src/HOL/Probability/Probability_Mass_Function.thy
 author hoelzl Mon Oct 06 16:27:07 2014 +0200 (2014-10-06) changeset 58587 5484f6079bcd child 58606 9c66f7c541fb permissions -rw-r--r--
add type for probability mass functions, i.e. discrete probability distribution
```     1 theory Probability_Mass_Function
```
```     2   imports Probability_Measure
```
```     3 begin
```
```     4
```
```     5 lemma sets_Pair: "{x} \<in> sets M1 \<Longrightarrow> {y} \<in> sets M2 \<Longrightarrow> {(x, y)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
```
```     6   using pair_measureI[of "{x}" M1 "{y}" M2] by simp
```
```     7
```
```     8 lemma finite_subset_card:
```
```     9   assumes X: "infinite X" shows "\<exists>A\<subseteq>X. finite A \<and> card A = n"
```
```    10 proof (induct n)
```
```    11   case (Suc n) then guess A .. note A = this
```
```    12   with X obtain x where "x \<in> X" "x \<notin> A"
```
```    13     by (metis subset_antisym subset_eq)
```
```    14   with A show ?case
```
```    15     by (intro exI[of _ "insert x A"]) auto
```
```    16 qed (simp cong: conj_cong)
```
```    17
```
```    18 lemma (in prob_space) countable_support:
```
```    19   "countable {x. measure M {x} \<noteq> 0}"
```
```    20 proof -
```
```    21   let ?m = "\<lambda>x. measure M {x}"
```
```    22   have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. inverse (real (Suc n)) < ?m x})"
```
```    23     by (auto intro!: measure_nonneg reals_Archimedean order_le_neq_trans)
```
```    24   have **: "\<And>n. finite {x. inverse (Suc n) < ?m x}"
```
```    25   proof (rule ccontr)
```
```    26     fix n assume "infinite {x. inverse (Suc n) < ?m x}" (is "infinite ?X")
```
```    27     then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"
```
```    28       by (metis finite_subset_card)
```
```    29     from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> 1 / Suc n \<le> ?m x"
```
```    30       by (auto simp: inverse_eq_divide)
```
```    31     { fix x assume "x \<in> X"
```
```    32       from *[OF this] have "?m x \<noteq> 0" by auto
```
```    33       then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }
```
```    34     note singleton_sets = this
```
```    35     have "1 < (\<Sum>x\<in>X. 1 / Suc n)"
```
```    36       by (simp add: `card X = Suc (Suc n)` real_eq_of_nat[symmetric] real_of_nat_Suc)
```
```    37     also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
```
```    38       by (rule setsum_mono) fact
```
```    39     also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
```
```    40       using singleton_sets `finite X`
```
```    41       by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)
```
```    42     finally show False
```
```    43       using prob_le_1[of "\<Union>x\<in>X. {x}"] by arith
```
```    44   qed
```
```    45   show ?thesis
```
```    46     unfolding * by (intro countable_UN countableI_type countable_finite[OF **])
```
```    47 qed
```
```    48
```
```    49 lemma measure_count_space: "measure (count_space A) X = (if X \<subseteq> A then card X else 0)"
```
```    50   unfolding measure_def
```
```    51   by (cases "finite X") (simp_all add: emeasure_notin_sets)
```
```    52
```
```    53 typedef 'a pmf = "{M :: 'a measure. prob_space M \<and> sets M = UNIV \<and> (AE x in M. measure M {x} \<noteq> 0)}"
```
```    54   morphisms measure_pmf Abs_pmf
```
```    55   apply (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"])
```
```    56   apply (auto intro!: prob_space_uniform_measure simp: measure_count_space)
```
```    57   apply (subst uniform_measure_def)
```
```    58   apply (simp add: AE_density AE_count_space split: split_indicator)
```
```    59   done
```
```    60
```
```    61 declare [[coercion measure_pmf]]
```
```    62
```
```    63 lemma prob_space_measure_pmf: "prob_space (measure_pmf p)"
```
```    64   using pmf.measure_pmf[of p] by auto
```
```    65
```
```    66 interpretation measure_pmf!: prob_space "measure_pmf M" for M
```
```    67   by (rule prob_space_measure_pmf)
```
```    68
```
```    69 locale pmf_as_measure
```
```    70 begin
```
```    71
```
```    72 setup_lifting type_definition_pmf
```
```    73
```
```    74 end
```
```    75
```
```    76 context
```
```    77 begin
```
```    78
```
```    79 interpretation pmf_as_measure .
