Theory Stopping_Time

theory Stopping_Time
imports Analysis
(* Author: Johannes Hölzl <hoelzl@in.tum.de> *)

section {* Stopping times *}

theory Stopping_Time
  imports "HOL-Analysis.Analysis"
begin

subsection ‹Stopping Time›

text ‹This is also called strong stopping time. Then stopping time is T with alternative is
  ‹T x < t› measurable.›

definition stopping_time :: "('t::linorder ⇒ 'a measure) ⇒ ('a ⇒ 't) ⇒ bool"
where
  "stopping_time F T = (∀t. Measurable.pred (F t) (λx. T x ≤ t))"

lemma stopping_time_cong: "(⋀t x. x ∈ space (F t) ⟹ T x = S x) ⟹ stopping_time F T = stopping_time F S"
  unfolding stopping_time_def by (intro arg_cong[where f=All] ext measurable_cong) simp

lemma stopping_timeD: "stopping_time F T ⟹ Measurable.pred (F t) (λx. T x ≤ t)"
  by (auto simp: stopping_time_def)

lemma stopping_timeD2: "stopping_time F T ⟹ Measurable.pred (F t) (λx. t < T x)"
  unfolding not_le[symmetric] by (auto intro: stopping_timeD Measurable.pred_intros_logic)

lemma stopping_timeI[intro?]: "(⋀t. Measurable.pred (F t) (λx. T x ≤ t)) ⟹ stopping_time F T"
  by (auto simp: stopping_time_def)

lemma measurable_stopping_time:
  fixes T :: "'a ⇒ 't::{linorder_topology, second_countable_topology}"
  assumes T: "stopping_time F T"
    and M: "⋀t. sets (F t) ⊆ sets M" "⋀t. space (F t) = space M"
  shows "T ∈ M →M borel"
proof (rule borel_measurableI_le)
  show "{x ∈ space M. T x ≤ t} ∈ sets M" for t
    using stopping_timeD[OF T] M by (auto simp: Measurable.pred_def)
qed

lemma stopping_time_const: "stopping_time F (λx. c)"
  by (auto simp: stopping_time_def)

lemma stopping_time_min:
  "stopping_time F T ⟹ stopping_time F S ⟹ stopping_time F (λx. min (T x) (S x))"
  by (auto simp: stopping_time_def min_le_iff_disj intro!: pred_intros_logic)

lemma stopping_time_max:
  "stopping_time F T ⟹ stopping_time F S ⟹ stopping_time F (λx. max (T x) (S x))"
  by (auto simp: stopping_time_def intro!: pred_intros_logic)

section ‹Filtration›

locale filtration =
  fixes Ω :: "'a set" and F :: "'t::{linorder_topology, second_countable_topology} ⇒ 'a measure"
  assumes space_F: "⋀i. space (F i) = Ω"
  assumes sets_F_mono: "⋀i j. i ≤ j ⟹ sets (F i) ≤ sets (F j)"
begin

subsection ‹$\sigma$-algebra of a Stopping Time›

definition pre_sigma :: "('a ⇒ 't) ⇒ 'a measure"
where
  "pre_sigma T = sigma Ω {A. ∀t. {ω∈A. T ω ≤ t} ∈ sets (F t)}"

lemma space_pre_sigma: "space (pre_sigma T) = Ω"
  unfolding pre_sigma_def using sets.space_closed[of "F _"]
  by (intro space_measure_of) (auto simp: space_F)

lemma measure_pre_sigma[simp]: "emeasure (pre_sigma T) = (λ_. 0)"
  by (simp add: pre_sigma_def emeasure_sigma)

lemma sigma_algebra_pre_sigma:
  assumes T: "stopping_time F T"
  shows "sigma_algebra Ω {A. ∀t. {ω∈A. T ω ≤ t} ∈ sets (F t)}"
  unfolding sigma_algebra_iff2
proof (intro sigma_algebra_iff2[THEN iffD2] conjI ballI allI impI CollectI)
  show "{A. ∀t. {ω ∈ A. T ω ≤ t} ∈ sets (F t)} ⊆ Pow Ω"
    using sets.space_closed[of "F _"] by (auto simp: space_F)
next
  fix A t assume "A ∈ {A. ∀t. {ω ∈ A. T ω ≤ t} ∈ sets (F t)}"
  then have "{ω ∈ space (F t). T ω ≤ t} - {ω ∈ A. T ω ≤ t} ∈ sets (F t)"
    using T stopping_timeD[measurable] by auto
  also have "{ω ∈ space (F t). T ω ≤ t} - {ω ∈ A. T ω ≤ t} = {ω ∈ Ω - A. T ω ≤ t}"
    by (auto simp: space_F)
  finally show "{ω ∈ Ω - A. T ω ≤ t} ∈ sets (F t)" .
next
  fix AA :: "nat ⇒ 'a set" and t assume "range AA ⊆ {A. ∀t. {ω ∈ A. T ω ≤ t} ∈ sets (F t)}"
  then have "(⋃i. {ω ∈ AA i. T ω ≤ t}) ∈ sets (F t)" for t
    by auto
  also have "(⋃i. {ω ∈ AA i. T ω ≤ t}) = {ω ∈ UNION UNIV AA. T ω ≤ t}"
    by auto
  finally show "{ω ∈ UNION UNIV AA. T ω ≤ t} ∈ sets (F t)" .
qed auto

