/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import probability.notation
import probability.hitting_time

/-!
# Martingales

A family of functions `f : ι → α → E` is a martingale with respect to a filtration `ℱ` if every
`f i` is integrable, `f` is adapted with respect to `ℱ` and for all `i ≤ j`,
`μ[f j | ℱ i] =ᵐ[μ] f i`. On the other hand, `f : ι → α → E` is said to be a supermartingale
with respect to the filtration `ℱ` if `f i` is integrable, `f` is adapted with resepct to `ℱ`
and for all `i ≤ j`, `μ[f j | ℱ i] ≤ᵐ[μ] f i`. Finally, `f : ι → α → E` is said to be a
submartingale with respect to the filtration `ℱ` if `f i` is integrable, `f` is adapted with
resepct to `ℱ` and for all `i ≤ j`, `f i ≤ᵐ[μ] μ[f j | ℱ i]`.

The definitions of filtration and adapted can be found in `probability.stopping`.

### Definitions

* `measure_theory.martingale f ℱ μ`: `f` is a martingale with respect to filtration `ℱ` and
  measure `μ`.
* `measure_theory.supermartingale f ℱ μ`: `f` is a supermartingale with respect to
  filtration `ℱ` and measure `μ`.
* `measure_theory.submartingale f ℱ μ`: `f` is a submartingale with respect to filtration `ℱ` and
  measure `μ`.

### Results

* `measure_theory.martingale_condexp f ℱ μ`: the sequence `λ i, μ[f | ℱ i, ℱ.le i])` is a
  martingale with respect to `ℱ` and `μ`.

-/

open topological_space filter
open_locale nnreal ennreal measure_theory probability_theory big_operators

namespace measure_theory

variables {α E ι : Type*} [preorder ι]
  {m0 : measurable_space α} {μ : measure α}
  [normed_add_comm_group E] [normed_space ℝ E] [complete_space E]
  {f g : ι → α → E} {ℱ : filtration ι m0}

/-- A family of functions `f : ι → α → E` is a martingale with respect to a filtration `ℱ` if `f`
is adapted with respect to `ℱ` and for all `i ≤ j`, `μ[f j | ℱ i] =ᵐ[μ] f i`. -/
def martingale (f : ι → α → E) (ℱ : filtration ι m0) (μ : measure α) : Prop :=
adapted ℱ f ∧ ∀ i j, i ≤ j → μ[f j | ℱ i] =ᵐ[μ] f i

/-- A family of integrable functions `f : ι → α → E` is a supermartingale with respect to a
filtration `ℱ` if `f` is adapted with respect to `ℱ` and for all `i ≤ j`,
`μ[f j | ℱ.le i] ≤ᵐ[μ] f i`. -/
def supermartingale [has_le E] (f : ι → α → E) (ℱ : filtration ι m0) (μ : measure α) : Prop :=
adapted ℱ f ∧ (∀ i j, i ≤ j → μ[f j | ℱ i] ≤ᵐ[μ] f i) ∧ ∀ i, integrable (f i) μ

/-- A family of integrable functions `f : ι → α → E` is a submartingale with respect to a
filtration `ℱ` if `f` is adapted with respect to `ℱ` and for all `i ≤ j`,
`f i ≤ᵐ[μ] μ[f j | ℱ.le i]`. -/
def submartingale [has_le E] (f : ι → α → E) (ℱ : filtration ι m0) (μ : measure α) : Prop :=
adapted ℱ f ∧ (∀ i j, i ≤ j → f i ≤ᵐ[μ] μ[f j | ℱ i]) ∧ ∀ i, integrable (f i) μ

variables (E)
lemma martingale_zero (ℱ : filtration ι m0) (μ : measure α) :
  martingale (0 : ι → α → E) ℱ μ :=
⟨adapted_zero E ℱ, λ i j hij, by { rw [pi.zero_apply, condexp_zero], simp, }⟩
variables {E}

namespace martingale

@[protected]
lemma adapted (hf : martingale f ℱ μ) : adapted ℱ f := hf.1

@[protected]
lemma strongly_measurable (hf : martingale f ℱ μ) (i : ι) : strongly_measurable[ℱ i] (f i) :=
hf.adapted i

lemma condexp_ae_eq (hf : martingale f ℱ μ) {i j : ι} (hij : i ≤ j) :
  μ[f j | ℱ i] =ᵐ[μ] f i :=
hf.2 i j hij

