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© Kompilation durch diese Firma

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Datei: List.v   Sprache: PVS

Original von: PVS©

(compute_sturm_tarski
 (der_prod_TCC1 0
  (der_prod_TCC1-1 nil 3619442978 ("" (subtype-tcc) nil nilnil nil))
 (der_prod_TCC2 0
  (der_prod_TCC2-1 nil 3619442978
   ("" (skeep)
    (("" (expand "polynomial_prod")
      (("" (assert)
        ((""
          (case "FORALL (abc:[nat->int],i1,i2:nat): rational_pred(sigma(i1,i2,abc)) AND integer_pred(sigma(i1,i2,abc))")
          (("1" (inst?)
            (("1" (hide 2)
              (("1" (skosimp*)
                (("1" (expand "poly_deriv") (("1" (propax) nil nil))
                  nil))
                nil))
              nil)
             ("2" (skosimp*) (("2" (assertnil nil)) nil))
            nil)
           ("2" (hide 2)
            (("2" (assert)
              (("2" (induct "i2")
                (("1" (skeep)
                  (("1" (expand "sigma")
                    (("1" (lift-if)
                      (("1" (assert)
                        (("1" (expand "sigma") (("1" (propax) nil nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (skeep)
                  (("2" (skeep)
                    (("2" (inst - "abc" "i1")
                      (("2" (flatten)
                        (("2" (expand "sigma" +)
                          (("2" (assert)
                            (("2" (lift-if)
                              (("2"
                                (assert)
                                (("2"
                                  (lemma "integers.closed_plus")
                                  (("2"
                                    (inst?)
                                    (("2"
                                      (assert)
                                      (("2"
                                        (lemma "rationals.closed_plus")
                                        (("2"
                                          (inst?)
                                          (("2" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                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)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (polynomial_prod const-decl "real" polynomials "reals/")
    (sigma def-decl "real" sigma "reals/")
    (T_high type-eq-decl nil sigma "reals/")
    (T_low type-eq-decl nil sigma "reals/")
    (<= const-decl "bool" reals nil)
    (OR const-decl "[bool, bool -> bool]" booleans 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)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (a skolem-const-decl "[nat -> int]" compute_sturm_tarski nil)
    (g skolem-const-decl "[nat -> int]" compute_sturm_tarski nil)
    (i skolem-const-decl "nat" compute_sturm_tarski nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (sequence type-eq-decl nil sequences nil)
    (poly_deriv const-decl "real" polynomials "reals/")
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (max const-decl "{p: real | p >= m AND p >= n}" real_defs nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (closed_plus formula-decl nil integers nil)
    (rat nonempty-type-eq-decl nil rationals nil)
    (closed_plus formula-decl nil rationals nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (pred type-eq-decl nil defined_types 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)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   nil))
 (finite_bij_real_remove_one 0
  (finite_bij_real_remove_one-1 nil 3594487196
   ("" (skeep)
    (("" (case "NOT m-1>=0")
      (("1" (hide 2)
        (("1" (expand "bijective?")
          (("1" (flatten)
            (("1" (expand "surjective?")
              (("1" (inst - "x")
                (("1" (assert) (("1" (skosimp*) nil nil)) nil)) nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (assert)
        (("2" (copy -2)
          (("2" (expand "bijective?" -1)
            (("2" (flatten)
              (("2" (copy -2)
                (("2" (expand "surjective?" -1)
                  (("2" (inst - "x")
                    (("2" (skolem -1 "ii")
                      (("2"
                        (inst +
                         "LAMBDA (i:below(m-1)): IF i)
                        (("1" (expand "bijective?" +)
                          (("1" (split +)
                            (("1" (expand "injective?" 1)
                              (("1"
                                (skeep)
                                (("1"
                                  (lift-if)
                                  (("1"
                                    (lift-if)
                                    (("1"
                                      (lift-if)
                                      (("1"
                                        (assert)
                                        (("1"
                                          (expand "injective?" -3)
                                          (("1"
                                            (ground)
                                            (("1"
                                              (inst - "x1" "x2")
                                              (("1" (assertnil nil))
                                              nil)
                                             ("2"
                                              (inst - "x1" "1+x2")
                                              (("2" (assertnil nil))
                                              nil)
                                             ("3"
                                              (inst - "1+x1" "x2")
                                              (("3" (assertnil nil))
                                              nil)
                                             ("4"
                                              (inst - "1+x1" "1+x2")
                                              (("4" (assertnil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (expand "surjective?")
