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Datei: vect_metric_space.prf   Sprache: Lisp

Untersuchung PVS©

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    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (sigma_algebra? const-decl "bool" subset_algebra_def nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (intersection const-decl "set" sets nil)
    (measurable_set nonempty-type-eq-decl nil measure_space_def nil)
    (measurable_set? const-decl "bool" measure_space_def nil)
    (m_monotone formula-decl nil measure_props nil)
    (subset? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (x_le const-decl "bool" extended_nnreal "extended_nnreal/")
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (mu_fin? const-decl "bool" measure_props nil))
   nil))
 (null_difference 0
  (null_difference-1 nil 3509948281
   ("" (skosimp)
    ((""
      (lemma "m_monotone" ("a" "difference[T](N1!1, N2!1)" "b" "N1!1"))
      (("" (typepred "N1!1")
        (("" (typepred "N2!1")
          (("" (split)
            (("1" (expand "null_set?")
              (("1" (expand "x_le")
                (("1" (flatten)
                  (("1" (assert)
                    (("1" (expand "mu_fin?")
                      (("1" (expand "mu")
                        (("1" (assert)
                          (("1" (flatten) (("1" (assertnil nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (hide-all-but 1) (("2" (grind) nil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((m formal-const-decl "measure_type" measure_theory nil)
    (measure_type nonempty-type-eq-decl nil generalized_measure_def
     nil)
    (measure? const-decl "bool" generalized_measure_def nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (S formal-const-decl "sigma_algebra" measure_theory nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (sigma_algebra? const-decl "bool" subset_algebra_def nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil measure_theory nil)
    (null_set nonempty-type-eq-decl nil measure_theory nil)
    (null_set? const-decl "bool" measure_theory nil)
    (difference const-decl "set" sets nil)
    (measurable_set nonempty-type-eq-decl nil measure_space_def nil)
    (measurable_set? const-decl "bool" measure_space_def nil)
    (set type-eq-decl nil sets nil)
    (m_monotone formula-decl nil measure_props nil)
    (measurable_difference application-judgement "measurable_set[T, S]"
     measure_theory nil)
    (subset? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (mu_fin? const-decl "bool" measure_props nil)
    (mu const-decl "nnreal" measure_props nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (x_le const-decl "bool" extended_nnreal "extended_nnreal/")
    (NOT const-decl "[bool -> bool]" booleans nil))
   nil))
 (null_IUnion 0
  (null_IUnion-1 nil 3509948281
   ("" (skosimp)
    (("" (typepred "m")
      (("" (lemma "disjoint_IUnion[T]" ("A" "NS!1"))
        (("" (skosimp)
          (("" (replace -5)
            (("" (case "forall (n:nat): null_set?(B!1(n))")
              (("1" (hide -3 -4 -5 -6)
                (("1" (expand "measure?")
                  (("1" (flatten)
                    (("1" (expand "measure_countably_additive?")
                      (("1" (expand "null_set?")
                        (("1" (expand "mu_fin?")
                          (("1" (expand "mu")
                            (("1" (assert)
                              (("1"
                                (rewrite "measurable_IUnion")
                                (("1"
                                  (inst -4 "B!1")
                                  (("1"
                                    (expand "x_eq")
                                    (("1"
                                      (flatten)
                                      (("1"
                                        (assert)
                                        (("1"
                                          (expand "o ")
                                          (("1"
                                            (expand "x_sum")
                                            (("1"
                                              (case-replace
                                               "FORALL (i:nat): m(B!1(i))`1")
                                              (("1"
                                                (case-replace
                                                 "series(LAMBDA i: m(B!1(i))`2)=lambda (n:nat): 0")
                                                (("1"
                                                  (hide -1 -2 -3 -4 -5)
                                                  (("1"
                                                    (case-replace
                                                     "convergence_sequences.convergent?(LAMBDA (n: nat): 0)")
                                                    (("1"
                                                      (assert)
                                                      (("1"
                                                        (lemma
                                                         "convergence_sequences.limit_lemma"
                                                         ("v"
                                                          "LAMBDA (n: nat): 0"))
                                                        (("1"
                                                          (replace -4)
                                                          (("1"
                                                            (hide-all-but
                                                             (-1 1))
                                                            (("1"
                                                              (expand
                                                               "convergence")
                                                              (("1"
                                                                (typepred
                                                                 "m(IUnion(B!1))`2")
                                                                (("1"
                                                                  (name-replace
                                                                   "BB"
                                                                   "m(IUnion(B!1))`2")
                                                                  (("1"
                                                                    (expand
                                                                     ">="
                                                                     -1)
                                                                    (("1"
                                                                      (expand
                                                                       "<="
                                                                       -1)
                                                                      (("1"
                                                                        (split)
                                                                        (("1"
                                                                          (inst
                                                                           -
                                                                           "BB")
                                                                          (("1"
                                                                            (skosimp)
                                                                            (("1"
                                                                              (inst
                                                                               -
                                                                               "n!1")
                                                                              (("1"
                                                                                (assert)
                                                                                (("1"
                                                                                  (grind)
                                                                                  nil
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil)
                                                                           ("2"
                                                                            (assert)
                                                                            nil
                                                                            nil))
                                                                          nil)
                                                                         ("2"
                                                                          (assert)
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (hide-all-but 1)
                                                      (("2"
                                                        (expand
                                                         "convergent?")
