Tests.StoppingTime

import AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingTime

namespace AutoSamplingTheory.Tests.StoppingTime

open MeasureTheory
open AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingTime
open scoped NNReal

#check AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingTime.IsChewiStoppingTime
#check IsChewiStoppingTime.min
#check IsChewiStoppingTime.min_const
#check stoppedProcess_stoppedProcess_inf
#check stoppedProcess_stoppedProcess_of_le

example {Ω : Type*} {m : MeasurableSpace Ω}
    (filtration : Filtration ℝ≥0 m) (t : ℝ≥0) :
    IsChewiStoppingTime filtration (fun _ : Ω => (t : WithTop ℝ≥0)) :=
  isChewiStoppingTime_const filtration t

example {Ω : Type*} {m : MeasurableSpace Ω}
    {filtration : Filtration ℝ≥0 m} {τ σ : Ω → WithTop ℝ≥0}
    (hτ : IsChewiStoppingTime filtration τ)
    (hσ : IsChewiStoppingTime filtration σ) :
    IsChewiStoppingTime filtration (τ ⊓ σ) :=
  hτ.min hσ

example {Ω : Type*} {m : MeasurableSpace Ω}
    {filtration : Filtration ℝ≥0 m} {τ : Ω → WithTop ℝ≥0}
    (hτ : IsChewiStoppingTime filtration τ) (T : ℝ≥0) :
    IsChewiStoppingTime filtration
      (τ ⊓ fun _ => (T : WithTop ℝ≥0)) :=
  hτ.min_const T

example {Ω β : Type*} (u : ℝ≥0 → Ω → β)
    (τ σ : Ω → WithTop ℝ≥0) :
    stoppedProcess (stoppedProcess u τ) σ =
      stoppedProcess u (σ ⊓ τ) :=
  stoppedProcess_stoppedProcess_inf u τ σ

example {Ω β : Type*} (u : ℝ≥0 → Ω → β)
    {τ σ : Ω → WithTop ℝ≥0} (hστ : σ ≤ τ) :
    stoppedProcess (stoppedProcess u τ) σ = stoppedProcess u σ :=
  stoppedProcess_stoppedProcess_of_le u hστ

end AutoSamplingTheory.Tests.StoppingTime