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Quellcode-Bibliothek

© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: asin.pvs   Sprache: Lisp

Untersuchung PVS©

(atan (real_abs_lt_pi2_TCC1 0
       (real_abs_lt_pi2_TCC1-1 nil 3323017154
        ("" (subtype-tcc) nil nil)
        ((minus_nzreal_is_nzreal application-judgement "nzreal"
          real_types nil)
         (nzreal_div_nzreal_is_nzreal application-judgement "nzreal"
          real_types nil)
         (posreal_div_posreal_is_posreal application-judgement
          "posreal" real_types nil)
         (real_lt_is_strict_total_order name-judgement
          "(strict_total_order?[real])" real_props nil))
        nil))
      (atan_strict_increasing 0
       (atan_strict_increasing-1 nil 3257226677
        ("" (expand "atan")
         (("" (lemma "atan_value_strict_increasing")
           ((""
             (lemma "extensionality_postulate"
              ("f" "atan_value" "g" "LAMBDA (x: real): atan_value(x)"))
             (("" (assertnil nil)) nil))
           nil))
         nil)
        nil shostak))
      (atan_bij 0
       (atan_bij-1 nil 3257310468
        ("" (expand "bijective?")
         (("" (split 1)
           (("1" (expand "injective?")
             (("1" (skolem 1 ("x" "y"))
               (("1" (flatten)
                 (("1" (lemma "atan_strict_increasing")
                   (("1" (expand "strict_increasing?")
                     (("1" (lemma "trich_lt" ("x" "x" "y" "y"))
                       (("1" (split -1)
                         (("1" (inst - "x" "y")
                           (("1" (assertnil nil)) nil)
                          ("2" (propax) nil nil)
                          ("3" (inst - "y" "x")
                           (("3" (assertnil nil)) nil))
                         nil))
                       nil))
                     nil))
                   nil))
                 nil))
               nil))
             nil)
            ("2" (expand "surjective?")
             (("2" (skosimp*)
               (("2" (typepred "y!1")
                 (("2"
                   (lemma "derivable_continuous2" ("f" "atan_value"))
                   (("2" (split -1)
                     (("1" (expand "atan")
                       (("1" (lemma "atan_value_strict_increasing")
                         (("1" (expand "strict_increasing?")
                           (("1" (lemma "trichotomy" ("x" "y!1"))
                             (("1" (split -1)
                               (("1"
                                 (lemma
                                  "trich_lt"
                                  ("x" "y!1" "y" "atan_value(1)"))
                                 (("1"
                                   (split -1)
                                   (("1"
                                     (lemma
                                      "intermediate1"
                                      ("g"
                                       "atan_value"
                                       "a"
                                       "0"
                                       "b"
                                       "1"
                                       "x"
                                       "y!1"))
                                     (("1"
                                       (rewrite "atan_value_0")
                                       (("1"
                                         (assert)
                                         (("1"
                                           (skosimp*)
                                           (("1"
                                             (inst + "c!1")
                                             nil
                                             nil))
                                           nil))
                                         nil))
                                       nil)
                                      ("2" (propax) nil nil))
                                     nil)
                                    ("2"
                                     (inst + "1")
                                     (("2" (assertnil nil))
                                     nil)
                                    ("3"
                                     (expand "abs")
                                     (("3"
                                       (inst-cp - "0" "1")
                                       (("3"
                                         (assert)
                                         (("3"
                                           (rewrite "atan_value_0")
                                           (("3"
                                             (lemma
                                              "intermediate1"
                                              ("a"
                                               "0"
                                               "b"
                                               "1"
                                               "x"
                                               "2*atan_value(1)-y!1"
                                               "g"
                                               "atan_value"))
                                             (("3"
                                               (rewrite "atan_value_0")
                                               (("3"
                                                 (assert)
                                                 (("3"
                                                   (skolem -1 ("c"))
                                                   (("3"
                                                     (flatten)
                                                     (("3"
                                                       (expand "<=" -1)
                                                       (("3"
                                                         (split -1)
                                                         (("1"
                                                           (lemma
                                                            "atan_inv_value"
                                                            ("px"
                                                             "1/c"))
                                                           (("1"
                                                             (inst
                                                              +
                                                              "1/c")
                                                             (("1"
                                                               (assert)
                                                               nil
                                                               nil)
                                                              ("2"
                                                               (assert)
                                                               nil
                                                               nil))
                                                             nil)
                                                            ("2"
                                                             (lemma
                                                              "posreal_div_posreal_is_posreal"
                                                              ("px"
                                                               "1"
                                                               "py"
                                                               "c"))
                                                             (("1"
                                                               (assert)
                                                               nil
                                                               nil)
                                                              ("2"
                                                               (assert)
                                                               nil
                                                               nil))
                                                             nil)
                                                            ("3"
                                                             (assert)
                                                             nil
                                                             nil))
                                                           nil)
                                                          ("2"
                                                           (replace
                                                            -1
                                                            *
                                                            rl)
                                                           (("2"
                                                             (rewrite
                                                              "atan_value_0")
                                                             (("2"
                                                               (assert)
                                                               nil
                                                               nil))
                                                             nil))
                                                           nil))
                                                         nil))
                                                       nil))
                                                     nil))
                                                   nil))
                                                 nil))
                                               nil))