                              (("2"
                                (skeep)
                                (("2"
                                  (typepred "y")
                                  (("2"
                                    (inst - "y")
                                    (("2"
                                      (assert)
                                      (("2"
                                        (skolem - "jj")
                                        (("2"
                                          (case "ii = jj")
                                          (("1"
                                            (expand "injective?")
                                            (("1"
                                              (inst - "ii" "jj")
                                              (("1" (assertnil nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (flatten)
                                            (("2"
                                              (assert)
                                              (("2"
                                                (case "jj)
                                                (("1"
                                                  (inst + "jj")
                                                  (("1"
                                                    (assert)
                                                    nil
                                                    nil)
                                                   ("2"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (inst + "jj-1")
                                                  (("1"
                                                    (assert)
                                                    nil
                                                    nil)
                                                   ("2"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (expand "injective?")
                          (("2" (skeep)
                            (("2" (inst - "1+i" "ii")
                              (("2" (assertnil nil)) nil))
                            nil))
                          nil)
                         ("3" (expand "injective?")
                          (("3" (skeep)
                            (("3" (inst - "i" "ii")
                              (("3" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nat nonempty-type-eq-decl nil naturalnumbers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (bijective? const-decl "bool" functions nil)
    (surjective? const-decl "bool" functions nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (x skolem-const-decl "real" compute_sturm_tarski nil)
    (A skolem-const-decl "set[real]" compute_sturm_tarski nil)
    (set type-eq-decl nil sets nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (jj skolem-const-decl "below(m)" compute_sturm_tarski nil)
    (injective? const-decl "bool" functions 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)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (< const-decl "bool" reals nil)
    (m skolem-const-decl "nat" compute_sturm_tarski nil)
    (below type-eq-decl nil naturalnumbers nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (ii skolem-const-decl "below(m)" compute_sturm_tarski nil)
    (/= const-decl "boolean" notequal nil)
    (bij skolem-const-decl "[below(m) -> (A)]" compute_sturm_tarski
     nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil))
   shostak))
 (finite_bij_real_remove_two 0
  (finite_bij_real_remove_two-1 nil 3594488379
   (""
    (case "FORALL (x, y: real, m: nat, A: set[real], bij: [below(m) -> (A)],i,j:below(m)):
        bijective?(bij) AND A(x) AND A(y) AND x /= y AND i<j AND bij(i)=x AND bij(j)=y IMPLIES
         m - 2 >= 0 AND
          (EXISTS (bijec: [below(m - 2) -> {r: (A) | r /= x AND r /= y}]):
             bijective?(bijec))")
    (("1" (skeep)
      (("1" (copy -2)
        (("1" (expand "bijective?" -1)
          (("1" (flatten)
            (("1" (expand "surjective?" -2)
              (("1" (inst-cp - "x")
                (("1" (inst - "y")
                  (("1" (skosimp*)
                    (("1" (case "x!1 < x!2")
                      (("1" (inst - "y" "x" "m" "A" "bij" "x!1" "x!2")
                        (("1" (assert)
                          (("1" (skosimp*)
                            (("1" (inst + "bijec!1")
                              (("1"
                                (assert)
                                (("1"
                                  (expand "bijective?" (-5 2))
                                  (("1"
                                    (flatten)
                                    (("1"
                                      (split)
                                      (("1"
                                        (expand "injective?" (-5 1))
                                        (("1" (propax) nil nil))
                                        nil)
                                       ("2"
                                        (expand "surjective?" (-6 1))
                                        (("2"
                                          (skosimp*)
                                          (("2"
                                            (inst - "y!1")
                                            nil
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (inst - "x" "y" "m" "A" "bij" "x!2" "x!1")
                        (("2" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil)
     ("2" (hide 2)
      (("2" (skeep)
        (("2" (case "NOT m-2>=0")
          (("1" (hide 3) (("1" (assertnil nil)) nil)
           ("2" (assert)
            (("2"
              (inst +
               "LAMBDA (ii:below(m-2)): IF ii)
              (("1" (expand "bijective?")
                (("1" (flatten)
                  (("1" (split)
                    (("1" (expand "injective?")