@[protected]
lemma integrable (hf : martingale f ℱ μ) (i : ι) : integrable (f i) μ :=
integrable_condexp.congr (hf.condexp_ae_eq (le_refl i))

lemma set_integral_eq [sigma_finite_filtration μ ℱ] (hf : martingale f ℱ μ) {i j : ι} (hij : i ≤ j)
  {s : set α} (hs : measurable_set[ℱ i] s) :
  ∫ x in s, f i x ∂μ = ∫ x in s, f j x ∂μ :=
begin
  rw ← @set_integral_condexp _ _ _ _ _ (ℱ i) m0 _ _ _ (ℱ.le i) _ (hf.integrable j) hs,
  refine set_integral_congr_ae (ℱ.le i s hs) _,
  filter_upwards [hf.2 i j hij] with _ heq _ using heq.symm,
end

lemma add (hf : martingale f ℱ μ) (hg : martingale g ℱ μ) : martingale (f + g) ℱ μ :=
begin
  refine ⟨hf.adapted.add hg.adapted, λ i j hij, _⟩,
  exact (condexp_add (hf.integrable j) (hg.integrable j)).trans
    ((hf.2 i j hij).add (hg.2 i j hij)),
end

lemma neg (hf : martingale f ℱ μ) : martingale (-f) ℱ μ :=
⟨hf.adapted.neg, λ i j hij, (condexp_neg (f j)).trans ((hf.2 i j hij).neg)⟩

lemma sub (hf : martingale f ℱ μ) (hg : martingale g ℱ μ) : martingale (f - g) ℱ μ :=
by { rw sub_eq_add_neg, exact hf.add hg.neg, }

lemma smul (c : ℝ) (hf : martingale f ℱ μ) : martingale (c • f) ℱ μ :=
begin
  refine ⟨hf.adapted.smul c, λ i j hij, _⟩,
  refine (condexp_smul c (f j)).trans ((hf.2 i j hij).mono (λ x hx, _)),
  rw [pi.smul_apply, hx, pi.smul_apply, pi.smul_apply],
end

lemma supermartingale [preorder E] (hf : martingale f ℱ μ) : supermartingale f ℱ μ :=
⟨hf.1, λ i j hij, (hf.2 i j hij).le, λ i, hf.integrable i⟩

lemma submartingale [preorder E] (hf : martingale f ℱ μ) : submartingale f ℱ μ :=
⟨hf.1, λ i j hij, (hf.2 i j hij).symm.le, λ i, hf.integrable i⟩

end martingale

lemma martingale_iff [partial_order E] : martingale f ℱ μ ↔
  supermartingale f ℱ μ ∧ submartingale f ℱ μ :=
⟨λ hf, ⟨hf.supermartingale, hf.submartingale⟩,
 λ ⟨hf₁, hf₂⟩, ⟨hf₁.1, λ i j hij, (hf₁.2.1 i j hij).antisymm (hf₂.2.1 i j hij)⟩⟩

lemma martingale_condexp (f : α → E) (ℱ : filtration ι m0) (μ : measure α)
  [sigma_finite_filtration μ ℱ] :
  martingale (λ i, μ[f | ℱ i]) ℱ μ :=
⟨λ i, strongly_measurable_condexp, λ i j hij, condexp_condexp_of_le (ℱ.mono hij) (ℱ.le j)⟩

namespace supermartingale

@[protected]
lemma adapted [has_le E] (hf : supermartingale f ℱ μ) : adapted ℱ f := hf.1

@[protected]
lemma strongly_measurable [has_le E] (hf : supermartingale f ℱ μ) (i : ι) :
  strongly_measurable[ℱ i] (f i) :=
hf.adapted i

@[protected]
lemma integrable [has_le E] (hf : supermartingale f ℱ μ) (i : ι) : integrable (f i) μ := hf.2.2 i

lemma condexp_ae_le [has_le E] (hf : supermartingale f ℱ μ) {i j : ι} (hij : i ≤ j) :
  μ[f j | ℱ i] ≤ᵐ[μ] f i :=
hf.2.1 i j hij

lemma set_integral_le [sigma_finite_filtration μ ℱ] {f : ι → α → ℝ} (hf : supermartingale f ℱ μ)
  {i j : ι} (hij : i ≤ j) {s : set α} (hs : measurable_set[ℱ i] s) :
  ∫ x in s, f j x ∂μ ≤ ∫ x in s, f i x ∂μ :=
begin
  rw ← set_integral_condexp (ℱ.le i) (hf.integrable j) hs,
  refine set_integral_mono_ae integrable_condexp.integrable_on (hf.integrable i).integrable_on _,
  filter_upwards [hf.2.1 i j hij] with _ heq using heq,
end

lemma add [preorder E] [covariant_class E E (+) (≤)]
  (hf : supermartingale f ℱ μ) (hg : supermartingale g ℱ μ) : supermartingale (f + g) ℱ μ :=
begin
  refine ⟨hf.1.add hg.1, λ i j hij, _, λ i, (hf.2.2 i).add (hg.2.2 i)⟩,
  refine (condexp_add (hf.integrable j) (hg.integrable j)).le.trans _,
  filter_upwards [hf.2.1 i j hij, hg.2.1 i j hij],
  intros,
  refine add_le_add _ _; assumption,
end