                                             nil))
                                           nil))
                                         nil))
                                       nil))
                                     nil))
                                   nil))
                                 nil)
                                ("2"
                                 (inst + "0")
                                 (("2"
                                   (rewrite "atan_value_0")
                                   (("2" (assertnil nil))
                                   nil))
                                 nil)
                                ("3"
                                 (lemma
                                  "trich_lt"
                                  ("x" "y!1" "y" "-atan_value(1)"))
                                 (("3"
                                   (expand "abs")
                                   (("3"
                                     (assert)
                                     (("3"
                                       (split -1)
                                       (("1"
                                         (lemma
                                          "intermediate1"
                                          ("a"
                                           "-1"
                                           "b"
                                           "0"
                                           "x"
                                           "-2*atan_value(1)-y!1"
                                           "g"
                                           "atan_value"))
                                         (("1"
                                           (rewrite
                                            "atan_neg_value"
                                            -1)
                                           (("1"
                                             (rewrite
                                              "atan_value_0"
                                              -1)
                                             (("1"
                                               (assert)
                                               (("1"
                                                 (skolem -1 ("c"))
                                                 (("1"
                                                   (flatten)
                                                   (("1"
                                                     (expand "<=" -2)
                                                     (("1"
                                                       (split -2)
                                                       (("1"
                                                         (lemma
                                                          "atan_inv_neg_value"
                                                          ("nx" "c"))
                                                         (("1"
                                                           (replace
                                                            -4
                                                            -1)
                                                           (("1"
                                                             (simplify
                                                              -1)
                                                             (("1"
                                                               (inst
                                                                +
                                                                "1/c")
                                                               (("1"
                                                                 (assert)
                                                                 nil
                                                                 nil))
                                                               nil))
                                                             nil))
                                                           nil)
                                                          ("2"
                                                           (assert)
                                                           nil
                                                           nil))
                                                         nil)
                                                        ("2"
                                                         (replace -1)
                                                         (("2"
                                                           (rewrite
                                                            "atan_value_0")
                                                           (("2"
                                                             (assert)
                                                             nil
                                                             nil))
                                                           nil))
                                                         nil))
                                                       nil))
                                                     nil))
                                                   nil))
                                                 nil))
                                               nil))
                                             nil))
                                           nil))
                                         nil)
                                        ("2"
                                         (inst + "-1")
                                         (("2"
                                           (rewrite "atan_neg_value")
                                           (("2" (assertnil nil))
                                           nil))
                                         nil)
                                        ("3"
                                         (lemma
                                          "intermediate1"
                                          ("a"
                                           "-1"
                                           "b"
                                           "0"
                                           "x"
                                           "y!1"
                                           "g"
                                           "atan_value"))
                                         (("3"
                                           (rewrite "atan_value_0")
                                           (("3"
                                             (rewrite "atan_neg_value")
                                             (("3"
                                               (assert)
                                               (("3"
                                                 (skolem -1 ("c"))
                                                 (("3"
                                                   (flatten -1)
                                                   (("3"
                                                     (inst + "c")
                                                     nil
                                                     nil))
                                                   nil))
                                                 nil))
                                               nil))
                                             nil))
                                           nil))
                                         nil))
                                       nil))
                                     nil))
                                   nil))
                                 nil))
                               nil))
                             nil))
                           nil))
                         nil))
                       nil)
                      ("2" (lemma "one_over_one_plus_t_sq_cont")
                       (("2"
                         (lemma "fundamental"
                          ("f" "atan_deriv_fn" "F" "atan_value" "a"
                           "0"))
                         (("2" (split -1)
                           (("1" (flatten) nil nil)
                            ("2" (propax) nil nil)
                            ("3" (expand "atan_value")
                             (("3" (propax) nil nil)) nil))
                           nil))
                         nil))
                       nil))
                     nil))
                   nil))
                 nil))
               nil))
             nil))
           nil))
         nil)
        ((strict_increasing? const-decl "bool" real_fun_preds
          "reals/"))
        shostak))
      (atan_0 0
       (atan_0-1 nil 3257226801
        ("" (expand "atan") (("" (rewrite "atan_value_0"nil nil))
         nil)
        nil shostak))
      (atan_inv 0
       (atan_inv-1 nil 3264783397
        ("" (skosimp*)
         (("" (rewrite "pi_value")
           (("" (expand "atan")
             (("" (lemma "atan_inv_value" ("px" "px!1"))
               (("" (assertnil nil)) nil))
             nil))
           nil))
         nil)
        nil shostak))
      (atan_inv_neg 0
       (atan_inv_neg-1 nil 3264783474
        ("" (skosimp*)
         (("" (rewrite "pi_value")
           (("" (expand "atan")
             (("" (lemma "atan_inv_neg_value" ("nx" "nx!1"))
               (("" (assertnil nil)) nil))
             nil))
           nil))
         nil)
        nil shostak))
      (acot_TCC1 0
       (acot_TCC1-1 nil 3257226194
        ("" (skolem 1 ("nzx"))
         (("" (lemma "atan_0")
           (("" (lemma "atan_strict_increasing")
             (("" (expand "strict_increasing?")
