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Datei: top.pvs Sprache: Unknown

Original von: PVS^©

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                                                    (("2"
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                                                              "pa"
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        (< const-decl "bool" reals nil)
        (real_ge_is_total_order name-judgement "(total_order?[real])"
         real_props nil)
        (div_mult_pos_ge1 formula-decl nil real_props nil)
        (NOT const-decl "[bool -> bool]" booleans nil)
        (* const-decl "[numfield, numfield -> numfield]" number_fields
           nil)
        (mult_divides1 application-judgement "(divides(n))" divides
         nil)
        (odd_times_odd_is_odd application-judgement "odd_int" integers
         nil)
        (posint_times_posint_is_posint application-judgement "posint"
         integers nil)
        (posreal_times_posreal_is_posreal application-judgement
         "posreal" real_types nil)
        (div_div1 formula-decl nil real_props nil)
        (nonzero_real nonempty-type-eq-decl nil reals nil)
        (div_mult_pos_gt1 formula-decl nil extra_real_props nil)
        (real_gt_is_strict_total_order name-judgement
         "(strict_total_order?[real])" real_props nil)
        (ne1x!1 skolem-const-decl "{r | r /= 1}" log nil)
        (posreal_div_posreal_is_posreal application-judgement "posreal"
         real_types nil)
        (number nonempty-type-decl nil numbers nil)
        (boolean nonempty-type-decl nil booleans nil)
        (number_field_pred const-decl "[number -> boolean]"
         number_fields nil)
        (number_field nonempty-type-from-decl nil number_fields nil)
        (real_pred const-decl "[number_field -> boolean]" reals nil)
        (real nonempty-type-from-decl nil reals nil)
        (bool nonempty-type-eq-decl nil booleans nil)
        (>= const-decl "bool" reals nil)
        (nonneg_real nonempty-type-eq-decl nil real_types nil)
        (> const-decl "bool" reals nil)
        (posreal nonempty-type-eq-decl nil real_types nil)
        (= const-decl "[T, T -> boolean]" equalities nil)
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        (log const-decl "real" log 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))
       shostak))
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      (log_inv-1 nil 3428820725
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        (("" (expand "log")
          (("" (case-replace "ne1x!1>1")
            (("1" (lemma "trich_lt" ("x" "y!1" "y" "1"))
              (("1" (split -1)
                (("1" (assert)
                  (("1" (rewrite "div_mult_pos_ge1" 1)
                    (("1" (assert) nil nil)) nil))
                  nil)
                 ("2" (replace -1)
                  (("2" (assert)
                    (("2" (rewrite "nn_log_1") (("2" (assert) nil nil))
                      nil))
                    nil))
                  nil)
                 ("3" (assert)
                  (("3" (rewrite "div_mult_pos_ge1" 1)
                    (("3" (assert) nil nil)) nil))
                  nil))
                nil))
              nil)
             ("2" (assert)
              (("2" (lemma "trich_lt" ("x" "y!1" "y" "1"))
                (("2" (split -1)
                  (("1" (assert)
                    (("1" (rewrite "div_mult_pos_le1" 2)
                      (("1" (rewrite "div_div1") nil nil)) nil))
                    nil)
                   ("2" (replace -1)
                    (("2" (assert)
                      (("2" (rewrite "nn_log_1")
                        (("2" (assert) nil nil)) nil))
                      nil))
                    nil)
                   ("3" (assert)
                    (("3" (rewrite "div_mult_pos_le1" 2)
                      (("3" (assert) nil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ((log const-decl "real" log nil)
        (div_mult_pos_le1 formula-decl nil real_props nil)
        (div_div1 formula-decl nil real_props nil)
        (nonzero_real nonempty-type-eq-decl nil reals nil)
        (trich_lt formula-decl nil real_props nil)
        (nn_log_1 formula-decl nil nn_log nil)
        (minus_even_is_even application-judgement "even_int" integers
         nil)
        (posrat_div_posrat_is_posrat application-judgement "posrat"
         rationals nil)
        (minus_real_is_real application-judgement "real" reals nil)
        (real_le_is_total_order name-judgement "(total_order?[real])"
         real_props nil)
        (posreal_div_posreal_is_posreal application-judgement "posreal"
         real_types nil)
        (real_ge_is_total_order name-judgement "(total_order?[real])"
         real_props nil)
        (real_gt_is_strict_total_order name-judgement
         "(strict_total_order?[real])" real_props nil)
        (real_lt_is_strict_total_order name-judgement
         "(strict_total_order?[real])" real_props nil)
        (posreal_times_posreal_is_posreal application-judgement
