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Datei: harmonic_polynomials.pvs Sprache: Latech

Original von: PVS^©

(harmonic_polynomials
(harmonic_poly_real_TCC1 0
  (harmonic_poly_real_TCC1-1 nil 3261618989
   ("" (skosimp*) (("" (inst + "1") (("" (assert) nil nil)) nil)) 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)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   shostak))
(harmonic_poly_real_TCC2 0
  (harmonic_poly_real_TCC2-1 nil 3261619116
   ("" (skosimp*)
    (("" (lemma "even_div2")
      (("" (inst?)
        (("" (assert)
          (("" (lemma "even_iff_not_odd")
            (("" (inst?) (("" (assert) nil nil)) nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((even_div2 formula-decl nil naturalnumbers nil)
    (nnrat_div_posrat_is_nnrat application-judgement "nonneg_rat"
     rationals nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (even_iff_not_odd formula-decl nil naturalnumbers 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))
(harmonic_poly_real_TCC3 0
  (harmonic_poly_real_TCC3-1 nil 3261619130
   ("" (skosimp*)
    (("" (lemma "even_div2")
      (("" (inst?)
        (("" (assert)
          (("" (lemma "odd_iff_not_even")
            (("" (inst?) (("" (assert) nil nil)) nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((even_div2 formula-decl nil naturalnumbers nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (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))
(harmonic_poly_real_TCC4 0
  (harmonic_poly_real_TCC4-1 nil 3261619140 ("" (grind) nil 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)
    (/= const-decl "boolean" notequal nil)
    (odd? const-decl "bool" integers nil))
   shostak))
(harmonic_poly_real_TCC5 0
  (harmonic_poly_real_TCC5-1 nil 3261619149 ("" (grind) nil 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)
    (/= const-decl "boolean" notequal nil)
    (odd? const-decl "bool" integers nil))
   shostak))
(harmonic_poly_real_TCC6 0
  (harmonic_poly_real_TCC6-1 nil 3261619159 ("" (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)
    (>= const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers 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)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (/= const-decl "boolean" notequal nil)
    (odd? const-decl "bool" 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))
   shostak))
(harmonic_poly_real_TCC7 0
  (harmonic_poly_real_TCC7-1 nil 3261620027 ("" (grind) nil 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)
    (/= const-decl "boolean" notequal nil)
    (odd? const-decl "bool" integers nil))
   shostak))
(harmonic_poly_imag_TCC1 0
  (harmonic_poly_imag_TCC1-1 nil 3261619175
   ("" (skosimp*)
    (("" (lemma "odd_div2" ("x" "i!1"))
      (("" (rewrite "even_or_odd") (("" (assert) nil nil)) nil)) nil))
    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)
    (odd_div2 formula-decl nil naturalnumbers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (rat_div_nzrat_is_rat application-judgement "rat" rationals nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (even_or_odd formula-decl nil naturalnumbers nil))
   shostak))
(harmonic_poly_imag_TCC2 0
  (harmonic_poly_imag_TCC2-1 nil 3261619184 ("" (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)
    (>= const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers 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)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (even? const-decl "bool" integers nil))
   shostak))
(harmonic_poly_imag_TCC3 0
  (harmonic_poly_imag_TCC3-1 nil 3261619194 ("" (grind) nil nil)
   ((even? const-decl "bool" integers nil)
    (/= const-decl "boolean" notequal nil))
   shostak))
(harmonic_poly_imag_TCC4 0
  (harmonic_poly_imag_TCC4-1 nil 3261619204 ("" (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)
    (>= const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers 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)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (even? const-decl "bool" integers nil)
    (/= const-decl "boolean" notequal nil))
   shostak))
(harmonic_poly_imag_TCC5 0
  (harmonic_poly_imag_TCC5-1 nil 3261620046 ("" (grind) nil nil)
   ((even? const-decl "bool" integers nil)
    (/= const-decl "boolean" notequal nil))
   shostak))
(harmonic_polynomial_real_n1 0
  (harmonic_polynomial_real_n1-1 nil 3261619224
   ("" (skolem 1 ("x" "y"))
    (("" (expand "harmonic_poly_real")
      (("" (expand "sigma")
        (("" (expand "sigma")
          (("" (rewrite "expt_x0")
            (("" (rewrite "expt_x1")
              (("" (rewrite "C_0")
                (("" (case "odd?(0)")
                  (("1" (expand "odd?") (("1" (propax) nil nil)) nil)
                   ("2" (assert)
                    (("2" (expand "sigma") (("2" (propax) nil nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((harmonic_poly_real const-decl "real" harmonic_polynomials nil)
    (expt_x1 formula-decl nil exponentiation nil)
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (odd? const-decl "bool" integers nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (sigma_0_neg formula-decl nil sigma_nat nil)
    (C_0 formula-decl nil binomial 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)
    (rat_exp application-judgement "rat" exponentiation nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (nzint_times_nzint_is_nzint application-judgement "nzint" integers
     nil)
    (nzreal_exp application-judgement "nzreal" exponentiation nil)
    (int_exp application-judgement "int" exponentiation nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields 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)
    (expt_x0 formula-decl nil exponentiation nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (sigma def-decl "real" sigma nil))
   shostak))
(harmonic_polynomial_imag_n1 0
  (harmonic_polynomial_imag_n1-1 nil 3261619351
   ("" (skolem 1 ("x" "y"))
    (("" (expand "harmonic_poly_imag")
      (("" (expand "sigma")
        (("" (expand "sigma")
          (("" (rewrite "expt_x1")
            (("" (rewrite "expt_x0")
              (("" (rewrite "C_n")
                (("" (case "even?(1)")
                  (("1" (expand "even?") (("1" (propax) nil nil)) nil)
                   ("2" (assert)
                    (("2" (expand "sigma") (("2" (propax) nil nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((int_minus_int_is_int application-judgement "int" integers nil)
    (harmonic_poly_imag const-decl "real" harmonic_polynomials nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (expt_x0 formula-decl nil exponentiation nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (even? const-decl "bool" integers nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (sigma_0_neg formula-decl nil sigma_nat nil)
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (C_n formula-decl nil binomial 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)