lemma add_martingale [preorder E] [covariant_class E E (+) (≤)]
  (hf : supermartingale f ℱ μ) (hg : martingale g ℱ μ) : supermartingale (f + g) ℱ μ :=
hf.add hg.supermartingale

lemma neg [preorder E] [covariant_class E E (+) (≤)]
  (hf : supermartingale f ℱ μ) : submartingale (-f) ℱ μ :=
begin
  refine ⟨hf.1.neg, λ i j hij, _, λ i, (hf.2.2 i).neg⟩,
  refine eventually_le.trans _ (condexp_neg (f j)).symm.le,
  filter_upwards [hf.2.1 i j hij] with _ _,
  simpa,
end

end supermartingale

namespace submartingale

@[protected]
lemma adapted [has_le E] (hf : submartingale f ℱ μ) : adapted ℱ f := hf.1

@[protected]
lemma strongly_measurable [has_le E] (hf : submartingale f ℱ μ) (i : ι) :
  strongly_measurable[ℱ i] (f i) :=
hf.adapted i

@[protected]
lemma integrable [has_le E] (hf : submartingale f ℱ μ) (i : ι) : integrable (f i) μ := hf.2.2 i

lemma ae_le_condexp [has_le E] (hf : submartingale f ℱ μ) {i j : ι} (hij : i ≤ j) :
  f i ≤ᵐ[μ] μ[f j | ℱ i] :=
hf.2.1 i j hij

lemma add [preorder E] [covariant_class E E (+) (≤)]
  (hf : submartingale f ℱ μ) (hg : submartingale g ℱ μ) : submartingale (f + g) ℱ μ :=
begin
  refine ⟨hf.1.add hg.1, λ i j hij, _, λ i, (hf.2.2 i).add (hg.2.2 i)⟩,
  refine eventually_le.trans _ (condexp_add (hf.integrable j) (hg.integrable j)).symm.le,
  filter_upwards [hf.2.1 i j hij, hg.2.1 i j hij],
  intros,
  refine add_le_add _ _; assumption,
end

lemma add_martingale [preorder E] [covariant_class E E (+) (≤)]
  (hf : submartingale f ℱ μ) (hg : martingale g ℱ μ) : submartingale (f + g) ℱ μ :=
hf.add hg.submartingale

lemma neg [preorder E] [covariant_class E E (+) (≤)]
  (hf : submartingale f ℱ μ) : supermartingale (-f) ℱ μ :=
begin
  refine ⟨hf.1.neg, λ i j hij, (condexp_neg (f j)).le.trans _, λ i, (hf.2.2 i).neg⟩,
  filter_upwards [hf.2.1 i j hij] with _ _,
  simpa,
end

/-- The converse of this lemma is `measure_theory.submartingale_of_set_integral_le`. -/
lemma set_integral_le [sigma_finite_filtration μ ℱ] {f : ι → α → ℝ} (hf : submartingale f ℱ μ)
  {i j : ι} (hij : i ≤ j) {s : set α} (hs : measurable_set[ℱ i] s) :
  ∫ x in s, f i x ∂μ ≤ ∫ x in s, f j x ∂μ :=
begin
  rw [← neg_le_neg_iff, ← integral_neg, ← integral_neg],
  exact supermartingale.set_integral_le hf.neg hij hs,
end

lemma sub_supermartingale [preorder E] [covariant_class E E (+) (≤)]
  (hf : submartingale f ℱ μ) (hg : supermartingale g ℱ μ) : submartingale (f - g) ℱ μ :=
by { rw sub_eq_add_neg, exact hf.add hg.neg }

lemma sub_martingale [preorder E] [covariant_class E E (+) (≤)]
  (hf : submartingale f ℱ μ) (hg : martingale g ℱ μ) : submartingale (f - g) ℱ μ :=
hf.sub_supermartingale hg.supermartingale

protected lemma sup {f g : ι → α → ℝ} (hf : submartingale f ℱ μ) (hg : submartingale g ℱ μ) :
  submartingale (f ⊔ g) ℱ μ :=
begin
  refine ⟨λ i, @strongly_measurable.sup _ _ _ _ (ℱ i) _ _ _ (hf.adapted i) (hg.adapted i),
    λ i j hij, _, λ i, integrable.sup (hf.integrable _) (hg.integrable _)⟩,
  refine eventually_le.sup_le _ _,
  { exact eventually_le.trans (hf.2.1 i j hij)
      (condexp_mono (hf.integrable _) (integrable.sup (hf.integrable j) (hg.integrable j))
      (eventually_of_forall (λ x, le_max_left _ _))) },
  { exact eventually_le.trans (hg.2.1 i j hij)
      (condexp_mono (hg.integrable _) (integrable.sup (hf.integrable j) (hg.integrable j))
      (eventually_of_forall (λ x, le_max_right _ _))) }
end

