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Datei: interm_value_thm.pvs   Sprache: Lisp

Original von: PVS©

(general_properties
 (seq_power_TCC1 0
  (seq_power_TCC1-1 nil 3529776145 ("" (subtype-tcc) nil nil)
   ((NOT const-decl "[bool -> bool]" booleans nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (real_ge_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_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers 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)
    (default const-decl "T" fseqs "structures/"))
   nil))
 (only_power_p_TCC1 0
  (only_power_p_TCC1-1 nil 3530711428 ("" (subtype-tcc) nil nilnil
   nil))
 (divides_element 0
  (divides_element-1 nil 3529776160
   ("" (skosimp)
    (("" (expand "divides")
      (("" (skosimp*)
        (("" (inst 1 "x!2 - x!3")
          (("" (replaces -2) (("" (assertnil nil)) nil)) nil))
        nil))
      nil))
    nil)
   ((divides const-decl "bool" divides nil)
    (int_minus_int_is_int application-judgement "int" integers 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)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (int_plus_int_is_int application-judgement "int" integers nil))
   shostak))
 (divides_rel_primes_TCC1 0
  (divides_rel_primes_TCC1-1 nil 3529776145 ("" (subtype-tcc) nil nil)
   ((divides const-decl "bool" divides nil)) nil))
 (divides_rel_primes 0
  (divides_rel_primes-1 nil 3529776181
   ("" (skosimp)
    (("" (expand "divides")
      (("" (skosimp)
        (("" (lemma "rel_prime_lem")
          (("" (inst -1 "x!1" "k!1")
            (("" (assert)
              (("" (expand "rel_prime")
                (("" (skosimp)
                  (("" (lemma "both_sides_times1")
                    (("" (inst -1 "y!1" "1" "m!1 * x!1 + n!1 * k!1")
                      (("1" (prop)
                        (("1" (hide (-2 -3 -5))
                          (("1" (assert)
                            (("1" (rewrite "identity_mult")
                              (("1"
                                (lemma "commutative_mult")
                                (("1"
                                  (inst -1 "k!1" "n!1")
                                  (("1"
                                    (replaces -1)
                                    (("1"
                                      (rewrite
                                       "associative_mult"
                                       :dir
                                       rl)
                                      (("1"
                                        (replaces -2)
                                        (("1"
                                          (inst 1 "n!1*x!2 + m!1*y!1")
                                          (("1" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (prop)
                        (("2" (inst 1 "0") (("2" (assertnil nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((divides const-decl "bool" divides nil)
    (rel_prime_lem formula-decl nil gcd "ints/")
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (/= const-decl "boolean" notequal nil)
    (y!1 skolem-const-decl "int" general_properties nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (identity_mult formula-decl nil number_fields nil)
    (associative_mult formula-decl nil number_fields nil)
    (commutative_mult formula-decl nil number_fields nil)
    (int_times_even_is_even application-judgement "even_int" integers
     nil)
    (both_sides_times1 formula-decl nil real_props nil)
    (rel_prime const-decl "bool" gcd "ints/")
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers 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))
   shostak))
 (divides_product 0
  (divides_product-1 nil 3530718323
   ("" (skosimp*)
    (("" (case "fs!1`length = 0")
      (("1" (grind) nil nil)
       ("2" (expand"divides" "product")
        (("2" (typepred "j!1")
          (("2" (typepred "fs!1`length")
            (("2" (lemma "product_middle")
              (("2" (inst?)