lemma sets_pre_sigma: "stopping_time F T ⟹ sets (pre_sigma T) = {A. ∀t. {ω∈A. T ω ≤ t} ∈ sets (F t)}"
  unfolding pre_sigma_def by (rule sigma_algebra.sets_measure_of_eq[OF sigma_algebra_pre_sigma])

lemma sets_pre_sigmaI: "stopping_time F T ⟹ (⋀t. {ω∈A. T ω ≤ t} ∈ sets (F t)) ⟹ A ∈ sets (pre_sigma T)"
  unfolding sets_pre_sigma by auto

lemma pred_pre_sigmaI:
  assumes T: "stopping_time F T"
  shows "(⋀t. Measurable.pred (F t) (λω. P ω ∧ T ω ≤ t)) ⟹ Measurable.pred (pre_sigma T) P"
  unfolding pred_def space_F space_pre_sigma by (intro sets_pre_sigmaI[OF T]) simp

lemma sets_pre_sigmaD: "stopping_time F T ⟹ A ∈ sets (pre_sigma T) ⟹ {ω∈A. T ω ≤ t} ∈ sets (F t)"
  unfolding sets_pre_sigma by auto

lemma stopping_time_le_const: "stopping_time F T ⟹ s ≤ t ⟹ Measurable.pred (F t) (λω. T ω ≤ s)"
  using stopping_timeD[of F T] sets_F_mono[of _ t] by (auto simp: pred_def space_F)

lemma measurable_stopping_time_pre_sigma:
  assumes T: "stopping_time F T" shows "T ∈ pre_sigma T →M borel"
proof (intro borel_measurableI_le sets_pre_sigmaI[OF T])
  fix t t'
  have "{ω∈space (F (min t' t)). T ω ≤ min t' t} ∈ sets (F (min t' t))"
    using T unfolding pred_def[symmetric] by (rule stopping_timeD)
  also have "… ⊆ sets (F t)"
    by (rule sets_F_mono) simp
  finally show "{ω ∈ {x ∈ space (pre_sigma T). T x ≤ t'}. T ω ≤ t} ∈ sets (F t)"
    by (simp add: space_pre_sigma space_F)
qed

lemma mono_pre_sigma:
  assumes T: "stopping_time F T" and S: "stopping_time F S"
    and le: "⋀ω. ω ∈ Ω ⟹ T ω ≤ S ω"
  shows "sets (pre_sigma T) ⊆ sets (pre_sigma S)"
  unfolding sets_pre_sigma[OF S] sets_pre_sigma[OF T]
proof safe
  interpret sigma_algebra Ω "{A. ∀t. {ω∈A. T ω ≤ t} ∈ sets (F t)}"
    using T by (rule sigma_algebra_pre_sigma)
  fix A t assume A: "∀t. {ω∈A. T ω ≤ t} ∈ sets (F t)"
  then have "A ⊆ Ω"
    using sets_into_space by auto
  from A have "{ω∈A. T ω ≤ t} ∩ {ω∈space (F t). S ω ≤ t} ∈ sets (F t)"
    using stopping_timeD[OF S] by (auto simp: pred_def)
  also have "{ω∈A. T ω ≤ t} ∩ {ω∈space (F t). S ω ≤ t} = {ω∈A. S ω ≤ t}"
    using ‹A ⊆ Ω› sets_into_space[of A] le by (auto simp: space_F intro: order_trans)
  finally show "{ω∈A. S ω ≤ t} ∈ sets (F t)"
    by auto
qed

lemma stopping_time_less_const:
  assumes T: "stopping_time F T" shows "Measurable.pred (F t) (λω. T ω < t)"
proof -
  guess D :: "'t set" by (rule countable_dense_setE)
  note D = this
  show ?thesis
  proof cases
    assume *: "∀t'<t. ∃t''. t' < t'' ∧ t'' < t"
    { fix t' assume "t' < t"
      with * have "{t' <..< t} ≠ {}"
        by fastforce
      with D(2)[OF _ this]
      have "∃d∈D. t'< d ∧ d < t"
        by auto }
    note ** = this

    show ?thesis
    proof (rule measurable_cong[THEN iffD2])
      show "T ω < t ⟷ (∃r∈{r∈D. r < t}. T ω ≤ r)" for ω
        by (auto dest: ** intro: less_imp_le)
      show "Measurable.pred (F t) (λw. ∃r∈{r ∈ D. r < t}. T w ≤ r)"
        by (intro measurable_pred_countable stopping_time_le_const[OF T] countable_Collect D) auto
    qed
  next
    assume "¬ (∀t'<t. ∃t''. t' < t'' ∧ t'' < t)"
    then obtain t' where t': "t' < t" "⋀t''. t'' < t ⟹ t'' ≤ t'"
      by (auto simp: not_less[symmetric])
    show ?thesis
    proof (rule measurable_cong[THEN iffD2])
      show "T ω < t ⟷ T ω ≤ t'" for ω
        using t' by auto
      show "Measurable.pred (F t) (λw. T w ≤ t')"
        using ‹t'<t› by (intro stopping_time_le_const[OF T]) auto
    qed
  qed
qed

