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Datei: real_orders.prf   Sprache: Lisp

Original von: PVS©

(nnreal_expt
 (nnreal_expt_TCC1 0
  (nnreal_expt_TCC1-1 nil 3427171289
   ("" (skosimp)
    (("" (case "y!1>0")
      (("1" (hide 1 2)
        (("1" (split)
          (("1" (expand "nonempty?")
            (("1" (expand "empty?")
              (("1" (expand "extend")
                (("1" (expand "member")
                  (("1" (lemma "density_positive" ("x" "0" "y" "y!1"))
                    (("1" (assert)
                      (("1" (skosimp)
                        (("1" (inst - "nn_rational_expt(x!1, r!1)")
                          (("1" (typepred "nn_rational_expt(x!1, r!1)")
                            (("1" (assert)
                              (("1"
                                (inst + "r!1")
                                (("1" (assertnil nil))
                                nil))
                              nil)
                             ("2" (assertnil nil))
                            nil)
                           ("2" (assertnil nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (expand "bounded_below?")
            (("2" (inst + "0")
              (("2" (expand "lower_bound?")
                (("2" (skosimp)
                  (("2" (typepred "s!1")
                    (("2" (expand "extend") (("2" (assertnil nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (assertnil nil))
      nil))
    nil)
   ((nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil) (> const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans 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)
    (empty? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_gt_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)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rat nonempty-type-eq-decl nil rationals nil)
    (r!1 skolem-const-decl "rat" nnreal_expt nil)
    (nn_rational_expt const-decl "nnreal" nn_rational_expt nil)
    (nnrat nonempty-type-eq-decl nil rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (density_positive formula-decl nil rational_props nil)
    (extend const-decl "R" extend nil)
    (nonempty? const-decl "bool" sets nil)
    (FALSE const-decl "bool" booleans nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (lower_bound? const-decl "bool" bounded_real_defs nil)
    (bounded_below? const-decl "bool" bounded_real_defs nil))
   nil))
 (nnreal_expt_TCC2 0
  (nnreal_expt_TCC2-1 nil 3427171289
   ("" (skosimp)
    ((""
      (typepred "glb(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= y!1 AND z = nn_rational_expt(x!1, q)}))")
      ((""
        (name-replace "GLB" "glb(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= y!1 AND z = nn_rational_expt(x!1, q)}))")
        (("" (expand "greatest_lower_bound?")
          (("" (flatten)
            (("" (inst - "0")
              (("" (assert)
                (("" (hide -1 4)
                  (("" (expand "lower_bound?")
                    (("" (skosimp)
                      (("" (typepred "s!1")
                        (("" (assert)
                          (("" (expand "extend")
                            (("" (propax) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nn_rational_expt const-decl "nnreal" nn_rational_expt nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (<= const-decl "bool" reals nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (nnrat nonempty-type-eq-decl nil rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (extend const-decl "R" extend nil)
    (FALSE const-decl "bool" booleans nil)
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (glb const-decl "{x | greatest_lower_bound?(x, SB)}"
     bounded_real_defs nil)
    (bounded_below? const-decl "bool" bounded_real_defs nil)
    (greatest_lower_bound? const-decl "bool" bounded_real_defs nil)
    (nonempty? const-decl "bool" sets nil)
    (set type-eq-decl nil sets 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)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (lower_bound? const-decl "bool" bounded_real_defs 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))
   nil))
 (nnreal_expt_TCC3 0
  (nnreal_expt_TCC3-1 nil 3427223359
   ("" (skosimp*)
    (("" (split)
      (("1" (expand "nonempty?")
        (("1" (expand "empty?")
          (("1" (expand "extend")
            (("1" (expand "member")
              (("1" (lemma "density_positive" ("x" "0" "y" "y!1"))
                (("1" (assert)
                  (("1" (skosimp)
                    (("1" (inst - "nn_rational_expt(x!1, r!1)")
                      (("1" (typepred "nn_rational_expt(x!1, r!1)")
                        (("1" (assert)
                          (("1" (inst + "r!1") (("1" (assertnil nil))
                            nil))
                          nil)
                         ("2" (assertnil nil))
                        nil)
                       ("2" (assertnil nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (expand "bounded_above?")
        (("2" (expand "extend")
          (("2" (expand "upper_bound?")
