Theory SPMF

(* Author: Andreas Lochbihler, ETH Zurich *)

section ‹Discrete subprobability distribution›

theory SPMF imports
  Probability_Mass_Function
  "HOL-Library.Complete_Partial_Order2"
  "HOL-Library.Rewrite"
begin

subsection ‹Auxiliary material›

lemma cSUP_singleton [simp]: "(SUP x{x}. f x :: _ :: conditionally_complete_lattice) = f x"
by (metis cSup_singleton image_empty image_insert)

subsubsection ‹More about extended reals›

lemma [simp]:
  shows ennreal_max_0: "ennreal (max 0 x) = ennreal x"
  and ennreal_max_0': "ennreal (max x 0) = ennreal x"
by(simp_all add: max_def ennreal_eq_0_iff)

lemma e2ennreal_0 [simp]: "e2ennreal 0 = 0"
by(simp add: zero_ennreal_def)

lemma enn2real_bot [simp]: "enn2real  = 0"
by(simp add: bot_ennreal_def)

lemma continuous_at_ennreal[continuous_intros]: "continuous F f  continuous F (λx. ennreal (f x))"
  unfolding continuous_def by auto

lemma ennreal_Sup:
  assumes *: "(SUP aA. ennreal a)  "
  and "A  {}"
  shows "ennreal (Sup A) = (SUP aA. ennreal a)"
proof (rule continuous_at_Sup_mono)
  obtain r where r: "ennreal r = (SUP aA. ennreal a)" "r  0"
    using * by(cases "(SUP aA. ennreal a)") simp_all
  then show "bdd_above A"
    by(auto intro!: SUP_upper bdd_aboveI[of _ r] simp add: ennreal_le_iff[symmetric])
qed (auto simp: mono_def continuous_at_imp_continuous_at_within continuous_at_ennreal ennreal_leI assms)

lemma ennreal_SUP:
  " (SUP aA. ennreal (f a))  ; A  {}   ennreal (SUP aA. f a) = (SUP aA. ennreal (f a))"
using ennreal_Sup[of "f ` A"] by (auto simp add: image_comp)

lemma ennreal_lt_0: "x < 0  ennreal x = 0"
by(simp add: ennreal_eq_0_iff)

subsubsection ‹More about typ'a option

lemma None_in_map_option_image [simp]: "None  map_option f ` A  None  A"
by auto

lemma Some_in_map_option_image [simp]: "Some x  map_option f ` A  (y. x = f y  Some y  A)"
by(auto intro: rev_image_eqI dest: sym)

lemma case_option_collapse: "case_option x (λ_. x) = (λ_. x)"
by(simp add: fun_eq_iff split: option.split)

lemma case_option_id: "case_option None Some = id"
by(rule ext)(simp split: option.split)

inductive ord_option :: "('a  'b  bool)  'a option  'b option  bool"
  for ord :: "'a  'b  bool"
where
  None: "ord_option ord None x"
| Some: "ord x y  ord_option ord (Some x) (Some y)"

inductive_simps ord_option_simps [simp]:
  "ord_option ord None x"
  "ord_option ord x None"
  "ord_option ord (Some x) (Some y)"
  "ord_option ord (Some x) None"

inductive_simps ord_option_eq_simps [simp]:
  "ord_option (=) None y"
  "ord_option (=) (Some x) y"

lemma ord_option_reflI: "(y. y  set_option x  ord y y)  ord_option ord x x"
by(cases x) simp_all

lemma reflp_ord_option: "reflp ord  reflp (ord_option ord)"
by(simp add: reflp_def ord_option_reflI)

lemma ord_option_trans:
  " ord_option ord x y; ord_option ord y z;
    a b c.  a  set_option x; b  set_option y; c  set_option z; ord a b; ord b c   ord a c 
   ord_option ord x z"
by(auto elim!: ord_option.cases)

lemma transp_ord_option: "transp ord  transp (ord_option ord)"
unfolding transp_def by(blast intro: ord_option_trans)

lemma antisymp_ord_option: "antisymp ord  antisymp (ord_option ord)"
by(auto intro!: antisympI elim!: ord_option.cases dest: antisympD)

lemma ord_option_chainD:
  "Complete_Partial_Order.chain (ord_option ord) Y
   Complete_Partial_Order.chain ord {x. Some x  Y}"
by(rule chainI)(auto dest: chainD)

definition lub_option :: "('a set  'b)  'a option set  'b option"
where "lub_option lub Y = (if Y  {None} then None else Some (lub {x. Some x  Y}))"

lemma map_lub_option: "map_option f (lub_option lub Y) = lub_option (f  lub) Y"
by(simp add: lub_option_def)

lemma lub_option_upper:
  assumes "Complete_Partial_Order.chain (ord_option ord) Y" "x  Y"
  and lub_upper: "Y x.  Complete_Partial_Order.chain ord Y; x  Y   ord x (lub Y)"
  shows "ord_option ord x (lub_option lub Y)"
using assms(1-2)
by(cases x)(auto simp add: lub_option_def intro: lub_upper[OF ord_option_chainD])

lemma lub_option_least:
  assumes Y: "Complete_Partial_Order.chain (ord_option ord) Y"
  and upper: "x. x  Y  ord_option ord x y"
  assumes lub_least: "Y y.  Complete_Partial_Order.chain ord Y; x. x  Y  ord x y   ord (lub Y) y"
  shows "ord_option ord (lub_option lub Y) y"
using Y
by(cases y)(auto 4 3 simp add: lub_option_def intro: lub_least[OF ord_option_chainD] dest: upper)

lemma lub_map_option: "lub_option lub (map_option f ` Y) = lub_option (lub  (`) f) Y"
apply(auto simp add: lub_option_def)
apply(erule notE)
apply(rule arg_cong[where f=lub])
apply(auto intro: rev_image_eqI dest: sym)
done

lemma ord_option_mono: " ord_option A x y; x y. A x y  B x y   ord_option B x y"
by(auto elim: ord_option.cases)

lemma ord_option_mono' [mono]:
  "(x y. A x y  B x y)  ord_option A x y  ord_option B x y"
by(blast intro: ord_option_mono)

lemma ord_option_compp: "ord_option (A OO B) = ord_option A OO ord_option B"
by(auto simp add: fun_eq_iff elim!: ord_option.cases intro: ord_option.intros)

lemma ord_option_inf: "inf (ord_option A) (ord_option B) = ord_option (inf A B)" (is "?lhs = ?rhs")
proof(rule antisym)
  show "?lhs  ?rhs" by(auto elim!: ord_option.cases)
qed(auto elim: ord_option_mono)

lemma ord_option_map2: "ord_option ord x (map_option f y) = ord_option (λx y. ord x (f y)) x y"
by(auto elim: ord_option.cases)

lemma ord_option_map1: "ord_option ord (map_option f x) y = ord_option (λx y. ord (f x) y) x y"
by(auto elim: ord_option.cases)

lemma option_ord_Some1_iff: "option_ord (Some x) y  y = Some x"
by(auto simp add: flat_ord_def)

subsubsection ‹A relator for sets that treats sets like predicates›

context includes lifting_syntax
begin

definition rel_pred :: "('a  'b  bool)  'a set  'b set  bool"
where "rel_pred R A B = (R ===> (=)) (λx. x  A) (λy. y  B)"

lemma rel_predI: "(R ===> (=)) (λx. x  A) (λy. y  B)  rel_pred R A B"
by(simp add: rel_pred_def)

lemma rel_predD: " rel_pred R A B; R x y   x  A  y  B"
by(simp add: rel_pred_def rel_fun_def)

lemma Collect_parametric: "((A ===> (=)) ===> rel_pred A) Collect Collect"
  ― ‹Declare this rule as @{attribute transfer_rule} only locally
      because it blows up the search space for @{method transfer}
      (in combination with @{thm [source] Collect_transfer})›
by(simp add: rel_funI rel_predI)

end

subsubsection ‹Monotonicity rules›

lemma monotone_gfp_eadd1: "monotone (≥) (≥) (λx. x + y :: enat)"
by(auto intro!: monotoneI)

lemma monotone_gfp_eadd2: "monotone (≥) (≥) (λy. x + y :: enat)"
by(auto intro!: monotoneI)

lemma mono2mono_gfp_eadd[THEN gfp.mono2mono2, cont_intro, simp]:
  shows monotone_eadd: "monotone (rel_prod (≥) (≥)) (≥) (λ(x, y). x + y :: enat)"
by(simp add: monotone_gfp_eadd1 monotone_gfp_eadd2)

lemma eadd_gfp_partial_function_mono [partial_function_mono]:
  " monotone (fun_ord (≥)) (≥) f; monotone (fun_ord (≥)) (≥) g 
   monotone (fun_ord (≥)) (≥) (λx. f x + g x :: enat)"
by(rule mono2mono_gfp_eadd)

lemma mono2mono_ereal[THEN lfp.mono2mono]:
  shows monotone_ereal: "monotone (≤) (≤) ereal"
by(rule monotoneI) simp

lemma mono2mono_ennreal[THEN lfp.mono2mono]:
  shows monotone_ennreal: "monotone (≤) (≤) ennreal"
by(rule monotoneI)(simp add: ennreal_leI)