               (("" (lemma "trichotomy" ("x" "nzx"))
                 (("" (split -1)
                   (("1" (inst - "0" "1/nzx")
                     (("1" (lemma "quotient_pos_lt" ("n0x" "nzx"))
                       (("1" (assertnil nil)) nil))
                     nil)
                    ("2" (assertnil nil)
                    ("3" (lemma "quotient_neg_lt" ("n0x" "nzx"))
                     (("3" (inst - "1/nzx" "0")
                       (("3" (assertnil nil)) nil))
                     nil))
                   nil))
                 nil))
               nil))
             nil))
           nil))
         nil)
        ((atan_0 formula-decl nil atan nil)
         (strict_increasing? const-decl "bool" real_fun_preds "reals/")
         (quotient_pos_lt formula-decl nil real_props nil)
         (nonzero_real nonempty-type-eq-decl nil reals nil)
         (real_gt_is_strict_total_order name-judgement
          "(strict_total_order?[real])" real_props nil)
         (minus_nzreal_is_nzreal application-judgement "nzreal"
          real_types nil)
         (real_lt_is_strict_total_order name-judgement
          "(strict_total_order?[real])" real_props nil)
         (numfield nonempty-type-eq-decl nil number_fields nil)
         (nznum nonempty-type-eq-decl nil number_fields nil)
         (/ const-decl "[numfield, nznum -> numfield]" number_fields
            nil)
         (nzreal_div_nzreal_is_nzreal application-judgement "nzreal"
          real_types nil)
         (quotient_neg_lt formula-decl nil real_props nil)
         (trichotomy formula-decl nil real_axioms 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)
         (/= const-decl "boolean" notequal nil)
         (nzreal nonempty-type-eq-decl nil reals nil)
         (atan_strict_increasing formula-decl nil atan nil))
        shostak))
      (atan_neg 0
       (atan_neg-1 nil 3257226889
        ("" (lemma "atan_neg_value")
         (("" (expand "atan") (("" (propax) nil nil)) nil)) nil)
        nil shostak))
      (acot_neg 0
       (acot_neg-1 nil 3257226968
        ("" (expand "acot")
         (("" (skolem 1 ("nzx"))
           (("" (lemma "atan_neg" ("x" "1/nzx"))
             (("" (assertnil nil)) nil))
           nil))
         nil)
        ((minus_real_is_real application-judgement "real" reals nil)
         (nzreal_div_nzreal_is_nzreal application-judgement "nzreal"
          real_types nil)
         (atan_neg formula-decl nil atan 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)
         (numfield nonempty-type-eq-decl nil number_fields nil)
         (/= const-decl "boolean" notequal nil)
         (nznum nonempty-type-eq-decl nil number_fields nil)
         (/ const-decl "[numfield, nznum -> numfield]" number_fields
            nil)
         (nzreal nonempty-type-eq-decl nil reals nil)
         (acot const-decl "nzreal" atan nil)
         (minus_nzreal_is_nzreal application-judgement "nzreal"
          real_types nil))
        shostak))
      (atan_minus_TCC1 0
       (atan_minus_TCC1-1 nil 3257273266
        ("" (skolem 1 ("x" "y"))
         (("" (assert) (("" (flatten) (("" (assertnil nil)) nil))
           nil))
         nil)
        ((real_times_real_is_real application-judgement "real" reals
          nil)
         (minus_odd_is_odd application-judgement "odd_int" integers
          nil)
         (real_lt_is_strict_total_order name-judgement
          "(strict_total_order?[real])" real_props nil)
         (real_plus_real_is_real application-judgement "real" reals
          nil))
        shostak))
      (atan_minus_TCC2 0
       (atan_minus_TCC2-1 nil 3257273266
        ("" (skolem 1 ("x" "y"))
         (("" (flatten) (("" (hide -1 -2) (("" (assertnil nil)) nil))
           nil))
         nil)
        ((real_plus_real_is_real application-judgement "real" reals
          nil)
         (minus_odd_is_odd application-judgement "odd_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))
        shostak))