protected lemma pos {f : ι → α → ℝ} (hf : submartingale f ℱ μ) :
  submartingale (f⁺) ℱ μ :=
hf.sup (martingale_zero _ _ _).submartingale

end submartingale

section submartingale

lemma submartingale_of_set_integral_le [is_finite_measure μ]
  {f : ι → α → ℝ} (hadp : adapted ℱ f) (hint : ∀ i, integrable (f i) μ)
  (hf : ∀ i j : ι, i ≤ j → ∀ s : set α, measurable_set[ℱ i] s →
    ∫ x in s, f i x ∂μ ≤ ∫ x in s, f j x ∂μ) :
  submartingale f ℱ μ :=
begin
  refine ⟨hadp, λ i j hij, _, hint⟩,
  suffices : f i ≤ᵐ[μ.trim (ℱ.le i)] μ[f j| ℱ i],
  { exact ae_le_of_ae_le_trim this },
  suffices : 0 ≤ᵐ[μ.trim (ℱ.le i)] μ[f j| ℱ i] - f i,
  { filter_upwards [this] with x hx,
    rwa ← sub_nonneg },
  refine ae_nonneg_of_forall_set_integral_nonneg_of_finite_measure
    ((integrable_condexp.sub (hint i)).trim _ (strongly_measurable_condexp.sub $ hadp i))
    (λ s hs, _),
  specialize hf i j hij s hs,
  rwa [← set_integral_trim _ (strongly_measurable_condexp.sub $ hadp i) hs,
    integral_sub' integrable_condexp.integrable_on (hint i).integrable_on, sub_nonneg,
    set_integral_condexp (ℱ.le i) (hint j) hs],
end

lemma submartingale_of_condexp_sub_nonneg [is_finite_measure μ]
  {f : ι → α → ℝ} (hadp : adapted ℱ f) (hint : ∀ i, integrable (f i) μ)
  (hf : ∀ i j, i ≤ j → 0 ≤ᵐ[μ] μ[f j - f i | ℱ i]) :
  submartingale f ℱ μ :=
begin
  refine ⟨hadp, λ i j hij, _, hint⟩,
  rw [← condexp_of_strongly_measurable (ℱ.le _) (hadp _) (hint _), ← eventually_sub_nonneg],
  exact eventually_le.trans (hf i j hij) (condexp_sub (hint _) (hint _)).le,
  apply_instance
end

lemma submartingale.condexp_sub_nonneg [is_finite_measure μ]
  {f : ι → α → ℝ} (hf : submartingale f ℱ μ) {i j : ι} (hij : i ≤ j) :
  0 ≤ᵐ[μ] μ[f j - f i | ℱ i] :=
begin
  refine eventually_le.trans _ (condexp_sub (hf.integrable _) (hf.integrable _)).symm.le,
  rw [eventually_sub_nonneg,
    condexp_of_strongly_measurable (ℱ.le _) (hf.adapted _) (hf.integrable _)],
  exact hf.2.1 i j hij,
  apply_instance
end

lemma submartingale_iff_condexp_sub_nonneg [is_finite_measure μ] {f : ι → α → ℝ} :
  submartingale f ℱ μ ↔ adapted ℱ f ∧ (∀ i, integrable (f i) μ) ∧ ∀ i j, i ≤ j →
  0 ≤ᵐ[μ] μ[f j - f i | ℱ i] :=
⟨λ h, ⟨h.adapted, h.integrable, λ i j, h.condexp_sub_nonneg⟩,
 λ ⟨hadp, hint, h⟩, submartingale_of_condexp_sub_nonneg hadp hint h⟩

end submartingale

namespace supermartingale

lemma sub_submartingale [preorder E] [covariant_class E E (+) (≤)]
  (hf : supermartingale f ℱ μ) (hg : submartingale g ℱ μ) : supermartingale (f - g) ℱ μ :=
by { rw sub_eq_add_neg, exact hf.add hg.neg }

lemma sub_martingale [preorder E] [covariant_class E E (+) (≤)]
  (hf : supermartingale f ℱ μ) (hg : martingale g ℱ μ) : supermartingale (f - g) ℱ μ :=
hf.sub_submartingale hg.submartingale

section

variables {F : Type*} [normed_lattice_add_comm_group F]
  [normed_space ℝ F] [complete_space F] [ordered_smul ℝ F]

lemma smul_nonneg {f : ι → α → F}
  {c : ℝ} (hc : 0 ≤ c) (hf : supermartingale f ℱ μ) :
  supermartingale (c • f) ℱ μ :=
begin
  refine ⟨hf.1.smul c, λ i j hij, _, λ i, (hf.2.2 i).smul c⟩,
  refine (condexp_smul c (f j)).le.trans _,
  filter_upwards [hf.2.1 i j hij] with _ hle,
  simp,
  exact smul_le_smul_of_nonneg hle hc,
end