subsubsection ‹Bijections›

lemma bi_unique_rel_set_bij_betw:
  assumes unique: "bi_unique R"
  and rel: "rel_set R A B"
  shows "f. bij_betw f A B  (xA. R x (f x))"
proof -
  from assms obtain f where f: "x. x  A  R x (f x)" and B: "x. x  A  f x  B"
    apply(atomize_elim)
    apply(fold all_conj_distrib)
    apply(subst choice_iff[symmetric])
    apply(auto dest: rel_setD1)
    done
  have "inj_on f A" by(rule inj_onI)(auto dest!: f dest: bi_uniqueDl[OF unique])
  moreover have "f ` A = B" using rel
    by(auto 4 3 intro: B dest: rel_setD2 f bi_uniqueDr[OF unique])
  ultimately have "bij_betw f A B" unfolding bij_betw_def ..
  thus ?thesis using f by blast
qed

lemma bij_betw_rel_setD: "bij_betw f A B  rel_set (λx y. y = f x) A B"
by(rule rel_setI)(auto dest: bij_betwE bij_betw_imp_surj_on[symmetric])

subsection ‹Subprobability mass function›

type_synonym 'a spmf = "'a option pmf"
translations (type) "'a spmf"  (type) "'a option pmf"

definition measure_spmf :: "'a spmf  'a measure"
where "measure_spmf p = distr (restrict_space (measure_pmf p) (range Some)) (count_space UNIV) the"

abbreviation spmf :: "'a spmf  'a  real"
where "spmf p x  pmf p (Some x)"

lemma space_measure_spmf: "space (measure_spmf p) = UNIV"
by(simp add: measure_spmf_def)

lemma sets_measure_spmf [simp, measurable_cong]: "sets (measure_spmf p) = sets (count_space UNIV)"
by(simp add: measure_spmf_def)

lemma measure_spmf_not_bot [simp]: "measure_spmf p  "
proof
  assume "measure_spmf p = "
  hence "space (measure_spmf p) = space " by simp
  thus False by(simp add: space_measure_spmf)
qed

lemma measurable_the_measure_pmf_Some [measurable, simp]:
  "the  measurable (restrict_space (measure_pmf p) (range Some)) (count_space UNIV)"
by(auto simp add: measurable_def sets_restrict_space space_restrict_space integral_restrict_space)

lemma measurable_spmf_measure1[simp]: "measurable (measure_spmf M) N = UNIV  space N"
by(auto simp: measurable_def space_measure_spmf)

lemma measurable_spmf_measure2[simp]: "measurable N (measure_spmf M) = measurable N (count_space UNIV)"
by(intro measurable_cong_sets) simp_all

lemma subprob_space_measure_spmf [simp, intro!]: "subprob_space (measure_spmf p)"
proof
  show "emeasure (measure_spmf p) (space (measure_spmf p))  1"
    by(simp add: measure_spmf_def emeasure_distr emeasure_restrict_space space_restrict_space measure_pmf.measure_le_1)
qed(simp add: space_measure_spmf)

interpretation measure_spmf: subprob_space "measure_spmf p" for p
by(rule subprob_space_measure_spmf)

lemma finite_measure_spmf [simp]: "finite_measure (measure_spmf p)"
by unfold_locales

lemma spmf_conv_measure_spmf: "spmf p x = measure (measure_spmf p) {x}"
by(auto simp add: measure_spmf_def measure_distr measure_restrict_space pmf.rep_eq space_restrict_space intro: arg_cong2[where f=measure])

lemma emeasure_measure_spmf_conv_measure_pmf:
  "emeasure (measure_spmf p) A = emeasure (measure_pmf p) (Some ` A)"
by(auto simp add: measure_spmf_def emeasure_distr emeasure_restrict_space space_restrict_space intro: arg_cong2[where f=emeasure])

lemma measure_measure_spmf_conv_measure_pmf:
  "measure (measure_spmf p) A = measure (measure_pmf p) (Some ` A)"
using emeasure_measure_spmf_conv_measure_pmf[of p A]
by(simp add: measure_spmf.emeasure_eq_measure measure_pmf.emeasure_eq_measure)

lemma emeasure_spmf_map_pmf_Some [simp]:
  "emeasure (measure_spmf (map_pmf Some p)) A = emeasure (measure_pmf p) A"
by(auto simp add: measure_spmf_def emeasure_distr emeasure_restrict_space space_restrict_space intro: arg_cong2[where f=emeasure])

lemma measure_spmf_map_pmf_Some [simp]:
  "measure (measure_spmf (map_pmf Some p)) A = measure (measure_pmf p) A"
using emeasure_spmf_map_pmf_Some[of p A] by(simp add: measure_spmf.emeasure_eq_measure measure_pmf.emeasure_eq_measure)

lemma nn_integral_measure_spmf: "(+ x. f x measure_spmf p) = + x. ennreal (spmf p x) * f x count_space UNIV"
  (is "?lhs = ?rhs")
proof -
  have "?lhs = + x. pmf p x * f (the x) count_space (range Some)"
    by(simp add: measure_spmf_def nn_integral_distr nn_integral_restrict_space nn_integral_measure_pmf nn_integral_count_space_indicator ac_simps times_ereal.simps(1)[symmetric] del: times_ereal.simps(1))
  also have " = + x. ennreal (spmf p (the x)) * f (the x) count_space (range Some)"
    by(rule nn_integral_cong) auto
  also have " = + x. spmf p (the (Some x)) * f (the (Some x)) count_space UNIV"
    by(rule nn_integral_bij_count_space[symmetric])(simp add: bij_betw_def)
  also have " = ?rhs" by simp
  finally show ?thesis .
qed

lemma integral_measure_spmf:
  assumes "integrable (measure_spmf p) f"
  shows "( x. f x measure_spmf p) =  x. spmf p x * f x count_space UNIV"
proof -
  have "integrable (count_space UNIV) (λx. spmf p x * f x)"
    using assms by(simp add: integrable_iff_bounded nn_integral_measure_spmf abs_mult ennreal_mult'')
  then show ?thesis using assms
    by(simp add: real_lebesgue_integral_def nn_integral_measure_spmf ennreal_mult'[symmetric])
qed

lemma emeasure_spmf_single: "emeasure (measure_spmf p) {x} = spmf p x"
by(simp add: measure_spmf.emeasure_eq_measure spmf_conv_measure_spmf)

lemma measurable_measure_spmf[measurable]:
  "(λx. measure_spmf (M x))  measurable (count_space UNIV) (subprob_algebra (count_space UNIV))"
by (auto simp: space_subprob_algebra)

lemma nn_integral_measure_spmf_conv_measure_pmf:
  assumes [measurable]: "f  borel_measurable (count_space UNIV)"
  shows "nn_integral (measure_spmf p) f = nn_integral (restrict_space (measure_pmf p) (range Some)) (f  the)"
by(simp add: measure_spmf_def nn_integral_distr o_def)

lemma measure_spmf_in_space_subprob_algebra [simp]:
  "measure_spmf p  space (subprob_algebra (count_space UNIV))"
by(simp add: space_subprob_algebra)

lemma nn_integral_spmf_neq_top: "(+ x. spmf p x count_space UNIV)  "
using nn_integral_measure_spmf[where f="λ_. 1", of p, symmetric] by simp

lemma SUP_spmf_neq_top': "(SUP pY. ennreal (spmf p x))  "
proof(rule neq_top_trans)
  show "(SUP pY. ennreal (spmf p x))  1" by(rule SUP_least)(simp add: pmf_le_1)
qed simp

lemma SUP_spmf_neq_top: "(SUP i. ennreal (spmf (Y i) x))  "
proof(rule neq_top_trans)
  show "(SUP i. ennreal (spmf (Y i) x))  1" by(rule SUP_least)(simp add: pmf_le_1)
qed simp

lemma SUP_emeasure_spmf_neq_top: "(SUP pY. emeasure (measure_spmf p) A)  "
proof(rule neq_top_trans)
  show "(SUP pY. emeasure (measure_spmf p) A)  1"
    by(rule SUP_least)(simp add: measure_spmf.subprob_emeasure_le_1)
qed simp

subsection ‹Support›

definition set_spmf :: "'a spmf  'a set"
where "set_spmf p = set_pmf p  set_option"

lemma set_spmf_rep_eq: "set_spmf p = {x. measure (measure_spmf p) {x}  0}"
proof -
  have "x :: 'a. the -` {x}  range Some = {Some x}" by auto
  then show ?thesis
    by(auto simp add: set_spmf_def set_pmf.rep_eq measure_spmf_def measure_distr measure_restrict_space space_restrict_space intro: rev_image_eqI)
qed

lemma in_set_spmf: "x  set_spmf p  Some x  set_pmf p"
by(simp add: set_spmf_def)

lemma AE_measure_spmf_iff [simp]: "(AE x in measure_spmf p. P x)  (xset_spmf p. P x)"
by(auto 4 3 simp add: measure_spmf_def AE_distr_iff AE_restrict_space_iff AE_measure_pmf_iff set_spmf_def cong del: AE_cong)

lemma spmf_eq_0_set_spmf: "spmf p x = 0  x  set_spmf p"
by(auto simp add: pmf_eq_0_set_pmf set_spmf_def intro: rev_image_eqI)

lemma in_set_spmf_iff_spmf: "x  set_spmf p  spmf p x  0"
by(auto simp add: set_spmf_def set_pmf_iff intro: rev_image_eqI)

lemma set_spmf_return_pmf_None [simp]: "set_spmf (return_pmf None) = {}"
by(auto simp add: set_spmf_def)

