ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Martingale.setIntegral_eq
theorem setIntegral_eq [SigmaFiniteFiltration μ ℱ] (hf : Martingale f ℱ μ) {i j : ι} (hij : i ≤ j)
    {s : Set Ω} (hs : MeasurableSet[ℱ i] s) : ∫ ω in s, f i ω ∂μ = ∫ ω in s, f j ω ∂μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [MeasureTheory.SigmaFiniteFiltration μ ℱ] (hf : MeasureTheory.Martingale f ℱ μ) {i j : ι} (hij : i ≤ j) {s : Set Ω} (hs : MeasurableSet s)
Goal: ∫ (ω : Ω) in s, f i ω ∂μ = ∫ (ω : Ω) in s, f j ω ∂μ
-- MeasureTheory.Martingale.add
theorem add (hf : Martingale f ℱ μ) (hg : Martingale g ℱ μ) : Martingale (f + g) ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f g : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} (hf : MeasureTheory.Martingale f ℱ μ) (hg : MeasureTheory.Martingale g ℱ μ)
Goal: MeasureTheory.Martingale (f + g) ℱ μ
-- MeasureTheory.Martingale.neg
theorem neg (hf : Martingale f ℱ μ) : Martingale (-f) ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} (hf : MeasureTheory.Martingale f ℱ μ)
Goal: MeasureTheory.Martingale (-f) ℱ μ
-- MeasureTheory.Martingale.sub
theorem sub (hf : Martingale f ℱ μ) (hg : Martingale g ℱ μ) : Martingale (f - g) ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f g : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} (hf : MeasureTheory.Martingale f ℱ μ) (hg : MeasureTheory.Martingale g ℱ μ)
Goal: MeasureTheory.Martingale (f - g) ℱ μ
-- MeasureTheory.Martingale.smul
theorem smul (c : ℝ) (hf : Martingale f ℱ μ) : Martingale (c • f) ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} (c : ℝ) (hf : MeasureTheory.Martingale f ℱ μ)
Goal: MeasureTheory.Martingale (c • f) ℱ μ
-- MeasureTheory.Martingale.supermartingale
theorem supermartingale [Preorder E] (hf : Martingale f ℱ μ) : Supermartingale f ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [Preorder E] (hf : MeasureTheory.Martingale f ℱ μ)
Goal: MeasureTheory.Supermartingale f ℱ μ
-- MeasureTheory.Martingale.submartingale
theorem submartingale [Preorder E] (hf : Martingale f ℱ μ) : Submartingale f ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [Preorder E] (hf : MeasureTheory.Martingale f ℱ μ)
Goal: MeasureTheory.Submartingale f ℱ μ
-- MeasureTheory.martingale_iff
theorem martingale_iff [PartialOrder E] :
    Martingale f ℱ μ ↔ Supermartingale f ℱ μ ∧ Submartingale f ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [PartialOrder E]
Goal: MeasureTheory.Martingale f ℱ μ ↔ MeasureTheory.Supermartingale f ℱ μ ∧ MeasureTheory.Submartingale f ℱ μ
-- MeasureTheory.martingale_condexp
theorem martingale_condexp (f : Ω → E) (ℱ : Filtration ι m0) (μ : Measure Ω)
    [SigmaFiniteFiltration μ ℱ] : Martingale (fun i => μ[f|ℱ i]) ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (f : Ω → E) (ℱ : MeasureTheory.Filtration ι m0) (μ : MeasureTheory.Measure Ω) [MeasureTheory.SigmaFiniteFiltration μ ℱ]
Goal: MeasureTheory.Martingale (fun i => μ[f|↑ℱ i]) ℱ μ
-- MeasureTheory.Supermartingale.adapted
protected theorem adapted [LE E] (hf : Supermartingale f ℱ μ) : Adapted ℱ f 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [LE E] (hf : MeasureTheory.Supermartingale f ℱ μ)
