∀ i_1 ∈ insert i s, Integrable (fun ω => rexp (t * X i_1 ω)) μ
this : ∀ i ∈ s, Integrable (fun ω => rexp (t * X i ω)) μ
h_rec : Integrable (fun ω => rexp (t * (∑ i ∈ s, X i) ω)) μ
⊢ IndepFun (X i) (∑ x ∈ s, X x) μ

tactic: exact (h_indep.indepFun_finset_sum_of_not_mem h_meas hi_notin_s).symm
state_before: case insert
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
i : ι
s : Finset ι
hi_notin_s : i ∉ s
h_int : ∀ i_1 ∈ insert i s, Integrable (fun ω => rexp (t * X i_1 ω)) μ
this : ∀ i ∈ s, Integrable (fun ω => rexp (t * X i ω)) μ
h_rec : Integrable (fun ω => rexp (t * (∑ i ∈ s, X i) ω)) μ
⊢ IndepFun (X i) (∑ x ∈ s, X x) μ
state_after: no goals