      (atan_minus_TCC3 0
       (atan_minus_TCC3-1 nil 3408966186 ("" (subtype-tcc) nil nil)
        ((boolean nonempty-type-decl nil booleans nil)
         (bool nonempty-type-eq-decl nil booleans nil)
         (NOT const-decl "[bool -> bool]" booleans nil)
         (number nonempty-type-decl nil numbers nil)
         (number_field_pred const-decl "[number -> boolean]"
          number_fields nil)
         (number_field nonempty-type-from-decl nil number_fields nil)
         (real_pred const-decl "[number_field -> boolean]" reals nil)
         (real nonempty-type-from-decl nil reals nil)
         (real_div_nzreal_is_real application-judgement "real" reals
          nil)
         (minus_odd_is_odd application-judgement "odd_int" integers
          nil)
         (real_lt_is_strict_total_order name-judgement
          "(strict_total_order?[real])" real_props nil)
         (/= const-decl "boolean" notequal nil)
         (real_times_real_is_real application-judgement "real" reals
          nil)
         (real_minus_real_is_real application-judgement "real" reals
          nil)
         (real_plus_real_is_real application-judgement "real" reals
          nil))
        nil))
      (atan_minus 0
       (atan_minus-1 nil 3257227268
        ("" (skolem 1 ("x" "y"))
         (("" (lemma "atan_value_minus" ("x" "x" "y" "y"))
           (("" (flatten -1)
             (("" (rewrite "pi_value")
               (("" (expand "atan")
                 (("" (replace -1)
                   (("" (replace -2) (("" (propax) nil nil)) nil))
                   nil))
                 nil))
               nil))
             nil))
           nil))
         nil)
        nil shostak))
      (atan_plus_TCC1 0
       (atan_plus_TCC1-1 nil 3257228828 ("" (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)
         (real_minus_real_is_real application-judgement "real" reals
          nil)
         (real_lt_is_strict_total_order name-judgement
          "(strict_total_order?[real])" real_props nil)
         (/= const-decl "boolean" notequal nil)
         (real_times_real_is_real application-judgement "real" reals
          nil))
        shostak))
      (atan_plus_TCC2 0
       (atan_plus_TCC2-1 nil 3257273266
        ("" (skolem 1 ("x" "y"))
         (("" (flatten) (("" (hide -1) (("" (assertnil nil)) nil))
           nil))
         nil)
        ((real_minus_real_is_real application-judgement "real" reals
          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)
         (real_gt_is_strict_total_order name-judgement
          "(strict_total_order?[real])" real_props nil))
        shostak))
      (atan_plus_TCC3 0
       (atan_plus_TCC3-1 nil 3257275668
        ("" (skolem 1 ("x" "y"))
         (("" (flatten) (("" (hide -1 -2) (("" (assertnil nil)) nil))
           nil))
         nil)
        ((real_minus_real_is_real application-judgement "real" reals
          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))
        shostak))
      (atan_plus 0
       (atan_plus-1 nil 3257227479
        ("" (skolem 1 ("x" "y"))
         (("" (lemma "atan_minus" ("x" "x" "y" "-y"))
           (("" (rewrite "atan_neg" -1)
             (("" (flatten)
               ((""
                 (lemma "both_sides_times_neg_lt1"
                  ("y" "x*y" "x" "1" "nz" "-1"))
                 ((""
                   (lemma "both_sides_times_neg_lt1"
                    ("x" "x*y" "y" "1" "nz" "-1"))
                   (("" (replace -1)
                     (("" (replace -2)
                       ((""
                         (lemma "both_sides_times_neg_lt1"
                          ("x" "y" "y" "0" "nz" "-1"))
                         ((""
                           (lemma "both_sides_times_neg_lt1"
                            ("y" "y" "x" "0" "nz" "-1"))
                           (("" (lemma "trichotomy" ("x" "y"))
                             (("" (split -1)
                               (("1"
                                 (assert)