                                                        (("2"
                                                          (inst + "0")
                                                          (("2"
                                                            (expand
                                                             "convergence")
                                                            (("2"
                                                              (skosimp)
                                                              (("2"
                                                                (inst
                                                                 +
                                                                 "0")
                                                                (("2"
                                                                  (skosimp)
                                                                  (("2"
                                                                    (grind)
                                                                    nil
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (hide-all-but
                                                   (-1 -2 1))
                                                  (("2"
                                                    (apply-extensionality
                                                     :hide?
                                                     t)
                                                    (("1"
                                                      (expand "series")
                                                      (("1"
                                                        (case-replace
                                                         "(LAMBDA (i:nat): m(B!1(i))`2) = lambda (i:nat): 0")
                                                        (("1"
                                                          (rewrite
                                                           "sigma_zero[nat]")
                                                          nil
                                                          nil)
                                                         ("2"
                                                          (apply-extensionality
                                                           :hide?
                                                           t)
                                                          (("1"
                                                            (inst
                                                             -2
                                                             "x!2")
                                                            (("1"
                                                              (flatten)
                                                              nil
                                                              nil))
                                                            nil)
                                                           ("2"
                                                            (skosimp)
                                                            (("2"
                                                              (inst
                                                               -2
                                                               "i!1")
                                                              (("2"
                                                                (flatten)
                                                                (("2"
                                                                  (expand
                                                                   "measurable_set?")
                                                                  (("2"
                                                                    (propax)
                                                                    nil
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("3"
                                                          (skosimp)
                                                          (("3"
                                                            (inst
                                                             -2
                                                             "i!1")
                                                            (("3"
                                                              (flatten)
                                                              (("3"
                                                                (expand
                                                                 "measurable_set?")
                                                                (("3"
                                                                  (propax)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (skosimp)
                                                      (("2"
                                                        (inst -2 "i!1")
                                                        (("2"
                                                          (expand
                                                           "measurable_set?")
                                                          (("2"
                                                            (flatten)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("3"
                                                  (skosimp)
                                                  (("3"
                                                    (inst -2 "i!1")
                                                    (("3"
                                                      (flatten)
                                                      (("3"
                                                        (expand
                                                         "measurable_set?")
                                                        (("3"
                                                          (propax)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (replace 1)
                                                (("2"
                                                  (assert)
                                                  (("2"
                                                    (skosimp)
                                                    (("2"
                                                      (inst - "i!1")
                                                      (("2"
                                                        (flatten)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("3"
                                                (skosimp)
                                                (("3"
                                                  (inst - "i!1")
                                                  (("3"
                                                    (flatten)
                                                    (("3"
                                                      (expand
                                                       "measurable_set?")
                                                      (("3"
                                                        (propax)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (expand
                                     "disjoint_indexed_measurable?")
                                    (("2"
                                      (skolem + "n!1")
                                      (("2"
                                        (inst - "n!1")
                                        (("2"
                                          (flatten)
                                          (("2"
                                            (assert)
                                            (("2"
                                              (expand
                                               "measurable_set?")