lemma smul_nonpos {f : ι → α → F}
  {c : ℝ} (hc : c ≤ 0) (hf : supermartingale f ℱ μ) :
  submartingale (c • f) ℱ μ :=
begin
  rw [← neg_neg c, (by { ext i x, simp } : - -c • f = -(-c • f))],
  exact (hf.smul_nonneg $ neg_nonneg.2 hc).neg,
end

end

end supermartingale

namespace submartingale

section

variables {F : Type*} [normed_lattice_add_comm_group F]
  [normed_space ℝ F] [complete_space F] [ordered_smul ℝ F]

lemma smul_nonneg {f : ι → α → F}
  {c : ℝ} (hc : 0 ≤ c) (hf : submartingale f ℱ μ) :
  submartingale (c • f) ℱ μ :=
begin
  rw [← neg_neg c, (by { ext i x, simp } : - -c • f = -(c • -f))],
  exact supermartingale.neg (hf.neg.smul_nonneg hc),
end

lemma smul_nonpos {f : ι → α → F}
  {c : ℝ} (hc : c ≤ 0) (hf : submartingale f ℱ μ) :
  supermartingale (c • f) ℱ μ :=
begin
  rw [← neg_neg c, (by { ext i x, simp } : - -c • f = -(-c • f))],
  exact (hf.smul_nonneg $ neg_nonneg.2 hc).neg,
end

end

end submartingale

section nat

variables {𝒢 : filtration ℕ m0}

lemma submartingale_of_set_integral_le_succ [is_finite_measure μ]
  {f : ℕ → α → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ)
  (hf : ∀ i, ∀ s : set α, measurable_set[𝒢 i] s → ∫ x in s, f i x ∂μ ≤ ∫ x in s, f (i + 1) x ∂μ) :
  submartingale f 𝒢 μ :=
begin
  refine submartingale_of_set_integral_le hadp hint (λ i j hij s hs, _),
  induction hij with k hk₁ hk₂,
  { exact le_rfl },
  { exact le_trans hk₂ (hf k s (𝒢.mono hk₁ _ hs)) }
end

lemma submartingale_nat [is_finite_measure μ]
  {f : ℕ → α → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ)
  (hf : ∀ i, f i ≤ᵐ[μ] μ[f (i + 1) | 𝒢 i]) :
  submartingale f 𝒢 μ :=
begin
  refine submartingale_of_set_integral_le_succ hadp hint (λ i s hs, _),
  have : ∫ x in s, f (i + 1) x ∂μ = ∫ x in s, μ[f (i + 1)|𝒢 i] x ∂μ :=
    (set_integral_condexp (𝒢.le i) (hint _) hs).symm,
  rw this,
  exact set_integral_mono_ae (hint i).integrable_on integrable_condexp.integrable_on (hf i),
end

lemma submartingale_of_condexp_sub_nonneg_nat [is_finite_measure μ]
  {f : ℕ → α → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ)
  (hf : ∀ i, 0 ≤ᵐ[μ] μ[f (i + 1) - f i | 𝒢 i]) :
  submartingale f 𝒢 μ :=
begin
  refine submartingale_nat hadp hint (λ i, _),
  rw [← condexp_of_strongly_measurable (𝒢.le _) (hadp _) (hint _), ← eventually_sub_nonneg],
  exact eventually_le.trans (hf i) (condexp_sub (hint _) (hint _)).le,
  apply_instance
end

namespace submartingale

lemma integrable_stopped_value [has_le E] {f : ℕ → α → E} (hf : submartingale f 𝒢 μ) {τ : α → ℕ}
  (hτ : is_stopping_time 𝒢 τ) {N : ℕ} (hbdd : ∀ x, τ x ≤ N) :
  integrable (stopped_value f τ) μ :=
integrable_stopped_value hτ hf.integrable hbdd

-- We may generalize the below lemma to functions taking value in a `normed_lattice_add_comm_group`.
-- Similarly, generalize `(super/)submartingale.set_integral_le`.