            (("2" (lemma "density_positive" ("x" "y!1" "y" "y!1+1"))
              (("2" (assert)
                (("2" (skosimp)
                  (("2" (inst + "nn_rational_expt(x!1, r!1)")
                    (("1" (skosimp)
                      (("1" (typepred "s!1")
                        (("1" (assert)
                          (("1" (skosimp)
                            (("1" (replace -2)
                              (("1"
                                (lemma
                                 "nn_rational_expt_increasing"
                                 ("q" "q!1" "r" "r!1" "x" "x!1"))
                                (("1" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (assertnil nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((empty? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props 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)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rat nonempty-type-eq-decl nil rationals nil)
    (r!1 skolem-const-decl "rat" nnreal_expt nil)
    (nn_rational_expt const-decl "nnreal" nn_rational_expt nil)
    (nnrat nonempty-type-eq-decl nil rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (density_positive formula-decl nil rational_props nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans 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)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (extend const-decl "R" extend nil)
    (nonempty? const-decl "bool" sets nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (nnreal_plus_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (posreal_plus_nnreal_is_posreal application-judgement "posreal"
     real_types nil)
    (nn_rational_expt_increasing formula-decl nil nn_rational_expt nil)
    (FALSE const-decl "bool" booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (<= const-decl "bool" reals nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (r!1 skolem-const-decl "rat" nnreal_expt nil)
    (upper_bound? const-decl "bool" bounded_real_defs nil)
    (bounded_above? const-decl "bool" bounded_real_defs nil))
   nil))
 (nnreal_expt_TCC4 0
  (nnreal_expt_TCC4-1 nil 3427223359
   ("" (skosimp*)
    ((""
      (typepred "lub(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= y!1 AND z = nn_rational_expt(x!1, q)}))")
      ((""
        (name-replace "LUB" "lub(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= y!1 AND z = nn_rational_expt(x!1, q)}))")
        (("" (expand "least_upper_bound?")
          (("" (flatten)
            (("" (hide -2)
              (("" (expand "upper_bound?")
                (("" (inst - "nn_rational_expt(x!1, 0)")
                  (("1" (typepred "nn_rational_expt(x!1, 0)")
                    (("1" (assertnil nil)) nil)
                   ("2" (expand "extend")
                    (("2" (typepred "nn_rational_expt(x!1, 0)")
                      (("2" (assert)
                        (("2" (inst + "0") (("2" (assertnil nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nn_rational_expt const-decl "nnreal" nn_rational_expt nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (<= const-decl "bool" reals nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (nnrat nonempty-type-eq-decl nil rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (extend const-decl "R" extend nil)
    (FALSE const-decl "bool" booleans nil)
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (lub const-decl "{x | least_upper_bound?(x, SA)}" bounded_real_defs
     nil)
    (bounded_above? const-decl "bool" bounded_real_defs nil)
    (least_upper_bound? const-decl "bool" bounded_real_defs nil)
    (nonempty? const-decl "bool" sets nil)
    (set type-eq-decl nil sets 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)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (y!1 skolem-const-decl "nnreal" nnreal_expt nil)
    (x!1 skolem-const-decl "nnreal" nnreal_expt nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (upper_bound? const-decl "bool" bounded_real_defs nil))
   nil))
 (nnreal_expt_rat_rew 0
  (nnreal_expt_rat_rew-1 nil 3427179456
   ("" (skosimp)
    (("" (case-replace "q!1=0")
      (("1" (expand "nnreal_expt")
        (("1" (rewrite "nn_rational_expt_nat_rew")
          (("1" (grind) nil nil)) nil))
        nil)
       ("2" (case "q!1>0")
        (("1" (hide 1)
          (("1" (expand "nnreal_expt")
            (("1" (case-replace "x!1=1")
              (("1" (rewrite "nn_rational_expt_1q"nil nil)
               ("2" (case-replace "x!1<1")
                (("1" (assert)
                  (("1" (hide 1)
                    (("1"
                      (typepred "glb(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= q!1 AND z = nn_rational_expt(x!1, q)}))")
                      (("1"
                        (name-replace "GLB"
                         "glb(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= q!1 AND z = nn_rational_expt(x!1, q)}))")
                        (("1" (expand "greatest_lower_bound?")