                      (("1" (skosimp*)
                        (("1" (lift-if)
                          (("1" (lift-if)
                            (("1" (lift-if)
                              (("1"
                                (lift-if)
                                (("1"
                                  (assert)
                                  (("1"
                                    (ground)
                                    (("1"
                                      (inst - "x1!1" "x2!1")
                                      (("1" (assertnil nil))
                                      nil)
                                     ("2"
                                      (inst - "x1!1" "1+x2!1")
                                      (("2" (assertnil nil))
                                      nil)
                                     ("3"
                                      (inst - "x1!1" "2+x2!1")
                                      (("3" (assertnil nil))
                                      nil)
                                     ("4"
                                      (inst - "1+x1!1" "x2!1")
                                      (("4" (assertnil nil))
                                      nil)
                                     ("5"
                                      (inst - "1+x1!1" "1+x2!1")
                                      (("5" (assertnil nil))
                                      nil)
                                     ("6"
                                      (inst - "1+x1!1" "2+x2!1")
                                      (("6" (assertnil nil))
                                      nil)
                                     ("7"
                                      (inst - "2+x1!1" "x2!1")
                                      (("7" (assertnil nil))
                                      nil)
                                     ("8"
                                      (inst - "2+x1!1" "1+x2!1")
                                      (("8" (assertnil nil))
                                      nil)
                                     ("9"
                                      (inst - "2+x1!1" "2+x2!1")
                                      (("9" (assertnil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (expand "surjective?")
                      (("2" (skosimp*)
                        (("2" (inst - "y!1")
                          (("2" (skosimp*)
                            (("2" (case "x!1 = i OR x!1 = j")
                              (("1" (ground) nil nil)
                               ("2"
                                (flatten)
                                (("2"
                                  (name "ii" "x!1")
                                  (("2"
                                    (replaces -1)
                                    (("2"
                                      (assert)
                                      (("2"
                                        (case "ii < i")
                                        (("1"
                                          (inst + "ii")
                                          (("1" (assertnil nil)
                                           ("2" (assertnil nil))
                                          nil)
                                         ("2"
                                          (case "ii-1)
                                          (("1"
                                            (inst + "ii-1")
                                            (("1" (assertnil nil)
                                             ("2" (assertnil nil))
                                            nil)
                                           ("2"
                                            (inst + "ii-2")
                                            (("1" (assertnil nil)
                                             ("2" (assertnil nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (skosimp*)
                (("2" (assert)
                  (("2" (expand "bijective?")
                    (("2" (flatten)
                      (("2" (assert)
                        (("2" (expand "injective?")
                          (("2" (inst-cp - "i" "2+ii!1")
                            (("2" (assert)
                              (("2"
                                (assert)
                                (("2"
                                  (inst-cp - "j" "2+ii!1")
                                  (("2" (assertnil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("3" (assert)
                (("3" (skeep)
                  (("3" (expand "bijective?")
                    (("3" (flatten)
                      (("3" (expand "injective?")
                        (("3" (inst-cp - "1+ii" "i")
                          (("3" (assert)
                            (("3" (assert)
                              (("3"
                                (inst-cp - "1+ii" "j")
                                (("3" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("4" (expand "bijective?")
                (("4" (flatten)
                  (("4" (expand "injective?")
