-- Mathlib/Probability/Moments/ComplexMGF.lean
/-- If two random variables have the same moment generating function then they have
the same `complexMGF` on the vertical strip `{z | z.re ∈ interior (integrableExpSet X μ)}`.

TODO: once we know that equal `mgf` implies equal distributions, we will be able to show that
the `complexMGF` are equal everywhere, not only on the strip.
This lemma will be used in the proof of the equality of distributions. -/
lemma eqOn_complexMGF_of_mgf' (hXY : mgf X μ = mgf Y μ') (hμμ' : μ = 0 ↔ μ' = 0) :
    Set.EqOn (complexMGF X μ) (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet X μ)}

Proof state:
case neg
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
hXY : mgf X μ = mgf Y μ'
hμμ' : μ = 0 ↔ μ' = 0
h_empty : (interior (integrableExpSet X μ)).Nonempty
⊢ Set.EqOn (complexMGF X μ) (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet X μ)}

Tactic:
obtain ⟨t, ht⟩ := h_empty

Result:
case neg.intro
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
hXY : mgf X μ = mgf Y μ'
hμμ' : μ = 0 ↔ μ' = 0
t : ℝ
ht : t ∈ interior (integrableExpSet X μ)
⊢ Set.EqOn (complexMGF X μ) (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet X μ)}

Proof state:
case neg.intro
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
hXY : mgf X μ = mgf Y μ'
hμμ' : μ = 0 ↔ μ' = 0
t : ℝ
ht : t ∈ interior (integrableExpSet Y μ')
hX : AnalyticOnNhd ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExpSet Y μ')}
hY : AnalyticOnNhd ℂ (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet Y μ')}
h_real : ∃ ns, Tendsto ns atTop (𝓝[≠] t) ∧ ∀ (n : ℕ), complexMGF X μ ↑(ns n) = complexMGF Y μ' ↑(ns n)
⊢ ∃ ns, Tendsto ns atTop (𝓝[≠] ↑t) ∧ ∀ (n : ℕ), complexMGF X μ (ns n) = complexMGF Y μ' (ns n)

Tactic:
obtain ⟨xs, hx_tendsto, hx_eq⟩ := h_real

Result:
case neg.intro.intro.intro
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
hXY : mgf X μ = mgf Y μ'
hμμ' : μ = 0 ↔ μ' = 0
t : ℝ
ht : t ∈ interior (integrableExpSet Y μ')
hX : AnalyticOnNhd ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExpSet Y μ')}
hY : AnalyticOnNhd ℂ (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet Y μ')}
xs : ℕ → ℝ
hx_tendsto : Tendsto xs atTop (𝓝[≠] t)
hx_eq : ∀ (n : ℕ), complexMGF X μ ↑(xs n) = complexMGF Y μ' ↑(xs n)
⊢ ∃ ns, Tendsto ns atTop (𝓝[≠] ↑t) ∧ ∀ (n : ℕ), complexMGF X μ (ns n) = complexMGF Y μ' (ns n)

Proof state:
case neg.intro
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
hXY : mgf X μ = mgf Y μ'
hμμ' : μ = 0 ↔ μ' = 0
t : ℝ
ht : t ∈ interior (integrableExpSet Y μ')
hX : AnalyticOnNhd ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExpSet Y μ')}
hY : AnalyticOnNhd ℂ (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet Y μ')}
⊢ ∃ᶠ (z : ℂ) in 𝓝[≠] ↑t, complexMGF X μ z = complexMGF Y μ' z

Tactic:
have h_real : ∃ᶠ (x : ℝ) in 𝓝[≠] t, complexMGF X μ x = complexMGF Y μ' x := by
  refine .of_forall fun y ↦ ?_
  rw [complexMGF_ofReal, complexMGF_ofReal, hXY]

Result:
case neg.intro
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
hXY : mgf X μ = mgf Y μ'
hμμ' : μ = 0 ↔ μ' = 0
t : ℝ
ht : t ∈ interior (integrableExpSet Y μ')
hX : AnalyticOnNhd ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExpSet Y μ')}
hY : AnalyticOnNhd ℂ (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet Y μ')}
h_real : ∃ᶠ (x : ℝ) in 𝓝[≠] t, complexMGF X μ ↑x = complexMGF Y μ' ↑x
⊢ ∃ᶠ (z : ℂ) in 𝓝[≠] ↑t, complexMGF X μ z = complexMGF Y μ' z