lemma countable_set_spmf [simp]: "countable (set_spmf p)"
by(simp add: set_spmf_def bind_UNION)

lemma spmf_eqI:
  assumes "i. spmf p i = spmf q i"
  shows "p = q"
proof(rule pmf_eqI)
  fix i
  show "pmf p i = pmf q i"
  proof(cases i)
    case (Some i')
    thus ?thesis by(simp add: assms)
  next
    case None
    have "ennreal (pmf p i) = measure (measure_pmf p) {i}" by(simp add: pmf_def)
    also have "{i} = space (measure_pmf p) - range Some"
      by(auto simp add: None intro: ccontr)
    also have "measure (measure_pmf p)  = ennreal 1 - measure (measure_pmf p) (range Some)"
      by(simp add: measure_pmf.prob_compl ennreal_minus[symmetric] del: space_measure_pmf)
    also have "range Some = (xset_spmf p. {Some x})  Some ` (- set_spmf p)"
      by auto
    also have "measure (measure_pmf p)  = measure (measure_pmf p) (xset_spmf p. {Some x})"
      by(rule measure_pmf.measure_zero_union)(auto simp add: measure_pmf.prob_eq_0 AE_measure_pmf_iff in_set_spmf_iff_spmf set_pmf_iff)
    also have "ennreal  = + x. measure (measure_pmf p) {Some x} count_space (set_spmf p)"
      unfolding measure_pmf.emeasure_eq_measure[symmetric]
      by(simp_all add: emeasure_UN_countable disjoint_family_on_def)
    also have " = + x. spmf p x count_space (set_spmf p)" by(simp add: pmf_def)
    also have " = + x. spmf q x count_space (set_spmf p)" by(simp add: assms)
    also have "set_spmf p = set_spmf q" by(auto simp add: in_set_spmf_iff_spmf assms)
    also have "(+ x. spmf q x count_space (set_spmf q)) = + x. measure (measure_pmf q) {Some x} count_space (set_spmf q)"
      by(simp add: pmf_def)
    also have " = measure (measure_pmf q) (xset_spmf q. {Some x})"
      unfolding measure_pmf.emeasure_eq_measure[symmetric]
      by(simp_all add: emeasure_UN_countable disjoint_family_on_def)
    also have " = measure (measure_pmf q) ((xset_spmf q. {Some x})  Some ` (- set_spmf q))"
      by(rule ennreal_cong measure_pmf.measure_zero_union[symmetric])+(auto simp add: measure_pmf.prob_eq_0 AE_measure_pmf_iff in_set_spmf_iff_spmf set_pmf_iff)
    also have "((xset_spmf q. {Some x})  Some ` (- set_spmf q)) = range Some" by auto
    also have "ennreal 1 - measure (measure_pmf q)  = measure (measure_pmf q) (space (measure_pmf q) - range Some)"
      by(simp add: one_ereal_def measure_pmf.prob_compl ennreal_minus[symmetric] del: space_measure_pmf)
    also have "space (measure_pmf q) - range Some = {i}"
      by(auto simp add: None intro: ccontr)
    also have "measure (measure_pmf q)  = pmf q i" by(simp add: pmf_def)
    finally show ?thesis by simp
  qed
qed

lemma integral_measure_spmf_restrict:
  fixes f ::  "'a  'b :: {banach, second_countable_topology}" shows
  "( x. f x measure_spmf M) = ( x. f x restrict_space (measure_spmf M) (set_spmf M))"
by(auto intro!: integral_cong_AE simp add: integral_restrict_space)

lemma nn_integral_measure_spmf':
  "(+ x. f x measure_spmf p) = + x. ennreal (spmf p x) * f x count_space (set_spmf p)"
by(auto simp add: nn_integral_measure_spmf nn_integral_count_space_indicator in_set_spmf_iff_spmf intro!: nn_integral_cong split: split_indicator)

subsection ‹Functorial structure›

abbreviation map_spmf :: "('a  'b)  'a spmf  'b spmf"
where "map_spmf f  map_pmf (map_option f)"

context begin
local_setup Local_Theory.map_background_naming (Name_Space.mandatory_path "spmf")

lemma map_comp: "map_spmf f (map_spmf g p) = map_spmf (f  g) p"
by(simp add: pmf.map_comp o_def option.map_comp)

lemma map_id0: "map_spmf id = id"
by(simp add: pmf.map_id option.map_id0)

lemma map_id [simp]: "map_spmf id p = p"
by(simp add: map_id0)

lemma map_ident [simp]: "map_spmf (λx. x) p = p"
by(simp add: id_def[symmetric])

end

lemma set_map_spmf [simp]: "set_spmf (map_spmf f p) = f ` set_spmf p"
by(simp add: set_spmf_def image_bind bind_image o_def Option.option.set_map)

lemma map_spmf_cong:
  " p = q; x. x  set_spmf q  f x = g x 
   map_spmf f p = map_spmf g q"
by(auto intro: pmf.map_cong option.map_cong simp add: in_set_spmf)

lemma map_spmf_cong_simp:
  " p = q; x. x  set_spmf q =simp=> f x = g x 
   map_spmf f p = map_spmf g q"
unfolding simp_implies_def by(rule map_spmf_cong)

lemma map_spmf_idI: "(x. x  set_spmf p  f x = x)  map_spmf f p = p"
by(rule map_pmf_idI map_option_idI)+(simp add: in_set_spmf)

lemma emeasure_map_spmf:
  "emeasure (measure_spmf (map_spmf f p)) A = emeasure (measure_spmf p) (f -` A)"
by(auto simp add: measure_spmf_def emeasure_distr measurable_restrict_space1 space_restrict_space emeasure_restrict_space intro: arg_cong2[where f=emeasure])

lemma measure_map_spmf: "measure (measure_spmf (map_spmf f p)) A = measure (measure_spmf p) (f -` A)"
using emeasure_map_spmf[of f p A] by(simp add: measure_spmf.emeasure_eq_measure)

lemma measure_map_spmf_conv_distr:
  "measure_spmf (map_spmf f p) = distr (measure_spmf p) (count_space UNIV) f"
by(rule measure_eqI)(simp_all add: emeasure_map_spmf emeasure_distr)

lemma spmf_map_pmf_Some [simp]: "spmf (map_pmf Some p) i = pmf p i"
by(simp add: pmf_map_inj')

lemma spmf_map_inj: " inj_on f (set_spmf M); x  set_spmf M   spmf (map_spmf f M) (f x) = spmf M x"
by(subst option.map(2)[symmetric, where f=f])(rule pmf_map_inj, auto simp add: in_set_spmf inj_on_def elim!: option.inj_map_strong[rotated])

lemma spmf_map_inj': "inj f  spmf (map_spmf f M) (f x) = spmf M x"
by(subst option.map(2)[symmetric, where f=f])(rule pmf_map_inj'[OF option.inj_map])

lemma spmf_map_outside: "x  f ` set_spmf M  spmf (map_spmf f M) x = 0"
unfolding spmf_eq_0_set_spmf by simp

lemma ennreal_spmf_map: "ennreal (spmf (map_spmf f p) x) = emeasure (measure_spmf p) (f -` {x})"
by(auto simp add: ennreal_pmf_map measure_spmf_def emeasure_distr emeasure_restrict_space space_restrict_space intro: arg_cong2[where f=emeasure])

lemma spmf_map: "spmf (map_spmf f p) x = measure (measure_spmf p) (f -` {x})"
using ennreal_spmf_map[of f p x] by(simp add: measure_spmf.emeasure_eq_measure)

lemma ennreal_spmf_map_conv_nn_integral:
  "ennreal (spmf (map_spmf f p) x) = integralN (measure_spmf p) (indicator (f -` {x}))"
by(auto simp add: ennreal_pmf_map measure_spmf_def emeasure_distr space_restrict_space emeasure_restrict_space intro: arg_cong2[where f=emeasure])

subsection ‹Monad operations›

subsubsection ‹Return›

abbreviation return_spmf :: "'a  'a spmf"
where "return_spmf x  return_pmf (Some x)"

lemma pmf_return_spmf: "pmf (return_spmf x) y = indicator {y} (Some x)"
by(fact pmf_return)

lemma measure_spmf_return_spmf: "measure_spmf (return_spmf x) = Giry_Monad.return (count_space UNIV) x"
by(rule measure_eqI)(simp_all add: measure_spmf_def emeasure_distr space_restrict_space emeasure_restrict_space indicator_def)

lemma measure_spmf_return_pmf_None [simp]: "measure_spmf (return_pmf None) = null_measure (count_space UNIV)"
by(rule measure_eqI)(auto simp add: measure_spmf_def emeasure_distr space_restrict_space emeasure_restrict_space indicator_eq_0_iff)

lemma set_return_spmf [simp]: "set_spmf (return_spmf x) = {x}"
by(auto simp add: set_spmf_def)

subsubsection ‹Bind›

definition bind_spmf :: "'a spmf  ('a  'b spmf)  'b spmf"
where "bind_spmf x f = bind_pmf x (λa. case a of None  return_pmf None | Some a'  f a')"

adhoc_overloading Monad_Syntax.bind bind_spmf

lemma return_None_bind_spmf [simp]: "return_pmf None  (f :: 'a  _) = return_pmf None"
by(simp add: bind_spmf_def bind_return_pmf)

lemma return_bind_spmf [simp]: "return_spmf x  f = f x"
by(simp add: bind_spmf_def bind_return_pmf)