```
```    80
```
```    81 lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" .
```
```    82
```
```    83 lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" .
```
```    84
```
```    85 lift_definition map_pmf :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf" is
```
```    86   "\<lambda>f M. distr M (count_space UNIV) f"
```
```    87 proof safe
```
```    88   fix M and f :: "'a \<Rightarrow> 'b"
```
```    89   let ?D = "distr M (count_space UNIV) f"
```
```    90   assume "prob_space M" and [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
```
```    91   interpret prob_space M by fact
```
```    92   from ae have "AE x in M. measure M (f -` {f x}) \<noteq> 0"
```
```    93   proof eventually_elim
```
```    94     fix x
```
```    95     have "measure M {x} \<le> measure M (f -` {f x})"
```
```    96       by (intro finite_measure_mono) auto
```
```    97     then show "measure M {x} \<noteq> 0 \<Longrightarrow> measure M (f -` {f x}) \<noteq> 0"
```
```    98       using measure_nonneg[of M "{x}"] by auto
```
```    99   qed
```
```   100   then show "AE x in ?D. measure ?D {x} \<noteq> 0"
```
```   101     by (simp add: AE_distr_iff measure_distr measurable_def)
```
```   102 qed (auto simp: measurable_def prob_space.prob_space_distr)
```
```   103
```
```   104 declare [[coercion set_pmf]]
```
```   105
```
```   106 lemma countable_set_pmf: "countable (set_pmf p)"
```
```   107   by transfer (metis prob_space.countable_support)
```
```   108
```
```   109 lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV"
```
```   110   by transfer metis
```
```   111
```
```   112 lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV"
```
```   113   using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp
```
```   114
```
```   115 lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N"
```
```   116   by (auto simp: measurable_def)
```
```   117
```
```   118 lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)"
```
```   119   by (intro measurable_cong_sets) simp_all
```
```   120
```
```   121 lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x"
```
```   122   by transfer (simp add: less_le measure_nonneg)
```
```   123
```
```   124 lemma pmf_nonneg: "0 \<le> pmf p x"
```
```   125   by transfer (simp add: measure_nonneg)
```
```   126
```
```   127 lemma emeasure_pmf_single:
```
```   128   fixes M :: "'a pmf"
```
```   129   shows "emeasure M {x} = pmf M x"
```
```   130   by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
```
```   131
```
```   132 lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M"
```
```   133   by transfer simp
```
```   134
```
```   135 lemma emeasure_pmf_single_eq_zero_iff:
```
```   136   fixes M :: "'a pmf"
```
```   137   shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M"
```
```   138   by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
```
```   139
```
```   140 lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)"
```
```   141 proof -
```
```   142   { fix y assume y: "y \<in> M" and P: "AE x in M. P x" "\<not> P y"
```
```   143     with P have "AE x in M. x \<noteq> y"
```
```   144       by auto
```
```   145     with y have False
```
```   146       by (simp add: emeasure_pmf_single_eq_zero_iff AE_iff_measurable[OF _ refl]) }
```
```   147   then show ?thesis
```
```   148     using AE_measure_pmf[of M] by auto
```
```   149 qed
```
```   150
```
```   151 lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
```
```   152 proof (transfer, elim conjE)
```
```   153   fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
```
```   154   assume "prob_space M" then interpret prob_space M .