                          (("1" (flatten)
                            (("1" (inst - "nn_rational_expt(x!1, q!1)")
                              (("1"
                                (split -2)
                                (("1"
                                  (expand "lower_bound?")
                                  (("1"
                                    (inst
                                     -
                                     "nn_rational_expt(x!1, q!1)")
                                    (("1" (assertnil nil)
                                     ("2"
                                      (expand "extend")
                                      (("2"
                                        (inst + "q!1")
                                        (("2" (assertnil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (hide -1 2)
                                  (("2"
                                    (expand "lower_bound?")
                                    (("2"
                                      (skosimp)
                                      (("2"
                                        (typepred "s!1")
                                        (("2"
                                          (expand "extend")
                                          (("2"
                                            (prop)
                                            (("2"
                                              (skosimp)
                                              (("2"
                                                (expand "<=" -2)
                                                (("2"
                                                  (replace -3)
                                                  (("2"
                                                    (hide -1 -3)
                                                    (("2"
                                                      (split)
                                                      (("1"
                                                        (case-replace
                                                         "x!1=0")
                                                        (("1"
                                                          (assert)
                                                          (("1"
                                                            (case-replace
                                                             "q!2=0")
                                                            (("1"
                                                              (assert)
                                                              (("1"
                                                                (rewrite
                                                                 "nn_rational_expt_0q")
                                                                (("1"
                                                                  (rewrite
                                                                   "nn_rational_expt_0q")
                                                                  (("1"
                                                                    (assert)
                                                                    nil
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil)
                                                             ("2"
                                                              (assert)
                                                              (("2"
                                                                (rewrite
                                                                 "nn_rational_expt_0q")
                                                                (("2"
                                                                  (rewrite
                                                                   "nn_rational_expt_0q")
                                                                  (("2"
                                                                    (assert)
                                                                    nil
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (case
                                                           "x!1>0")
                                                          (("1"
                                                            (hide 1)
                                                            (("1"
                                                              (lemma
                                                               "nn_rational_expt_decreasing"
                                                               ("q"
                                                                "q!2"
                                                                "r"
                                                                "q!1"
                                                                "x"
                                                                "x!1"))
                                                              (("1"
                                                                (assert)
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil)
                                                           ("2"
                                                            (assert)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (assert)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (assert)
                  (("2" (case "x!1>1")
                    (("1" (hide 1 2)
                      (("1"
                        (typepred "lub(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= q!1 AND z = nn_rational_expt(x!1, q)}))")
                        (("1"
                          (name-replace "LUB"
                           "lub(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= q!1 AND z = nn_rational_expt(x!1, q)}))")
                          (("1" (expand "least_upper_bound?")
                            (("1" (flatten)
                              (("1"
                                (inst - "nn_rational_expt(x!1, q!1)")
                                (("1"
                                  (split -2)
                                  (("1"
                                    (expand "upper_bound?")
                                    (("1"
                                      (inst
                                       -
                                       "nn_rational_expt(x!1, q!1)")
                                      (("1" (assertnil nil)
                                       ("2"
                                        (expand "extend")
                                        (("2"
                                          (inst + "q!1")
                                          (("2" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (hide -1 2)
                                    (("2"
                                      (expand "upper_bound?")