                                              (("2" (propax) nil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (skolem + "n!1")
                                  (("2"
                                    (inst - "n!1")
                                    (("2" (flatten) nil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide 2 -5)
                (("2" (induct "n")
                  (("1" (replace -3) (("1" (assertnil nil)) nil)
                   ("2" (skosimp)
                    (("2" (inst -5 "j!1")
                      (("2" (inst -3 "j!1")
                        (("2" (replace -3 * rl)
                          (("2"
                            (lemma "null_difference"
                             ("N1" "NS!1(j!1+1)" "N2"
                              "IUnion(j!1, B!1)"))
                            (("1" (replace -6 -1 rl)
                              (("1" (propax) nil nil)) nil)
                             ("2" (replace -3)
                              (("2"
                                (hide-all-but 1)
                                (("2"
                                  (case
                                   "forall (n:nat): null_set?(IUnion(n, NS!1))")
                                  (("1" (inst - "j!1"nil nil)
                                   ("2"
                                    (hide 2)
                                    (("2"
                                      (induct "n")
                                      (("1"
                                        (rewrite "IUnion_0_def")
                                        (("1" (assertnil nil))
                                        nil)
                                       ("2"
                                        (skosimp)
                                        (("2"
                                          (rewrite "IUnion_n_def" 1)
                                          (("2"
                                            (rewrite "null_union")
                                            nil
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((m formal-const-decl "measure_type" measure_theory nil)
    (measure_type nonempty-type-eq-decl nil generalized_measure_def
     nil)
    (measure? const-decl "bool" generalized_measure_def nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (S formal-const-decl "sigma_algebra" measure_theory nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (sigma_algebra? const-decl "bool" subset_algebra_def nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (T formal-type-decl nil measure_theory nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (sigma_algebra_IUnion_rew application-judgement "(S)"
     measure_theory nil)
    (measure_countably_additive? const-decl "bool"
     generalized_measure_def nil)
    (mu_fin? const-decl "bool" measure_props nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (B!1 skolem-const-decl "sequence[set[T]]" measure_theory nil)
    (disjoint_indexed_measurable? const-decl "bool"
     generalized_measure_def nil)
    (disjoint_indexed_measurable nonempty-type-eq-decl nil
     generalized_measure_def nil)
    (O const-decl "T3" function_props nil)
    (sigma_nnreal application-judgement "nnreal" sigma_nat "reals/")
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (T_low type-eq-decl nil sigma "reals/")
    (T_high type-eq-decl nil sigma "reals/")
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (sigma_zero formula-decl nil sigma "reals/")
    (int_abs_is_nonneg application-judgement "{j: nonneg_int | j >= i}"
     real_defs nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (convergence const-decl "bool" convergence_sequences "analysis/")
    (<= const-decl "bool" reals nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (> const-decl "bool" reals nil)
    (BB skolem-const-decl "nnreal" measure_theory nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (IUnion const-decl "set[T]" indexed_sets nil)
    (limit_lemma formula-decl nil convergence_sequences "analysis/")
    (convergent? const-decl "bool" convergence_sequences "analysis/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (series const-decl "sequence[real]" series "series/")
    (x_sum const-decl "extended_nnreal" extended_nnreal
     "extended_nnreal/")
    (x_eq const-decl "bool" extended_nnreal "extended_nnreal/")
    (measurable_IUnion judgement-tcc nil measure_space_def nil)
    (measurable_set? const-decl "bool" measure_space_def nil)
    (measurable_set nonempty-type-eq-decl nil measure_space_def nil)
    (mu const-decl "nnreal" measure_props nil)
    (pred type-eq-decl nil defined_types nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (IUnion_0_def formula-decl nil nat_indexed_sets "sets_aux/")
    (IUnion_n_def formula-decl nil nat_indexed_sets "sets_aux/")
    (int_minus_int_is_int application-judgement "int" integers nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (null_union judgement-tcc nil measure_theory nil)
    (IUnion const-decl "set[T]" nat_indexed_sets "sets_aux/")
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (null_difference judgement-tcc nil measure_theory nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (null_set nonempty-type-eq-decl nil measure_theory nil)
    (null_set? const-decl "bool" measure_theory nil)
    (sequence type-eq-decl nil sequences nil)
    (set type-eq-decl nil sets nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (disjoint_IUnion formula-decl nil nat_indexed_sets "sets_aux/"))
   nil))
 (null_Union 0
  (null_Union-1 nil 3423545392
   ("" (skosimp)
    (("" (lemma "Union_IUnion" ("XS" "Z!1"))
      (("" (assert)
        (("" (skosimp)
          (("" (replace -1)
            (("" (rewrite "null_IUnion")
              (("" (hide -1 -4 2)
                (("" (skolem + ("n!1"))
                  (("" (inst - "n!1")
                    (("" (split -1)
                      (("1" (rewrite "emptyset_is_empty?")