```
```   155   show "M = density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))"
```
```   156   proof (rule measure_eqI)
```
```   157     fix A :: "'a set"
```
```   158     have "(\<integral>\<^sup>+ x. ereal (measure M {x}) * indicator A x \<partial>count_space UNIV) =
```
```   159       (\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
```
```   160       by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
```
```   161     also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))"
```
```   162       by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
```
```   163     also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})"
```
```   164       by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
```
```   165          (auto simp: disjoint_family_on_def)
```
```   166     also have "\<dots> = emeasure M A"
```
```   167       using ae by (intro emeasure_eq_AE) auto
```
```   168     finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))) A"
```
```   169       using emeasure_space_1 by (simp add: emeasure_density)
```
```   170   qed simp
```
```   171 qed
```
```   172
```
```   173 lemma set_pmf_not_empty: "set_pmf M \<noteq> {}"
```
```   174   using AE_measure_pmf[of M] by (intro notI) simp
```
```   175
```
```   176 lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0"
```
```   177   by transfer simp
```
```   178
```
```   179 lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
```
```   180 proof -
```
```   181   have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
```
```   182     by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
```
```   183   then show ?thesis
```
```   184     using measure_pmf.emeasure_space_1 by simp
```
```   185 qed
```
```   186
```
```   187 lemma map_pmf_id[simp]: "map_pmf id = id"
```
```   188   by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
```
```   189
```
```   190 lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
```
```   191   by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def)
```
```   192
```
```   193 lemma map_pmf_cong:
```
```   194   assumes "p = q"
```
```   195   shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g q"
```
```   196   unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] map_pmf.rep_eq
```
```   197   by (auto simp add: emeasure_distr AE_measure_pmf_iff intro!: emeasure_eq_AE measure_eqI)
```
```   198
```
```   199 lemma pmf_set_map:
```
```   200   fixes f :: "'a \<Rightarrow> 'b"
```
```   201   shows "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
```
```   202 proof (rule, transfer, clarsimp simp add: measure_distr measurable_def)
```
```   203   fix f :: "'a \<Rightarrow> 'b" and M :: "'a measure"
```
```   204   assume "prob_space M" and ae: "AE x in M. measure M {x} \<noteq> 0" and [simp]: "sets M = UNIV"
```
```   205   interpret prob_space M by fact
```
```   206   show "{x. measure M (f -` {x}) \<noteq> 0} = f ` {x. measure M {x} \<noteq> 0}"
```
```   207   proof safe
```
```   208     fix x assume "measure M (f -` {x}) \<noteq> 0"
```
```   209     moreover have "measure M (f -` {x}) = measure M {y. f y = x \<and> measure M {y} \<noteq> 0}"
```
```   210       using ae by (intro finite_measure_eq_AE) auto
```
```   211     ultimately have "{y. f y = x \<and> measure M {y} \<noteq> 0} \<noteq> {}"
```
```   212       by (metis measure_empty)
```
```   213     then show "x \<in> f ` {x. measure M {x} \<noteq> 0}"
```
```   214       by auto
```
```   215   next
```
```   216     fix x assume "measure M {x} \<noteq> 0"
```
```   217     then have "0 < measure M {x}"
```
```   218       using measure_nonneg[of M "{x}"] by auto
```
```   219     also have "measure M {x} \<le> measure M (f -` {f x})"
```
```   220       by (intro finite_measure_mono) auto
```
```   221     finally show "measure M (f -` {f x}) = 0 \<Longrightarrow> False"
```
```   222       by simp
```
```   223   qed
```
```   224 qed
```
```   225
```
```   226 context
```
```   227   fixes f :: "'a \<Rightarrow> real"
```
```   228   assumes nonneg: "\<And>x. 0 \<le> f x"
```
```   229   assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
```
```   230 begin
```
```   231
```
```   232 lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ereal \<circ> f)"
```
```   233 proof (intro conjI)
```
```   234   have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
```
```   235     by (simp split: split_indicator)
```
```   236   show "AE x in density (count_space UNIV) (ereal \<circ> f).
