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Datei: sincos_def.prf   Sprache: Lisp

Original von: PVS©
(sincos_def
 (cos_ub 0
  (cos_ub-1 nil 3265619589
   ("" (skosimp*)
    (("" (typepred "cos(x!1)") (("" (propax) nil nil)) nil)) nil)
   ((cos const-decl "real" trig_basic nil)
    (- const-decl "[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)
    (cos_range application-judgement "trig_range" trig_basic nil))
   shostak))
 (sin_ub 0
  (sin_ub-2 nil 3307184568
   ("" (skolem 1 ("px"))
    (("" (case "px<=1")
      (("1" (expand "sin")
        (("1" (lemma "floor_def" ("x" "px/(2*pi)"))
          (("1" (rewrite "div_mult_pos_lt1")
            (("1" (rewrite "div_mult_pos_le2")
              (("1" (flatten)
                (("1" (lemma "trichotomy" ("x" "floor(px / (2 * pi))"))
                  (("1" (split -1)
                    (("1" (case "floor(px / (2 * pi)) =1")
                      (("1" (lemma "pi_bnds")
                        (("1" (flatten) (("1" (assert) nil nil)) nil))
                        nil)
                       ("2" (case "floor(px / (2 * pi))>1")
                        (("1" (lemma "pi_bnds")
                          (("1" (flatten)
                            (("1" (assert)
                              (("1"
                                (lemma
                                 "both_sides_times_pos_lt1"
                                 ("x"
                                  "1"
                                  "y"
                                  "floor(px / (2 * pi))"
                                  "pz"
                                  "2*pi"))
                                (("1" (assert) nil nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (assert) nil nil))
                        nil))
                      nil)
                     ("2" (replace -1)
                      (("2" (hide -1 -2 -3)
                        (("2" (simplify 1)
                          (("2" (lemma "pi_bnds")
                            (("2" (flatten)
                              (("2"
                                (lemma "phases_sin" ("x" "px"))
                                (("2"
                                  (split -1)
                                  (("1"
                                    (rewrite "sin_q1")
                                    (("1"
                                      (hide -1)
                                      (("1"
                                        (lemma "sin_value_derivable2")
                                        (("1"
                                          (lemma "deriv_sin_value")
                                          (("1"
                                            (lemma
                                             "identity_derivable_fun[real_abs_lt_pi]")
                                            (("1"
                                              (lemma
                                               "deriv_id_fun[real_abs_lt_pi]")
                                              (("1"
                                                (expand "I")
                                                (("1"
                                                  (expand "const_fun")
                                                  (("1"
                                                    (lemma
                                                     "diff_derivable_fun"
                                                     ("f1"
                                                      "LAMBDA (x: real_abs_lt_pi): x"
                                                      "f2"
                                                      "LAMBDA (x: real_abs_lt_pi): sin_value(x)"))
                                                    (("1"
                                                      (lemma
                                                       "deriv_diff_fun"
                                                       ("ff1"
                                                        "LAMBDA (x: real_abs_lt_pi): x"
                                                        "ff2"
                                                        "LAMBDA (x: real_abs_lt_pi): sin_value(x)"))
                                                      (("1"
                                                        (replace -3 -1)
                                                        (("1"
                                                          (replace
                                                           -5
                                                           -1)
                                                          (("1"
                                                            (expand
                                                             "-")
                                                            (("1"
                                                              (assert)
                                                              (("1"
                                                                (lemma
                                                                 "identity_derivable_fun[{x:nnreal|x)
                                                                (("1"
                                                                  (lemma
                                                                   "deriv_id_fun[{x:nnreal|x)
                                                                  (("1"
                                                                    (expand
                                                                     "I")
                                                                    (("1"
                                                                      (expand
                                                                       "const_fun")
                                                                      (("1"
                                                                        (lemma
                                                                         "composition_derivable_fun[{x: nnreal | x < pi / 2},real_abs_lt_pi]"
                                                                         ("f"
                                                                          "LAMBDA (x: {x: nnreal | x < pi / 2}): x"
                                                                          "g"
                                                                          "LAMBDA (x_1: real_abs_lt_pi): x_1 - sin_value(x_1)"))
                                                                        (("1"
                                                                          (replace
                                                                           -3)
                                                                          (("1"
                                                                            (replace
                                                                             -5)
                                                                            (("1"
                                                                              (expand
                                                                               "o")
                                                                              (("1"
                                                                                (lemma
                                                                                 "deriv_comp_fun[{x: nnreal | x < pi / 2},real_abs_lt_pi]"
                                                                                 ("ff"
                                                                                  "LAMBDA (x: {x: nnreal | x < pi / 2}): x"
                                                                                  "gg"
                                                                                  "LAMBDA (x_1: real_abs_lt_pi): x_1 - sin_value(x_1)"))
                                                                                (("1"
                                                                                  (expand
                                                                                   "o")
                                                                                  (("1"
                                                                                    (replace
                                                                                     -3
                                                                                     -1)
                                                                                    (("1"
                                                                                      (replace
                                                                                       -5
                                                                                       -1)
                                                                                      (("1"
                                                                                        (expand
                                                                                         "*"
                                                                                         -1)
                                                                                        (("1"
                                                                                          (lemma
                                                                                           "minimum_derivative[{x: nnreal | x < pi / 2}]"
                                                                                           ("g"
                                                                                            "LAMBDA (x_1: {x: nnreal | x < pi / 2}): x_1 - sin_value(x_1)"