                        (("1" (replace -1)
                          (("1" (rewrite "null_emptyset"nil nil))
                          nil))
                        nil)
                       ("2" (expand "every")
                        (("2" (inst - "YS!1(n!1)"nil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((T formal-type-decl nil measure_theory nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (Union_IUnion formula-decl nil countable_indexed_sets "sets_aux/")
    (null_IUnion judgement-tcc nil measure_theory nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (set type-eq-decl nil sets nil)
    (null_set? const-decl "bool" measure_theory nil)
    (null_set nonempty-type-eq-decl nil measure_theory nil)
    (sequence type-eq-decl nil sequences nil)
    (null_emptyset judgement-tcc nil measure_theory nil)
    (finite_emptyset name-judgement "finite_set" finite_sets nil)
    (finite_emptyset name-judgement "finite_set[T]" countable_props
     "sets_aux/")
    (finite_emptyset name-judgement "finite_set[T]" countable_setofsets
     "sets_aux/")
    (null_emptyset name-judgement "null_set" measure_theory nil)
    (subset_algebra_emptyset name-judgement "(S)" measure_theory nil)
    (emptyset_is_empty? formula-decl nil sets_lemmas nil)
    (Z!1 skolem-const-decl "setofsets[T]" measure_theory nil)
    (YS!1 skolem-const-decl "sequence[set[T]]" measure_theory nil)
    (n!1 skolem-const-decl "nat" measure_theory nil)
    (every const-decl "bool" sets nil)
    (Union_surjective name-judgement
     "(surjective?[setofsets[T], set[T]])" sets_lemmas nil))
   shostak))
 (negligible_union 0
  (negligible_union-1 nil 3509948281
   ("" (skosimp)
    (("" (typepred "E2!1")
      (("" (typepred "E1!1")
        (("" (expand "negligible_set?")
          (("" (skosimp*)
            (("" (inst + "union(X!1,X!2)")
              (("" (rewrite "null_union")
                (("" (hide -1 -3)
                  (("" (expand "subset?")
                    (("" (skosimp)
                      (("" (inst - "x!1")
                        (("" (inst - "x!1") (("" (grind) nil nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((negligible nonempty-type-eq-decl nil measure_theory nil)
    (negligible_set? const-decl "bool" measure_theory nil)
    (set type-eq-decl nil sets nil)
    (T formal-type-decl nil measure_theory nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (union const-decl "set" sets nil)
    (member const-decl "bool" sets nil)
    (subset? const-decl "bool" sets nil)
    (null_union judgement-tcc nil measure_theory nil)
    (null_set? const-decl "bool" measure_theory nil)
    (null_set nonempty-type-eq-decl nil measure_theory nil))
   nil))
 (negligible_intersection 0
  (negligible_intersection-1 nil 3509948281
   ("" (skosimp)
    (("" (typepred "E2!1")
      (("" (typepred "E1!1")
        (("" (expand "negligible_set?")
          (("" (skosimp*)
            (("" (inst + "X!1")
              (("" (assert)
                (("" (hide -1 -3)
                  (("" (expand "subset?")
                    (("" (skosimp)
                      (("" (inst - "x!1")
                        (("" (inst - "x!1") (("" (grind) nil nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((negligible nonempty-type-eq-decl nil measure_theory nil)
    (negligible_set? const-decl "bool" measure_theory nil)
    (set type-eq-decl nil sets nil)
    (T formal-type-decl nil measure_theory nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (member const-decl "bool" sets nil)
    (intersection const-decl "set" sets nil)
    (subset? const-decl "bool" sets nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil))
   nil))
 (negligible_IUnion 0
  (negligible_IUnion-1 nil 3509948281
   ("" (skosimp)
    (("" (case "forall (n:nat): exists N: subset?(ES!1(n),N)")
      (("1" (expand "negligible_set?")
        (("1"
          (name "NNS"
                "lambda (n:nat): choose[null_set]({ N | subset?(ES!1(n),N)})")
          (("1" (lemma "null_IUnion" ("NS" "NNS"))
            (("1" (inst + "IUnion(NNS)")
              (("1" (assert)
                (("1" (hide -2)
                  (("1" (expand "subset?" 1)
                    (("1" (skosimp)
                      (("1" (expand "member")
                        (("1" (expand "IUnion")
                          (("1" (hide -1)
                            (("1" (skosimp)
                              (("1"
                                (inst + "i!1")
                                (("1"
                                  (expand "NNS")
                                  (("1"
                                    (lemma
                                     "choose_member[null_set]"
                                     ("a"
                                      "{N | subset?(ES!1(i!1), N)}"))
                                    (("1"
                                      (split)
                                      (("1"
                                        (expand "member")
                                        (("1"
                                          (expand "subset?" -1 1)
                                          (("1"
                                            (expand "member")
                                            (("1"
                                              (inst - "x!1")
                                              (("1" (assertnil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (hide 1)
                                        (("2"
                                          (expand "empty?")