Goal: MeasureTheory.Adapted ℱ f
-- MeasureTheory.Supermartingale.stronglyMeasurable
protected theorem stronglyMeasurable [LE E] (hf : Supermartingale f ℱ μ) (i : ι) :
    StronglyMeasurable[ℱ i] (f i) 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [LE E] (hf : MeasureTheory.Supermartingale f ℱ μ) (i : ι)
Goal: MeasureTheory.StronglyMeasurable (f i)
-- MeasureTheory.Supermartingale.integrable
protected theorem integrable [LE E] (hf : Supermartingale f ℱ μ) (i : ι) : Integrable (f i) μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [LE E] (hf : MeasureTheory.Supermartingale f ℱ μ) (i : ι)
Goal: MeasureTheory.Integrable (f i) μ
-- MeasureTheory.Supermartingale.condexp_ae_le
theorem condexp_ae_le [LE E] (hf : Supermartingale f ℱ μ) {i j : ι} (hij : i ≤ j) :
    μ[f j|ℱ i] ≤ᵐ[μ] f i 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [LE E] (hf : MeasureTheory.Supermartingale f ℱ μ) {i j : ι} (hij : i ≤ j)
Goal: μ[f j|↑ℱ i] ≤ᵐ[μ] f i
-- MeasureTheory.Supermartingale.setIntegral_le
theorem setIntegral_le [SigmaFiniteFiltration μ ℱ] {f : ι → Ω → ℝ} (hf : Supermartingale f ℱ μ)
    {i j : ι} (hij : i ≤ j) {s : Set Ω} (hs : MeasurableSet[ℱ i] s) :
    ∫ ω in s, f j ω ∂μ ≤ ∫ ω in s, f i ω ∂μ 
Arguments: {Ω : Type u_1} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ι m0} [MeasureTheory.SigmaFiniteFiltration μ ℱ] {f : ι → Ω → ℝ} (hf : MeasureTheory.Supermartingale f ℱ μ) {i j : ι} (hij : i ≤ j) {s : Set Ω} (hs : MeasurableSet s)
Goal: ∫ (ω : Ω) in s, f j ω ∂μ ≤ ∫ (ω : Ω) in s, f i ω ∂μ
-- MeasureTheory.Supermartingale.add
theorem add [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)] (hf : Supermartingale f ℱ μ)
    (hg : Supermartingale g ℱ μ) : Supermartingale (f + g) ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f g : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [Preorder E] [CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hf : MeasureTheory.Supermartingale f ℱ μ) (hg : MeasureTheory.Supermartingale g ℱ μ)
Goal: MeasureTheory.Supermartingale (f + g) ℱ μ
-- MeasureTheory.Supermartingale.add_martingale
theorem add_martingale [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)]
    (hf : Supermartingale f ℱ μ) (hg : Martingale g ℱ μ) : Supermartingale (f + g) ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f g : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [Preorder E] [CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hf : MeasureTheory.Supermartingale f ℱ μ) (hg : MeasureTheory.Martingale g ℱ μ)
Goal: MeasureTheory.Supermartingale (f + g) ℱ μ
-- MeasureTheory.Supermartingale.neg
theorem neg [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)] (hf : Supermartingale f ℱ μ) :
    Submartingale (-f) ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [Preorder E] [CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hf : MeasureTheory.Supermartingale f ℱ μ)
Goal: MeasureTheory.Submartingale (-f) ℱ μ
-- MeasureTheory.Submartingale.adapted
protected theorem adapted [LE E] (hf : Submartingale f ℱ μ) : Adapted ℱ f 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [LE E] (hf : MeasureTheory.Submartingale f ℱ μ)
Goal: MeasureTheory.Adapted ℱ f