/-- Given a submartingale `f` and bounded stopping times `τ` and `π` such that `τ ≤ π`, the
expectation of `stopped_value f τ` is less than or equal to the expectation of `stopped_value f π`.
This is the forward direction of the optional stopping theorem. -/
lemma expected_stopped_value_mono [sigma_finite_filtration μ 𝒢]
  {f : ℕ → α → ℝ} (hf : submartingale f 𝒢 μ) {τ π : α → ℕ}
  (hτ : is_stopping_time 𝒢 τ) (hπ : is_stopping_time 𝒢 π) (hle : τ ≤ π)
  {N : ℕ} (hbdd : ∀ x, π x ≤ N) :
  μ[stopped_value f τ] ≤ μ[stopped_value f π] :=
begin
  rw [← sub_nonneg, ← integral_sub', stopped_value_sub_eq_sum' hle hbdd],
  { simp only [finset.sum_apply],
    have : ∀ i, measurable_set[𝒢 i] {x : α | τ x ≤ i ∧ i < π x},
    { intro i,
      refine (hτ i).inter _,
      convert (hπ i).compl,
      ext x,
      simpa },
    rw integral_finset_sum,
    { refine finset.sum_nonneg (λ i hi, _),
      rw [integral_indicator (𝒢.le _ _ (this _)), integral_sub', sub_nonneg],
      { exact hf.set_integral_le (nat.le_succ i) (this _) },
      { exact (hf.integrable _).integrable_on },
      { exact (hf.integrable _).integrable_on } },
    intros i hi,
    exact integrable.indicator (integrable.sub (hf.integrable _) (hf.integrable _))
      (𝒢.le _ _ (this _)) },
  { exact hf.integrable_stopped_value hπ hbdd },
  { exact hf.integrable_stopped_value hτ (λ x, le_trans (hle x) (hbdd x)) }
end

end submartingale

/-- The converse direction of the optional stopping theorem, i.e. an adapted integrable process `f`
is a submartingale if for all bounded stopping times `τ` and `π` such that `τ ≤ π`, the
stopped value of `f` at `τ` has expectation smaller than its stopped value at `π`. -/
lemma submartingale_of_expected_stopped_value_mono [is_finite_measure μ]
  {f : ℕ → α → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ)
  (hf : ∀ τ π : α → ℕ, is_stopping_time 𝒢 τ → is_stopping_time 𝒢 π → τ ≤ π → (∃ N, ∀ x, π x ≤ N) →
    μ[stopped_value f τ] ≤ μ[stopped_value f π]) :
  submartingale f 𝒢 μ :=
begin
  refine submartingale_of_set_integral_le hadp hint (λ i j hij s hs, _),
  classical,
  specialize hf (s.piecewise (λ _, i) (λ _, j)) _
    (is_stopping_time_piecewise_const hij hs)
    (is_stopping_time_const 𝒢 j) (λ x, (ite_le_sup _ _ _).trans (max_eq_right hij).le)
    ⟨j, λ x, le_rfl⟩,
  rwa [stopped_value_const, stopped_value_piecewise_const,
    integral_piecewise (𝒢.le _ _ hs) (hint _).integrable_on (hint _).integrable_on,
    ← integral_add_compl (𝒢.le _ _ hs) (hint j), add_le_add_iff_right] at hf,
end

/-- **The optional stopping theorem** (fair game theorem): an adapted integrable process `f`
is a submartingale if and only if for all bounded stopping times `τ` and `π` such that `τ ≤ π`, the
stopped value of `f` at `τ` has expectation smaller than its stopped value at `π`. -/
lemma submartingale_iff_expected_stopped_value_mono [is_finite_measure μ]
  {f : ℕ → α → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ) :
  submartingale f 𝒢 μ ↔
  ∀ τ π : α → ℕ, is_stopping_time 𝒢 τ → is_stopping_time 𝒢 π → τ ≤ π → (∃ N, ∀ x, π x ≤ N) →
    μ[stopped_value f τ] ≤ μ[stopped_value f π] :=
⟨λ hf _ _ hτ hπ hle ⟨N, hN⟩, hf.expected_stopped_value_mono hτ hπ hle hN,
 submartingale_of_expected_stopped_value_mono hadp hint⟩

section maximal

open finset

lemma smul_le_stopped_value_hitting [is_finite_measure μ]
  {f : ℕ → α → ℝ} (hsub : submartingale f 𝒢 μ) {ε : ℝ≥0} (n : ℕ) :
  ε • μ {x | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k x)} ≤
  ennreal.of_real (∫ x in {x | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k x)},
    stopped_value f (hitting f {y : ℝ | ↑ε ≤ y} 0 n) x ∂μ) :=
begin
  have hn : set.Icc 0 n = {k | k ≤ n},
  { ext x, simp },
  have : ∀ x, ((ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k x)) →
    (ε : ℝ) ≤ stopped_value f (hitting f {y : ℝ | ↑ε ≤ y} 0 n) x,
  { intros x hx,
    simp_rw [le_sup'_iff, mem_range, nat.lt_succ_iff] at hx,
    refine stopped_value_hitting_mem _,
    simp only [set.mem_set_of_eq, exists_prop, hn],
    exact let ⟨j, hj₁, hj₂⟩ := hx in ⟨j, hj₁, hj₂⟩ },
  have h := set_integral_ge_of_const_le (measurable_set_le measurable_const
    (finset.measurable_range_sup'' (λ n _, (hsub.strongly_measurable n).measurable.le (𝒢.le n))))
    (measure_ne_top _ _) this
    (integrable.integrable_on (integrable_stopped_value (hitting_is_stopping_time
     hsub.adapted measurable_set_Ici) hsub.integrable hitting_le)),
  rw [ennreal.le_of_real_iff_to_real_le, ennreal.to_real_smul],
  { exact h },
  { exact ennreal.mul_ne_top (by simp) (measure_ne_top _ _) },
  { exact le_trans (mul_nonneg ε.coe_nonneg ennreal.to_real_nonneg) h }
end

/-- **Doob's maximal inequality**: Given a non-negative submartingale `f`, for all `ε : ℝ≥0`,
we have `ε • μ {ε ≤ f* n} ≤ ∫ x in {ε ≤ f* n}, f n` where `f* n x = max_{k ≤ n}, f k x`.