                    (("4" (skeep)
                      (("4" (inst-cp - "ii" "i")
                        (("4" (assert)
                          (("4" (assert)
                            (("4" (inst - "ii" "j")
                              (("4" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((NOT const-decl "[bool -> bool]" booleans nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (m skolem-const-decl "nat" compute_sturm_tarski nil)
    (i skolem-const-decl "below(m)" compute_sturm_tarski nil)
    (A skolem-const-decl "set[real]" compute_sturm_tarski nil)
    (bij skolem-const-decl "[below(m) -> (A)]" compute_sturm_tarski
     nil)
    (x skolem-const-decl "real" compute_sturm_tarski nil)
    (y skolem-const-decl "real" compute_sturm_tarski nil)
    (j skolem-const-decl "below(m)" compute_sturm_tarski nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (ii skolem-const-decl "below(m)" compute_sturm_tarski nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (surjective? const-decl "bool" functions nil)
    (y skolem-const-decl "real" compute_sturm_tarski nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (injective? const-decl "bool" functions nil)
    (x skolem-const-decl "real" compute_sturm_tarski nil)
    (A skolem-const-decl "set[real]" compute_sturm_tarski 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)
    (set type-eq-decl nil sets nil) (< const-decl "bool" reals nil)
    (below type-eq-decl nil naturalnumbers nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (bijective? const-decl "bool" functions nil)
    (/= const-decl "boolean" notequal nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields
       nil))
   shostak))
 (computed_sturm_spec_TCC1 0
  (computed_sturm_spec_TCC1-1 nil 3593788550
   ("" (skeep)
    (("" (assert)
      (("" (expand "der_prod")
        (("" (expand "polynomial_prod")
          (("" (expand "max")
            (("" (expand "sigma")
              (("" (expand "sigma")
                (("" (expand "poly_deriv") (("" (assertnil nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((int_minus_int_is_int application-judgement "int" integers nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (polynomial_prod const-decl "real" polynomials "reals/")
    (sigma def-decl "real" sigma "reals/")
    (poly_deriv const-decl "real" polynomials "reals/")
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (max const-decl "{p: real | p >= m AND p >= n}" real_defs nil)
    (der_prod const-decl "int" compute_sturm_tarski nil))
   nil))
 (computed_sturm_spec_TCC2 0
  (computed_sturm_spec_TCC2-1 nil 3593788550 ("" (subtype-tcc) 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)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props 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)
    (listn_0 name-judgement "listn[real](0)" polynomial_division
     "Sturm/")
    (listn_0 name-judgement "listn[int](0)" gcd_coeff "Sturm/")
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (adjusted_remainder const-decl "list[int]" polynomial_pseudo_divide
     "Sturm/")
    (mod const-decl "{k | abs(k) < abs(j)}" mod nil)
    (nonzero_version const-decl "list[int]" gcd_coeff "Sturm/")
    (array2list const-decl
                "{l: listn(n) | FORALL (i: below(n)): a(i) = nth(l, i)}"
                array2list "structures/")
    (nth def-decl "T" list_props nil)
    (below type-eq-decl nil nat_types nil)
    (< const-decl "bool" reals nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (length def-decl "nat" list_props 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)
    (der_prod const-decl "int" compute_sturm_tarski nil)
    (is_neg_remainder_list? const-decl "bool" remainder_sequence
     "Sturm/")
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (every adt-def-decl "boolean" list_adt nil)
    (PRED type-eq-decl nil defined_types nil)
    (list type-decl nil list_adt nil)
    (/= const-decl "boolean" notequal nil)
    (remainder_seq const-decl
     "{crem: (is_neg_remainder_list?(g, n, h, m)) |
         length(crem) > 1 AND length(nth(crem, 0)) = 0}"
     remainder_sequence "Sturm/")
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil))
   nil))
 (computed_sturm_spec_TCC3 0
  (computed_sturm_spec_TCC3-1 nil 3593788550 ("" (subtype-tcc) 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)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (array2list const-decl
                "{l: listn(n) | FORALL (i: below(n)): a(i) = nth(l, i)}"
                array2list "structures/")
    (nonzero_version const-decl "list[int]" gcd_coeff "Sturm/")
    (mod const-decl "{k | abs(k) < abs(j)}" mod nil)
    (adjusted_remainder const-decl "list[int]" polynomial_pseudo_divide
     "Sturm/")
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (rat_max application-judgement "{s: rat | s >= q AND s >= r}"
     real_defs nil)
    (int_max application-judgement "{k: int | i <= k AND j <= k}"
     real_defs nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (listn_0 name-judgement "listn[int](0)" gcd_coeff "Sturm/")
    (listn_0 name-judgement "listn[real](0)" polynomial_division
     "Sturm/")
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nth def-decl "T" list_props nil)
    (below type-eq-decl nil nat_types nil)
    (< const-decl "bool" reals nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (length def-decl "nat" list_props 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)
    (der_prod const-decl "int" compute_sturm_tarski nil)
    (is_neg_remainder_list? const-decl "bool" remainder_sequence
     "Sturm/")
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (every adt-def-decl "boolean" list_adt nil)
    (PRED type-eq-decl nil defined_types nil)
    (list type-decl nil list_adt nil)
    (/= const-decl "boolean" notequal nil)
    (remainder_seq const-decl
     "{crem: (is_neg_remainder_list?(g, n, h, m)) |
         length(crem) > 1 AND length(nth(crem, 0)) = 0}"
     remainder_sequence "Sturm/")
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil))
   nil))
 (computed_sturm_spec_TCC4 0
  (computed_sturm_spec_TCC4-1 nil 3593788550 ("" (subtype-tcc) nil nil)
   ((/= const-decl "boolean" notequal nil)
    (remainder_seq const-decl
     "{crem: (is_neg_remainder_list?(g, n, h, m)) |
         length(crem) > 1 AND length(nth(crem, 0)) = 0}"
     remainder_sequence "Sturm/")