lemma bind_return_spmf [simp]: "x  return_spmf = x"
proof -
  have "a :: 'a option. (case a of None  return_pmf None | Some a'  return_spmf a') = return_pmf a"
    by(simp split: option.split)
  then show ?thesis
    by(simp add: bind_spmf_def bind_return_pmf')
qed

lemma bind_spmf_assoc [simp]:
  fixes x :: "'a spmf" and f :: "'a  'b spmf" and g :: "'b  'c spmf"
  shows "(x  f)  g = x  (λy. f y  g)"
by(auto simp add: bind_spmf_def bind_assoc_pmf fun_eq_iff bind_return_pmf split: option.split intro: arg_cong[where f="bind_pmf x"])

lemma pmf_bind_spmf_None: "pmf (p  f) None = pmf p None +  x. pmf (f x) None measure_spmf p"
  (is "?lhs = ?rhs")
proof -
  let ?f = "λx. pmf (case x of None  return_pmf None | Some x  f x) None"
  have "?lhs =  x. ?f x measure_pmf p"
    by(simp add: bind_spmf_def pmf_bind)
  also have " =  x. ?f None * indicator {None} x + ?f x * indicator (range Some) x measure_pmf p"
    by(rule Bochner_Integration.integral_cong)(auto simp add: indicator_def)
  also have " = ( x. ?f None * indicator {None} x measure_pmf p) + ( x. ?f x * indicator (range Some) x measure_pmf p)"
    by(rule Bochner_Integration.integral_add)(auto 4 3 intro: integrable_real_mult_indicator measure_pmf.integrable_const_bound[where B=1] simp add: AE_measure_pmf_iff pmf_le_1)
  also have " = pmf p None +  x. indicator (range Some) x * pmf (f (the x)) None measure_pmf p"
    by(auto simp add: measure_measure_pmf_finite indicator_eq_0_iff intro!: Bochner_Integration.integral_cong)
  also have " = ?rhs" unfolding measure_spmf_def
    by(subst integral_distr)(auto simp add: integral_restrict_space)
  finally show ?thesis .
qed

lemma spmf_bind: "spmf (p  f) y =  x. spmf (f x) y measure_spmf p"
unfolding measure_spmf_def
by(subst integral_distr)(auto simp add: bind_spmf_def pmf_bind integral_restrict_space indicator_eq_0_iff intro!: Bochner_Integration.integral_cong split: option.split)

lemma ennreal_spmf_bind: "ennreal (spmf (p  f) x) = + y. spmf (f y) x measure_spmf p"
by(auto simp add: bind_spmf_def ennreal_pmf_bind nn_integral_measure_spmf_conv_measure_pmf nn_integral_restrict_space intro: nn_integral_cong split: split_indicator option.split)

lemma measure_spmf_bind_pmf: "measure_spmf (p  f) = measure_pmf p  measure_spmf  f"
  (is "?lhs = ?rhs")
proof(rule measure_eqI)
  show "sets ?lhs = sets ?rhs"
    by(simp add: sets_bind[where N="count_space UNIV"] space_measure_spmf)
next
  fix A :: "'a set"
  have "emeasure ?lhs A = + x. emeasure (measure_spmf (f x)) A measure_pmf p"
    by(simp add: measure_spmf_def emeasure_distr space_restrict_space emeasure_restrict_space bind_spmf_def)
  also have " = emeasure ?rhs A"
    by(simp add: emeasure_bind[where N="count_space UNIV"] space_measure_spmf space_subprob_algebra)
  finally show "emeasure ?lhs A = emeasure ?rhs A" .
qed

lemma measure_spmf_bind: "measure_spmf (p  f) = measure_spmf p  measure_spmf  f"
  (is "?lhs = ?rhs")
proof(rule measure_eqI)
  show "sets ?lhs = sets ?rhs"
    by(simp add: sets_bind[where N="count_space UNIV"] space_measure_spmf)
next
  fix A :: "'a set"
  let ?A = "the -` A  range Some"
  have "emeasure ?lhs A = + x. emeasure (measure_pmf (case x of None  return_pmf None | Some x  f x)) ?A measure_pmf p"
    by(simp add: measure_spmf_def emeasure_distr space_restrict_space emeasure_restrict_space bind_spmf_def)
  also have " =  + x. emeasure (measure_pmf (f (the x))) ?A * indicator (range Some) x measure_pmf p"
    by(rule nn_integral_cong)(auto split: option.split simp add: indicator_def)
  also have " = + x. emeasure (measure_spmf (f x)) A measure_spmf p"
    by(simp add: measure_spmf_def nn_integral_distr nn_integral_restrict_space emeasure_distr space_restrict_space emeasure_restrict_space)
  also have " = emeasure ?rhs A"
    by(simp add: emeasure_bind[where N="count_space UNIV"] space_measure_spmf space_subprob_algebra)
  finally show "emeasure ?lhs A = emeasure ?rhs A" .
qed

lemma map_spmf_bind_spmf: "map_spmf f (bind_spmf p g) = bind_spmf p (map_spmf f  g)"
by(auto simp add: bind_spmf_def map_bind_pmf fun_eq_iff split: option.split intro: arg_cong2[where f=bind_pmf])

lemma bind_map_spmf: "map_spmf f p  g = p  g  f"
by(simp add: bind_spmf_def bind_map_pmf o_def cong del: option.case_cong_weak)

lemma spmf_bind_leI:
  assumes "y. y  set_spmf p  spmf (f y) x  r"
  and "0  r"
  shows "spmf (bind_spmf p f) x  r"
proof -
  have "ennreal (spmf (bind_spmf p f) x) = + y. spmf (f y) x measure_spmf p" by(rule ennreal_spmf_bind)
  also have "  + y. r measure_spmf p" by(rule nn_integral_mono_AE)(simp add: assms)
  also have "  r" using assms measure_spmf.emeasure_space_le_1
    by(auto simp add: measure_spmf.emeasure_eq_measure intro!: mult_left_le)
  finally show ?thesis using assms(2) by(simp)
qed

lemma map_spmf_conv_bind_spmf: "map_spmf f p = (p  (λx. return_spmf (f x)))"
by(simp add: map_pmf_def bind_spmf_def)(rule bind_pmf_cong, simp_all split: option.split)

lemma bind_spmf_cong:
  " p = q; x. x  set_spmf q  f x = g x 
   bind_spmf p f = bind_spmf q g"
by(auto simp add: bind_spmf_def in_set_spmf intro: bind_pmf_cong option.case_cong)

lemma bind_spmf_cong_simp:
  " p = q; x. x  set_spmf q =simp=> f x = g x 
   bind_spmf p f = bind_spmf q g"
by(simp add: simp_implies_def cong: bind_spmf_cong)

lemma set_bind_spmf: "set_spmf (M  f) = set_spmf M  (set_spmf  f)"
by(auto simp add: set_spmf_def bind_spmf_def bind_UNION split: option.splits)

lemma bind_spmf_const_return_None [simp]: "bind_spmf p (λ_. return_pmf None) = return_pmf None"
by(simp add: bind_spmf_def case_option_collapse)

lemma bind_commute_spmf:
  "bind_spmf p (λx. bind_spmf q (f x)) = bind_spmf q (λy. bind_spmf p (λx. f x y))"
  (is "?lhs = ?rhs")
proof -
  let ?f = "λx y. case x of None  return_pmf None | Some a  (case y of None  return_pmf None | Some b  f a b)"
  have "?lhs = p  (λx. q  ?f x)"
    unfolding bind_spmf_def by(rule bind_pmf_cong[OF refl])(simp split: option.split)
  also have " = q  (λy. p  (λx. ?f x y))" by(rule bind_commute_pmf)
  also have " = ?rhs" unfolding bind_spmf_def
    by(rule bind_pmf_cong[OF refl])(auto split: option.split, metis bind_spmf_const_return_None bind_spmf_def)
  finally show ?thesis .
qed

subsection ‹Relator›

abbreviation rel_spmf :: "('a  'b  bool)  'a spmf  'b spmf  bool"
where "rel_spmf R  rel_pmf (rel_option R)"

lemma rel_pmf_mono:
  "rel_pmf A f g; x y. A x y  B x y   rel_pmf B f g"
using pmf.rel_mono[of A B] by(simp add: le_fun_def)

lemma rel_spmf_mono:
  "rel_spmf A f g; x y. A x y  B x y   rel_spmf B f g"
apply(erule rel_pmf_mono)
using option.rel_mono[of A B] by(simp add: le_fun_def)

lemma rel_spmf_mono_strong:
  " rel_spmf A f g; x y.  A x y; x  set_spmf f; y  set_spmf g   B x y   rel_spmf B f g"
apply(erule pmf.rel_mono_strong)
apply(erule option.rel_mono_strong)
apply(auto simp add: in_set_spmf)
done

lemma rel_spmf_reflI: "(x. x  set_spmf p  P x x)  rel_spmf P p p"
by(rule rel_pmf_reflI)(auto simp add: set_spmf_def intro: rel_option_reflI)

lemma rel_spmfI [intro?]:
  " x y. (x, y)  set_spmf pq  P x y; map_spmf fst pq = p; map_spmf snd pq = q 
   rel_spmf P p q"
by(rule rel_pmf.intros[where pq="map_pmf (λx. case x of None  (None, None) | Some (a, b)  (Some a, Some b)) pq"])
  (auto simp add: pmf.map_comp o_def in_set_spmf split: option.splits intro: pmf.map_cong)

