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Datei: shift.prf Sprache: Lisp

Original von: PVS^©

(shift
(cauchy_div2n_TCC1 0
  (cauchy_div2n_TCC1-1 nil 3251040549 ("" (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)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   shostak))
(cauchy_div2n_TCC2 0
  (cauchy_div2n_TCC2-1 nil 3251040561
   ("" (skosimp*)
    (("" (typepred "cx!1")
      (("" (expand "cauchy_real?")
        (("" (skosimp*)
          (("" (inst 1 "x!1/2^n!1")
            (("" (expand "cauchy_prop")
              (("" (skosimp*)
                (("" (case "p!1>=n!1")
                  (("1" (replace -1 1)
                    (("1" (inst -2 "p!1-n!1")
                      (("1"
                        (lemma "expt_div"
                         ("n0x" "2" "i" "p!1" "j" "n!1"))
                        (("1" (replace -1 -3 rl)
                          (("1" (grind) nil nil)) nil))
                        nil)
                       ("2" (assert) nil nil))
                      nil))
                    nil)
                   ("2" (replace 1 2)
                    (("2" (name "RR" "round(cx!1(p!1)/2^n!1)")
                      (("2" (inst -2 "p!1")
                        (("2" (lemma "expt_pos" ("px" "2" "i" "n!1"))
                          (("2"
                            (lemma "lemma_A2"
                             ("r" "RR" "p" "cx!1(p!1)" "q" "2^n!1"))
                            (("2" (replace -3)
                              (("2"
                                (skosimp*)
                                (("2"
                                  (lemma
                                   "div_mult_pos_lt1"
                                   ("z"
                                    "x!1*2^p!1"
                                    "py"
                                    "2^n!1"
                                    "x"
                                    "RR+1"))
                                  (("2"
                                    (lemma
                                     "div_mult_pos_lt2"
                                     ("x"
                                      "RR-1"
                                      "z"
                                      "x!1*2^p!1"
                                      "py"
                                      "2^n!1"))
                                    (("2"
                                      (lemma
                                       "both_sides_expt_gt1_ge"
                                       ("gt1x" "2" "i" "n!1" "j" "1"))
                                      (("2"
                                        (rewrite "expt_x1")
                                        (("2" (assert) nil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((cauchy_real nonempty-type-eq-decl nil cauchy nil)
    (cauchy_real? const-decl "bool" cauchy nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals 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)
    (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)
    (cauchy_prop const-decl "bool" cauchy 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 "[numfield, numfield -> numfield]" number_fields nil)
    (p!1 skolem-const-decl "nat" shift nil)
    (n!1 skolem-const-decl "nat" shift nil)
    (nnrat_exp application-judgement "nnrat" exponentiation nil)
    (posrat_exp application-judgement "posrat" exponentiation nil)
    (expt def-decl "real" exponentiation nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (minus_int_is_int application-judgement "int" integers nil)
    (posnat_expt application-judgement "posnat" exponentiation nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (nzreal nonempty-type-eq-decl nil reals nil)
    (expt_div formula-decl nil exponentiation nil)
    (rat_div_nzrat_is_rat application-judgement "rat" rationals nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (round const-decl "int" prelude_aux nil)
    (expt_pos formula-decl nil exponentiation nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (rat_minus_rat_is_rat application-judgement "rat" rationals nil)
    (rat_plus_rat_is_rat application-judgement "rat" rationals nil)
    (div_mult_pos_lt1 formula-decl nil real_props nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (both_sides_expt_gt1_ge formula-decl nil exponentiation nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (posrat_times_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (rat_times_rat_is_rat application-judgement "rat" rationals nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (expt_x1 formula-decl nil exponentiation nil)
    (div_mult_pos_lt2 formula-decl nil real_props nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (lemma_A2 formula-decl nil appendix nil)
    (^ const-decl "real" exponentiation nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/= const-decl "boolean" notequal nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (posint_exp application-judgement "posint" exponentiation nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil))
   shostak))
(cauchy_mul2n_TCC1 0
  (cauchy_mul2n_TCC1-1 nil 3251041439
   ("" (skosimp*)
    (("" (typepred "cx!1")
      (("" (expand "cauchy_real?")
        (("" (skosimp*)
          (("" (inst 1 "x!1*2^n!1")
            (("" (expand "cauchy_prop")
              (("" (skosimp*)
                (("" (inst - "p!1+n!1")
                  ((""
                    (lemma "expt_plus" ("n0x" "2" "i" "p!1" "j" "n!1"))
                    (("" (grind) nil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((cauchy_real nonempty-type-eq-decl nil cauchy nil)
    (cauchy_real? const-decl "bool" cauchy nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals 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)
    (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)
    (posint_exp application-judgement "posint" exponentiation nil)