    (rat_exp application-judgement "rat" exponentiation nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (nzint_times_nzint_is_nzint application-judgement "nzint" integers
     nil)
    (nzreal_exp application-judgement "nzreal" exponentiation nil)
    (int_exp application-judgement "int" exponentiation 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)
    (expt_x1 formula-decl nil exponentiation nil)
    (sigma def-decl "real" sigma nil))
   shostak))
(harmonic_polynomial_real_rec 0
  (harmonic_polynomial_real_rec-1 nil 3261620367
   ("" (skolem 1 ("n" "x" "y"))
    (("" (expand "harmonic_poly_real")
      (("" (expand "harmonic_poly_imag")
        ((""
          (name "F1" "LAMBDA (i: nat):
              IF i > 1 + n OR odd?(i) THEN 0
              ELSE C(1 + n, i) * (-1) ^ (i / 2) *
                    (IF i = 0 THEN x ^ (1 + n)
                     ELSIF i = 1 + n THEN y ^ (1 + n)
                     ELSE (x ^ (1 - i + n)) * y ^ i
                     ENDIF)
              ENDIF")
          (("1" (replace -1)
            (("1"
              (lemma "sigma_scal[nat]"
               ("low" "0" "high" "n" "a" "x" "F" " LAMBDA (i: nat):
                IF i > n OR odd?(i) THEN 0
                ELSE C(n, i) * (-1) ^ (i / 2) *
                      (IF i = 0 THEN x ^ n
                       ELSIF i = n THEN y ^ n
                       ELSE x ^ (n - i) * y ^ i
                       ENDIF)
                ENDIF"))
              (("1" (replace -1 1 rl)
                (("1" (hide -1)
                  (("1"
                    (name "F2" "LAMBDA (i_1: nat):
               x *
                (LAMBDA (i: nat):
                   IF i > n OR odd?(i) THEN 0
                   ELSE C(n, i) * (-1) ^ (i / 2) *
                         (IF i = 0 THEN x ^ n
                          ELSIF i = n THEN y ^ n
                          ELSE x ^ (n - i) * y ^ i
                          ENDIF)
                   ENDIF)
                    (i_1)")
                    (("1" (replace -1)
                      (("1"
                        (name "F3" "LAMBDA (i: nat):
                 IF i > n OR even?(i) THEN 0
                 ELSE C(n, i) * (-1) ^ ((i - 1) / 2) *
                       (IF i = n THEN y ^ n ELSE x ^ (n - i) * y ^ i ENDIF)
                 ENDIF")
                        (("1" (replace -1)
                          (("1"
                            (lemma "sigma_scal[nat]"
                             ("low" "0" "high" "n" "F" "F3" "a" "y"))
                            (("1" (replace -1 1 rl)
                              (("1"
                                (case
                                 "sigma(0, n, F2) = sigma(0, n+1, F2)")
                                (("1"
                                  (replace -1 1)
                                  (("1"
                                    (case
                                     "FORALL (n:nat): sigma(0, n, LAMBDA (i: nat): y * F3(i)) = sigma(0,n+1, LAMBDA (i:nat): IF i = 0 THEN 0 ELSE y*F3(i-1) ENDIF)")
                                    (("1"
                                      (inst - "n")
                                      (("1"
                                        (replace -1 1)
                                        (("1"
                                          (lemma
                                           "sigma_minus[nat]"
                                           ("low"
                                            "0"
                                            "high"
                                            "n+1"
                                            "F"
                                            "F2"
                                            "G"
                                            "LAMBDA (i: nat): IF i = 0 THEN 0 ELSE y * F3(i - 1) ENDIF"))
                                          (("1"
                                            (replace -1 1)
                                            (("1"
                                              (hide -1 -2 -3 -4)
                                              (("1"
                                                (lemma
                                                 "sigma_eq[nat]"
                                                 ("low"
                                                  "0"
                                                  "high"
                                                  "n+1"
                                                  "F"
                                                  "F1"
                                                  "G"
                                                  "LAMBDA (i_1: nat):
               F2(i_1) -
                (LAMBDA (i: nat): IF i = 0 THEN 0 ELSE y * F3(i - 1) ENDIF)
                    (i_1)"))
                                                (("1"
                                                  (split -1)
                                                  (("1"
                                                    (assert)
                                                    nil
                                                    nil)
                                                   ("2"
                                                    (hide 2)
                                                    (("2"
                                                      (skosimp*)
                                                      (("2"
                                                        (typepred
                                                         "n!1")
                                                        (("2"
                                                          (expand "F1")
                                                          (("2"
                                                            (hide
                                                             -5
                                                             -4
                                                             -3)
                                                            (("2"
                                                              (expand
                                                               "F2")
                                                              (("2"
                                                                (expand
                                                                 "F3")
                                                                (("2"
                                                                  (case
                                                                   "n!1=0")
                                                                  (("1"
                                                                    (replace
                                                                     -1)
                                                                    (("1"