```
```   237     measure (density (count_space UNIV) (ereal \<circ> f)) {x} \<noteq> 0"
```
```   238     by (simp add: AE_density nonneg emeasure_density measure_def nn_integral_cmult_indicator)
```
```   239   show "prob_space (density (count_space UNIV) (ereal \<circ> f))"
```
```   240     by default (simp add: emeasure_density prob)
```
```   241 qed simp
```
```   242
```
```   243 lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
```
```   244 proof transfer
```
```   245   have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
```
```   246     by (simp split: split_indicator)
```
```   247   fix x show "measure (density (count_space UNIV) (ereal \<circ> f)) {x} = f x"
```
```   248     by transfer (simp add: measure_def emeasure_density nn_integral_cmult_indicator nonneg)
```
```   249 qed
```
```   250
```
```   251 end
```
```   252
```
```   253 lemma embed_pmf_transfer:
```
```   254   "rel_fun (eq_onp (\<lambda>f::'a \<Rightarrow> real. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ereal (f x) \<partial>count_space UNIV) = 1)) pmf_as_measure.cr_pmf (\<lambda>f. density (count_space UNIV) (ereal \<circ> f)) embed_pmf"
```
```   255   by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)
```
```   256
```
```   257 lemma td_pmf_embed_pmf:
```
```   258   "type_definition pmf embed_pmf {f::'a \<Rightarrow> real. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ereal (f x) \<partial>count_space UNIV) = 1}"
```
```   259   unfolding type_definition_def
```
```   260 proof safe
```
```   261   fix p :: "'a pmf"
```
```   262   have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1"
```
```   263     using measure_pmf.emeasure_space_1[of p] by simp
```
```   264   then show *: "(\<integral>\<^sup>+ x. ereal (pmf p x) \<partial>count_space UNIV) = 1"
```
```   265     by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)
```
```   266
```
```   267   show "embed_pmf (pmf p) = p"
```
```   268     by (intro measure_pmf_inject[THEN iffD1])
```
```   269        (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
```
```   270 next
```
```   271   fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
```
```   272   then show "pmf (embed_pmf f) = f"
```
```   273     by (auto intro!: pmf_embed_pmf)
```
```   274 qed (rule pmf_nonneg)
```
```   275
```
```   276 end
```
```   277
```
```   278 locale pmf_as_function
```
```   279 begin
```
```   280
```
```   281 setup_lifting td_pmf_embed_pmf
```
```   282
```
```   283 end
```
```   284
```
```   285 (*
```
```   286
```
```   287 definition
```
```   288   "rel_pmf P d1 d2 \<longleftrightarrow> (\<exists>p3. (\<forall>(x, y) \<in> set_pmf p3. P x y) \<and> map_pmf fst p3 = d1 \<and> map_pmf snd p3 = d2)"
```
```   289
```
```   290 lift_definition pmf_join :: "real \<Rightarrow> 'a pmf \<Rightarrow> 'a pmf \<Rightarrow> 'a pmf" is
```
```   291   "\<lambda>p M1 M2. density (count_space UNIV) (\<lambda>x. p * measure M1 {x} + (1 - p) * measure M2 {x})"
```
```   292 sorry
```
```   293
```
```   294 lift_definition pmf_single :: "'a \<Rightarrow> 'a pmf" is
```
```   295   "\<lambda>x. uniform_measure (count_space UNIV) {x}"
```
```   296 sorry
```
```   297
```
```   298 bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: pmf_rel
```
```   299 proof -
```
```   300   show "map_pmf id = id" by (rule map_pmf_id)
```
```   301   show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose)
```
```   302   show "\<And>f g::'a \<Rightarrow> 'b. \<And>p. (\<And>x. x \<in> set_pmf p \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g p"
```
```   303     by (intro map_pmg_cong refl)