                (("2" (inst -1 "j!1")
                  (("2" (assert)
                    (("2" (replaces -1)
                      (("2"
                        (inst 2
                         "product(0, j!1 - 1, fs!1`seq) * product(1 + j!1, length(fs!1) - 1, fs!1`seq)")
                        (("1" (assertnil nil) ("2" (grind) nil nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((fseq type-eq-decl nil fseqs "structures/")
    (barray type-eq-decl nil fseqs "structures/")
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers 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)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (below type-eq-decl nil naturalnumbers nil)
    (< const-decl "bool" reals nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (product_middle formula-decl nil product "reals/")
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (product def-decl "real" product "reals/")
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (j!1 skolem-const-decl "below(fs!1`length)" general_properties nil)
    (fs!1 skolem-const-decl "fseq[posnat]" general_properties nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (prod_nnr application-judgement "nnreal" product_nat "reals/")
    (prod_nat application-judgement "nat" product_nat "reals/")
    (prod_pr application-judgement "posreal" product_nat "reals/")
    (prod_posnat application-judgement "posnat" product_nat "reals/")
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (T_high type-eq-decl nil product "reals/")
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (T_low type-eq-decl nil product "reals/")
    (real_times_real_is_real application-judgement "real" reals nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (divides const-decl "bool" divides nil)
    (product const-decl "posnat" product_fseq_posnat "reals/"))
   shostak))
 (product_power_TCC1 0
  (product_power_TCC1-1 nil 3529776145 ("" (subtype-tcc) nil nilnil
   nil))
 (product_power 0
  (product_power-1 nil 3529776213
   ("" (induct "i")
    (("1" (typepred "i!1") (("1" (propax) nil nil)) nil)
     ("2" (assertnil nil)
     ("3" (skosimp*)
      (("3" (case-replace "j!1 = 0" :hide? T)
        (("1" (hide (-1 -2)) (("1" (grind) nil nil)) nil)
         ("2" (assert)
          (("2" (inst -1 "p!1")
            (("2" (rewrite "expt_plus")
              (("2" (rewrite "expt_x1")
                (("2"
                  (case-replace
                   "seq_power(p!1, 1 + j!1) = seq_power(p!1, j!1) o fseq1(p!1)"
                   :hide? T)
                  (("1" (lemma "product_fseq_concat1")
                    (("1" (inst?) (("1" (assertnil nil)) nil)) nil)
                   ("2" (hide 3)
                    (("2" (decompose-equality 1)
                      (("1" (grind) nil nil)
                       ("2" (decompose-equality 1)
                        (("2" (grind) nil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nzreal nonempty-type-eq-decl nil reals nil)
    (expt_plus formula-decl nil exponentiation nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (fseq1 const-decl "fseq" fseqs "structures/")
    (O const-decl "fseq" fseqs "structures/")
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (product_fseq_concat1 formula-decl nil product_fseq_posnat
     "reals/")
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (prod_posnat application-judgement "posnat" product_nat "reals/")
    (prod_pr application-judgement "posreal" product_nat "reals/")
    (prod_nat application-judgement "nat" product_nat "reals/")
    (expt_x1 formula-decl nil exponentiation nil)
    (posint_exp application-judgement "posint" exponentiation nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (even_plus_odd_is_odd application-judgement "odd_int" integers nil)
    (posnat_expt application-judgement "posnat" exponentiation nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (product_nat_nat application-judgement "posnat" product_fseq_posnat
     "reals/")
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (prod_nnr application-judgement "nnreal" product_nat "reals/")
    (product def-decl "real" product "reals/")
    (default const-decl "T" fseqs "structures/")
    (expt def-decl "real" exponentiation nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (seq_power const-decl "fseq" general_properties nil)
    (product const-decl "posnat" product_fseq_posnat "reals/")
    (fseq type-eq-decl nil fseqs "structures/")
    (barray type-eq-decl nil fseqs "structures/")
    (^ const-decl "real" exponentiation nil)
    (/= const-decl "boolean" notequal nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (pred type-eq-decl nil defined_types 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))
   shostak))
 (product_only_power_TCC1 0
  (product_only_power_TCC1-1 nil 3530711428 ("" (subtype-tcc) nil nil)
   ((^ const-decl "real" exponentiation nil)
    (only_power_p const-decl "bool" general_properties nil))
   nil))
 (product_only_power 0
  (product_only_power-1 nil 3530711533
   ("" (measure-induct+ "fs`length" "fs")
    (("" (skosimp*)
      (("" (case "x!1`length = 0")
        (("1" (hide (-2 -3))
          (("1" (inst 1 "0")
            (("1" (expand "product")
              (("1" (assert) (("1" (rewrite "expt_x0"nil nil)) nil))
              nil))
            nil))
          nil)
         ("2" (inst -1 "fseqs_ops_def[posnat,default].rest(x!1)")
          (("2" (inst -1 "p!1")
            (("2" (prop)
              (("1" (skosimp)
                (("1"
                  (case "x!1 = fseq1(x!1`seq(0)) o fseqs_ops_def[posnat, default].rest(x!1)")
                  (("1" (replace -1 2)
                    (("1" (expand "only_power_p")
                      (("1" (inst -3 "0")
                        (("1" (skosimp)
                          (("1" (inst 2 "n!1 + i!1")
                            (("1" (rewrite "product_fseq_concat")
                              (("1"
                                (hide -1)
                                (("1"
                                  (replaces -1)
                                  (("1"
                                    (replaces -1)
                                    (("1"
                                      (expand "product")
                                      (("1"
                                        (expand "fseq1")
                                        (("1"
                                          (assert)
                                          (("1"
                                            (rewrite "expt_plus")
                                            nil
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (hide (-1 -2 3)) (("2" (assertnil nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (hide (-1 -2 3))
                    (("2" (decompose-equality 1)
                      (("1" (grind) nil nil)
                       ("2"
                        (lemma
                         "fseqs_ops_def[posnat, default].add_first_rest")
                        (("2" (inst?)