                                 (("1"
                                   (replace -6 1)
                                   (("1"
                                     (replace -8 1)
                                     (("1" (propax) nil nil))
                                     nil))
                                   nil))
                                 nil)
                                ("2" (assertnil nil)
                                ("3"
                                 (assert)
                                 (("3"
                                   (replace -6 1)
                                   (("3"
                                     (replace -7 1)
                                     (("3" (propax) nil nil))
                                     nil))
                                   nil))
                                 nil))
                               nil))
                             nil))
                           nil))
                         nil))
                       nil))
                     nil))
                   nil))
                 nil))
               nil))
             nil))
           nil))
         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)
         (atan_minus formula-decl nil atan nil)
         (minus_real_is_real application-judgement "real" reals nil)
         (real_gt_is_strict_total_order name-judgement
          "(strict_total_order?[real])" real_props nil)
         (real_lt_is_strict_total_order name-judgement
          "(strict_total_order?[real])" real_props 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)
         (odd_times_odd_is_odd application-judgement "odd_int" integers
          nil)
         (real_div_nzreal_is_real application-judgement "real" reals
          nil)
         (trichotomy formula-decl nil real_axioms nil)
         (* const-decl "[numfield, numfield -> numfield]" number_fields
            nil)
         (negreal nonempty-type-eq-decl nil real_types nil)
         (< const-decl "bool" reals nil)
         (nonpos_real nonempty-type-eq-decl nil real_types nil)
         (<= const-decl "bool" reals nil)
         (bool nonempty-type-eq-decl nil booleans nil)
         (both_sides_times_neg_lt1 formula-decl nil real_props nil)
         (minus_odd_is_odd application-judgement "odd_int" integers
          nil)
         (minus_nzreal_is_nzreal application-judgement "nzreal"
          real_types nil)
         (real_plus_real_is_real application-judgement "real" reals
          nil)
         (real_minus_real_is_real application-judgement "real" reals
          nil)
         (real_times_real_is_real application-judgement "real" reals
          nil)
         (atan_neg formula-decl nil atan nil))
        shostak))
      (atan_sub_swap_TCC1 0
       (atan_sub_swap_TCC1-1 nil 3514303177 ("" (subtype-tcc) nil nil)
        ((/= const-decl "boolean" notequal nil)) nil))
      (atan_sub_swap_TCC2 0
       (atan_sub_swap_TCC2-1 nil 3514303177 ("" (subtype-tcc) nil nil)
        ((/= const-decl "boolean" notequal nil)) nil))
      (atan_sub_swap_TCC3 0
       (atan_sub_swap_TCC3-1 nil 3514303177 ("" (subtype-tcc) nil nil)
        ((boolean nonempty-type-decl nil booleans nil)
         (bool nonempty-type-eq-decl nil booleans nil)
         (NOT const-decl "[bool -> bool]" booleans nil)
         (number nonempty-type-decl nil numbers nil)
         (number_field_pred const-decl "[number -> boolean]"
          number_fields nil)
         (number_field nonempty-type-from-decl nil number_fields nil)
         (real_pred const-decl "[number_field -> boolean]" reals nil)
         (real nonempty-type-from-decl nil reals nil)
         (/= const-decl "boolean" notequal nil)
         (real_times_real_is_real application-judgement "real" reals
          nil))
        nil))
      (atan_sub_swap 0
       (atan_sub_swap-1 nil 3514303178
        ("" (skeep)
         (("" (lemma "atan_minus")
           (("" (inst?)