```
```   304
```
```   305   show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
```
```   306     by (rule pmf_set_map)
```
```   307
```
```   308   { fix p :: "'s pmf"
```
```   309     have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq"
```
```   310       by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
```
```   311          (auto intro: countable_set_pmf inj_on_to_nat_on)
```
```   312     also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
```
```   313       by (metis Field_natLeq card_of_least natLeq_Well_order)
```
```   314     finally show "(card_of (set_pmf p), natLeq) \<in> ordLeq" . }
```
```   315
```
```   316   show "\<And>R. pmf_rel R =
```
```   317          (BNF_Util.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO
```
```   318          BNF_Util.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)"
```
```   319      by (auto simp add: fun_eq_iff pmf_rel_def BNF_Util.Grp_def OO_def)
```
```   320
```
```   321   { let ?f = "map_pmf fst" and ?s = "map_pmf snd"
```
```   322     fix R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and A assume "\<And>x y. (x, y) \<in> set_pmf A \<Longrightarrow> R x y"
```
```   323     fix S :: "'b \<Rightarrow> 'c \<Rightarrow> bool" and B assume "\<And>y z. (y, z) \<in> set_pmf B \<Longrightarrow> S y z"
```
```   324     assume "?f B = ?s A"
```
```   325     have "\<exists>C. (\<forall>(x, z)\<in>set_pmf C. \<exists>y. R x y \<and> S y z) \<and> ?f C = ?f A \<and> ?s C = ?s B"
```
```   326       sorry }
```
```   327 oops
```
```   328   then show "\<And>R::'a \<Rightarrow> 'b \<Rightarrow> bool. \<And>S::'b \<Rightarrow> 'c \<Rightarrow> bool. pmf_rel R OO pmf_rel S \<le> pmf_rel (R OO S)"
```
```   329       by (auto simp add: subset_eq pmf_rel_def fun_eq_iff OO_def Ball_def)
```
```   330 qed (fact natLeq_card_order natLeq_cinfinite)+
```
```   331
```
```   332 notepad
```
```   333 begin
```
```   334   fix x y :: "nat \<Rightarrow> real"
```
```   335   def IJz \<equiv> "rec_nat ((0, 0), \<lambda>_. 0) (\<lambda>n ((I, J), z).
```
```   336     let a = x I - (\<Sum>j<J. z (I, j)) ; b = y J - (\<Sum>i<I. z (i, J)) in
```
```   337       ((if a \<le> b then I + 1 else I, if b \<le> a then J + 1 else J), z((I, J) := min a b)))"
```
```   338   def I == "fst \<circ> fst \<circ> IJz" def J == "snd \<circ> fst \<circ> IJz" def z == "snd \<circ> IJz"
```
```   339   let ?a = "\<lambda>n. x (I n) - (\<Sum>j<J n. z n (I n, j))" and ?b = "\<lambda>n. y (J n) - (\<Sum>i<I n. z n (i, J n))"
```
```   340   have IJz_0[simp]: "\<And>p. z 0 p = 0" "I 0 = 0" "J 0 = 0"
```
```   341     by (simp_all add: I_def J_def z_def IJz_def)
```
```   342   have z_Suc[simp]: "\<And>n. z (Suc n) = (z n)((I n, J n) := min (?a n) (?b n))"
```
```   343     by (simp add: z_def I_def J_def IJz_def Let_def split_beta)
```
```   344   have I_Suc[simp]: "\<And>n. I (Suc n) = (if ?a n \<le> ?b n then I n + 1 else I n)"
```
```   345     by (simp add: z_def I_def J_def IJz_def Let_def split_beta)
```
```   346   have J_Suc[simp]: "\<And>n. J (Suc n) = (if ?b n \<le> ?a n then J n + 1 else J n)"
```
```   347     by (simp add: z_def I_def J_def IJz_def Let_def split_beta)
```
```   348
```
```   349   { fix N have "\<And>p. z N p \<noteq> 0 \<Longrightarrow> \<exists>n<N. p = (I n, J n)"
```
```   350       by (induct N) (auto simp add: less_Suc_eq split: split_if_asm) }
```
```   351
```
```   352   { fix i n assume "i < I n"
```
```   353     then have "(\<Sum>j. z n (i, j)) = x i"
```
```   354     oops
```
```   355 *)
```
```   356
```
```   357 end
```
```   358
```