                          (("2" (replace -1 1 rl)
                            (("2" (hide -1)
                              (("2"
                                (decompose-equality 1)
                                (("2"
                                  (expand*
                                   "add"
                                   "first"
                                   "fseq1"
                                   "rest"
                                   "o"
                                   "^")
                                  (("2"
                                    (lift-if)
                                    (("2"
                                      (lift-if)
                                      (("2"
                                        (lift-if)
                                        (("2"
                                          (lift-if)
                                          (("2"
                                            (lift-if)
                                            (("2"
                                              (lift-if)
                                              (("2"
                                                (lift-if)
                                                (("2"
                                                  (lift-if)
                                                  (("2"
                                                    (lift-if)
                                                    (("2"
                                                      (grind)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide 3)
                (("2" (expand "only_power_p")
                  (("2" (skosimp)
                    (("2" (inst -1 "j!1 + 1")
                      (("1" (skosimp)
                        (("1" (inst 1 "i!1")
                          (("1"
                            (case-replace
                             "fseqs_ops_def[posnat, default].rest(x!1)`seq(j!1) = x!1`seq(j!1 + 1)")
                            (("1" (hide (-1 2))
                              (("1"
                                (expand "rest")
                                (("1"
                                  (expand "^")
                                  (("1"
                                    (lift-if)
                                    (("1"
                                      (prop)
                                      (("1" (assertnil nil)
                                       ("2"
                                        (typepred "j!1")
                                        (("2" (grind) nil nil))
                                        nil)
                                       ("3"
                                        (typepred "j!1")
                                        (("3"
                                          (expand "rest")
                                          (("3"
                                            (expand "^")
                                            (("3" (assertnil nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (hide 2)
                        (("2" (typepred "j!1")
                          (("2" (expand "rest") (("2" (grind) nil nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("3" (hide (-1 3)) (("3" (grind) nil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((rest const-decl "fseq" fseqs_ops_def "structures/")
    (fseq type-eq-decl nil fseqs_def "structures/")
    (barray type-eq-decl nil fseqs_def "structures/")
    (default const-decl "T" fseqs "structures/")
    (O const-decl "fseq" fseqs "structures/")
    (fseq1 const-decl "fseq" fseqs "structures/")
    (product_fseq_concat formula-decl nil product_fseq_posnat "reals/")
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (product_nat_nat application-judgement "posnat" product_fseq_posnat
     "reals/")
    (product_eq_arg formula-decl nil product "reals/")
    (expt_plus formula-decl nil exponentiation nil)
    (nzreal nonempty-type-eq-decl nil reals nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (below type-eq-decl nil naturalnumbers nil)
    (x!1 skolem-const-decl "fseq[posnat]" general_properties nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (^ const-decl "fseq" fseqs_def "structures/")
    (empty_seq const-decl "fsq" fseqs_def "structures/")
    (int_minus_int_is_int application-judgement "int" integers nil)
    (int_plus_int_is_int application-judgement "int" integers 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)
    (ne_fseq type-eq-decl nil fseqs_def "structures/")
    (empty_seq_fseq name-judgement "fseq[T, default]" fseqs_ops_def
     "structures/")
    (insert const-decl "fseq" fseqs_ops_def "structures/")
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (add const-decl "fseq" fseqs_ops_def "structures/")
    (first const-decl "T" fseqs_ops_def "structures/")