                                      (("2"
                                        (skosimp)
                                        (("2"
                                          (typepred "s!1")
                                          (("2"
                                            (expand "extend")
                                            (("2"
                                              (prop)
                                              (("2"
                                                (skosimp)
                                                (("2"
                                                  (replace -3)
                                                  (("2"
                                                    (hide -1 -3)
                                                    (("2"
                                                      (expand "<=" -1)
                                                      (("2"
                                                        (split -1)
                                                        (("1"
                                                          (lemma
                                                           "nn_rational_expt_increasing"
                                                           ("q"
                                                            "q!2"
                                                            "r"
                                                            "q!1"
                                                            "x"
                                                            "x!1"))
                                                          (("1"
                                                            (assert)
                                                            nil
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (assert)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (assertnil nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (assertnil nil))
        nil))
      nil))
    nil)
   ((number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (= const-decl "[T, T -> boolean]" equalities 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)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nnrat nonempty-type-eq-decl nil rationals nil)
    (nn_rational_expt_nat_rew formula-decl nil nn_rational_expt nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (nnreal type-eq-decl nil real_types nil)
    (nnreal_exp application-judgement "nnreal" exponentiation nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nnreal_expt application-judgement "nnreal" exponentiation nil)
    (^ const-decl "real" exponentiation nil)
    (expt def-decl "real" exponentiation nil)
    (nnreal_expt const-decl "nnreal" nnreal_expt nil)
    (nn_rational_expt_1q formula-decl nil nn_rational_expt nil)
    (upper_bound? const-decl "bool" bounded_real_defs nil)
    (nn_rational_expt_increasing formula-decl nil nn_rational_expt nil)
    (least_upper_bound? const-decl "bool" bounded_real_defs nil)
    (bounded_above? const-decl "bool" bounded_real_defs nil)
    (lub const-decl "{x | least_upper_bound?(x, SA)}" bounded_real_defs
     nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nn_rational_expt const-decl "nnreal" nn_rational_expt nil)
    (<= const-decl "bool" reals nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (extend const-decl "R" extend nil)
    (FALSE const-decl "bool" booleans nil)
    (glb const-decl "{x | greatest_lower_bound?(x, SB)}"
     bounded_real_defs nil)
    (bounded_below? const-decl "bool" bounded_real_defs nil)
    (greatest_lower_bound? const-decl "bool" bounded_real_defs nil)
    (nonempty? const-decl "bool" sets nil)
    (set type-eq-decl nil sets nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (nn_rational_expt_decreasing formula-decl nil nn_rational_expt nil)
    (nn_rational_expt_0q formula-decl nil nn_rational_expt nil)
    (lower_bound? const-decl "bool" bounded_real_defs nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (x!1 skolem-const-decl "nnreal" nnreal_expt nil)
    (q!1 skolem-const-decl "nnrat" nnreal_expt nil)
    (< const-decl "bool" reals nil) (> const-decl "bool" reals nil))
   shostak))
 (nnreal_expt_nat_rew_TCC1 0
  (nnreal_expt_nat_rew_TCC1-1 nil 3428161604 ("" (subtype-tcc) nil nil)
   ((/= const-decl "boolean" notequal nil)) nil))
 (nnreal_expt_nat_rew 0
  (nnreal_expt_nat_rew-1 nil 3428161605
   ("" (skosimp)
    (("" (rewrite "nnreal_expt_rat_rew")
      (("" (rewrite "nn_rational_expt_nat_rew"nil nil)) nil))
    nil)
   ((nnreal_expt_rat_rew formula-decl nil nnreal_expt nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans 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)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nnrat nonempty-type-eq-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (nnreal type-eq-decl nil real_types nil)
    (nn_rational_expt_nat_rew formula-decl nil nn_rational_expt nil))
   shostak))
 (nnreal_expt_root_rew 0
  (nnreal_expt_root_rew-1 nil 3428161631
   ("" (skosimp)
    (("" (rewrite "nnreal_expt_rat_rew")
      (("" (rewrite "nn_rational_expt_root_rew"nil nil)) nil))
    nil)
   ((posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (nnreal_expt_rat_rew formula-decl nil nnreal_expt nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans 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)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nnrat nonempty-type-eq-decl nil rationals nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (/= const-decl "boolean" notequal nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nnreal type-eq-decl nil real_types nil)
    (nn_rational_expt_root_rew formula-decl nil nn_rational_expt nil))
   shostak))
 (nnreal_expt_0a 0
  (nnreal_expt_0a-1 nil 3427304632
   ("" (skosimp)
    (("" (case-replace "a!1=0")
      (("1" (rewrite "nnreal_expt_rat_rew")
        (("1" (rewrite "nn_rational_expt_0q"nil nil)) nil)
       ("2" (assert)
        (("2" (case "a!1>0")
          (("1" (hide 1)
            (("1" (expand "nnreal_expt")
              (("1"
                (typepred "glb(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= a!1 AND z = nn_rational_expt(0, q)}))")
                (("1"
                  (name-replace "GLB"
                   "glb(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= a!1 AND z = nn_rational_expt(0, q)}))")
                  (("1" (expand "greatest_lower_bound?")
                    (("1" (flatten)
                      (("1" (expand "lower_bound?")