         "posreal" real_types nil)
        (div_mult_pos_ge1 formula-decl nil real_props nil)
        (/= const-decl "boolean" notequal nil)
        (posreal nonempty-type-eq-decl nil real_types nil)
        (nonneg_real nonempty-type-eq-decl nil real_types nil)
        (>= const-decl "bool" reals nil)
        (> const-decl "bool" reals nil)
        (bool nonempty-type-eq-decl nil booleans nil)
        (real nonempty-type-from-decl nil reals nil)
        (real_pred const-decl "[number_field -> boolean]" reals nil)
        (number_field nonempty-type-from-decl nil number_fields nil)
        (number_field_pred const-decl "[number -> boolean]"
         number_fields nil)
        (boolean nonempty-type-decl nil booleans nil)
        (number nonempty-type-decl nil numbers nil))
       shostak))
     (log_mult 0
      (log_mult-1 nil 3428820389
       ("" (skosimp)
        (("" (case "ne1x!1>1")
          (("1" (case "x!1>=1")
            (("1" (case "y!1>=1")
              (("1"
                (lemma "le_times_le_pos"
                 ("nnx" "1" "y" "x!1" "nnz" "1" "w" "y!1"))
                (("1" (expand "log")
                  (("1" (assert)
                    (("1" (rewrite "nn_log_plus") nil nil)) nil))
                  nil))
                nil)
               ("2" (case "y!1<1")
                (("1" (hide 1)
                  (("1" (case "x!1*y!1>=1")
                    (("1" (expand "log")
                      (("1" (assert)
                        (("1"
                          (lemma "nn_log_plus"
                           ("ne1x" "ne1x!1" "x" "x!1 * y!1" "y"
                            "1/y!1"))
                          (("1" (assert) nil nil)) nil))
                        nil))
                      nil)
                     ("2" (case "x!1*y!1<1")
                      (("1" (hide 1)
                        (("1" (expand "log")
                          (("1" (assert)
                            (("1"
                              (lemma "nn_log_plus"
                               ("ne1x"
                                "ne1x!1"
                                "x"
                                "x!1"
                                "y"
                                "1 / (x!1 * y!1)"))
                              (("1"
                                (case-replace
                                 "x!1 * (1 / (x!1 * y!1))=1/y!1")
                                (("1" (assert) nil nil)
                                 ("2" (assert) nil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (assert) nil nil))
                      nil))
                    nil))
                  nil)
                 ("2" (assert) nil nil))
                nil))
              nil)
             ("2" (case "x!1<1")
              (("1" (hide 1)
                (("1" (case "y!1>=1")
                  (("1" (case "x!1 * y!1>=1")
                    (("1" (expand "log")
                      (("1" (assert)
                        (("1"
                          (lemma "nn_log_plus"
                           ("ne1x" "ne1x!1" "x" "x!1*y!1" "y" "1/x!1"))
                          (("1" (assert) nil nil)) nil))
                        nil))
                      nil)
                     ("2" (expand "log")
                      (("2" (assert)
                        (("2"
                          (lemma "nn_log_plus"
                           ("ne1x" "ne1x!1" "x" "y!1" "y"
                            "1 / (x!1 * y!1)"))
                          (("2"
                            (case-replace
                             "y!1 * (1 / (x!1 * y!1))=1/x!1")
                            (("1" (assert) nil nil)
                             ("2" (assert) nil nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2"
                    (lemma "lt_times_lt_pos1"
                     ("px" "x!1" "y" "1" "nnz" "y!1" "w" "1"))
                    (("2" (expand "log")
                      (("2" (assert)
                        (("2"
                          (lemma "nn_log_plus"
                           ("ne1x" "ne1x!1" "x" "1/x!1" "y" "1/y!1"))
                          (("2" (assert) nil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (assert) nil nil))
              nil))
            nil)
           ("2" (case "ne1x!1<1")
            (("1" (hide 1)
              (("1" (case "x!1<=1")
                (("1" (case "y!1<=1")
                  (("1"
                    (lemma "le_times_le_pos"
                     ("nnx" "x!1" "y" "1" "nnz" "y!1" "w" "1"))
                    (("1" (expand "log")
                      (("1" (assert)
                        (("1" (rewrite "nn_log_plus") nil nil)) nil))
                      nil))
                    nil)
                   ("2" (case "y!1>1")
                    (("1" (hide 1)
                      (("1" (case "x!1 * y!1<=1")
                        (("1" (expand "log")
                          (("1" (assert)
                            (("1"
                              (lemma "nn_log_plus"
                               ("ne1x"
                                "ne1x!1"