                                                                                            "x"
                                                                                            "0"
                                                                                            "y1"
                                                                                            "px"))
                                                                                          (("1"
                                                                                            (split
                                                                                             -1)
                                                                                            (("1"
                                                                                              (simplify
                                                                                               -1)
                                                                                              (("1"
                                                                                                (rewrite
                                                                                                 "sin_value_0")
                                                                                                (("1"
                                                                                                  (assert)
                                                                                                  nil
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil)
                                                                                             ("2"
                                                                                              (replace
                                                                                               -1
                                                                                               1)
                                                                                              (("2"
                                                                                                (assert)
                                                                                                (("2"
                                                                                                  (expand
                                                                                                   "abs")
                                                                                                  (("2"
                                                                                                    (rewrite
                                                                                                     "cos_value_0")
                                                                                                    (("2"
                                                                                                      (assert)
                                                                                                      nil
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil)
                                                                                             ("3"
                                                                                              (assert)
                                                                                              nil
                                                                                              nil)
                                                                                             ("4"
                                                                                              (replace
                                                                                               -1
                                                                                               1)
                                                                                              (("4"
                                                                                                (hide-all-but
                                                                                                 1)
                                                                                                (("4"
                                                                                                  (skosimp*)
                                                                                                  (("4"
                                                                                                    (expand
                                                                                                     "abs")
                                                                                                    (("4"
                                                                                                      (lemma
                                                                                                       "cos_value_strict_decreasing")
                                                                                                      (("4"
                                                                                                        (expand
                                                                                                         "strict_decreasing?")
                                                                                                        (("4"
                                                                                                          (inst
                                                                                                           -
                                                                                                           "0"
                                                                                                           "y!1")
                                                                                                          (("4"
                                                                                                            (rewrite
                                                                                                             "cos_value_0")
                                                                                                            (("4"
                                                                                                              (assert)
                                                                                                              (("4"
                                                                                                                (lemma
                                                                                                                 "posreal_times_posreal_is_posreal"
                                                                                                                 ("px"
                                                                                                                  "1-cos_value(y!1)"
                                                                                                                  "py"
                                                                                                                  "y!1"))
                                                                                                                (("4"
                                                                                                                  (assert)
                                                                                                                  nil
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil)
                                                                                           ("2"
                                                                                            (propax)
                                                                                            nil
                                                                                            nil)
                                                                                           ("3"
                                                                                            (skosimp*)
                                                                                            (("3"
                                                                                              (assert)
                                                                                              (("3"
                                                                                                (expand
                                                                                                 "abs")
                                                                                                (("3"
                                                                                                  (assert)
                                                                                                  nil
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil)
                                                                                 ("2"
                                                                                  (propax)
                                                                                  nil
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil)
                                                                         ("2"