    (add_first_rest formula-decl nil fseqs_ops_def "structures/")
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (j!1 skolem-const-decl
     "below(fseqs_ops_def[posnat, default].rest(x!1)`length)"
     general_properties nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (expt_x0 formula-decl nil exponentiation nil)
    (prod_nnr application-judgement "nnreal" product_nat "reals/")
    (prod_nat application-judgement "nat" product_nat "reals/")
    (prod_pr application-judgement "posreal" product_nat "reals/")
    (prod_posnat application-judgement "posnat" product_nat "reals/")
    (posint_exp application-judgement "posint" exponentiation nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (^ const-decl "real" exponentiation nil)
    (/= const-decl "boolean" notequal nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (product const-decl "posnat" product_fseq_posnat "reals/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (only_power_p const-decl "bool" general_properties nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (wf_nat formula-decl nil naturalnumbers nil)
    (< const-decl "bool" reals nil)
    (naturalnumber type-eq-decl nil naturalnumbers nil)
    (fseq type-eq-decl nil fseqs "structures/")
    (barray type-eq-decl nil fseqs "structures/")
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers 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)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (measure_induction formula-decl nil measure_induction nil)
    (well_founded? const-decl "bool" orders nil)
    (pred type-eq-decl nil defined_types nil))
   shostak))
 (divides_power 0
  (divides_power-1 nil 3530367448
   ("" (skosimp)
    (("" (expand "divides")
      (("" (case-replace "i!1 = 1" :hide? T)
        (("1" (inst 1 "1") (("1" (grind) nil nil)) nil)
         ("2" (inst 2 "p!1 ^ (i!1 - 1)")
          (("1" (lemma "expt_plus")
            (("1" (inst -1 "1" "i!1 - 1" "p!1") (("1" (grind) nil nil))
              nil))
            nil)
           ("2" (typepred "p!1" "i!1") (("2" (grind) nil nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((divides const-decl "bool" divides nil)
    (posrat_exp application-judgement "posrat" exponentiation nil)
    (nnrat_exp application-judgement "nnrat" exponentiation nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (i!1 skolem-const-decl "posnat" general_properties nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (p!1 skolem-const-decl "posnat" general_properties nil)
    (/= const-decl "boolean" notequal nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (nzreal nonempty-type-eq-decl nil reals nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (posrat_times_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (expt_plus formula-decl nil exponentiation nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (posint_exp application-judgement "posint" exponentiation nil)
    (posnat_expt application-judgement "posnat" exponentiation nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (^ const-decl "real" exponentiation nil)
    (expt def-decl "real" exponentiation nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers 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)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil))
   shostak))
 (divides_prime_power_TCC1 0
  (divides_prime_power_TCC1-1 nil 3529776145 ("" (subtype-tcc) nil nil)
   ((/= const-decl "boolean" notequal nil)
    (divides const-decl "bool" divides nil)
    (prime? const-decl "bool" primes "ints/"))
   nil))
 (divides_prime_power_TCC2 0
  (divides_prime_power_TCC2-1 nil 3529776145 ("" (subtype-tcc) nil nil)
   ((^ const-decl "real" exponentiation nil)
    (divides const-decl "bool" divides nil)
    (/= const-decl "boolean" notequal nil)
    (prime? const-decl "bool" primes "ints/"))
   nil))
 (divides_prime_power 0
  (divides_prime_power-1 nil 3529776240
   ("" (skosimp*)
    (("" (case "j!1=0")
      (("1" (replaces -1)
        (("1" (inst 1 "0")
          (("1" (rewrite "expt_x0")
            (("1" (rewrite "one_div_one") (("1" (assertnil nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (expand "divides")
        (("2" (skosimp)
          (("2" (lemma "prime_factors")
            (("2" (copy -1)
              (("2" (inst -1 "q!1")
                (("2" (inst -2 "x!1")
                  (("1" (skosimp*)
                    (("1" (lemma "product_power")
                      (("1" (inst?)
                        (("1" (copy -7)
                          (("1" (replace -2 -1)
                            (("1" (replace -3 -1)
                              (("1"
                                (replace -5 -1)
                                (("1"
                                  (lemma "product_fseq_concat")
                                  (("1"
                                    (inst?)