                                "x"
                                "x!1 * y!1"
                                "y"
                                "1/y!1"))
                              (("1"
                                (case-replace
                                 "x!1 * y!1 * (1 / y!1)=x!1")
                                (("1" (assert) nil nil)
                                 ("2" (assert) nil nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (expand "log")
                          (("2" (assert)
                            (("2"
                              (lemma "nn_log_plus"
                               ("ne1x"
                                "ne1x!1"
                                "y"
                                "1/(x!1*y!1)"
                                "x"
                                "x!1"))
                              (("2"
                                (case-replace
                                 "x!1 * (1 / (x!1 * y!1))=1/y!1")
                                (("1" (assert) nil nil)
                                 ("2" (assert) nil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (assert) nil nil))
                    nil))
                  nil)
                 ("2" (case "x!1>1")
                  (("1" (hide 1)
                    (("1" (case "y!1<=1")
                      (("1" (case "x!1*y!1<=1")
                        (("1" (expand "log")
                          (("1" (assert)
                            (("1"
                              (lemma "nn_log_plus"
                               ("ne1x"
                                "ne1x!1"
                                "x"
                                "x!1 * y!1"
                                "y"
                                "1/x!1"))
                              (("1" (assert) nil nil)) nil))
                            nil))
                          nil)
                         ("2" (expand "log")
                          (("2" (assert)
                            (("2"
                              (lemma "nn_log_plus"
                               ("ne1x"
                                "ne1x!1"
                                "x"
                                "y!1"
                                "y"
                                "1/(x!1*y!1)"))
                              (("2" (assert) nil nil)) nil))
                            nil))
                          nil))
                        nil)
                       ("2"
                        (lemma "lt_times_lt_pos1"
                         ("px" "1" "y" "x!1" "nnz" "1" "w" "y!1"))
                        (("2" (expand "log")
                          (("2" (assert)
                            (("2"
                              (lemma "nn_log_plus"
                               ("ne1x"
                                "ne1x!1"
                                "x"
                                "1/x!1"
                                "y"
                                "1/y!1"))
                              (("2" (assert) nil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (assert) nil nil))
                  nil))
                nil))
              nil)
             ("2" (assert) nil nil))
            nil))
          nil))
        nil)
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        (nonneg_real nonempty-type-eq-decl nil real_types nil)
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        (number_field_pred const-decl "[number -> boolean]"
         number_fields nil)
        (boolean nonempty-type-decl nil booleans nil)
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        (log const-decl "real" log nil)
        (<= const-decl "bool" reals nil)
        (< const-decl "bool" reals nil)
        (AND const-decl "[bool, bool -> bool]" booleans nil)
        (OR const-decl "[bool, bool -> bool]" booleans nil)
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        (real_gt_is_strict_total_order name-judgement
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        (real_le_is_total_order name-judgement "(total_order?[real])"
         real_props nil)
        (posreal_times_posreal_is_posreal application-judgement
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        (real_plus_real_is_real application-judgement "real" reals nil)
        (le_times_le_pos formula-decl nil real_props nil)
        (= const-decl "[T, T -> boolean]" equalities nil)
        (nznum nonempty-type-eq-decl nil number_fields nil)
        (/ const-decl "[numfield, nznum -> numfield]" number_fields
           nil)
        (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
         real_types nil)
        (minus_real_is_real application-judgement "real" reals nil)
        (posreal_div_posreal_is_posreal application-judgement "posreal"
         real_types nil)
        (* const-decl "[numfield, numfield -> numfield]" number_fields
           nil)
        (numfield nonempty-type-eq-decl nil number_fields nil))
       shostak))
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      (log_div-1 nil 3428826839