lemma rel_spmfE [elim?, consumes 1, case_names rel_spmf]:
  assumes "rel_spmf P p q"
  obtains pq where
    "x y. (x, y)  set_spmf pq  P x y"
    "p = map_spmf fst pq"
    "q = map_spmf snd pq"
using assms
proof(cases rule: rel_pmf.cases[consumes 1, case_names rel_pmf])
  case (rel_pmf pq)
  let ?pq = "map_pmf (λ(a, b). case (a, b) of (Some x, Some y)  Some (x, y) | _  None) pq"
  have "x y. (x, y)  set_spmf ?pq  P x y"
    by(auto simp add: in_set_spmf split: option.split_asm dest: rel_pmf(1))
  moreover
  have "x. (x, None)  set_pmf pq  x = None" by(auto dest!: rel_pmf(1))
  then have "p = map_spmf fst ?pq" using rel_pmf(2)
    by(auto simp add: pmf.map_comp split_beta intro!: pmf.map_cong split: option.split)
  moreover
  have "y. (None, y)  set_pmf pq  y = None" by(auto dest!: rel_pmf(1))
  then have "q = map_spmf snd ?pq" using rel_pmf(3)
    by(auto simp add: pmf.map_comp split_beta intro!: pmf.map_cong split: option.split)
  ultimately show thesis ..
qed

lemma rel_spmf_simps:
  "rel_spmf R p q  (pq. ((x, y)set_spmf pq. R x y)  map_spmf fst pq = p  map_spmf snd pq = q)"
by(auto intro: rel_spmfI elim!: rel_spmfE)

lemma spmf_rel_map:
  shows spmf_rel_map1: "R f x. rel_spmf R (map_spmf f x) = rel_spmf (λx. R (f x)) x"
  and spmf_rel_map2: "R x g y. rel_spmf R x (map_spmf g y) = rel_spmf (λx y. R x (g y)) x y"
by(simp_all add: fun_eq_iff pmf.rel_map option.rel_map[abs_def])

lemma spmf_rel_conversep: "rel_spmf R¯¯ = (rel_spmf R)¯¯"
by(simp add: option.rel_conversep pmf.rel_conversep)

lemma spmf_rel_eq: "rel_spmf (=) = (=)"
by(simp add: pmf.rel_eq option.rel_eq)

context includes lifting_syntax
begin

lemma bind_spmf_parametric [transfer_rule]:
  "(rel_spmf A ===> (A ===> rel_spmf B) ===> rel_spmf B) bind_spmf bind_spmf"
unfolding bind_spmf_def[abs_def] by transfer_prover

lemma return_spmf_parametric: "(A ===> rel_spmf A) return_spmf return_spmf"
by transfer_prover

lemma map_spmf_parametric: "((A ===> B) ===> rel_spmf A ===> rel_spmf B) map_spmf map_spmf"
by transfer_prover

lemma rel_spmf_parametric:
  "((A ===> B ===> (=)) ===> rel_spmf A ===> rel_spmf B ===> (=)) rel_spmf rel_spmf"
by transfer_prover

lemma set_spmf_parametric [transfer_rule]:
  "(rel_spmf A ===> rel_set A) set_spmf set_spmf"
unfolding set_spmf_def[abs_def] by transfer_prover

lemma return_spmf_None_parametric:
  "(rel_spmf A) (return_pmf None) (return_pmf None)"
by simp

end

lemma rel_spmf_bindI:
  " rel_spmf R p q; x y. R x y  rel_spmf P (f x) (g y) 
   rel_spmf P (p  f) (q  g)"
by(fact bind_spmf_parametric[THEN rel_funD, THEN rel_funD, OF _ rel_funI])

lemma rel_spmf_bind_reflI:
  "(x. x  set_spmf p  rel_spmf P (f x) (g x))  rel_spmf P (p  f) (p  g)"
by(rule rel_spmf_bindI[where R="λx y. x = y  x  set_spmf p"])(auto intro: rel_spmf_reflI)

lemma rel_pmf_return_pmfI: "P x y  rel_pmf P (return_pmf x) (return_pmf y)"
by(rule rel_pmf.intros[where pq="return_pmf (x, y)"])(simp_all)

context includes lifting_syntax
begin

text ‹We do not yet have a relator for typ'a measure, so we combine constmeasure and constmeasure_pmf
lemma measure_pmf_parametric:
  "(rel_pmf A ===> rel_pred A ===> (=)) (λp. measure (measure_pmf p)) (λq. measure (measure_pmf q))"
proof(rule rel_funI)+
  fix p q X Y
  assume "rel_pmf A p q" and "rel_pred A X Y"
  from this(1) obtain pq where A: "x y. (x, y)  set_pmf pq  A x y"
    and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto
  show "measure p X = measure q Y" unfolding p q measure_map_pmf
    by(rule measure_pmf.finite_measure_eq_AE)(auto simp add: AE_measure_pmf_iff dest!: A rel_predD[OF rel_pred _ _ _])
qed

lemma measure_spmf_parametric:
  "(rel_spmf A ===> rel_pred A ===> (=)) (λp. measure (measure_spmf p)) (λq. measure (measure_spmf q))"
unfolding measure_measure_spmf_conv_measure_pmf[abs_def]
apply(rule rel_funI)+
apply(erule measure_pmf_parametric[THEN rel_funD, THEN rel_funD])
apply(auto simp add: rel_pred_def rel_fun_def elim: option.rel_cases)
done

end

subsection ‹From typ'a pmf to typ'a spmf

definition spmf_of_pmf :: "'a pmf  'a spmf"
where "spmf_of_pmf = map_pmf Some"

lemma set_spmf_spmf_of_pmf [simp]: "set_spmf (spmf_of_pmf p) = set_pmf p"
by(auto simp add: spmf_of_pmf_def set_spmf_def bind_image o_def)

lemma spmf_spmf_of_pmf [simp]: "spmf (spmf_of_pmf p) x = pmf p x"
by(simp add: spmf_of_pmf_def)

lemma pmf_spmf_of_pmf_None [simp]: "pmf (spmf_of_pmf p) None = 0"
using ennreal_pmf_map[of Some p None] by(simp add: spmf_of_pmf_def)

lemma emeasure_spmf_of_pmf [simp]: "emeasure (measure_spmf (spmf_of_pmf p)) A = emeasure (measure_pmf p) A"
by(simp add: emeasure_measure_spmf_conv_measure_pmf spmf_of_pmf_def inj_vimage_image_eq)

lemma measure_spmf_spmf_of_pmf [simp]: "measure_spmf (spmf_of_pmf p) = measure_pmf p"
by(rule measure_eqI) simp_all

lemma map_spmf_of_pmf [simp]: "map_spmf f (spmf_of_pmf p) = spmf_of_pmf (map_pmf f p)"
by(simp add: spmf_of_pmf_def pmf.map_comp o_def)

lemma rel_spmf_spmf_of_pmf [simp]: "rel_spmf R (spmf_of_pmf p) (spmf_of_pmf q) = rel_pmf R p q"
by(simp add: spmf_of_pmf_def pmf.rel_map)

lemma spmf_of_pmf_return_pmf [simp]: "spmf_of_pmf (return_pmf x) = return_spmf x"
by(simp add: spmf_of_pmf_def)

lemma bind_spmf_of_pmf [simp]: "bind_spmf (spmf_of_pmf p) f = bind_pmf p f"
by(simp add: spmf_of_pmf_def bind_spmf_def bind_map_pmf)

lemma set_spmf_bind_pmf: "set_spmf (bind_pmf p f) = Set.bind (set_pmf p) (set_spmf  f)"
unfolding bind_spmf_of_pmf[symmetric] by(subst set_bind_spmf) simp

lemma spmf_of_pmf_bind: "spmf_of_pmf (bind_pmf p f) = bind_pmf p (λx. spmf_of_pmf (f x))"
by(simp add: spmf_of_pmf_def map_bind_pmf)

lemma bind_pmf_return_spmf: "p  (λx. return_spmf (f x)) = spmf_of_pmf (map_pmf f p)"
by(simp add: map_pmf_def spmf_of_pmf_bind)

subsection ‹Weight of a subprobability›

abbreviation weight_spmf :: "'a spmf  real"
where "weight_spmf p  measure (measure_spmf p) (space (measure_spmf p))"

lemma weight_spmf_def: "weight_spmf p = measure (measure_spmf p) UNIV"
by(simp add: space_measure_spmf)

lemma weight_spmf_le_1: "weight_spmf p  1"
by(simp add: measure_spmf.subprob_measure_le_1)

lemma weight_return_spmf [simp]: "weight_spmf (return_spmf x) = 1"
by(simp add: measure_spmf_return_spmf measure_return)

lemma weight_return_pmf_None [simp]: "weight_spmf (return_pmf None) = 0"
by(simp)

lemma weight_map_spmf [simp]: "weight_spmf (map_spmf f p) = weight_spmf p"
by(simp add: weight_spmf_def measure_map_spmf)

lemma weight_spmf_of_pmf [simp]: "weight_spmf (spmf_of_pmf p) = 1"
using measure_pmf.prob_space[of p] by(simp add: spmf_of_pmf_def weight_spmf_def)

lemma weight_spmf_nonneg: "weight_spmf p  0"
by(fact measure_nonneg)