                                                                          (skosimp*)
                                                                          (("2"
                                                                            (assert)
                                                                            (("2"
                                                                              (expand
                                                                               "abs")
                                                                              (("2"
                                                                                (assert)
                                                                                nil
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil)
                                                                 ("2"
                                                                  (hide-all-but
                                                                   1)
                                                                  (("2"
                                                                    (skosimp*)
                                                                    (("2"
                                                                      (case
                                                                       "x!1=0")
                                                                      (("1"
                                                                        (inst
                                                                         +
                                                                         "pi/4")
                                                                        (("1"
                                                                          (assert)
                                                                          nil
                                                                          nil))
                                                                        nil)
                                                                       ("2"
                                                                        (inst
                                                                         +
                                                                         "0")
                                                                        (("2"
                                                                          (assert)
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil)
                                                                 ("3"
                                                                  (hide-all-but
                                                                   1)
                                                                  (("3"
                                                                    (skosimp*)
                                                                    (("3"
                                                                      (typepred
                                                                       "x!1")
                                                                      (("3"
                                                                        (typepred
                                                                         "y!1")
                                                                        (("3"
                                                                          (assert)
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (propax)
                                                        nil
                                                        nil)
                                                       ("3"
                                                        (propax)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("2"
                                              (hide-all-but 1)
                                              (("2"
                                                (skosimp*)
                                                (("2"
                                                  (case "x!1=0")
                                                  (("1"
                                                    (inst + "pi/4")
                                                    (("1"
                                                      (assert)
                                                      nil
                                                      nil)
                                                     ("2"
                                                      (expand "abs" 1)
                                                      (("2"
                                                        (assert)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (inst + "0")
                                                    (("1"
                                                      (assert)
                                                      nil
                                                      nil)
                                                     ("2"
                                                      (expand "abs" 1)
                                                      (("2"
                                                        (assert)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("3"
                                              (hide-all-but 1)
                                              (("3"
                                                (skosimp*)
                                                (("3"
                                                  (typepred "x!1")
                                                  (("3"
                                                    (typepred "y!1")
                                                    (("3"
                                                      (name-replace
                                                       "K1"
                                                       "pi/2")
                                                      (("3"
                                                        (grind)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (rewrite "phase_sin_q2")
                                    (("2"
                                      (flatten)
                                      (("2" (assert) nil nil))
                                      nil))
                                    nil)
                                   ("3"
                                    (rewrite "phase_sin_q3")
                                    (("3"
                                      (flatten)
                                      (("3" (assert) nil nil))
                                      nil))
                                    nil)
                                   ("4"
                                    (rewrite "phase_sin_q4")
                                    (("4"
                                      (flatten)
                                      (("4" (assert) nil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("3" (assert) nil nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (typepred "sin(px)") (("2" (assert) nil nil)) nil))
      nil))
    nil)
   nil nil)
  (sin_ub-1 nil 3265693281
   ("" (skolem 1 ("px"))
    (("" (case "px<=1")
      (("1" (expand "sin")
        (("1" (lemma "floor_def" ("x" "px/(2*pi)"))
          (("1" (rewrite "div_mult_pos_lt1")
            (("1" (rewrite "div_mult_pos_le2")
              (("1" (flatten)
                (("1" (lemma "trichotomy" ("x" "floor(px / (2 * pi))"))
                  (("1" (split -1)
                    (("1" (case "floor(px / (2 * pi)) =1")
                      (("1" (lemma "pi_bnds")
                        (("1" (flatten) (("1" (assert) nil nil)) nil))
                        nil)
                       ("2" (case "floor(px / (2 * pi))>1")
                        (("1" (lemma "pi_bnds")
                          (("1" (flatten)
                            (("1" (assert)
                              (("1"
                                (lemma
                                 "both_sides_times_pos_lt1"
                                 ("x"
                                  "1"
                                  "y"
                                  "floor(px / (2 * pi))"
                                  "pz"
                                  "2*pi"))
                                (("1" (assert) nil nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (assert) nil nil))
                        nil))
                      nil)
                     ("2" (replace -1)
                      (("2" (hide -1 -2 -3)
                        (("2" (simplify 1)
                          (("2" (lemma "pi_bnds")
                            (("2" (flatten)
                              (("2"
                                (lemma "phases_sin" ("x" "px"))
                                (("1"
                                  (split -1)
                                  (("1"
                                    (rewrite "sin_q1")
                                    (("1"
                                      (hide -1)
                                      (("1"
                                        (lemma "sin_value_derivable2")
                                        (("1"
                                          (lemma "deriv_sin_value")
                                          (("1"
                                            (lemma
                                             "identity_derivable_fun[real_abs_lt_pi]")
                                            (("1"
                                              (lemma
                                               "deriv_id_fun[real_abs_lt_pi]")
                                              (("1"