                                          (("2"
                                            (inst -2 "i!1")
                                            (("2"
                                              (skosimp)
                                              (("2"
                                                (inst - "N!1")
                                                (("2"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (skosimp)
            (("2" (inst - "n!1")
              (("2" (skosimp)
                (("2" (expand "nonempty?")
                  (("2" (expand "empty?")
                    (("2" (inst - "N!1") (("2" (assertnil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (hide 2)
        (("2" (skosimp)
          (("2" (typepred "ES!1(n!1)")
            (("2" (expand "negligible_set?")
              (("2" (skosimp) (("2" (inst + "X!1"nil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((sequence type-eq-decl nil sequences nil)
    (negligible nonempty-type-eq-decl nil measure_theory nil)
    (negligible_set? const-decl "bool" measure_theory nil)
    (subset? const-decl "bool" sets nil)
    (null_set nonempty-type-eq-decl nil measure_theory nil)
    (null_set? const-decl "bool" measure_theory nil)
    (set type-eq-decl nil sets nil)
    (T formal-type-decl nil measure_theory nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (choose const-decl "(p)" sets nil)
    (nonempty? const-decl "bool" sets nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (IUnion const-decl "set[T]" indexed_sets nil)
    (NNS skolem-const-decl "[n: nat -> ({N | subset?(ES!1(n), N)})]"
     measure_theory nil)
    (empty? const-decl "bool" sets nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (choose_member formula-decl nil sets_lemmas nil)
    (member const-decl "bool" sets nil)
    (null_IUnion application-judgement "null_set" measure_theory nil)
    (null_IUnion judgement-tcc nil measure_theory nil)
    (X!1 skolem-const-decl "set[T]" measure_theory nil)
    (NOT const-decl "[bool -> bool]" booleans nil))
   nil))
 (negligible_Union 0
  (negligible_Union-1 nil 3423653542
   ("" (skosimp)
    (("" (lemma "Union_IUnion" ("XS" "Z!1"))
      (("" (assert)
        (("" (skosimp)
          (("" (replace -1)
            (("" (rewrite "negligible_IUnion")
              (("" (hide -1 -4 2)
                (("" (skolem + ("n!1"))
                  (("" (inst - "n!1")
                    (("" (split -1)
                      (("1" (rewrite "emptyset_is_empty?")
                        (("1" (replace -1)
                          (("1" (hide -1 -2)
                            (("1" (expand "negligible_set?")
                              (("1"
                                (inst + "emptyset[T]")
                                (("1" (grind) nil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (expand "every")
                        (("2" (inst - "YS!1(n!1)"nil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((T formal-type-decl nil measure_theory nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (Union_IUnion formula-decl nil countable_indexed_sets "sets_aux/")
    (negligible_IUnion judgement-tcc nil measure_theory nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (set type-eq-decl nil sets nil)
    (negligible_set? const-decl "bool" measure_theory nil)
    (negligible nonempty-type-eq-decl nil measure_theory nil)
    (sequence type-eq-decl nil sequences nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (subset? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (emptyset const-decl "set" sets nil)
    (finite_emptyset name-judgement "finite_set" finite_sets nil)
    (finite_emptyset name-judgement "finite_set[T]" countable_props
     "sets_aux/")
    (finite_emptyset name-judgement "finite_set[T]" countable_setofsets
     "sets_aux/")
    (null_emptyset name-judgement "null_set" measure_theory nil)
    (subset_algebra_emptyset name-judgement "(S)" measure_theory nil)
    (emptyset_is_empty? formula-decl nil sets_lemmas nil)
    (Z!1 skolem-const-decl "setofsets[T]" measure_theory nil)
    (YS!1 skolem-const-decl "sequence[set[T]]" measure_theory nil)
    (n!1 skolem-const-decl "nat" measure_theory nil)
    (every const-decl "bool" sets nil)
    (Union_surjective name-judgement
     "(surjective?[setofsets[T], set[T]])" sets_lemmas nil))
   shostak))
 (negligible_subset 0
  (negligible_subset-1 nil 3395735108
   ("" (skosimp)
    (("" (typepred "E!1")
      (("" (expand "negligible_set?")
        (("" (skosimp)
          (("" (inst + "X!2")
            (("" (assert)
              (("" (hide -1)
                (("" (expand "subset?")
                  (("" (skosimp)
                    (("" (inst - "x!1")
                      (("" (inst - "x!1") (("" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
--> --------------------

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--> --------------------

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