-- MeasureTheory.Submartingale.stronglyMeasurable
protected theorem stronglyMeasurable [LE E] (hf : Submartingale f ℱ μ) (i : ι) :
    StronglyMeasurable[ℱ i] (f i) 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [LE E] (hf : MeasureTheory.Submartingale f ℱ μ) (i : ι)
Goal: MeasureTheory.StronglyMeasurable (f i)
-- MeasureTheory.Submartingale.integrable
protected theorem integrable [LE E] (hf : Submartingale f ℱ μ) (i : ι) : Integrable (f i) μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [LE E] (hf : MeasureTheory.Submartingale f ℱ μ) (i : ι)
Goal: MeasureTheory.Integrable (f i) μ
-- MeasureTheory.Submartingale.ae_le_condexp
theorem ae_le_condexp [LE E] (hf : Submartingale f ℱ μ) {i j : ι} (hij : i ≤ j) :
    f i ≤ᵐ[μ] μ[f j|ℱ i] 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [LE E] (hf : MeasureTheory.Submartingale f ℱ μ) {i j : ι} (hij : i ≤ j)
Goal: f i ≤ᵐ[μ] μ[f j|↑ℱ i]
-- MeasureTheory.Submartingale.add
theorem add [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)] (hf : Submartingale f ℱ μ)
    (hg : Submartingale g ℱ μ) : Submartingale (f + g) ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f g : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [Preorder E] [CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hf : MeasureTheory.Submartingale f ℱ μ) (hg : MeasureTheory.Submartingale g ℱ μ)
Goal: MeasureTheory.Submartingale (f + g) ℱ μ
-- MeasureTheory.Submartingale.add_martingale
theorem add_martingale [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)] (hf : Submartingale f ℱ μ)
    (hg : Martingale g ℱ μ) : Submartingale (f + g) ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f g : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [Preorder E] [CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hf : MeasureTheory.Submartingale f ℱ μ) (hg : MeasureTheory.Martingale g ℱ μ)
Goal: MeasureTheory.Submartingale (f + g) ℱ μ
-- MeasureTheory.Submartingale.neg
theorem neg [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)] (hf : Submartingale f ℱ μ) :
    Supermartingale (-f) ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [Preorder E] [CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hf : MeasureTheory.Submartingale f ℱ μ)
Goal: MeasureTheory.Supermartingale (-f) ℱ μ
-- MeasureTheory.Submartingale.setIntegral_le
theorem setIntegral_le [SigmaFiniteFiltration μ ℱ] {f : ι → Ω → ℝ} (hf : Submartingale f ℱ μ)
    {i j : ι} (hij : i ≤ j) {s : Set Ω} (hs : MeasurableSet[ℱ i] s) :
    ∫ ω in s, f i ω ∂μ ≤ ∫ ω in s, f j ω ∂μ 
Arguments: {Ω : Type u_1} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ι m0} [MeasureTheory.SigmaFiniteFiltration μ ℱ] {f : ι → Ω → ℝ} (hf : MeasureTheory.Submartingale f ℱ μ) {i j : ι} (hij : i ≤ j) {s : Set Ω} (hs : MeasurableSet s)
Goal: ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f j ω ∂μ
-- MeasureTheory.Submartingale.sub_supermartingale
theorem sub_supermartingale [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)]
    (hf : Submartingale f ℱ μ) (hg : Supermartingale g ℱ μ) : Submartingale (f - g) ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f g : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [Preorder E] [CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hf : MeasureTheory.Submartingale f ℱ μ) (hg : MeasureTheory.Supermartingale g ℱ μ)
