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© Kompilation durch diese Firma

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Datei: cyclic_group.pvs   Sprache: Lisp

Original von: PVS©
(poly_families
 (polynomial_prod_int 0
  (polynomial_prod_int-1 nil 3619796120
   ("" (skeep)
    (("" (expand "polynomial_prod")
      ((""
        (case "FORALL (jj:nat): rational_pred(sigma(max(i - k, 0), jj,
                          LAMBDA (k_1: nat):
                            IF k_1 <= i THEN a(k_1) * g(i - k_1)
                            ELSE 0
                            ENDIF))
       AND
       integer_pred(sigma(max(i - k, 0), jj,
                          LAMBDA (k_1: nat):
                            IF k_1 <= i THEN a(k_1) * g(i - k_1)
                            ELSE 0
                            ENDIF))")
        (("1" (inst?) nil nil)
         ("2" (hide 2)
          (("2" (induct "jj")
            (("1" (grind)
              (("1" (grind)
                (("1" (expand "sigma")
                  (("1" (expand "sigma") (("1" (propax) nil nil)) nil))
                  nil))
                nil))
              nil)
             ("2" (grind)
              (("2" (expand "sigma") (("2" (ground) nil nil)) nil))
              nil)
             ("3" (skeep)
              (("3" (assert)
                (("3" (expand "sigma" +)
                  (("3" (lift-if)
                    (("3" (ground)
                      (("1" (assert)
                        (("1" (rewrite "rationals.closed_plus" +) nil
                          nil))
                        nil)
                       ("2" (rewrite "integers.closed_plus" +) nil
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("4" (skosimp*) (("4" (assert) nil nil)) nil)
             ("5" (skosimp*) (("5" (assert) nil nil)) nil)
             ("6" (skosimp*) (("6" (assert) nil nil)) nil)
             ("7" (skosimp*) (("7" (assert) nil nil)) nil)
             ("8" (skosimp*) (("8" (assert) nil nil)) nil)
             ("9" (skosimp*) (("9" (assert) nil nil)) nil)
             ("10" (assert) nil nil))
            nil))
          nil)
         ("3" (skosimp*) (("3" (assert) nil nil)) nil)
         ("4" (skosimp*) (("4" (assert) nil nil)) nil)
         ("5" (skosimp*) (("5" (assert) nil nil)) nil)
         ("6" (skosimp*) (("6" (assert) nil nil)) nil))
        nil))
      nil))
    nil)
   ((polynomial_prod const-decl "real" polynomials "reals/")
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (rat nonempty-type-eq-decl nil rationals nil)
    (closed_plus formula-decl nil rationals nil)
    (closed_plus formula-decl nil integers nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (real_plus_real_is_real application-judgement "real" reals 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)
    (nat_induction formula-decl nil naturalnumbers nil)
    (pred type-eq-decl nil defined_types nil)
    (g skolem-const-decl "[nat -> int]" poly_families nil)
    (a skolem-const-decl "[nat -> int]" poly_families nil)
    (k skolem-const-decl "nat" poly_families nil)
    (i skolem-const-decl "nat" poly_families nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans 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)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (int_max application-judgement "{k: int | i <= k AND j <= k}"
     real_defs nil)
    (rat_max application-judgement "{s: rat | s >= q AND s >= r}"
     real_defs 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)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (T_low type-eq-decl nil sigma "reals/")
    (T_high type-eq-decl nil sigma "reals/")
    (sigma def-decl "real" sigma "reals/")
    (max const-decl "{p: real | p >= m AND p >= n}" real_defs nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields
       nil))
   shostak))
 (prod_polynomials_TCC1 0
  (prod_polynomials_TCC1-1 nil 3619793663
   ("" (skeep) (("" (assert) nil nil)) nil)
   ((real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   nil))
 (prod_polynomials_TCC2 0
  (prod_polynomials_TCC2-1 nil 3619793663
   ("" (induct "k")
    (("1" (grind) nil nil)
     ("2" (skeep)
      (("2" (skeep)
        (("2" (inst - "GP" "KF")
          (("2" (expand "sigma" +)
            (("2" (expand "*" + 1)
              (("2" (expand "*" + 3)
                (("2" (assert)
                  (("2" (flatten)
                    (("2" (assert)
                      (("2" (split)
                        (("1" (rewrite "rationals.closed_plus") nil
                          nil)
                         ("2" (rewrite "integers.closed_plus") nil nil)
                         ("3" (assert)
                          (("3" (expand "*" + 1)
                            (("3" (assert) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    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)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (closed_plus formula-decl nil integers 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)
    (rat nonempty-type-eq-decl nil rationals nil)
    (closed_plus formula-decl nil rationals nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (* const-decl "[T -> real]" real_fun_ops "reals/")
    (sigma def-decl "real" sigma "reals/")
    (T_high type-eq-decl nil sigma "reals/")
    (T_low type-eq-decl nil sigma "reals/")
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (subrange type-eq-decl nil integers nil)
    (<= const-decl "bool" reals nil)
    (AND 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))
   nil))
 (prod_polynomials_TCC3 0
  (prod_polynomials_TCC3-1 nil 3619793663
   ("" (skeep) (("" (assert) nil nil)) nil)
   ((real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   nil))
 (prod_polynomials_TCC4 0
  (prod_polynomials_TCC4-1 nil 3619793663 ("" (grind) nil nil)
   ((boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (>= const-decl "bool" 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)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (integer nonempty-type-from-decl nil integers 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))
   nil))
 (prod_polynomials_TCC5 0
  (prod_polynomials_TCC5-1 nil 3619793663
   ("" (skosimp*) (("" (assert) nil nil)) nil)
   ((int_minus_int_is_int application-judgement "int" integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   nil))
 (prod_polynomials_TCC6 0
  (prod_polynomials_TCC6-1 nil 3619793663
   ("" (skeep) (("" (skeep) (("" (assert) nil nil)) nil)) nil)
   ((real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (int_minus_int_is_int application-judgement "int" integers nil))
   nil))
 (prod_polynomials_TCC7 0
  (prod_polynomials_TCC7-1 nil 3619793663
   ("" (skeep)
    (("" (skeep)
      (("" (skeep)
        (("" (skeep)
          (("" (assert)
            (("" (lemma "prod_polynomials_TCC2")
              (("" (inst?) (("" (flatten) (("" (assert) nil nil)) nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((prod_polynomials_TCC2 subtype-tcc nil poly_families nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (subrange type-eq-decl nil integers nil)
    (<= const-decl "bool" reals nil)
    (AND const-decl "[bool, bool -> bool]" booleans 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)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (int_minus_int_is_int application-judgement "int" integers 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))
   nil))
 (prod_polynomials_TCC8 0
  (prod_polynomials_TCC8-1 nil 3619793663
   ("" (skeep*)
    ((""
      (case "FORALL (dd:nat): rational_pred(sigma(0, dd, KF * GP)) AND
                                                         integer_pred(sigma(0, dd, KF * GP)) AND
                                                          sigma(0, dd, KF * GP) >= 0")
      (("1" (label "dd65" -1)
        (("1" (copy -1)
          (("1" (label "dd64" -1)
            (("1" (hide "dd64")
              (("1" (case "k = 0")
                (("1" (replaces -1)
                  (("1" (expand "sigma" +)
                    (("1" (expand "sigma" +)
                      (("1" (expand "product" +)
                        (("1" (expand "product" +)
                          (("1" (assert)
                            (("1" (lift-if)
                              (("1"
                                (split -)
                                (("1"
                                  (flatten)
                                  (("1"
                                    (replace -1)
                                    (("1"
                                      (replace -7)
                                      (("1"
                                        (replace -2)
                                        (("1"
                                          (assert)
                                          (("1"
                                            (expand "*")