             (("" (assert)
               ((""
                 (case-replace
                  "(x / y - y / x) / (1 + x / y * (y / x)) = (x ^ 2 - y ^ 2) / (2 * (x * y))")
                 (("" (hide -1 4)
                   (("" (expand "^")
                     (("" (expand "expt")
                       (("" (expand "expt")
                         (("" (expand "expt") (("" (assertnil nil))
                           nil))
                         nil))
                       nil))
                     nil))
                   nil))
                 nil))
               nil))
             nil))
           nil))
         nil)
        ((atan_minus formula-decl nil atan nil)
         (real_lt_is_strict_total_order name-judgement
          "(strict_total_order?[real])" real_props nil)
         (minus_nzreal_is_nzreal application-judgement "nzreal"
          real_types nil)
         (expt def-decl "real" exponentiation nil)
         (^ const-decl "real" exponentiation nil)
         (>= const-decl "bool" reals nil)
         (OR const-decl "[bool, bool -> bool]" booleans 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)
         (* const-decl "[numfield, numfield -> numfield]" number_fields
            nil)
         (+ const-decl "[numfield, numfield -> numfield]" number_fields
            nil)
         (- const-decl "[numfield, numfield -> numfield]" number_fields
            nil)
         (= const-decl "[T, T -> boolean]" equalities 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)
         (real nonempty-type-from-decl nil reals nil)
         (real_pred const-decl "[number_field -> boolean]" reals nil)
         (number_field nonempty-type-from-decl nil number_fields nil)
         (number_field_pred const-decl "[number -> boolean]"
          number_fields nil)
         (boolean nonempty-type-decl nil booleans nil)
         (number nonempty-type-decl nil numbers nil)
         (real_div_nzreal_is_real application-judgement "real" reals
          nil)
         (real_times_real_is_real application-judgement "real" reals
          nil)
         (real_minus_real_is_real application-judgement "real" reals
          nil)
         (real_plus_real_is_real application-judgement "real" reals
          nil)
         (minus_odd_is_odd application-judgement "odd_int" integers
          nil))
        shostak))
      (atan_1 0
       (atan_1-1 nil 3259650470
        ("" (lemma "atan_plus" ("x" "1/5" "y" "1/5"))
         (("" (assert)
           (("" (case "(2 * (1 / 5) / (1 - 1 / 5 * (1 / 5))) = 5/12")
             (("1" (replace -1)
               (("1" (lemma "atan_plus" ("x" "5/12" "y" "5/12"))
                 (("1" (assert)
                   (("1"
                     (case "(2 * (5 / 12) / (1 - 5 / 12 * (5 / 12))) = 120/119")
                     (("1" (replace -1)
                       (("1" (hide -1 -3)
                         (("1" (replace -2 -1 rl)
                           (("1" (simplify -1)
                             (("1" (replace -1 1)
                               (("1"
                                 (lemma
                                  "atan_minus"
                                  ("x" "120/119" "y" "1/239"))
                                 (("1" (assertnil nil))
                                 nil))
                               nil))
                             nil))
                           nil))
                         nil))
                       nil)
                      ("2" (hide-all-but 1)
                       (("2"
                         (lemma "cross_mult"
                          ("x" "2*(5/12)" "n0x"
                           "(1 - 5 / 12 * (5 / 12))" "y" "120" "n0y"
                           "119"))
                         (("2" (replace -1)
                           (("2" (hide -1) (("2" (assertnil nil))
                             nil))
                           nil))
                         nil))
                       nil))
                     nil))
                   nil))
                 nil))
               nil)
              ("2" (hide-all-but 1) (("2" (assertnil nil)) nil))
             nil))
           nil))
         nil)
        ((rat_minus_rat_is_rat application-judgement "rat" rationals
          nil)
         (posrat_plus_nnrat_is_posrat application-judgement "posrat"
          rationals nil)
         (nzrat_div_nzrat_is_nzrat application-judgement "nzrat"
          rationals nil)
         (real_plus_real_is_real application-judgement "real" reals
          nil)
         (posrat_times_posrat_is_posrat application-judgement "posrat"
          rationals nil)
         (real_lt_is_strict_total_order name-judgement
          "(strict_total_order?[real])" real_props nil)
         (minus_nzreal_is_nzreal application-judgement "nzreal"
          real_types nil)
         (real_times_real_is_real application-judgement "real" reals
          nil)
         (real_minus_real_is_real application-judgement "real" reals
          nil)
         (atan_minus formula-decl nil atan nil)
         (minus_odd_is_odd application-judgement "odd_int" integers
          nil)
         (rat_div_nzrat_is_rat application-judgement "rat" rationals
          nil)
         (cross_mult formula-decl nil real_props nil)
         (nonzero_real nonempty-type-eq-decl nil reals nil)
         (rat_times_rat_is_rat application-judgement "rat" rationals
          nil)
         (- const-decl "[numfield, numfield -> numfield]" number_fields
            nil)
         (* const-decl "[numfield, numfield -> numfield]" number_fields
            nil)
         (= const-decl "[T, T -> boolean]" equalities nil)
         (posrat_div_posrat_is_posrat application-judgement "posrat"
          rationals nil)
         (atan_plus formula-decl nil atan 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)
         (numfield nonempty-type-eq-decl nil number_fields nil)
         (/= const-decl "boolean" notequal nil)
         (nznum nonempty-type-eq-decl nil number_fields nil)
         (/ const-decl "[numfield, nznum -> numfield]" number_fields
            nil))
        shostak))
      (atan_bnds 0
       (atan_bnds-1 nil 3259651388
        ("" (skolem 1 ("x"))
         (("" (lemma "deriv_atan_fun")
           (("" (flatten)
             (("" (lemma "atan_0")
               (("" (lemma "mean_value[nnreal]" ("a" "0"))
                 (("1" (inst - "x" "_")
                   (("1"
                     (lemma "restrict2_derivable[nnreal,real]"
                      ("f" "atan"))
                     (("1"
                       (lemma "restrict2_deriv[nnreal,real]"
                        ("f" "atan"))
                       (("1" (expand "restrict2")
                         (("1" (replace -6 -1)
                           (("1" (simplify -1)
                             (("1"
                               (lemma
                                "extensionality"
                                ("f"
                                 "LAMBDA (a: nnreal): 1 / (1 + a * a)"
                                 "g"
                                 "deriv[nnreal](LAMBDA (u: nnreal): atan(u))"))
                               (("1"
                                 (simplify -1)
                                 (("1"
                                   (replace -2 -1)
                                   (("1"
                                     (hide -2 -6 -7)
                                     (("1"
                                       (lemma
                                        "identity_derivable_fun[nnreal]")
                                       (("1"
                                         (lemma "deriv_id_fun[nnreal]")
                                         (("1"
                                           (lemma
                                            "deriv_const_fun[nnreal]"
                                            ("b" "1"))
                                           (("1"
                                             (expand "I")
                                             (("1"
                                               (expand "const_fun")
                                               (("1"
                                                 (split 1)
                                                 (("1"
                                                   (lemma
                                                    "prod_derivable_fun"
                                                    ("f1"
                                                     "LAMBDA (x: nnreal): x"
                                                     "f2"
                                                     "LAMBDA (x: nnreal): x"))
                                                   (("1"
                                                     (assert)
                                                     (("1"
                                                       (lemma
                                                        "deriv_prod_fun"
                                                        ("ff1"
                                                         "LAMBDA (x: nnreal): x"
                                                         "ff2"
                                                         "LAMBDA (x: nnreal): x"))
                                                       (("1"
                                                         (replace
                                                          -4
                                                          -1)
                                                         (("1"
                                                           (expand "*")
                                                           (("1"
                                                             (expand
                                                              "+"
                                                              -1)
                                                             (("1"
                                                               (lemma
                                                                "sum_derivable_fun"
                                                                ("f1"
                                                                 "LAMBDA (x: nnreal): 1"
                                                                 "f2"