Goal: MeasureTheory.Submartingale (f - g) ℱ μ
-- MeasureTheory.Submartingale.sub_martingale
theorem sub_martingale [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)] (hf : Submartingale f ℱ μ)
    (hg : Martingale g ℱ μ) : Submartingale (f - g) ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f g : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [Preorder E] [CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hf : MeasureTheory.Submartingale f ℱ μ) (hg : MeasureTheory.Martingale g ℱ μ)
Goal: MeasureTheory.Submartingale (f - g) ℱ μ
-- MeasureTheory.Submartingale.sup
protected theorem sup {f g : ι → Ω → ℝ} (hf : Submartingale f ℱ μ) (hg : Submartingale g ℱ μ) :
    Submartingale (f ⊔ g) ℱ μ 
Arguments: {Ω : Type u_1} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ι m0} {f g : ι → Ω → ℝ} (hf : MeasureTheory.Submartingale f ℱ μ) (hg : MeasureTheory.Submartingale g ℱ μ)
Goal: MeasureTheory.Submartingale (f ⊔ g) ℱ μ
-- MeasureTheory.Submartingale.pos
protected theorem pos {f : ι → Ω → ℝ} (hf : Submartingale f ℱ μ) : Submartingale (f⁺) ℱ μ 
Arguments: {Ω : Type u_1} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ι m0} {f : ι → Ω → ℝ} (hf : MeasureTheory.Submartingale f ℱ μ)
Goal: MeasureTheory.Submartingale f⁺ ℱ μ
-- MeasureTheory.submartingale_of_setIntegral_le
theorem submartingale_of_setIntegral_le [IsFiniteMeasure μ] {f : ι → Ω → ℝ} (hadp : Adapted ℱ f)
    (hint : ∀ i, Integrable (f i) μ) (hf : ∀ i j : ι,
      i ≤ j → ∀ s : Set Ω, MeasurableSet[ℱ i] s → ∫ ω in s, f i ω ∂μ ≤ ∫ ω in s, f j ω ∂μ) :
    Submartingale f ℱ μ 
Arguments: {Ω : Type u_1} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ι m0} [MeasureTheory.IsFiniteMeasure μ] {f : ι → Ω → ℝ} (hadp : MeasureTheory.Adapted ℱ f) (hint : ∀ (i : ι), MeasureTheory.Integrable (f i) μ) (hf : ∀ (i j : ι), i ≤ j → ∀ (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f j ω ∂μ)
Goal: MeasureTheory.Submartingale f ℱ μ
-- MeasureTheory.submartingale_of_condexp_sub_nonneg
theorem submartingale_of_condexp_sub_nonneg [IsFiniteMeasure μ] {f : ι → Ω → 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 
Arguments: {Ω : Type u_1} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ι m0} [MeasureTheory.IsFiniteMeasure μ] {f : ι → Ω → ℝ} (hadp : MeasureTheory.Adapted ℱ f) (hint : ∀ (i : ι), MeasureTheory.Integrable (f i) μ) (hf : ∀ (i j : ι), i ≤ j → 0 ≤ᵐ[μ] μ[f j - f i|↑ℱ i])
Goal: MeasureTheory.Submartingale f ℱ μ
-- MeasureTheory.Submartingale.condexp_sub_nonneg
theorem Submartingale.condexp_sub_nonneg {f : ι → Ω → ℝ} (hf : Submartingale f ℱ μ) {i j : ι}
    (hij : i ≤ j) : 0 ≤ᵐ[μ] μ[f j - f i|ℱ i] 
Arguments: {Ω : Type u_1} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ι m0} {f : ι → Ω → ℝ} (hf : MeasureTheory.Submartingale f ℱ μ) {i j : ι} (hij : i ≤ j)
Goal: 0 ≤ᵐ[μ] μ[f j - f i|↑ℱ i]
-- MeasureTheory.submartingale_iff_condexp_sub_nonneg
theorem submartingale_iff_condexp_sub_nonneg [IsFiniteMeasure μ] {f : ι → Ω → 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] 