lemma (in finite_measure) integrable_weight_spmf [simp]:
  "(λx. weight_spmf (f x))  borel_measurable M  integrable M (λx. weight_spmf (f x))"
by(rule integrable_const_bound[where B=1])(simp_all add: weight_spmf_nonneg weight_spmf_le_1)

lemma weight_spmf_eq_nn_integral_spmf: "weight_spmf p = + x. spmf p x count_space UNIV"
by(simp add: measure_measure_spmf_conv_measure_pmf space_measure_spmf measure_pmf.emeasure_eq_measure[symmetric] nn_integral_pmf[symmetric] embed_measure_count_space[symmetric] inj_on_def nn_integral_embed_measure measurable_embed_measure1)

lemma weight_spmf_eq_nn_integral_support:
  "weight_spmf p = + x. spmf p x count_space (set_spmf p)"
unfolding weight_spmf_eq_nn_integral_spmf
by(auto simp add: nn_integral_count_space_indicator in_set_spmf_iff_spmf intro!: nn_integral_cong split: split_indicator)

lemma pmf_None_eq_weight_spmf: "pmf p None = 1 - weight_spmf p"
proof -
  have "weight_spmf p = + x. spmf p x count_space UNIV" by(rule weight_spmf_eq_nn_integral_spmf)
  also have " = + x. ennreal (pmf p x) * indicator (range Some) x count_space UNIV"
    by(simp add: nn_integral_count_space_indicator[symmetric] embed_measure_count_space[symmetric] nn_integral_embed_measure measurable_embed_measure1)
  also have " + pmf p None = + x. ennreal (pmf p x) * indicator (range Some) x + ennreal (pmf p None) * indicator {None} x count_space UNIV"
    by(subst nn_integral_add)(simp_all add: max_def)
  also have " = + x. pmf p x count_space UNIV"
    by(rule nn_integral_cong)(auto split: split_indicator)
  also have " = 1" by (simp add: nn_integral_pmf)
  finally show ?thesis by(simp add: ennreal_plus[symmetric] del: ennreal_plus)
qed

lemma weight_spmf_conv_pmf_None: "weight_spmf p = 1 - pmf p None"
by(simp add: pmf_None_eq_weight_spmf)

lemma weight_spmf_le_0: "weight_spmf p  0  weight_spmf p = 0"
by(rule measure_le_0_iff)

lemma weight_spmf_lt_0: "¬ weight_spmf p < 0"
by(simp add: not_less weight_spmf_nonneg)

lemma spmf_le_weight: "spmf p x  weight_spmf p"
proof -
  have "ennreal (spmf p x)  weight_spmf p"
    unfolding weight_spmf_eq_nn_integral_spmf by(rule nn_integral_ge_point) simp
  then show ?thesis by simp
qed

lemma weight_spmf_eq_0: "weight_spmf p = 0  p = return_pmf None"
by(auto intro!: pmf_eqI simp add: pmf_None_eq_weight_spmf split: split_indicator)(metis not_Some_eq pmf_le_0_iff spmf_le_weight)

lemma weight_bind_spmf: "weight_spmf (x  f) = lebesgue_integral (measure_spmf x) (weight_spmf  f)"
unfolding weight_spmf_def
by(simp add: measure_spmf_bind o_def measure_spmf.measure_bind[where N="count_space UNIV"])

lemma rel_spmf_weightD: "rel_spmf A p q  weight_spmf p = weight_spmf q"
by(erule rel_spmfE) simp

lemma rel_spmf_bij_betw:
  assumes f: "bij_betw f (set_spmf p) (set_spmf q)"
  and eq: "x. x  set_spmf p  spmf p x = spmf q (f x)"
  shows "rel_spmf (λx y. f x = y) p q"
proof -
  let ?f = "map_option f"

  have weq: "ennreal (weight_spmf p) = ennreal (weight_spmf q)"
    unfolding weight_spmf_eq_nn_integral_support
    by(subst nn_integral_bij_count_space[OF f, symmetric])(rule nn_integral_cong_AE, simp add: eq AE_count_space)
  then have "None  set_pmf p  None  set_pmf q"
    by(simp add: pmf_None_eq_weight_spmf set_pmf_iff)
  with f have "bij_betw (map_option f) (set_pmf p) (set_pmf q)"
    apply(auto simp add: bij_betw_def in_set_spmf inj_on_def intro: option.expand)
    apply(rename_tac [!] x)
    apply(case_tac [!] x)
    apply(auto iff: in_set_spmf)
    done
  then have "rel_pmf (λx y. ?f x = y) p q"
    by(rule rel_pmf_bij_betw)(case_tac x, simp_all add: weq[simplified] eq in_set_spmf pmf_None_eq_weight_spmf)
  thus ?thesis by(rule pmf.rel_mono_strong)(auto intro!: rel_optionI simp add: Option.is_none_def)
qed

subsection ‹From density to spmfs›

context fixes f :: "'a  real" begin

definition embed_spmf :: "'a spmf"
where "embed_spmf = embed_pmf (λx. case x of None  1 - enn2real (+ x. ennreal (f x) count_space UNIV) | Some x'  max 0 (f x'))"

context
  assumes prob: "(+ x. ennreal (f x) count_space UNIV)  1"
begin

lemma nn_integral_embed_spmf_eq_1:
  "(+ x. ennreal (case x of None  1 - enn2real (+ x. ennreal (f x) count_space UNIV) | Some x'  max 0 (f x')) count_space UNIV) = 1"
  (is "?lhs = _" is "(+ x. ?f x ?M) = _")
proof -
  have "?lhs = + x. ?f x * indicator {None} x + ?f x * indicator (range Some) x ?M"
    by(rule nn_integral_cong)(auto split: split_indicator)
  also have " = (1 - enn2real (+ x. ennreal (f x) count_space UNIV)) + + x. ?f x * indicator (range Some) x ?M"
    (is "_ = ?None + ?Some")
    by(subst nn_integral_add)(simp_all add: AE_count_space max_def le_diff_eq real_le_ereal_iff one_ereal_def[symmetric] prob split: option.split)
  also have "?Some = + x. ?f x count_space (range Some)"
    by(simp add: nn_integral_count_space_indicator)
  also have "count_space (range Some) = embed_measure (count_space UNIV) Some"
    by(simp add: embed_measure_count_space)
  also have "(+ x. ?f x ) = + x. ennreal (f x) count_space UNIV"
    by(subst nn_integral_embed_measure)(simp_all add: measurable_embed_measure1)
  also have "?None +  = 1" using prob
    by(auto simp add: ennreal_minus[symmetric] ennreal_1[symmetric] ennreal_enn2real_if top_unique simp del: ennreal_1)(simp add: diff_add_self_ennreal)
  finally show ?thesis .
qed

lemma pmf_embed_spmf_None: "pmf embed_spmf None = 1 - enn2real (+ x. ennreal (f x) count_space UNIV)"
unfolding embed_spmf_def
apply(subst pmf_embed_pmf)
  subgoal using prob by(simp add: field_simps enn2real_leI split: option.split)
 apply(rule nn_integral_embed_spmf_eq_1)
apply simp
done

lemma spmf_embed_spmf [simp]: "spmf embed_spmf x = max 0 (f x)"
unfolding embed_spmf_def
apply(subst pmf_embed_pmf)
  subgoal using prob by(simp add: field_simps enn2real_leI split: option.split)
 apply(rule nn_integral_embed_spmf_eq_1)
apply simp
done

end

end

lemma embed_spmf_K_0[simp]: "embed_spmf (λ_. 0) = return_pmf None" (is "?lhs = ?rhs")
by(rule spmf_eqI)(simp add: zero_ereal_def[symmetric])

subsection ‹Ordering on spmfs›

text constrel_pmf does not preserve a ccpo structure. Counterexample by Saheb-Djahromi:
  Take prefix order over bool llist› and
  the set range (λn :: nat. uniform (llist_n n))› where llist_n› is the set
  of all llist›s of length n› and uniform› returns a uniform distribution over
  the given set. The set forms a chain in ord_pmf lprefix›, but it has not an upper bound.
  Any upper bound may contain only infinite lists in its support because otherwise it is not greater
  than the n+1›-st element in the chain where n› is the length of the finite list.
  Moreover its support must contain all infinite lists, because otherwise there is a finite list
  all of whose finite extensions are not in the support - a contradiction to the upper bound property.
  Hence, the support is uncountable, but pmf's only have countable support.