In some literature, the Doob's maximal inequality refers to what we call Doob's Lp inequality
(which is a corollary of this lemma and will be proved in an upcomming PR). -/
lemma maximal_ineq [is_finite_measure μ]
  {f : ℕ → α → ℝ} (hsub : submartingale f 𝒢 μ) (hnonneg : 0 ≤ f) {ε : ℝ≥0} (n : ℕ) :
  ε • μ {x | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k x)} ≤
  ennreal.of_real (∫ x in {x | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k x)},
    f n x ∂μ) :=
begin
  suffices : ε • μ {x | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k x)} +
    ennreal.of_real (∫ x in {x | ((range (n + 1)).sup' nonempty_range_succ (λ k, f k x)) < ε},
      f n x ∂μ) ≤ ennreal.of_real (μ[f n]),
  { have hadd : ennreal.of_real (∫ (x : α), f n x ∂μ) =
      ennreal.of_real (∫ (x : α) in
        {x : α | ↑ε ≤ ((range (n + 1)).sup' nonempty_range_succ (λ k, f k x))}, f n x ∂μ) +
      ennreal.of_real (∫ (x : α) in
        {x : α | ((range (n + 1)).sup' nonempty_range_succ (λ k, f k x)) < ↑ε}, f n x ∂μ),
    { rw [← ennreal.of_real_add, ← integral_union],
      { conv_lhs { rw ← integral_univ },
        convert rfl,
        ext x,
        change (ε : ℝ) ≤ _ ∨ _ < (ε : ℝ) ↔ _,
        simp only [le_or_lt, true_iff] },
      { rintro x ⟨hx₁ : _ ≤ _, hx₂ : _ < _⟩,
        exact (not_le.2 hx₂) hx₁ },
      { exact (measurable_set_lt (finset.measurable_range_sup''
          (λ n _, (hsub.strongly_measurable n).measurable.le (𝒢.le n))) measurable_const) },
      exacts [(hsub.integrable _).integrable_on, (hsub.integrable _).integrable_on,
        integral_nonneg (hnonneg _), integral_nonneg (hnonneg _)] },
    rwa [hadd, ennreal.add_le_add_iff_right ennreal.of_real_ne_top] at this },
  calc ε • μ {x | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k x)}
    + ennreal.of_real (∫ x in {x | ((range (n + 1)).sup' nonempty_range_succ (λ k, f k x)) < ε},
        f n x ∂μ)
    ≤ ennreal.of_real (∫ x in {x | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k x)},
        stopped_value f (hitting f {y : ℝ | ↑ε ≤ y} 0 n) x ∂μ)
    + ennreal.of_real (∫ x in {x | ((range (n + 1)).sup' nonempty_range_succ (λ k, f k x)) < ε},
        stopped_value f (hitting f {y : ℝ | ↑ε ≤ y} 0 n) x ∂μ) :
    begin
      refine add_le_add (smul_le_stopped_value_hitting hsub _)
        (ennreal.of_real_le_of_real (set_integral_mono_on (hsub.integrable n).integrable_on
        (integrable.integrable_on (integrable_stopped_value
          (hitting_is_stopping_time hsub.adapted measurable_set_Ici) hsub.integrable hitting_le))
        (measurable_set_lt (finset.measurable_range_sup''
          (λ n _, (hsub.strongly_measurable n).measurable.le (𝒢.le n))) measurable_const) _)),
      intros x hx,
      rw set.mem_set_of_eq at hx,
      have : hitting f {y : ℝ | ↑ε ≤ y} 0 n x = n,
      { simp only [hitting, set.mem_set_of_eq, exists_prop, pi.coe_nat, nat.cast_id,
          ite_eq_right_iff, forall_exists_index, and_imp],
        intros m hm hεm,
        exact false.elim ((not_le.2 hx)
          ((le_sup'_iff _).2 ⟨m, mem_range.2 (nat.lt_succ_of_le hm.2), hεm⟩)) },
      simp_rw [stopped_value, this],
    end
    ... = ennreal.of_real (∫ x, stopped_value f (hitting f {y : ℝ | ↑ε ≤ y} 0 n) x ∂μ) :
    begin
      rw [← ennreal.of_real_add, ← integral_union],
      { conv_rhs { rw ← integral_univ },
        convert rfl,
        ext x,
        change _ ↔ (ε : ℝ) ≤ _ ∨ _ < (ε : ℝ),
        simp only [le_or_lt, iff_true] },
      { rintro x ⟨hx₁ : _ ≤ _, hx₂ : _ < _⟩,
        exact (not_le.2 hx₂) hx₁ },
      { exact (measurable_set_lt (finset.measurable_range_sup''
          (λ n _, (hsub.strongly_measurable n).measurable.le (𝒢.le n))) measurable_const) },
      { exact (integrable.integrable_on (integrable_stopped_value
          (hitting_is_stopping_time hsub.adapted measurable_set_Ici) hsub.integrable hitting_le)) },
      { exact (integrable.integrable_on (integrable_stopped_value
          (hitting_is_stopping_time hsub.adapted measurable_set_Ici) hsub.integrable hitting_le)) },
      exacts [integral_nonneg (λ x, hnonneg _ _), integral_nonneg (λ x, hnonneg _ _)],
    end
    ... ≤ ennreal.of_real (μ[f n]) :
    begin
      refine ennreal.of_real_le_of_real _,
      rw ← stopped_value_const f n,
      exact hsub.expected_stopped_value_mono
        (hitting_is_stopping_time hsub.adapted measurable_set_Ici)
        (is_stopping_time_const _ _) (λ x, hitting_le x) (λ x, le_rfl : ∀ x, n ≤ n),
    end
end