Arguments: {Ω : Type u_1} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ι m0} [MeasureTheory.IsFiniteMeasure μ] {f : ι → Ω → ℝ}
Goal: MeasureTheory.Submartingale f ℱ μ ↔
  MeasureTheory.Adapted ℱ f ∧
    (∀ (i : ι), MeasureTheory.Integrable (f i) μ) ∧ ∀ (i j : ι), i ≤ j → 0 ≤ᵐ[μ] μ[f j - f i|↑ℱ i]
-- MeasureTheory.Supermartingale.sub_submartingale
theorem sub_submartingale [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)]
    (hf : Supermartingale f ℱ μ) (hg : Submartingale g ℱ μ) : Supermartingale (f - g) ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f g : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [Preorder E] [CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hf : MeasureTheory.Supermartingale f ℱ μ) (hg : MeasureTheory.Submartingale g ℱ μ)
Goal: MeasureTheory.Supermartingale (f - g) ℱ μ
-- MeasureTheory.Supermartingale.sub_martingale
theorem sub_martingale [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)]
    (hf : Supermartingale f ℱ μ) (hg : Martingale g ℱ μ) : Supermartingale (f - g) ℱ μ 
Arguments: {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f g : ι → Ω → E} {ℱ : MeasureTheory.Filtration ι m0} [Preorder E] [CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (hf : MeasureTheory.Supermartingale f ℱ μ) (hg : MeasureTheory.Martingale g ℱ μ)
Goal: MeasureTheory.Supermartingale (f - g) ℱ μ
-- MeasureTheory.Supermartingale.smul_nonneg
theorem smul_nonneg {f : ι → Ω → F} {c : ℝ} (hc : 0 ≤ c) (hf : Supermartingale f ℱ μ) :
    Supermartingale (c • f) ℱ μ 
Arguments: {Ω : Type u_1} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ι m0} {F : Type u_4} [NormedLatticeAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] [OrderedSMul ℝ F] {f : ι → Ω → F} {c : ℝ} (hc : 0 ≤ c) (hf : MeasureTheory.Supermartingale f ℱ μ)
Goal: MeasureTheory.Supermartingale (c • f) ℱ μ
-- MeasureTheory.Supermartingale.smul_nonpos
theorem smul_nonpos {f : ι → Ω → F} {c : ℝ} (hc : c ≤ 0) (hf : Supermartingale f ℱ μ) :
    Submartingale (c • f) ℱ μ 
Arguments: {Ω : Type u_1} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ι m0} {F : Type u_4} [NormedLatticeAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] [OrderedSMul ℝ F] {f : ι → Ω → F} {c : ℝ} (hc : c ≤ 0) (hf : MeasureTheory.Supermartingale f ℱ μ)
Goal: MeasureTheory.Submartingale (c • f) ℱ μ
-- MeasureTheory.Submartingale.smul_nonneg
theorem smul_nonneg {f : ι → Ω → F} {c : ℝ} (hc : 0 ≤ c) (hf : Submartingale f ℱ μ) :
    Submartingale (c • f) ℱ μ 
Arguments: {Ω : Type u_1} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ι m0} {F : Type u_4} [NormedLatticeAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] [OrderedSMul ℝ F] {f : ι → Ω → F} {c : ℝ} (hc : 0 ≤ c) (hf : MeasureTheory.Submartingale f ℱ μ)
Goal: MeasureTheory.Submartingale (c • f) ℱ μ
-- MeasureTheory.Submartingale.smul_nonpos
theorem smul_nonpos {f : ι → Ω → F} {c : ℝ} (hc : c ≤ 0) (hf : Submartingale f ℱ μ) :
    Supermartingale (c • f) ℱ μ 
Arguments: {Ω : Type u_1} {ι : Type u_3} [Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ι m0} {F : Type u_4} [NormedLatticeAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] [OrderedSMul ℝ F] {f : ι → Ω → F} {c : ℝ} (hc : c ≤ 0) (hf : MeasureTheory.Submartingale f ℱ μ)