                                                                      (rewrite
                                                                       "C_0")
                                                                      (("1"
                                                                        (rewrite
                                                                         "C_0")
                                                                        (("1"
                                                                          (rewrite
                                                                           "expt_x0")
                                                                          (("1"
                                                                            (case
                                                                             "odd?(0)")
                                                                            (("1"
                                                                              (expand
                                                                               "odd?")
                                                                              (("1"
                                                                                (propax)
                                                                                nil
                                                                                nil))
                                                                              nil)
                                                                             ("2"
                                                                              (assert)
                                                                              (("2"
                                                                                (expand
                                                                                 "^")
                                                                                (("2"
                                                                                  (expand
                                                                                   "expt"
                                                                                   2
                                                                                   1)
                                                                                  (("2"
                                                                                    (assert)
                                                                                    nil
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil)
                                                                   ("2"
                                                                    (assert)
                                                                    (("2"
                                                                      (case
                                                                       "n!1= 1+n")
                                                                      (("1"
                                                                        (assert)
                                                                        (("1"
                                                                          (replace
                                                                           -1)
                                                                          (("1"
                                                                            (rewrite
                                                                             "C_n")
                                                                            (("1"
                                                                              (rewrite
                                                                               "C_n")
                                                                              (("1"
                                                                                (case
                                                                                 "even?(n)")
                                                                                (("1"
                                                                                  (case
                                                                                   "odd?(n+1)")
                                                                                  (("1"
                                                                                    (assert)
                                                                                    nil
                                                                                    nil)
                                                                                   ("2"
                                                                                    (hide-all-but
                                                                                     (-1
                                                                                      1))
                                                                                    (("2"
                                                                                      (expand
                                                                                       "even?")
                                                                                      (("2"
                                                                                        (skosimp*)
                                                                                        (("2"
                                                                                          (replace
                                                                                           -1)
                                                                                          (("2"
                                                                                            (expand
                                                                                             "odd?")
                                                                                            (("2"
                                                                                              (inst
                                                                                               +
                                                                                               "j!1")
                                                                                              nil
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil)
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