                                                                 "LAMBDA (x: nnreal): x*x"))
                                                               (("1"
                                                                 (lemma
                                                                  "const_derivable_fun[nnreal]"
                                                                  ("b"
                                                                   "1"))
                                                                 (("1"
                                                                   (expand
                                                                    "const_fun")
                                                                   (("1"
                                                                     (assert)
                                                                     (("1"
                                                                       (lemma
                                                                        "deriv_sum_fun"
                                                                        ("ff1"
                                                                         "LAMBDA (x: nnreal): 1"
                                                                         "ff2"
                                                                         "LAMBDA (x: nnreal): x*x"))
                                                                       (("1"
                                                                         (replace
                                                                          -4
                                                                          -1)
                                                                         (("1"
                                                                           (replace
                                                                            -6
                                                                            -1)
                                                                           (("1"
                                                                             (expand
                                                                              "+")
                                                                             (("1"
                                                                               (lemma
                                                                                "div_derivable_fun"
                                                                                ("f"
                                                                                 "LAMBDA (x: nnreal): x"
                                                                                 "g"
                                                                                 "LAMBDA (x: nnreal): 1 + x*x"))
                                                                               (("1"
                                                                                 (assert)
                                                                                 (("1"
                                                                                   (expand
                                                                                    "/"
                                                                                    -1)
                                                                                   (("1"
                                                                                     (lemma
                                                                                      "deriv_div_fun"
                                                                                      ("ff"
                                                                                       "LAMBDA (x: nnreal): x"
                                                                                       "gg"
                                                                                       "LAMBDA (x: nnreal): 1 + x*x"))
                                                                                     (("1"
                                                                                       (replace
                                                                                        -3
                                                                                        -1)
                                                                                       (("1"
                                                                                         (replace
                                                                                          -9
                                                                                          -1)
                                                                                         (("1"
                                                                                           (expand
                                                                                            "/")
                                                                                           (("1"
                                                                                             (expand
                                                                                              "-"
                                                                                              -1)
                                                                                             (("1"
                                                                                               (expand
                                                                                                "*"
                                                                                                -1)
                                                                                               (("1"
                                                                                                 (lemma
                                                                                                  "diff_derivable_fun"
                                                                                                  ("f1"
                                                                                                   "LAMBDA (u: nnreal): atan(u)"
                                                                                                   "f2"
                                                                                                   "LAMBDA (x: nnreal): x/(1+x*x)"))
                                                                                                 (("1"
                                                                                                   (assert)
                                                                                                   (("1"
                                                                                                     (lemma
                                                                                                      "deriv_diff_fun"
                                                                                                      ("ff1"
                                                                                                       "LAMBDA (u: nnreal): atan(u)"
                                                                                                       "ff2"
                                                                                                       "LAMBDA (x: nnreal): x/(1+x*x)"))
                                                                                                     (("1"
                                                                                                       (expand
                                                                                                        "-")
                                                                                                       (("1"
                                                                                                         (replace
                                                                                                          -3
                                                                                                          -1)
                                                                                                         (("1"
                                                                                                           (replace
                                                                                                            -13
                                                                                                            -1
                                                                                                            rl)
                                                                                                           (("1"
                                                                                                             (simplify
                                                                                                              -1)
                                                                                                             (("1"
                                                                                                               (inst
                                                                                                                -15
                                                                                                                "LAMBDA (x_1: nnreal): atan(x_1) - x_1 / (1 + x_1 * x_1)")
                                                                                                               (("1"
                                                                                                                 (assert)
                                                                                                                 (("1"