                                            (("1"
                                              (assert)
                                              (("1"
                                                (replace -1)
                                                (("1"
                                                  (assert)
                                                  (("1"
                                                    (ground)
                                                    (("1"
                                                      (skeep)
                                                      (("1"
                                                        (hide -)
                                                        (("1"
                                                          (grind)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (hide-all-but -2)
                                                      (("2"
                                                        (grind)
                                                        nil
                                                        nil))
                                                      nil)
                                                     ("3"
                                                      (hide -)
                                                      (("3"
                                                        (grind)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (flatten)
                                  (("2"
                                    (assert)
                                    (("2"
                                      (split -)
                                      (("1"
                                        (flatten)
                                        (("1"
                                          (replace -1)
                                          (("1"
                                            (assert)
                                            (("1"
                                              (expand "*")
                                              (("1"
                                                (assert)
                                                (("1"
                                                  (replace -7)
                                                  (("1"
                                                    (replace -2)
                                                    (("1"
                                                      (assert)
                                                      (("1"
                                                        (ground)
                                                        (("1"
                                                          (replace -1)
                                                          (("1"
                                                            (expand
                                                             "^"
                                                             +)
                                                            (("1"
                                                              (expand
                                                               "expt"
                                                               +)
                                                              (("1"
                                                                (expand
                                                                 "expt"
                                                                 +)
                                                                (("1"
                                                                  (skeep)
                                                                  (("1"
                                                                    (rewrite
                                                                     "poly_eq")
                                                                    (("1"
                                                                      (hide
                                                                       2)
                                                                      (("1"
                                                                        (skosimp*)
                                                                        (("1"
                                                                          (ground)
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil)
                                                                     ("2"
                                                                      (hide
                                                                       2)
                                                                      (("2"
                                                                        (skosimp*)
                                                                        (("2"
                                                                          (ground)
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (inst - "0")
                                                          (("2"
                                                            (replace
                                                             -3)
                                                            (("2"
                                                              (assert)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("3"
                                                          (replace -1)
                                                          (("3"
                                                            (skosimp*)
                                                            (("3"
                                                              (lift-if)
                                                              (("3"
                                                                (ground)
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (case "GP(0)=1")
                                        (("1" (assert) nil nil)
                                         ("2"
                                          (assert)
                                          (("2"
                                            (case "NOT GP(0)=2")
                                            (("1" (assert) nil nil)
                                             ("2"
                                              (replace -1)
                                              (("2"
                                                (replace -7)
                                                (("2"
                                                  (replace -2)
                                                  (("2"
                                                    (expand "*")
                                                    (("2"
                                                      (ground)
                                                      (("1"
                                                        (lemma
                                                         "polynomial_prod_int")
                                                        (("1"
                                                          (inst?)
                                                          nil
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (expand "^" +)
                                                        (("2"
                                                          (expand
                                                           "expt")
                                                          (("2"
                                                            (expand
                                                             "expt")
                                                            (("2"
                                                              (expand
                                                               "expt")
                                                              (("2"
                                                                (skeep)
                                                                (("2"
                                                                  (replace
                                                                   -1)
                                                                  (("2"
                                                                    (rewrite
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                                                                     :dir
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