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        (("" (lemma "log_inv" ("ne1x" "ne1x!1" "y" "y!1"))
          ((""
            (lemma "log_mult" ("ne1x" "ne1x!1" "x" "x!1" "y" "1/y!1"))
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          nil))
        nil)
       ((/= const-decl "boolean" notequal nil)
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        (number_field_pred const-decl "[number -> boolean]"
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        (boolean nonempty-type-decl nil booleans nil)
        (number nonempty-type-decl nil numbers nil)
        (log_inv formula-decl nil log nil)
        (real_minus_real_is_real application-judgement "real" reals
         nil)
        (minus_real_is_real application-judgement "real" reals nil)
        (real_plus_real_is_real application-judgement "real" reals nil)
        (posreal_times_posreal_is_posreal application-judgement
         "posreal" real_types nil)
        (posreal_div_posreal_is_posreal application-judgement "posreal"
         real_types nil)
        (log_mult formula-decl nil log 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))
       shostak))
     (log_xz_TCC1 0
      (log_xz_TCC1-1 nil 3428954667 ("" (subtype-tcc) nil nil) nil
       nil))
     (log_xz_TCC2 0
      (log_xz_TCC2-1 nil 3428954667
       ("" (skosimp) (("" (rewrite "real_expt_pos") nil nil)) nil)
       ((real_expt_pos formula-decl nil real_expt nil)
        (number nonempty-type-decl nil numbers nil)
        (boolean nonempty-type-decl nil booleans nil)
        (number_field_pred const-decl "[number -> boolean]"
         number_fields nil)
        (number_field nonempty-type-from-decl nil number_fields nil)
        (real_pred const-decl "[number_field -> boolean]" reals nil)
        (real nonempty-type-from-decl nil reals nil)
        (bool nonempty-type-eq-decl nil booleans nil)
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        (nonneg_real nonempty-type-eq-decl nil real_types nil)
        (> const-decl "bool" reals nil)
        (posreal nonempty-type-eq-decl nil real_types nil))
       nil))
     (log_xz 0
      (log_xz-1 nil 3428954668
       ("" (skosimp)
        (("" (case-replace "x!1 ^ z!1=ne1x!1^(z!1*log(ne1x!1)(x!1))")
          (("1" (rewrite "log_expt") nil nil)
           ("2" (hide 2)
            (("2"
              (lemma "real_expt_times"
               ("x" "ne1x!1" "a" "log(ne1x!1)(x!1)" "b" "z!1"))
              (("1" (rewrite "expt_log" -1) (("1" (assert) nil nil))
                nil)
               ("2" (rewrite "expt_log" 1) (("2" (assert) nil nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ((real_gt_is_strict_total_order name-judgement
         "(strict_total_order?[real])" real_props nil)
        (real_ge_is_total_order name-judgement "(total_order?[real])"
         real_props nil)
        (real_times_real_is_real application-judgement "real" reals
         nil)
        (number nonempty-type-decl nil numbers nil)
        (boolean nonempty-type-decl nil booleans nil)
        (= const-decl "[T, T -> boolean]" equalities nil)
        (nnreal type-eq-decl nil real_types nil)
        (number_field_pred const-decl "[number -> boolean]"
         number_fields nil)
        (number_field nonempty-type-from-decl nil number_fields nil)
        (real_pred const-decl "[number_field -> boolean]" reals nil)
        (real nonempty-type-from-decl nil reals nil)
        (bool nonempty-type-eq-decl nil booleans nil)
        (OR const-decl "[bool, bool -> bool]" booleans nil)
        (>= const-decl "bool" reals nil)
        (> const-decl "bool" reals nil)
        (^ const-decl "nnreal" real_expt nil)
        (nonneg_real nonempty-type-eq-decl nil real_types nil)
        (posreal nonempty-type-eq-decl nil real_types nil)
        (/= const-decl "boolean" notequal nil)
        (numfield nonempty-type-eq-decl nil number_fields nil)
        (* const-decl "[numfield, numfield -> numfield]" number_fields
           nil)
        (log const-decl "real" log nil)
        (log_expt formula-decl nil log nil)
        (real_expt_times formula-decl nil real_expt nil)
        (expt_log formula-decl nil log nil))
       shostak))
     (log_increasing_TCC1 0
      (log_increasing_TCC1-1 nil 3428835559 ("" (subtype-tcc) nil nil)
       nil nil))
     (log_increasing 0
      (log_increasing-1 nil 3428835694
       ("" (skosimp)
        (("" (case "log(gt1x!1)(x!1) = log(gt1x!1)(y!1)")
          (("1" (lemma "expt_log" ("ne1x" "gt1x!1" "y" "x!1"))
            (("1" (lemma "expt_log" ("ne1x" "gt1x!1" "y" "y!1"))
              (("1" (assert) nil nil)) nil))
            nil)
           ("2"
            (lemma "real_expt_increasing"
             ("gt1x" "gt1x!1" "a" "log(gt1x!1)(y!1)" "b"
              "log(gt1x!1)(x!1)"))
            (("2" (rewrite "expt_log")
              (("2" (rewrite "expt_log") (("2" (assert) nil nil)) nil))
              nil))
            nil))
          nil))
        nil)
       ((log const-decl "real" log nil)
        (/= const-decl "boolean" notequal nil)