  However, if all chains in the ccpo are finite, then it should preserve the ccpo structure.
›

abbreviation ord_spmf :: "('a  'a  bool)  'a spmf  'a spmf  bool"
where "ord_spmf ord  rel_pmf (ord_option ord)"

locale ord_spmf_syntax begin
notation ord_spmf (infix "ı" 60)
end

lemma ord_spmf_map_spmf1: "ord_spmf R (map_spmf f p) = ord_spmf (λx. R (f x)) p"
by(simp add: pmf.rel_map[abs_def] ord_option_map1[abs_def])

lemma ord_spmf_map_spmf2: "ord_spmf R p (map_spmf f q) = ord_spmf (λx y. R x (f y)) p q"
by(simp add: pmf.rel_map ord_option_map2)

lemma ord_spmf_map_spmf12: "ord_spmf R (map_spmf f p) (map_spmf f q) = ord_spmf (λx y. R (f x) (f y)) p q"
by(simp add: pmf.rel_map ord_option_map1[abs_def] ord_option_map2)

lemmas ord_spmf_map_spmf = ord_spmf_map_spmf1 ord_spmf_map_spmf2 ord_spmf_map_spmf12

context fixes ord :: "'a  'a  bool" (structure) begin
interpretation ord_spmf_syntax .

lemma ord_spmfI:
  " x y. (x, y)  set_spmf pq  ord x y; map_spmf fst pq = p; map_spmf snd pq = q 
   p  q"
by(rule rel_pmf.intros[where pq="map_pmf (λx. case x of None  (None, None) | Some (a, b)  (Some a, Some b)) pq"])
  (auto simp add: pmf.map_comp o_def in_set_spmf split: option.splits intro: pmf.map_cong)

lemma ord_spmf_None [simp]: "return_pmf None  x"
by(rule rel_pmf.intros[where pq="map_pmf (Pair None) x"])(auto simp add: pmf.map_comp o_def)

lemma ord_spmf_reflI: "(x. x  set_spmf p  ord x x)  p  p"
by(rule rel_pmf_reflI ord_option_reflI)+(auto simp add: in_set_spmf)

lemma rel_spmf_inf:
  assumes "p  q"
  and "q  p"
  and refl: "reflp ord"
  and trans: "transp ord"
  shows "rel_spmf (inf ord ord¯¯) p q"
proof -
  from p  q q  p
  have "rel_pmf (inf (ord_option ord) (ord_option ord)¯¯) p q"
    by(rule rel_pmf_inf)(blast intro: reflp_ord_option transp_ord_option refl trans)+
  also have "inf (ord_option ord) (ord_option ord)¯¯ = rel_option (inf ord ord¯¯)"
    by(auto simp add: fun_eq_iff elim: ord_option.cases option.rel_cases)
  finally show ?thesis .
qed

end

lemma ord_spmf_return_spmf2: "ord_spmf R p (return_spmf y)  (xset_spmf p. R x y)"
by(auto simp add: rel_pmf_return_pmf2 in_set_spmf ord_option.simps intro: ccontr)

lemma ord_spmf_mono: " ord_spmf A p q; x y. A x y  B x y   ord_spmf B p q"
by(erule rel_pmf_mono)(erule ord_option_mono)

lemma ord_spmf_compp: "ord_spmf (A OO B) = ord_spmf A OO ord_spmf B"
by(simp add: ord_option_compp pmf.rel_compp)

lemma ord_spmf_bindI:
  assumes pq: "ord_spmf R p q"
  and fg: "x y. R x y  ord_spmf P (f x) (g y)"
  shows "ord_spmf P (p  f) (q  g)"
unfolding bind_spmf_def using pq
by(rule rel_pmf_bindI)(auto split: option.split intro: fg)

lemma ord_spmf_bind_reflI:
  "(x. x  set_spmf p  ord_spmf R (f x) (g x))
   ord_spmf R (p  f) (p  g)"
by(rule ord_spmf_bindI[where R="λx y. x = y  x  set_spmf p"])(auto intro: ord_spmf_reflI)

lemma ord_pmf_increaseI:
  assumes le: "x. spmf p x  spmf q x"
  and refl: "x. x  set_spmf p  R x x"
  shows "ord_spmf R p q"
proof(rule rel_pmf.intros)
  define pq where "pq = embed_pmf
    (λ(x, y). case x of Some x'  (case y of Some y'  if x' = y' then spmf p x' else 0 | None  0)
      | None  (case y of None  pmf q None | Some y'  spmf q y' - spmf p y'))"
     (is "_ = embed_pmf ?f")
  have nonneg: "xy. ?f xy  0"
    by(clarsimp simp add: le field_simps split: option.split)
  have integral: "(+ xy. ?f xy count_space UNIV) = 1" (is "nn_integral ?M _ = _")
  proof -
    have "(+ xy. ?f xy count_space UNIV) =
      + xy. ennreal (?f xy) * indicator {(None, None)} xy +
             ennreal (?f xy) * indicator (range (λx. (None, Some x))) xy +
             ennreal (?f xy) * indicator (range (λx. (Some x, Some x))) xy ?M"
      by(rule nn_integral_cong)(auto split: split_indicator option.splits if_split_asm)
    also have " = (+ xy. ?f xy * indicator {(None, None)} xy ?M) +
        (+ xy. ennreal (?f xy) * indicator (range (λx. (None, Some x))) xy ?M) +
        (+ xy. ennreal (?f xy) * indicator (range (λx. (Some x, Some x))) xy ?M)"
      (is "_ = ?None + ?Some2 + ?Some")
      by(subst nn_integral_add)(simp_all add: nn_integral_add AE_count_space le_diff_eq le split: option.split)
    also have "?None = pmf q None" by simp
    also have "?Some2 = + x. ennreal (spmf q x) - spmf p x count_space UNIV"
      by(simp add: nn_integral_count_space_indicator[symmetric] embed_measure_count_space[symmetric] inj_on_def nn_integral_embed_measure measurable_embed_measure1 ennreal_minus)
    also have " = (+ x. spmf q x count_space UNIV) - (+ x. spmf p x count_space UNIV)"
      (is "_ = ?Some2' - ?Some2''")
      by(subst nn_integral_diff)(simp_all add: le nn_integral_spmf_neq_top)
    also have "?Some = + x. spmf p x count_space UNIV"
      by(simp add: nn_integral_count_space_indicator[symmetric] embed_measure_count_space[symmetric] inj_on_def nn_integral_embed_measure measurable_embed_measure1)
    also have "pmf q None + (?Some2' - ?Some2'') +  = pmf q None + ?Some2'"
      by(auto simp add: diff_add_self_ennreal le intro!: nn_integral_mono)
    also have " = + x. ennreal (pmf q x) * indicator {None} x + ennreal (pmf q x) * indicator (range Some) x count_space UNIV"
      by(subst nn_integral_add)(simp_all add: nn_integral_count_space_indicator[symmetric] embed_measure_count_space[symmetric] nn_integral_embed_measure measurable_embed_measure1)
    also have " = + x. pmf q x count_space UNIV"
      by(rule nn_integral_cong)(auto split: split_indicator)
    also have " = 1" by(simp add: nn_integral_pmf)
    finally show ?thesis .
  qed
  note f = nonneg integral

  { fix x y
    assume "(x, y)  set_pmf pq"
    hence "?f (x, y)  0" unfolding pq_def by(simp add: set_embed_pmf[OF f])
    then show "ord_option R x y"
      by(simp add: spmf_eq_0_set_spmf refl split: option.split_asm if_split_asm) }

  have weight_le: "weight_spmf p  weight_spmf q"
    by(subst ennreal_le_iff[symmetric])(auto simp add: weight_spmf_eq_nn_integral_spmf intro!: nn_integral_mono le)

  show "map_pmf fst pq = p"
  proof(rule pmf_eqI)
    fix i
    have "ennreal (pmf (map_pmf fst pq) i) = (+ y. pmf pq (i, y) count_space UNIV)"
      unfolding pq_def ennreal_pmf_map
      apply(simp add: embed_pmf.rep_eq[OF f] o_def emeasure_density nn_integral_count_space_indicator[symmetric])
      apply(subst pmf_embed_pmf[OF f])
      apply(rule nn_integral_bij_count_space[symmetric])
      apply(auto simp add: bij_betw_def inj_on_def)
      done
    also have " = pmf p i"
    proof(cases i)
      case (Some x)
      have "(+ y. pmf pq (Some x, y) count_space UNIV) = + y. pmf p (Some x) * indicator {Some x} y count_space UNIV"
        by(rule nn_integral_cong)(simp add: pq_def pmf_embed_pmf[OF f] split: option.split)
      then show ?thesis using Some by simp
    next
      case None
      have "(+ y. pmf pq (None, y) count_space UNIV) =
            (+ y. ennreal (pmf pq (None, Some (the y))) * indicator (range Some) y +
                   ennreal (pmf pq (None, None)) * indicator {None} y count_space UNIV)"
        by(rule nn_integral_cong)(auto split: split_indicator)
      also have " = (+ y. ennreal (pmf pq (None, Some (the y))) count_space (range Some)) + pmf pq (None, None)"
        by(subst nn_integral_add)(simp_all add: nn_integral_count_space_indicator)
      also have " = (+ y. ennreal (spmf q y) - ennreal (spmf p y) count_space UNIV) + pmf q None"
        by(simp add: pq_def pmf_embed_pmf[OF f] embed_measure_count_space[symmetric] nn_integral_embed_measure measurable_embed_measure1 ennreal_minus)
      also have "(+ y. ennreal (spmf q y) - ennreal (spmf p y) count_space UNIV) =
                 (+ y. spmf q y count_space UNIV) - (+ y. spmf p y count_space UNIV)"
        by(subst nn_integral_diff)(simp_all add: AE_count_space le nn_integral_spmf_neq_top split: split_indicator)
      also have " = pmf p None - pmf q None"
        by(simp add: pmf_None_eq_weight_spmf weight_spmf_eq_nn_integral_spmf[symmetric] ennreal_minus)
      also have " = ennreal (pmf p None) - ennreal (pmf q None)" by(simp add: ennreal_minus)
      finally show ?thesis using None weight_le
        by(auto simp add: diff_add_self_ennreal pmf_None_eq_weight_spmf intro: ennreal_leI)
    qed
    finally show "pmf (map_pmf fst pq) i = pmf p i" by simp
  qed