                        (("1" (inst - "0")
                          (("1" (inst - "0")
                            (("1" (split -2)
                              (("1" (assertnil nil)
                               ("2"
                                (skosimp)
                                (("2"
                                  (typepred "s!1")
                                  (("2"
                                    (expand "extend")
                                    (("2" (assertnil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (expand "extend")
                            (("2" (hide -1 2)
                              (("2"
                                (lemma
                                 "density_positive"
                                 ("x" "0" "y" "a!1"))
                                (("2"
                                  (assert)
                                  (("2"
                                    (skosimp)
                                    (("2"
                                      (inst + "r!1")
                                      (("1"
                                        (assert)
                                        (("1"
                                          (rewrite
                                           "nn_rational_expt_0q")
                                          nil
                                          nil))
                                        nil)
                                       ("2" (assertnil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (assertnil nil))
          nil))
        nil))
      nil))
    nil)
   ((number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (= const-decl "[T, T -> boolean]" equalities 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)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (nn_rational_expt_0q formula-decl nil nn_rational_expt nil)
    (nnrat nonempty-type-eq-decl nil rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (nnreal_expt_rat_rew formula-decl nil nnreal_expt nil)
    (> const-decl "bool" reals nil)
    (nnreal_expt const-decl "nnreal" nnreal_expt nil)
    (a!1 skolem-const-decl "nnreal" nnreal_expt nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_gt_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)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (rat nonempty-type-eq-decl nil rationals nil)
    (r!1 skolem-const-decl "rat" nnreal_expt nil)
    (density_positive formula-decl nil rational_props nil)
    (lower_bound? const-decl "bool" bounded_real_defs nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (set type-eq-decl nil sets nil)
    (nonempty? const-decl "bool" sets nil)
    (greatest_lower_bound? const-decl "bool" bounded_real_defs nil)
    (bounded_below? const-decl "bool" bounded_real_defs nil)
    (glb const-decl "{x | greatest_lower_bound?(x, SB)}"
     bounded_real_defs nil)
    (FALSE const-decl "bool" booleans nil)
    (extend const-decl "R" extend nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (nn_rational_expt const-decl "nnreal" nn_rational_expt nil))
   shostak))
 (nnreal_expt_1a 0
  (nnreal_expt_1a-1 nil 3427304898
   ("" (skosimp)
    (("" (expand "nnreal_expt") (("" (propax) nil nil)) nil)) nil)
   ((nnreal_expt const-decl "nnreal" nnreal_expt nil)) shostak))
 (nnreal_expt_x1 0
  (nnreal_expt_x1-1 nil 3427304910
   ("" (skosimp)
    (("" (rewrite "nnreal_expt_rat_rew")
      (("" (rewrite "nn_rational_expt_nat_rew")
        (("" (ground) (("" (grind) nil nil)) nil)) nil))
      nil))
    nil)
   ((nnreal_expt_rat_rew formula-decl nil nnreal_expt nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans 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)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nnrat nonempty-type-eq-decl nil rationals nil)
    (nnreal type-eq-decl nil real_types nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nnreal_expt application-judgement "nnreal" exponentiation nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (^ const-decl "real" exponentiation nil)
    (expt def-decl "real" exponentiation nil)
    (nnreal_exp application-judgement "nnreal" exponentiation 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)
    (nn_rational_expt_nat_rew formula-decl nil nn_rational_expt nil))
   shostak))
 (nnreal_expt_pos 0
  (nnreal_expt_pos-1 nil 3427306409
   ("" (skosimp)
    (("" (expand "nnreal_expt")
      (("" (case-replace "a!1=0")
        (("1" (assertnil nil)
         ("2" (assert)
          (("2" (case-replace "px!1=1")
            (("1" (assertnil nil)
             ("2" (assert)
              (("2" (case "px!1<1")
                (("1" (assert)
                  (("1" (case "a!1>0")
                    (("1" (hide 1 2)
                      (("1"
                        (typepred "glb(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= a!1 AND z = nn_rational_expt(px!1, q)}))
")
                        (("1"
                          (name-replace "GLB"
                           "glb(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= a!1 AND z = nn_rational_expt(px!1, q)}))")
                          (("1" (expand "extend")
                            (("1" (expand "greatest_lower_bound?")
                              (("1"
                                (flatten)
                                (("1"
                                  (expand "lower_bound?")