        (posreal nonempty-type-eq-decl nil real_types nil)
        (> const-decl "bool" reals nil)
        (nonneg_real nonempty-type-eq-decl nil real_types nil)
        (>= const-decl "bool" reals nil)
        (bool nonempty-type-eq-decl nil booleans 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)
        (= const-decl "[T, T -> boolean]" equalities nil)
        (boolean nonempty-type-decl nil booleans nil)
        (number nonempty-type-decl nil numbers nil)
        (real_lt_is_strict_total_order name-judgement
         "(strict_total_order?[real])" real_props nil)
        (expt_log formula-decl nil log nil)
        (real_expt_increasing formula-decl nil real_expt nil))
       shostak))
     (log_decreasing_TCC1 0
      (log_decreasing_TCC1-1 nil 3428835559 ("" (subtype-tcc) nil nil)
       nil nil))
     (log_decreasing 0
      (log_decreasing-1 nil 3428835560
       ("" (skosimp)
        (("" (case "log(lt1x!1)(x!1) = log(lt1x!1)(y!1)")
          (("1" (lemma "expt_log" ("ne1x" "lt1x!1" "y" "x!1"))
            (("1" (replace -1 -3 rl)
              (("1" (lemma "expt_log" ("ne1x" "lt1x!1" "y" "y!1"))
                (("1" (replace -1 -4 rl)
                  (("1" (replace -3) (("1" (assert) nil nil)) nil))
                  nil))
                nil))
              nil))
            nil)
           ("2"
            (lemma "real_expt_decreasing"
             ("lt1x" "lt1x!1" "a" "log(lt1x!1)(x!1)" "b"
              "log(lt1x!1)(y!1)"))
            (("2" (rewrite "expt_log")
              (("2" (rewrite "expt_log") (("2" (assert) nil nil)) nil))
              nil))
            nil))
          nil))
        nil)
       ((< const-decl "bool" reals nil) (log const-decl "real" log nil)
        (/= const-decl "boolean" notequal nil)
        (posreal nonempty-type-eq-decl nil real_types nil)
        (> const-decl "bool" reals nil)
        (nonneg_real nonempty-type-eq-decl nil real_types nil)
        (>= const-decl "bool" reals nil)
        (bool nonempty-type-eq-decl nil booleans nil)
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        (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)
        (= const-decl "[T, T -> boolean]" equalities nil)
        (boolean nonempty-type-decl nil booleans nil)
        (number nonempty-type-decl nil numbers nil)
        (real_lt_is_strict_total_order name-judgement
         "(strict_total_order?[real])" real_props nil)
        (expt_log formula-decl nil log nil)
        (real_gt_is_strict_total_order name-judgement
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        (real_gt_is_strict_total_order name-judgement
         "(strict_total_order?[real])" real_props nil)
        (log_increasing formula-decl nil log nil)
        (log_decreasing formula-decl nil log nil)
        (< const-decl "bool" reals nil))
       shostak))
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                       ("lt1x" "ne1x!1" "x" "1" "y" "y!1"))
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         "(strict_total_order?[real])" real_props nil)
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                (("1" (lemma "expt_log" ("ne1x" "y!1" "y" "z!1"))
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                           ("2" (assert) nil nil))
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                         ("2" (assert)
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                      (("1" (lemma "log_neg" ("ne1x" "x!1" "y" "z!1"))
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                             ("px" "x!1" "y" "y!1" "na"
                              "log(x!1)(z!1)"))
                            (("1"
                              (lemma "real_expt_increasing"
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                                "b"
                                "log(x!1)(z!1)"
                                "gt1x"
                                "y!1"))
                              (("1"
                                (rewrite "expt_log")
                                (("1"
                                  (rewrite "expt_log")
                                  (("1" (assert) nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (lemma "log_pos" ("ne1x" "y!1" "y" "z!1"))
                        (("2" (assert)
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                            (lemma "real_expt_strict_increasing"
                             ("x" "x!1" "y" "y!1" "pa"
                              "log(y!1)(z!1)"))
                            (("2"
                              (lemma "real_expt_decreasing"
                               ("a"
                                "log(y!1)(z!1)"
                                "b"
                                "log(x!1)(z!1)"
                                "lt1x"
                                "x!1"))
                              (("2"
--> --------------------

--> maximum size reached

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

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