    (int_minus_int_is_int application-judgement "int" integers nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil))
   nil))
 (computed_sturm_spec_TCC5 0
  (computed_sturm_spec_TCC5-1 nil 3593790499 ("" (subtype-tcc) 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)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props 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)
    (listn_0 name-judgement "listn[real](0)" polynomial_division
     "Sturm/")
    (listn_0 name-judgement "listn[int](0)" gcd_coeff "Sturm/")
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (rat_max application-judgement "{s: rat | s >= q AND s >= r}"
     real_defs nil)
    (adjusted_remainder const-decl "list[int]" polynomial_pseudo_divide
     "Sturm/")
    (mod const-decl "{k | abs(k) < abs(j)}" mod nil)
    (nonzero_version const-decl "list[int]" gcd_coeff "Sturm/")
    (array2list const-decl
                "{l: listn(n) | FORALL (i: below(n)): a(i) = nth(l, i)}"
                array2list "structures/")
    (nth def-decl "T" list_props nil)
    (below type-eq-decl nil nat_types nil)
    (< const-decl "bool" reals nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (length def-decl "nat" list_props 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)
    (der_prod const-decl "int" compute_sturm_tarski nil)
    (is_neg_remainder_list? const-decl "bool" remainder_sequence
     "Sturm/")
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (every adt-def-decl "boolean" list_adt nil)
    (PRED type-eq-decl nil defined_types nil)
    (list type-decl nil list_adt nil)
    (/= const-decl "boolean" notequal nil)
    (remainder_seq const-decl
     "{crem: (is_neg_remainder_list?(g, n, h, m)) |
         length(crem) > 1 AND length(nth(crem, 0)) = 0}"
     remainder_sequence "Sturm/")
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil))
   nil))
 (computed_sturm_spec 0
  (computed_sturm_spec-1 nil 3593788956
   ("" (skeep)
    ((""
      (name "sl" "remainder_seq(a,
                                       n,
                                       der_prod(a, n, g, k),
                                       n - 1 + k)")
      (("" (replace -1)
        ((""
          (name "P" "LAMBDA (j: nat):
                                                                             IF j < length(sl)
                                                                               THEN list2array[int](0)(nth(sl, length(sl) - 1 - j))
                                                                             ELSE (LAMBDA (i: nat): 0)
                                                                             ENDIF")
          (("1" (replaces -1)
            (("1"
              (name "N" " LAMBDA (j: nat):
                                                                                               IF j < length(sl)
                                                                                                 THEN max(length(nth(sl, length(sl) - 1 - j)) - 1, 0)
                                                                                               ELSE 0
                                                                                               ENDIF")
              (("1" (name "M" "length(sl)-1")
                (("1" (replace -1)
                  (("1" (assert)
                    (("1" (replace -1)
                      (("1" (replace -2)
                        (("1" (replace -3)
                          (("1" (split +)
                            (("1" (copy 2)
                              (("1"
                                (hide 3)
                                (("1"
                                  (expand
                                   "constructed_sturm_sequence?")
                                  (("1"
                                    (invoke (case "NOT %1") (! 2 1))
                                    (("1"
                                      (hide 3)
                                      (("1"
                                        (skeep)
                                        (("1"
                                          (expand "P" -1)
                                          (("1"
                                            (expand "N" -1)
                                            (("1"
                                              (assert)
                                              (("1"
                                                (expand "max" -1)
                                                (("1"
                                                  (lift-if)
                                                  (("1"
                                                    (assert)
                                                    (("1"
                                                      (split -)
                                                      (("1"
                                                        (flatten)
                                                        (("1"
                                                          (case
                                                           "NOT length(sl)-1-i = 0")
                                                          (("1"
                                                            (assert)
                                                            (("1"
                                                              (typepred
                                                               "sl")
                                                              (("1"
                                                                (hide
                                                                 -1)
                                                                (("1"
                                                                  (expand
                                                                   "is_neg_remainder_list?")
                                                                  (("1"
                                                                    (flatten)
                                                                    (("1"
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--> --------------------

--> maximum size reached

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

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