                                    (("1"
                                      (replace -1 -2 rl)
                                      (("1"
                                        (hide -1)
                                        (("1"
                                          (lemma "product_sort")
                                          (("1"
                                            (inst -1 "fs!1 o fs!2")
                                            (("1"
                                              (replace -1 -2 rl)
                                              (("1"
                                                (hide -1)
                                                (("1"
                                                  (lemma
                                                   "Fundamental_Theorem_Arithmetic")
                                                  (("1"
                                                    (inst
                                                     -1
                                                     "seq_power(p!1, j!1)"
                                                     "sort(fs!1 o fs!2)"
                                                     "product(seq_power(p!1, j!1))")
                                                    (("1"
                                                      (assert)
                                                      (("1"
                                                        (prop)
                                                        (("1"
                                                          (case
                                                           "length(fs!1) = 0")
                                                          (("1"
                                                            (hide-all-but
                                                             (-1 -5 2))
                                                            (("1"
                                                              (expand
                                                               "product")
                                                              (("1"
                                                                (assert)
                                                                (("1"
                                                                  (inst
                                                                   1
                                                                   "0")
                                                                  (("1"
                                                                    (rewrite
                                                                     "expt_x0")
                                                                    (("1"
                                                                      (assert)
                                                                      nil
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil)
                                                           ("2"
                                                            (case
                                                             "fs!1 = seq_power(p!1,length(fs!1))")
                                                            (("1"
                                                              (replace
                                                               -1
                                                               -5)
                                                              (("1"
                                                                (hide-all-but
                                                                 (-2
                                                                  -5
                                                                  1
                                                                  3))
                                                                (("1"
                                                                  (lemma
                                                                   "product_power")
                                                                  (("1"
                                                                    (inst
                                                                     -1
                                                                     "length(fs!1)"
                                                                     "p!1")
                                                                    (("1"
                                                                      (replace
                                                                       -1
                                                                       -3
                                                                       rl)
                                                                      (("1"
                                                                        (inst?)
                                                                        (("1"
                                                                          (assert)
                                                                          (("1"
                                                                            (hide
                                                                             (-1
                                                                              -3))
                                                                            (("1"
                                                                              (case
                                                                               "length(seq_power(p!1, j!1)) = length(sort(fs!1 o fs!2))")
                                                                              (("1"
                                                                                (hide
                                                                                 -2)
                                                                                (("1"
                                                                                  (grind)
                                                                                  nil
                                                                                  nil))
                                                                                nil)
                                                                               ("2"
                                                                                (replaces
                                                                                 -1)
                                                                                nil
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil)
                                                             ("2"
                                                              (hide-all-but
                                                               (-1 1))
                                                              (("2"
                                                                (decompose-equality
                                                                 1)
                                                                (("1"
                                                                  (hide
                                                                   -1)
                                                                  (("1"
                                                                    (grind)
                                                                    nil
                                                                    nil))
                                                                  nil)
                                                                 ("2"
                                                                  (decompose-equality
                                                                   1)
                                                                  (("2"
                                                                    (case
                                                                     "x!2 >= fs!1`length")
                                                                    (("1"
                                                                      (hide
                                                                       -2)
                                                                      (("1"
                                                                        (typepred
                                                                         "fs!1`seq")
                                                                        (("1"
                                                                          (inst?)
                                                                          (("1"
                                                                            (assert)
                                                                            (("1"
                                                                              (grind)
                                                                              nil
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil)
                                                                     ("2"
                                                                      (lemma
                                                                       "sort_fseq_in?")
                                                                      (("2"
                                                                        (inst
                                                                         -1
                                                                         "fs!1 o fs!2"
                                                                         "fs!1`seq(x!2)")
                                                                        (("2"
                                                                          (prop)
                                                                          (("1"
                                                                            (expand
                                                                             "in?")
                                                                            (("1"
                                                                              (skosimp*)
                                                                              (("1"
                                                                                (replace
                                                                                 -3
                                                                                 -2
                                                                                 rl)
                                                                                (("1"
                                                                                  (replace
                                                                                   -2
                                                                                   2)
                                                                                  (("1"
                                                                                    (expand
                                                                                     "seq_power"
                                                                                     2)
                                                                                    (("1"
                                                                                      (lift-if)
                                                                                      (("1"
                                                                                        (lift-if)
                                                                                        (("1"
                                                                                          (lift-if)
                                                                                          (("1"
                                                                                            (prop)
                                                                                            (("1"
                                                                                              (assert)
                                                                                              nil
                                                                                              nil)
                                                                                             ("2"
                                                                                              (typepred
                                                                                               "ii!2")
                                                                                              (("2"
                                                                                                (replace
                                                                                                 -5
                                                                                                 -1
                                                                                                 rl)
                                                                                                (("2"
                                                                                                  (hide-all-but
                                                                                                   (-1
                                                                                                    2))
                                                                                                  (("2"
                                                                                                    (grind)
                                                                                                    nil
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil)
                                                                           ("2"
                                                                            (hide
                                                                             (1
                                                                              4))
                                                                            (("2"
                                                                              (expand
                                                                               "in?")