                                                                                                                   (skolem
                                                                                                                    -
                                                                                                                    "c")
                                                                                                                   (("1"
                                                                                                                     (expand
                                                                                                                      "deriv"
                                                                                                                      -1)
                                                                                                                     (("1"
                                                                                                                       (lemma
                                                                                                                        "congruence"
                                                                                                                        ("f"
                                                                                                                         "(LAMBDA (x: nnreal):
         deriv(LAMBDA (x_1: nnreal): atan(x_1) - x_1 / (1 + x_1 * x_1), x))"
                                                                                                                         "g"
                                                                                                                         "(LAMBDA (x_1: nnreal):
          1 / (1 + x_1 * x_1) -
           (1 - x_1 * x_1) / (1 + 2 * (x_1 * x_1) + x_1 * x_1 * x_1 * x_1))"
                                                                                                                         "x1"
                                                                                                                         "c"
                                                                                                                         "x2"
                                                                                                                         "c"))
                                                                                                                       (("1"
                                                                                                                         (assert)
                                                                                                                         (("1"
                                                                                                                           (replace
                                                                                                                            -1
                                                                                                                            -16)
                                                                                                                           (("1"
                                                                                                                             (replace
                                                                                                                              -17
                                                                                                                              -16)
                                                                                                                             (("1"
                                                                                                                               (hide-all-but
                                                                                                                                (-16
                                                                                                                                 1))
                                                                                                                               (("1"
                                                                                                                                 (flatten)
                                                                                                                                 (("1"
                                                                                                                                   (lemma
                                                                                                                                    "posreal_times_posreal_is_posreal"
                                                                                                                                    ("px"
                                                                                                                                     "c"
                                                                                                                                     "py"
                                                                                                                                     "c"))
                                                                                                                                   (("1"
                                                                                                                                     (lemma
                                                                                                                                      "posreal_times_posreal_is_posreal"
                                                                                                                                      ("px"
                                                                                                                                       "1+c*c"
                                                                                                                                       "py"
                                                                                                                                       "1+c*c"))
                                                                                                                                     (("1"
                                                                                                                                       (lemma
                                                                                                                                        "minus_div1"
                                                                                                                                        ("x"
                                                                                                                                         "1"
                                                                                                                                         "n0x"
                                                                                                                                         "1+c*c"
                                                                                                                                         "y"
                                                                                                                                         "1-c*c"
                                                                                                                                         "n0y"
                                                                                                                                         "(1 + c * c) * (1 + c * c)"))
                                                                                                                                       (("1"
                                                                                                                                         (replace
                                                                                                                                          -1
                                                                                                                                          -6)
                                                                                                                                         (("1"
                                                                                                                                           (lemma
                                                                                                                                            "div_cancel1"
                                                                                                                                            ("x"
                                                                                                                                             "2*(c*c)/((1 + c * c) * (1 + c * c))"
                                                                                                                                             "n0z"
                                                                                                                                             "1+c*c"))
                                                                                                                                           (("1"
                                                                                                                                             (name
--> --------------------

--> maximum size reached

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

¤ Dauer der Verarbeitung: 0.67 Sekunden  (vorverarbeitet)  ¤





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