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (cauchy_prop const-decl "bool" cauchy nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (posnat_expt application-judgement "posnat" exponentiation nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (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)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (nzreal nonempty-type-eq-decl nil reals nil)
    (expt_plus formula-decl nil exponentiation nil)
    (^ const-decl "real" exponentiation nil)
    (/= const-decl "boolean" notequal nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (real_times_real_is_real application-judgement "real" reals nil))
   shostak))
(lemma_div2n 0
  (lemma_div2n-1 nil 3251034397
   ("" (expand "cauchy_prop")
    (("" (expand "cauchy_div2n")
      (("" (expand "div2n")
        (("" (skosimp*)
          (("" (case "p!1>=n!1")
            (("1" (replace -1 1)
              (("1" (inst -2 "p!1-n!1")
                (("1"
                  (lemma "expt_div" ("n0x" "2" "i" "p!1" "j" "n!1"))
                  (("1" (replace -1 -3 rl) (("1" (grind) nil nil))
                    nil))
                  nil)
                 ("2" (assert) nil nil))
                nil))
              nil)
             ("2" (replace 1 2)
              (("2" (name "RR" "round(cx!1(p!1)/2^n!1)")
                (("2" (inst -2 "p!1")
                  (("2" (lemma "expt_pos" ("px" "2" "i" "n!1"))
                    (("2"
                      (lemma "lemma_A2"
                       ("r" "RR" "p" "cx!1(p!1)" "q" "2^n!1"))
                      (("2" (replace -3)
                        (("2" (skosimp*)
                          (("2"
                            (lemma "div_mult_pos_lt1"
                             ("z" "x!1*2^p!1" "py" "2^n!1" "x" "RR+1"))
                            (("2"
                              (lemma "div_mult_pos_lt2"
                               ("x"
                                "RR-1"
                                "z"
                                "x!1*2^p!1"
                                "py"
                                "2^n!1"))
                              (("2"
                                (replace -1)
                                (("2"
                                  (replace -2)
                                  (("2"
                                    (lemma
                                     "both_sides_expt_gt1_ge"
                                     ("gt1x" "2" "i" "n!1" "j" "1"))
                                    (("2"
                                      (rewrite "expt_x1")
                                      (("2" (assert) nil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((cauchy_div2n const-decl "cauchy_real" shift nil)
    (lemma_A2 formula-decl nil appendix nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (div_mult_pos_lt2 formula-decl nil real_props nil)
    (expt_x1 formula-decl nil exponentiation nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (rat_times_rat_is_rat application-judgement "rat" rationals nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (posrat_times_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (both_sides_expt_gt1_ge formula-decl nil exponentiation nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (div_mult_pos_lt1 formula-decl nil real_props nil)
    (rat_plus_rat_is_rat application-judgement "rat" rationals nil)
    (rat_minus_rat_is_rat application-judgement "rat" rationals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (expt_pos formula-decl nil exponentiation nil)
    (rat_div_nzrat_is_rat application-judgement "rat" rationals nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (round const-decl "int" prelude_aux nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (cauchy_real? const-decl "bool" cauchy nil)
    (cauchy_real nonempty-type-eq-decl nil cauchy nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (expt_div formula-decl nil exponentiation nil)
    (/= const-decl "boolean" notequal nil)
    (nzreal nonempty-type-eq-decl nil reals nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (posnat_expt application-judgement "posnat" exponentiation nil)
    (minus_int_is_int application-judgement "int" integers nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (expt def-decl "real" exponentiation nil)
    (^ const-decl "real" exponentiation nil)
    (posrat_exp application-judgement "posrat" exponentiation nil)
    (nnrat_exp application-judgement "nnrat" exponentiation nil)
    (n!1 skolem-const-decl "nat" shift nil)
    (p!1 skolem-const-decl "nat" shift nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (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)
    (div2n const-decl "real" shift nil)
    (cauchy_prop const-decl "bool" cauchy nil)
    (posint_exp application-judgement "posint" exponentiation nil))
   shostak))
(lemma_mul2n 0
  (lemma_mul2n-1 nil 3251034151
   ("" (expand "cauchy_prop")
    (("" (expand "cauchy_mul2n")
      (("" (expand "mul2n")
        (("" (skosimp*)
          (("" (inst - "n!1+p!1")
            (("" (lemma "expt_plus" ("n0x" "2" "i" "n!1" "j" "p!1"))
              (("" (grind) nil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((cauchy_mul2n const-decl "cauchy_real" shift nil)
    (expt_plus formula-decl nil exponentiation nil)
    (/= const-decl "boolean" notequal nil)
    (nzreal nonempty-type-eq-decl nil reals nil)
    (^ const-decl "real" exponentiation nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (int_minus_int_is_int application-judgement "int" integers 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)
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (posnat_expt application-judgement "posnat" exponentiation nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields 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)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (mul2n const-decl "real" shift nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (cauchy_prop const-decl "bool" cauchy nil)
    (posint_exp application-judgement "posint" exponentiation nil))
   shostak)))

¤ Dauer der Verarbeitung: 0.46 Sekunden (vorverarbeitet) ¤

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