lemma stopping_time_eq_const: "stopping_time F T ⟹ Measurable.pred (F t) (λω. T ω = t)"
  unfolding eq_iff using stopping_time_less_const[of T t]
  by (intro pred_intros_logic stopping_time_le_const) (auto simp: not_less[symmetric] )

lemma stopping_time_less:
  assumes T: "stopping_time F T" and S: "stopping_time F S"
  shows "Measurable.pred (pre_sigma T) (λω. T ω < S ω)"
proof (rule pred_pre_sigmaI[OF T])
  fix t
  obtain D :: "'t set"
    where [simp]: "countable D" and semidense_D: "⋀x y. x < y ⟹ (∃b∈D. x ≤ b ∧ b < y)"
    using countable_separating_set_linorder2 by auto
  show "Measurable.pred (F t) (λω. T ω < S ω ∧ T ω ≤ t)"
  proof (rule measurable_cong[THEN iffD2])
    let ?f = "λω. if T ω = t then ¬ S ω ≤ t else ∃s∈{s∈D. s ≤ t}. T ω ≤ s ∧ ¬ (S ω ≤ s)"
    { fix ω assume "T ω ≤ t" "T ω ≠ t" "T ω < S ω"
      then have "T ω < min t (S ω)"
        by auto
      then obtain r where "r ∈ D" "T ω ≤ r" "r < min t (S ω)"
        by (metis semidense_D)
      then have "∃s∈{s∈D. s ≤ t}. T ω ≤ s ∧ s < S ω"
        by auto }
    then show "(T ω < S ω ∧ T ω ≤ t) = ?f ω" for ω
      by (auto simp: not_le)
    show "Measurable.pred (F t) ?f"
      by (intro pred_intros_logic measurable_If measurable_pred_countable countable_Collect
                stopping_time_le_const predE stopping_time_eq_const T S)
         auto
  qed
qed

end

lemma stopping_time_SUP_enat:
  fixes T :: "nat ⇒ ('a ⇒ enat)"
  shows "(⋀i. stopping_time F (T i)) ⟹ stopping_time F (SUP i. T i)"
  unfolding stopping_time_def SUP_apply SUP_le_iff by (auto intro!: pred_intros_countable)

lemma less_eSuc_iff: "a < eSuc b ⟷ (a ≤ b ∧ a ≠ ∞)"
  by (cases a) auto

lemma stopping_time_Inf_enat:
  fixes F :: "enat ⇒ 'a measure"
  assumes F: "filtration Ω F"
  assumes P: "⋀i. Measurable.pred (F i) (P i)"
  shows "stopping_time F (λω. Inf {i. P i ω})"
proof (rule stopping_timeI, cases)
  fix t :: enat assume "t = ∞" then show "Measurable.pred (F t) (λω. Inf {i. P i ω} ≤ t)"
    by auto
next
  fix t :: enat assume "t ≠ ∞"
  moreover
  { fix i ω assume "Inf {i. P i ω} ≤ t"
    with ‹t ≠ ∞› have "(∃i≤t. P i ω)"
      unfolding Inf_le_iff by (cases t) (auto elim!: allE[of _ "eSuc t"] simp: less_eSuc_iff) }
  ultimately have *: "⋀ω. Inf {i. P i ω} ≤ t ⟷ (∃i∈{..t}. P i ω)"
    by (auto intro!: Inf_lower2)
  show "Measurable.pred (F t) (λω. Inf {i. P i ω} ≤ t)"
    unfolding * using filtration.sets_F_mono[OF F, of _ t] P
    by (intro pred_intros_countable_bounded) (auto simp: pred_def filtration.space_F[OF F])
qed

lemma stopping_time_Inf_nat:
  fixes F :: "nat ⇒ 'a measure"
  assumes F: "filtration Ω F"
  assumes P: "⋀i. Measurable.pred (F i) (P i)" and wf: "⋀i ω. ω ∈ Ω ⟹ ∃n. P n ω"
  shows "stopping_time F (λω. Inf {i. P i ω})"
  unfolding stopping_time_def
proof (intro allI, subst measurable_cong)
  fix t ω assume "ω ∈ space (F t)"
  then have "ω ∈ Ω"
    using filtration.space_F[OF F] by auto
  from wf[OF this] have "((LEAST n. P n ω) ≤ t) = (∃i≤t. P i ω)"
    by (rule LeastI2_wellorder_ex) auto
  then show "(Inf {i. P i ω} ≤ t) = (∃i∈{..t}. P i ω)"
    by (simp add: Inf_nat_def Bex_def)
next
  fix t from P show "Measurable.pred (F t) (λw. ∃i∈{..t}. P i w)"
    using filtration.sets_F_mono[OF F, of _ t]
    by (intro pred_intros_countable_bounded) (auto simp: pred_def filtration.space_F[OF F])
qed

end