                                  (("1"
                                    (lemma
                                     "density_positive"
                                     ("x" "a!1" "y" "a!1+1"))
                                    (("1"
                                      (assert)
                                      (("1"
                                        (skosimp)
                                        (("1"
                                          (inst
                                           -4
                                           "nn_rational_expt(px!1, r!1)")
                                          (("1"
                                            (split -4)
                                            (("1"
                                              (lemma
                                               "nn_rational_expt_pos"
                                               ("px" "px!1" "q" "r!1"))
                                              (("1" (assertnil nil)
                                               ("2" (assertnil nil))
                                              nil)
                                             ("2"
                                              (skosimp)
                                              (("2"
                                                (typepred "s!1")
                                                (("2"
                                                  (assert)
                                                  (("2"
                                                    (skosimp)
                                                    (("2"
                                                      (lemma
                                                       "nn_rational_expt_decreasing"
                                                       ("q"
                                                        "q!1"
                                                        "r"
                                                        "r!1"
                                                        "x"
                                                        "px!1"))
                                                      (("2"
                                                        (assert)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil)
                                           ("2" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (assertnil nil))
                    nil))
                  nil)
                 ("2" (assert)
                  (("2" (case "a!1>0&px!1>1")
                    (("1" (flatten)
                      (("1" (hide 1 2 3)
                        (("1"
                          (typepred
                           "lub(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= a!1 AND z = nn_rational_expt(px!1, q)}))")
                          (("1"
                            (name-replace "LUB"
                             "lub(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= a!1 AND z = nn_rational_expt(px!1, q)}))")
                            (("1" (expand "least_upper_bound?")
                              (("1"
                                (flatten)
                                (("1"
                                  (expand "extend")
                                  (("1"
                                    (expand "upper_bound?")
                                    (("1"
                                      (lemma
                                       "density_positive"
                                       ("x" "0" "y" "a!1"))
                                      (("1"
                                        (assert)
                                        (("1"
                                          (skosimp)
                                          (("1"
                                            (inst
                                             -
                                             "nn_rational_expt(px!1, r!1)")
                                            (("1"
                                              (lemma
                                               "nn_rational_expt_pos"
                                               ("px" "px!1" "q" "r!1"))
                                              (("1" (assertnil nil)
                                               ("2" (assertnil nil))
                                              nil)
                                             ("2"
                                              (typepred
                                               "nn_rational_expt(px!1, r!1)")
                                              (("1"
                                                (assert)
                                                (("1"
                                                  (inst + "r!1")
                                                  (("1"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2" (assertnil nil))
                                              nil)
                                             ("3" (assertnil nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (assertnil nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nnreal_expt const-decl "nnreal" nnreal_expt nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (lub const-decl "{x | least_upper_bound?(x, SA)}" bounded_real_defs
     nil)
    (bounded_above? const-decl "bool" bounded_real_defs nil)
    (least_upper_bound? const-decl "bool" bounded_real_defs nil)
    (a!1 skolem-const-decl "nnreal" nnreal_expt nil)
    (px!1 skolem-const-decl "posreal" nnreal_expt nil)
    (r!1 skolem-const-decl "rat" nnreal_expt nil)
    (upper_bound? const-decl "bool" bounded_real_defs nil)
    (lower_bound? const-decl "bool" bounded_real_defs nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (rat nonempty-type-eq-decl nil rationals nil)
    (r!1 skolem-const-decl "rat" nnreal_expt nil)
    (posreal_plus_nnreal_is_posreal application-judgement "posreal"
     real_types nil)
    (nn_rational_expt_decreasing formula-decl nil nn_rational_expt nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (nn_rational_expt_pos formula-decl nil nn_rational_expt nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (density_positive formula-decl nil rational_props nil)