Goal: MeasureTheory.Supermartingale (c • f) ℱ μ
-- MeasureTheory.submartingale_of_setIntegral_le_succ
theorem submartingale_of_setIntegral_le_succ [IsFiniteMeasure μ] {f : ℕ → Ω → ℝ}
    (hadp : Adapted 𝒢 f) (hint : ∀ i, Integrable (f i) μ)
    (hf : ∀ i, ∀ s : Set Ω, MeasurableSet[𝒢 i] s → ∫ ω in s, f i ω ∂μ ≤ ∫ ω in s, f (i + 1) ω ∂μ) :
    Submartingale f 𝒢 μ 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {𝒢 : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {f : ℕ → Ω → ℝ} (hadp : MeasureTheory.Adapted 𝒢 f) (hint : ∀ (i : ℕ), MeasureTheory.Integrable (f i) μ) (hf : ∀ (i : ℕ) (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f (i + 1) ω ∂μ)
Goal: MeasureTheory.Submartingale f 𝒢 μ
-- MeasureTheory.supermartingale_of_setIntegral_succ_le
theorem supermartingale_of_setIntegral_succ_le [IsFiniteMeasure μ] {f : ℕ → Ω → ℝ}
    (hadp : Adapted 𝒢 f) (hint : ∀ i, Integrable (f i) μ)
    (hf : ∀ i, ∀ s : Set Ω, MeasurableSet[𝒢 i] s → ∫ ω in s, f (i + 1) ω ∂μ ≤ ∫ ω in s, f i ω ∂μ) :
    Supermartingale f 𝒢 μ 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {𝒢 : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {f : ℕ → Ω → ℝ} (hadp : MeasureTheory.Adapted 𝒢 f) (hint : ∀ (i : ℕ), MeasureTheory.Integrable (f i) μ) (hf : ∀ (i : ℕ) (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f (i + 1) ω ∂μ ≤ ∫ (ω : Ω) in s, f i ω ∂μ)
Goal: MeasureTheory.Supermartingale f 𝒢 μ
-- MeasureTheory.martingale_of_setIntegral_eq_succ
theorem martingale_of_setIntegral_eq_succ [IsFiniteMeasure μ] {f : ℕ → Ω → ℝ} (hadp : Adapted 𝒢 f)
    (hint : ∀ i, Integrable (f i) μ)
    (hf : ∀ i, ∀ s : Set Ω, MeasurableSet[𝒢 i] s → ∫ ω in s, f i ω ∂μ = ∫ ω in s, f (i + 1) ω ∂μ) :
    Martingale f 𝒢 μ 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {𝒢 : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {f : ℕ → Ω → ℝ} (hadp : MeasureTheory.Adapted 𝒢 f) (hint : ∀ (i : ℕ), MeasureTheory.Integrable (f i) μ) (hf : ∀ (i : ℕ) (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ = ∫ (ω : Ω) in s, f (i + 1) ω ∂μ)
Goal: MeasureTheory.Martingale f 𝒢 μ
-- MeasureTheory.submartingale_nat
theorem submartingale_nat [IsFiniteMeasure μ] {f : ℕ → Ω → 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 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {𝒢 : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {f : ℕ → Ω → ℝ} (hadp : MeasureTheory.Adapted 𝒢 f) (hint : ∀ (i : ℕ), MeasureTheory.Integrable (f i) μ) (hf : ∀ (i : ℕ), f i ≤ᵐ[μ] μ[f (i + 1)|↑𝒢 i])
Goal: MeasureTheory.Submartingale f 𝒢 μ
-- MeasureTheory.supermartingale_nat
theorem supermartingale_nat [IsFiniteMeasure μ] {f : ℕ → Ω → ℝ} (hadp : Adapted 𝒢 f)
    (hint : ∀ i, Integrable (f i) μ) (hf : ∀ i, μ[f (i + 1)|𝒢 i] ≤ᵐ[μ] f i) :
    Supermartingale f 𝒢 μ 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {𝒢 : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {f : ℕ → Ω → ℝ} (hadp : MeasureTheory.Adapted 𝒢 f) (hint : ∀ (i : ℕ), MeasureTheory.Integrable (f i) μ) (hf : ∀ (i : ℕ), μ[f (i + 1)|↑𝒢 i] ≤ᵐ[μ] f i)
Goal: MeasureTheory.Supermartingale f 𝒢 μ