Proof state:
case neg.intro
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
hXY : mgf X μ = mgf Y μ'
hμμ' : μ = 0 ↔ μ' = 0
t : ℝ
ht : t ∈ interior (integrableExpSet Y μ')
hX : AnalyticOnNhd ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExpSet Y μ')}
hY : AnalyticOnNhd ℂ (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet Y μ')}
h_real : ∃ᶠ (x : ℝ) in 𝓝[≠] t, complexMGF X μ ↑x = complexMGF Y μ' ↑x
⊢ ∃ᶠ (z : ℂ) in 𝓝[≠] ↑t, complexMGF X μ z = complexMGF Y μ' z

Tactic:
rw [frequently_iff_seq_forall] at h_real ⊢

Result:
case neg.intro
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
hXY : mgf X μ = mgf Y μ'
hμμ' : μ = 0 ↔ μ' = 0
t : ℝ
ht : t ∈ interior (integrableExpSet Y μ')
hX : AnalyticOnNhd ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExpSet Y μ')}
hY : AnalyticOnNhd ℂ (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet Y μ')}
h_real : ∃ ns, Tendsto ns atTop (𝓝[≠] t) ∧ ∀ (n : ℕ), complexMGF X μ ↑(ns n) = complexMGF Y μ' ↑(ns n)
⊢ ∃ ns, Tendsto ns atTop (𝓝[≠] ↑t) ∧ ∀ (n : ℕ), complexMGF X μ (ns n) = complexMGF Y μ' (ns n)

Proof state:
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
hXY : mgf X μ = mgf Y μ'
hμμ' : μ = 0 ↔ μ' = 0
⊢ Set.EqOn (complexMGF X μ) (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet X μ)}

Tactic:
by_cases h_empty : interior (integrableExpSet X μ) = ∅

Result:
case pos
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
hXY : mgf X μ = mgf Y μ'
hμμ' : μ = 0 ↔ μ' = 0
h_empty : interior (integrableExpSet X μ) = ∅
⊢ Set.EqOn (complexMGF X μ) (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet X μ)}

case neg
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
hXY : mgf X μ = mgf Y μ'
hμμ' : μ = 0 ↔ μ' = 0
h_empty : ¬interior (integrableExpSet X μ) = ∅
⊢ Set.EqOn (complexMGF X μ) (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet X μ)}

Proof state:
case neg.intro
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
hXY : mgf X μ = mgf Y μ'
hμμ' : μ = 0 ↔ μ' = 0
t : ℝ
ht : t ∈ interior (integrableExpSet Y μ')
hX : AnalyticOnNhd ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExpSet Y μ')}
hY : AnalyticOnNhd ℂ (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet Y μ')}
⊢ Set.EqOn (complexMGF X μ) (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet Y μ')}

Tactic:
refine AnalyticOnNhd.eqOn_of_preconnected_of_frequently_eq hX hY
  (convex_integrableExpSet.interior.linear_preimage reLm).isPreconnected
  (z₀ := (t : ℂ)) (by simp [ht]) ?_

Result:
case neg.intro
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
hXY : mgf X μ = mgf Y μ'
hμμ' : μ = 0 ↔ μ' = 0
t : ℝ
ht : t ∈ interior (integrableExpSet Y μ')
hX : AnalyticOnNhd ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExpSet Y μ')}
hY : AnalyticOnNhd ℂ (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet Y μ')}
⊢ ∃ᶠ (z : ℂ) in 𝓝[≠] ↑t, complexMGF X μ z = complexMGF Y μ' z

Proof state:
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
hXY : mgf X μ = mgf Y μ'
hμμ' : μ = 0 ↔ μ' = 0
t : ℝ
ht : t ∈ interior (integrableExpSet Y μ')
hX : AnalyticOnNhd ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExpSet Y μ')}
hY : AnalyticOnNhd ℂ (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet Y μ')}
⊢ ↑t ∈ {z | z.re ∈ interior (integrableExpSet Y μ')}

Tactic:
simp [ht]

Result:
no goals