    (nnreal_plus_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (set type-eq-decl nil sets nil)
    (nonempty? const-decl "bool" sets nil)
    (greatest_lower_bound? const-decl "bool" bounded_real_defs nil)
    (bounded_below? const-decl "bool" bounded_real_defs nil)
    (glb const-decl "{x | greatest_lower_bound?(x, SB)}"
     bounded_real_defs nil)
    (FALSE const-decl "bool" booleans nil)
    (extend const-decl "R" extend nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (nnrat nonempty-type-eq-decl nil rationals nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (nn_rational_expt const-decl "nnreal" nn_rational_expt nil)
    (< const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans 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)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil))
   shostak))
 (nnreal_expt_is_0 0
  (nnreal_expt_is_0-1 nil 3427304987
   ("" (skosimp)
    (("" (case-replace "a!1=0")
      (("1" (assert)
        (("1" (rewrite "nnreal_expt_rat_rew")
          (("1" (rewrite "nn_rational_expt_is_0"nil nil)) nil))
        nil)
       ("2" (assert)
        (("2" (case "a!1>0")
          (("1" (hide 1)
            (("1" (split)
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                  (("1" (hide 1)
                    (("1" (assert)
                      (("1"
                        (lemma "nnreal_expt_pos"
                         ("px" "x!1" "a" "a!1"))
                        (("1" (assertnil nil)) nil))
                      nil))
                    nil)
                   ("2" (assertnil nil))
                  nil))
                nil)
               ("2" (flatten)
                (("2" (replace -1)
                  (("2" (rewrite "nnreal_expt_0a"nil nil)) nil))
                nil))
              nil))
            nil)
           ("2" (assertnil nil))
          nil))
        nil))
      nil))
    nil)
   ((number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (= const-decl "[T, T -> boolean]" equalities 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)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (nnreal_expt_rat_rew formula-decl nil nnreal_expt nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (nnrat nonempty-type-eq-decl nil rationals nil)
    (nn_rational_expt_is_0 formula-decl nil nn_rational_expt nil)
    (> const-decl "bool" reals nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nnreal_expt_pos formula-decl nil nnreal_expt nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (nnreal_expt_0a formula-decl nil nnreal_expt nil))
   shostak))
 (nnreal_expt_gt1 0
  (nnreal_expt_gt1-1 nil 3427306444
   ("" (skosimp)
    (("" (split)
      (("1" (flatten)
        (("1" (case "x!1<=1")
          (("1" (hide 1)
            (("1" (expand "<=")
              (("1" (split)
                (("1" (assert)
                  (("1" (case-replace "x!1=0")
                    (("1" (rewrite "nnreal_expt_0a")
                      (("1" (assertnil nil)) nil)
                     ("2" (case "x!1>0")
                      (("1" (hide 1)
                        (("1" (expand "nnreal_expt")
                          (("1"
                            (typepred
                             "glb(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= pa!1 AND z = nn_rational_expt(x!1, q)}))")
                            (("1"
                              (name-replace "GLB"
                               "glb(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= pa!1 AND z = nn_rational_expt(x!1, q)}))")
                              (("1"
                                (expand "extend")
                                (("1"
                                  (expand "greatest_lower_bound?")
                                  (("1"
                                    (flatten)
                                    (("1"
                                      (expand "lower_bound?")
                                      (("1"
                                        (inst
                                         -
                                         "nn_rational_expt(x!1, 0)")
                                        (("1"
                                          (rewrite
                                           "nn_rational_expt_nat_rew")
                                          (("1"
                                            (rewrite "expt_x0")
                                            (("1" (assertnil nil))
                                            nil))
                                          nil)
                                         ("2"
                                          (typepred
                                           "nn_rational_expt(x!1, 0)")
                                          (("2"
                                            (assert)
                                            (("2"
                                              (inst + "0")
                                              (("2" (assertnil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (assertnil nil))
                      nil))
                    nil))
                  nil)
                 ("2" (replace -1)
                  (("2" (rewrite "nnreal_expt_1a")
                    (("2" (assertnil nil)) nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (assertnil nil))
          nil))
        nil)
       ("2" (flatten)
        (("2" (assert)
          (("2" (expand "nnreal_expt")
            (("2"
              (typepred "lub(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= pa!1 AND z = nn_rational_expt(x!1, q)}))")
              (("2"
                (name-replace "LUB"
                 "lub(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= pa!1 AND z = nn_rational_expt(x!1, q)}))")
                (("2" (expand "extend")
                  (("2" (expand "least_upper_bound?")
                    (("2" (flatten)
                      (("2"
                        (lemma "density_positive" ("x" "0" "y" "pa!1"))
                        (("2" (assert)
                          (("2" (skosimp)
                            (("2" (expand "upper_bound?")