                                                (expand "I")
                                                (("1"
                                                  (expand "const_fun")
                                                  (("1"
                                                    (lemma
                                                     "diff_derivable_fun"
                                                     ("f1"
                                                      "LAMBDA (x: real_abs_lt_pi): x"
                                                      "f2"
                                                      "LAMBDA (x: real_abs_lt_pi): sin_value(x)"))
                                                    (("1"
                                                      (lemma
                                                       "deriv_diff_fun"
                                                       ("ff1"
                                                        "LAMBDA (x: real_abs_lt_pi): x"
                                                        "ff2"
                                                        "LAMBDA (x: real_abs_lt_pi): sin_value(x)"))
                                                      (("1"
                                                        (replace -3 -1)
                                                        (("1"
                                                          (replace
                                                           -5
                                                           -1)
                                                          (("1"
                                                            (expand
                                                             "-")
                                                            (("1"
                                                              (assert)
                                                              (("1"
                                                                (lemma
                                                                 "identity_derivable_fun[{x:nnreal|x)
                                                                (("1"
                                                                  (lemma
                                                                   "deriv_id_fun[{x:nnreal|x)
                                                                  (("1"
                                                                    (expand
                                                                     "I")
                                                                    (("1"
                                                                      (expand
                                                                       "const_fun")
                                                                      (("1"
                                                                        (lemma
                                                                         "composition_derivable_fun[{x: nnreal | x < pi / 2},real_abs_lt_pi]"
                                                                         ("f"
                                                                          "LAMBDA (x: {x: nnreal | x < pi / 2}): x"
                                                                          "g"
                                                                          "LAMBDA (x_1: real_abs_lt_pi): x_1 - sin_value(x_1)"))
                                                                        (("1"
                                                                          (replace
                                                                           -3)
                                                                          (("1"
                                                                            (replace
                                                                             -5)
                                                                            (("1"
                                                                              (expand
                                                                               "o")
                                                                              (("1"
                                                                                (lemma
                                                                                 "deriv_comp_fun[{x: nnreal | x < pi / 2},real_abs_lt_pi]"
                                                                                 ("ff"
                                                                                  "LAMBDA (x: {x: nnreal | x < pi / 2}): x"
                                                                                  "gg"
                                                                                  "LAMBDA (x_1: real_abs_lt_pi): x_1 - sin_value(x_1)"))
                                                                                (("1"
                                                                                  (expand
                                                                                   "o")
                                                                                  (("1"
                                                                                    (replace
                                                                                     -3
                                                                                     -1)
                                                                                    (("1"
                                                                                      (replace
                                                                                       -5
                                                                                       -1)
                                                                                      (("1"
                                                                                        (expand
                                                                                         "*"
                                                                                         -1)
                                                                                        (("1"
                                                                                          (lemma
                                                                                           "minimum_derivative[{x: nnreal | x < pi / 2}]"
                                                                                           ("g"
                                                                                            "LAMBDA (x_1: {x: nnreal | x < pi / 2}): x_1 - sin_value(x_1)"
                                                                                            "x"
                                                                                            "0"
                                                                                            "y"
                                                                                            "px"))
                                                                                          (("1"
                                                                                            (split
                                                                                             -1)
                                                                                            (("1"
                                                                                              (simplify
                                                                                               -1)
                                                                                              (("1"
                                                                                                (rewrite
                                                                                                 "sin_value_0")
                                                                                                (("1"
                                                                                                  (assert)
                                                                                                  nil
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil)
                                                                                             ("2"
                                                                                              (replace
                                                                                               -1
                                                                                               1)
                                                                                              (("2"
                                                                                                (assert)
                                                                                                (("2"
                                                                                                  (expand
                                                                                                   "abs")
                                                                                                  (("2"
                                                                                                    (rewrite
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--> --------------------

--> maximum size reached

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

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