                                                                              (("2"
                                                                                (inst?)
                                                                                (("1"
                                                                                  (expand
                                                                                   "o"
                                                                                   1)
                                                                                  (("1"
                                                                                    (lift-if)
                                                                                    (("1"
                                                                                      (prop)
                                                                                      (("1"
                                                                                        (assert)
                                                                                        nil
                                                                                        nil)
                                                                                       ("2"
                                                                                        (assert)
                                                                                        nil
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil)
                                                                                 ("2"
                                                                                  (hide
                                                                                   -)
                                                                                  (("2"
                                                                                    (grind)
                                                                                    nil
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil)
                                                             ("3"
                                                              (assert)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (hide-all-but
                                                           (-7 1))
                                                          (("2"
                                                            (expand
                                                             "ordered_list_of_primes?")
                                                            (("2"
                                                              (prop)
                                                              (("1"
                                                                (expand
                                                                 "list_of_primes?")
                                                                (("1"
                                                                  (skosimp)
                                                                  (("1"
                                                                    (expand
                                                                     "seq_power")
                                                                    (("1"
                                                                      (prop)
                                                                      (("1"
                                                                        (typepred
                                                                         "i!1")
                                                                        (("1"
                                                                          (hide
                                                                           (-2
                                                                            2))
                                                                          (("1"
                                                                            (grind)
                                                                            nil
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil)
                                                               ("2"
                                                                (expand
                                                                 "non_decreasing?")
                                                                (("2"
                                                                  (skosimp*)
                                                                  (("2"
                                                                    (expand
                                                                     "seq_power")
                                                                    (("2"
                                                                      (lift-if)
                                                                      (("2"
                                                                        (lift-if)
                                                                        (("2"
                                                                          (lift-if)
                                                                          (("2"
                                                                            (prop)
                                                                            (("1"
                                                                              (assert)
                                                                              nil
                                                                              nil)
                                                                             ("2"
                                                                              (typepred
                                                                               "j!2")
                                                                              (("2"
                                                                                (hide
                                                                                 (-4
                                                                                  2))
                                                                                (("2"
                                                                                  (grind)
                                                                                  nil
                                                                                  nil))
                                                                                nil))
                                                                              nil)
                                                                             ("3"
                                                                              (typepred
                                                                               "i!1")
                                                                              (("3"
                                                                                (hide
                                                                                 (-3
                                                                                  1))
                                                                                (("3"
                                                                                  (grind)
                                                                                  nil
                                                                                  nil))
                                                                                nil))
                                                                              nil)
                                                                             ("4"
                                                                              (typepred
                                                                               "i!1")
                                                                              (("4"
                                                                                (hide
                                                                                 (-3
                                                                                  2))
                                                                                (("4"
                                                                                  (grind)
                                                                                  nil
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("3"
                                                          (hide
                                                           (-1
                                                            -2
                                                            -3
                                                            -5
                                                            -8
                                                            3))
                                                          (("3"
                                                            (expand
                                                             "ordered_list_of_primes?")
                                                            (("3"
                                                              (prop)
                                                              (("1"
                                                                (hide
                                                                 (-2
                                                                  -4
                                                                  -5
                                                                  2))
                                                                (("1"
                                                                  (expand
                                                                   "list_of_primes?")
                                                                  (("1"
                                                                    (skosimp)
                                                                    (("1"
                                                                      (lemma
                                                                       "sort_fseq_in")
                                                                      (("1"
                                                                        (inst
                                                                         -1
                                                                         "fs!1 o fs!2"
                                                                         "i!1")
                                                                        (("1"
                                                                          (expand
                                                                           "in?")
                                                                          (("1"
                                                                            (skosimp)
                                                                            (("1"
                                                                              (replaces
                                                                               -1)
                                                                              (("1"
                                                                                (expand
--> --------------------

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

¤ Dauer der Verarbeitung: 0.94 Sekunden  (vorverarbeitet)  ¤





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