                              (("2"
                                (inst - "nn_rational_expt(x!1, r!1)")
                                (("1"
                                  (lemma
                                   "nn_rational_expt_gt1"
                                   ("x" "x!1" "pq" "r!1"))
                                  (("1" (assertnil nil)
                                   ("2" (assertnil nil))
                                  nil)
                                 ("2"
                                  (typepred
                                   "nn_rational_expt(x!1, r!1)")
                                  (("1"
                                    (assert)
                                    (("1"
                                      (inst + "r!1")
                                      (("1" (assertnil nil))
                                      nil))
                                    nil)
                                   ("2" (assertnil nil))
                                  nil)
                                 ("3" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil) (<= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans 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)
    (nnreal_expt_1a formula-decl nil nnreal_expt nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nnreal_expt const-decl "nnreal" nnreal_expt nil)
    (lower_bound? const-decl "bool" bounded_real_defs 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)
    (nn_rational_expt_nat_rew formula-decl nil nn_rational_expt nil)
    (nnreal_exp application-judgement "nnreal" exponentiation nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (expt_x0 formula-decl nil exponentiation nil)
    (pa!1 skolem-const-decl "posreal" nnreal_expt nil)
    (x!1 skolem-const-decl "nnreal" nnreal_expt nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (set type-eq-decl nil sets nil)
    (nonempty? const-decl "bool" sets nil)
    (greatest_lower_bound? const-decl "bool" bounded_real_defs nil)
    (bounded_below? const-decl "bool" bounded_real_defs nil)
    (glb const-decl "{x | greatest_lower_bound?(x, SB)}"
     bounded_real_defs nil)
    (FALSE const-decl "bool" booleans nil)
    (extend const-decl "R" extend nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (nnrat nonempty-type-eq-decl nil rationals nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (nn_rational_expt const-decl "nnreal" nn_rational_expt nil)
    (nnreal_expt_0a formula-decl nil nnreal_expt nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (lub const-decl "{x | least_upper_bound?(x, SA)}" bounded_real_defs
     nil)
    (bounded_above? const-decl "bool" bounded_real_defs nil)
    (least_upper_bound? const-decl "bool" bounded_real_defs nil)
    (upper_bound? const-decl "bool" bounded_real_defs nil)
    (nn_rational_expt_gt1 formula-decl nil nn_rational_expt nil)
    (nonneg_rat nonempty-type-eq-decl nil rationals nil)
    (posrat nonempty-type-eq-decl nil rationals nil)
    (r!1 skolem-const-decl "rat" nnreal_expt nil)
    (rat nonempty-type-eq-decl nil rationals nil)
    (density_positive formula-decl nil rational_props nil))
   shostak))
 (nnreal_expt_lt1 0
  (nnreal_expt_lt1-1 nil 3427306586
   ("" (skosimp)
    (("" (split)
      (("1" (flatten)
        (("1" (case "1<=x!1")
          (("1" (hide 1)
            (("1" (expand "<=")
              (("1" (split -1)
                (("1" (assert)
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                      (typepred "lub(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= pa!1 AND z = nn_rational_expt(x!1, q)}))")
                      (("1"
                        (name-replace "LUB"
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                        (("1" (expand "extend")
                          (("1" (expand "least_upper_bound?")
                            (("1" (flatten)
                              (("1"
                                (expand "upper_bound?")
                                (("1"
                                  (inst - "nn_rational_expt(x!1, 0)")
                                  (("1"
                                    (rewrite
                                     "nn_rational_expt_nat_rew")
                                    (("1"
                                      (rewrite "expt_x0")
                                      (("1" (assertnil nil))
                                      nil))
                                    nil)
                                   ("2"
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                                     "nn_rational_expt(x!1, 0)")
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                                        (("2" (assertnil nil))
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                              nil))
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                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (replace -1 * rl)
                  (("2" (rewrite "nnreal_expt_1a")
                    (("2" (assertnil nil)) nil))
                  nil))
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              nil))
            nil)
           ("2" (assertnil nil))
          nil))
        nil)
       ("2" (flatten)
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            (("2"
              (typepred "glb(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= pa!1 AND z = nn_rational_expt(x!1, q)}))")
              (("2"
                (name-replace "GLB"
                 "glb(extend[real, nnreal, bool, FALSE]
              ({z | EXISTS q: q <= pa!1 AND z = nn_rational_expt(x!1, q)}))")
--> --------------------

--> maximum size reached

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

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