  show "map_pmf snd pq = q"
  proof(rule pmf_eqI)
    fix i
    have "ennreal (pmf (map_pmf snd pq) i) = (+ x. pmf pq (x, i) count_space UNIV)"
      unfolding pq_def ennreal_pmf_map
      apply(simp add: embed_pmf.rep_eq[OF f] o_def emeasure_density nn_integral_count_space_indicator[symmetric])
      apply(subst pmf_embed_pmf[OF f])
      apply(rule nn_integral_bij_count_space[symmetric])
      apply(auto simp add: bij_betw_def inj_on_def)
      done
    also have " = ennreal (pmf q i)"
    proof(cases i)
      case None
      have "(+ x. pmf pq (x, None) count_space UNIV) = + x. pmf q None * indicator {None :: 'a option} x count_space UNIV"
        by(rule nn_integral_cong)(simp add: pq_def pmf_embed_pmf[OF f] split: option.split)
      then show ?thesis using None by simp
    next
      case (Some y)
      have "(+ x. pmf pq (x, Some y) count_space UNIV) =
        (+ x. ennreal (pmf pq (x, Some y)) * indicator (range Some) x +
               ennreal (pmf pq (None, Some y)) * indicator {None} x count_space UNIV)"
        by(rule nn_integral_cong)(auto split: split_indicator)
      also have " = (+ x. ennreal (pmf pq (x, Some y)) * indicator (range Some) x count_space UNIV) + pmf pq (None, Some y)"
        by(subst nn_integral_add)(simp_all)
      also have " = (+ x. ennreal (spmf p y) * indicator {Some y} x count_space UNIV) + (spmf q y - spmf p y)"
        by(auto simp add: pq_def pmf_embed_pmf[OF f] one_ereal_def[symmetric] simp del: nn_integral_indicator_singleton intro!: arg_cong2[where f="(+)"] nn_integral_cong split: option.split)
      also have " = spmf q y" by(simp add: ennreal_minus[symmetric] le)
      finally show ?thesis using Some by simp
    qed
    finally show "pmf (map_pmf snd pq) i = pmf q i" by simp
  qed
qed

lemma ord_spmf_eq_leD:
  assumes "ord_spmf (=) p q"
  shows "spmf p x  spmf q x"
proof(cases "x  set_spmf p")
  case False
  thus ?thesis by(simp add: in_set_spmf_iff_spmf)
next
  case True
  from assms obtain pq
    where pq: "x y. (x, y)  set_pmf pq  ord_option (=) x y"
    and p: "p = map_pmf fst pq"
    and q: "q = map_pmf snd pq" by cases auto
  have "ennreal (spmf p x) = integralN pq (indicator (fst -` {Some x}))"
    using p by(simp add: ennreal_pmf_map)
  also have " = integralN pq (indicator {(Some x, Some x)})"
    by(rule nn_integral_cong_AE)(auto simp add: AE_measure_pmf_iff split: split_indicator dest: pq)
  also have "  integralN pq (indicator (snd -` {Some x}))"
    by(rule nn_integral_mono) simp
  also have " = ennreal (spmf q x)" using q by(simp add: ennreal_pmf_map)
  finally show ?thesis by simp
qed

lemma ord_spmf_eqD_set_spmf: "ord_spmf (=) p q  set_spmf p  set_spmf q"
by(rule subsetI)(drule_tac x=x in ord_spmf_eq_leD, auto simp add: in_set_spmf_iff_spmf)

lemma ord_spmf_eqD_emeasure:
  "ord_spmf (=) p q  emeasure (measure_spmf p) A  emeasure (measure_spmf q) A"
by(auto intro!: nn_integral_mono split: split_indicator dest: ord_spmf_eq_leD simp add: nn_integral_measure_spmf nn_integral_indicator[symmetric])

lemma ord_spmf_eqD_measure_spmf: "ord_spmf (=) p q  measure_spmf p  measure_spmf q"
  by (subst le_measure) (auto simp: ord_spmf_eqD_emeasure)

subsection ‹CCPO structure for the flat ccpo termord_option (=)

context fixes Y :: "'a spmf set" begin

definition lub_spmf :: "'a spmf"
where "lub_spmf = embed_spmf (λx. enn2real (SUP p  Y. ennreal (spmf p x)))"
  ― ‹We go through typennreal to have a sensible definition even if termY is empty.›

lemma lub_spmf_empty [simp]: "SPMF.lub_spmf {} = return_pmf None"
by(simp add: SPMF.lub_spmf_def bot_ereal_def)

context assumes chain: "Complete_Partial_Order.chain (ord_spmf (=)) Y" begin

lemma chain_ord_spmf_eqD: "Complete_Partial_Order.chain (≤) ((λp x. ennreal (spmf p x)) ` Y)"
  (is "Complete_Partial_Order.chain _ (?f ` _)")
proof(rule chainI)
  fix f g
  assume "f  ?f ` Y" "g  ?f ` Y"
  then obtain p q where f: "f = ?f p" "p  Y" and g: "g = ?f q" "q  Y" by blast
  from chain p  Y q  Y have "ord_spmf (=) p q  ord_spmf (=) q p" by(rule chainD)
  thus "f  g  g  f"
  proof
    assume "ord_spmf (=) p q"
    hence "x. spmf p x  spmf q x" by(rule ord_spmf_eq_leD)
    hence "f  g" unfolding f g by(auto intro: le_funI)
    thus ?thesis ..
  next
    assume "ord_spmf (=) q p"
    hence "x. spmf q x  spmf p x" by(rule ord_spmf_eq_leD)
    hence "g  f" unfolding f g by(auto intro: le_funI)
    thus ?thesis ..
  qed
qed

lemma ord_spmf_eq_pmf_None_eq:
  assumes le: "ord_spmf (=) p q"
  and None: "pmf p None = pmf q None"
  shows "p = q"
proof(rule spmf_eqI)
  fix i
  from le have le': "x. spmf p x  spmf q x" by(rule ord_spmf_eq_leD)
  have "(+ x. ennreal (spmf q x) - spmf p x count_space UNIV) =
     (+ x. spmf q x count_space UNIV) - (+ x. spmf p x count_space UNIV)"
    by(subst nn_integral_diff)(simp_all add: AE_count_space le' nn_integral_spmf_neq_top)
  also have " = (1 - pmf q None) - (1 - pmf p None)" unfolding pmf_None_eq_weight_spmf
    by(simp add: weight_spmf_eq_nn_integral_spmf[symmetric] ennreal_minus)
  also have " = 0" using None by simp
  finally have "x. spmf q x  spmf p x"
    by(simp add: nn_integral_0_iff_AE AE_count_space ennreal_minus ennreal_eq_0_iff)
  with le' show "spmf p i = spmf q i" by(rule antisym)
qed

lemma ord_spmf_eqD_pmf_None:
  assumes "ord_spmf (=) x y"
  shows "pmf x None  pmf y None"
using assms
apply cases
apply(clarsimp simp only: ennreal_le_iff[symmetric, OF pmf_nonneg] ennreal_pmf_map)
apply(fastforce simp add: AE_measure_pmf_iff intro!: nn_integral_mono_AE)
done

text ‹
  Chains on typ'a spmf maintain countable support.
  Thanks to Johannes Hölzl for the proof idea.
›
lemma spmf_chain_countable: "countable (pY. set_spmf p)"
proof(cases "Y = {}")
  case Y: False
  show ?thesis
  proof(cases "xY. yY. ord_spmf (=) y x")
    case True
    then obtain x where x: "x  Y" and upper: "y. y  Y  ord_spmf (=) y x" by blast
    hence "(xY. set_spmf x)  set_spmf x" by(auto dest: ord_spmf_eqD_set_spmf)
    thus ?thesis by(rule countable_subset) simp
  next
    case False
    define N :: "'a option pmf  real" where "N p = pmf p None" for p

    have N_less_imp_le_spmf: " x  Y; y  Y; N y < N x   ord_spmf (=) x y" for x y
      using chainD[OF chain, of x y] ord_spmf_eqD_pmf_None[of x y] ord_spmf_eqD_pmf_None[of y x]
      by (auto simp: N_def)
    have N_eq_imp_eq: " x  Y; y  Y; N y = N x   x = y" for x y
      using chainD[OF chain, of x y] by(auto simp add: N_def dest: ord_spmf_eq_pmf_None_eq)

    have NC: "N ` Y  {}" "bdd_below (N ` Y)"
      using Y  {} by(auto intro!: bdd_belowI[of _ 0] simp: N_def)
    have NC_less: "Inf (N ` Y) < N x" if "x  Y" for x unfolding cInf_less_iff[OF NC]
    proof(rule ccontr)
      assume **: "¬ (yN ` Y. y < N x)"
      { fix y
        assume "y  Y"
        with ** consider "N x < N y" | "N x = N y" by(auto simp add: not_less le_less)
        hence "ord_spmf (=) y x" using y  Y x  Y
          by cases(auto dest: N_less_imp_le_spmf N_eq_imp_eq intro: ord_spmf_reflI) }
      with False x  Y show False by blast
    qed

    from NC have "Inf (N ` Y)  closure (N ` Y)" by (intro closure_contains_Inf)
    then obtain X' where "n. X' n  N ` Y" and X': "X'  Inf (N ` Y)"
      unfolding closure_sequential by auto
    then obtain X where