end maximal

lemma submartingale.sum_mul_sub [is_finite_measure μ] {R : ℝ} {ξ f : ℕ → α → ℝ}
  (hf : submartingale f 𝒢 μ) (hξ : adapted 𝒢 ξ)
  (hbdd : ∀ n x, ξ n x ≤ R) (hnonneg : ∀ n x, 0 ≤ ξ n x) :
  submartingale (λ n : ℕ, ∑ k in finset.range n, ξ k * (f (k + 1) - f k)) 𝒢 μ :=
begin
  have hξbdd : ∀ i, ∃ (C : ℝ), ∀ (x : α), |ξ i x| ≤ C :=
    λ i, ⟨R, λ x, (abs_of_nonneg (hnonneg i x)).trans_le (hbdd i x)⟩,
  have hint : ∀ m, integrable (∑ k in finset.range m, ξ k * (f (k + 1) - f k)) μ :=
    λ m, integrable_finset_sum' _
      (λ i hi, integrable.bdd_mul ((hf.integrable _).sub (hf.integrable _))
      hξ.strongly_measurable.ae_strongly_measurable (hξbdd _)),
  have hadp : adapted 𝒢 (λ (n : ℕ), ∑ (k : ℕ) in finset.range n, ξ k * (f (k + 1) - f k)),
  { intro m,
    refine finset.strongly_measurable_sum' _ (λ i hi, _),
    rw finset.mem_range at hi,
    exact (hξ.strongly_measurable_le hi.le).mul
      ((hf.adapted.strongly_measurable_le (nat.succ_le_of_lt hi)).sub
      (hf.adapted.strongly_measurable_le hi.le)) },
  refine submartingale_of_condexp_sub_nonneg_nat hadp hint (λ i, _),
  simp only [← finset.sum_Ico_eq_sub _ (nat.le_succ _), finset.sum_apply, pi.mul_apply,
    pi.sub_apply, nat.Ico_succ_singleton, finset.sum_singleton],
  exact eventually_le.trans (eventually_le.mul_nonneg (eventually_of_forall (hnonneg _))
    (hf.condexp_sub_nonneg (nat.le_succ _))) (condexp_strongly_measurable_mul (hξ _)
    (((hf.integrable _).sub (hf.integrable _)).bdd_mul
      hξ.strongly_measurable.ae_strongly_measurable (hξbdd _))
    ((hf.integrable _).sub (hf.integrable _))).symm.le,
end

/-- Given a discrete submartingale `f` and a predictable process `ξ` (i.e. `ξ (n + 1)` is adapted)
the process defined by `λ n, ∑ k in finset.range n, ξ (k + 1) * (f (k + 1) - f k)` is also a
submartingale. -/
lemma submartingale.sum_mul_sub' [is_finite_measure μ] {R : ℝ} {ξ f : ℕ → α → ℝ}
  (hf : submartingale f 𝒢 μ) (hξ : adapted 𝒢 (λ n, ξ (n + 1)))
  (hbdd : ∀ n x, ξ n x ≤ R) (hnonneg : ∀ n x, 0 ≤ ξ n x) :
  submartingale (λ n : ℕ, ∑ k in finset.range n, ξ (k + 1) * (f (k + 1) - f k)) 𝒢 μ :=
hf.sum_mul_sub hξ (λ n, hbdd _) (λ n, hnonneg _)

end nat

end measure_theory