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Datei: mod_lems.prf   Sprache: Lisp

Original von: PVS©
(exp_term
 (exp_term_TCC1 0
  (exp_term_TCC1-1 nil 3320227930
   ("" (skosimp) (("" (assert) nil nil)) nil)
   ((real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   shostak))
 (exp_term_power_TCC1 0
  (exp_term_power_TCC1-1 nil 3320473217
   ("" (skosimp) (("" (assert) nil nil)) nil)
   ((real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   shostak))
 (exp_term_power 0
  (exp_term_power-1 nil 3320469433
   ("" (induct "m")
    (("1" (skosimp)
      (("1" (rewrite "expt_x0")
        (("1" (expand "exp_term")
          (("1" (expand "abs") (("1" (assert) nil nil)) nil)) nil))
        nil))
      nil)
     ("2" (skosimp*)
      (("2" (inst - "a!1" "n!1")
        (("2" (assert)
          (("2" (case "abs(exp_term(a!1)(n!1)) /= 0")
            (("1" (rewrite "expt_plus" 2)
              (("1" (rewrite "expt_x1" 2)
                (("1"
                  (case "abs(exp_term(a!1)(j!1 * n!1 + n!1)) <= abs(exp_term(a!1)(n!1)) * abs(exp_term(a!1)(j!1 * n!1))")
                  (("1" (assert)
                    (("1"
                      (lemma "both_sides_times_pos_le1"
                       ("pz" "abs(exp_term(a!1)(n!1))" "x"
                        "abs(exp_term(a!1)(j!1 * n!1))" "y"
                        "abs(exp_term(a!1)(n!1)) ^ j!1"))
                      (("1" (assert) nil nil)) nil))
                    nil)
                   ("2" (hide 3 -2)
                    (("2" (expand "exp_term")
                      (("2" (case-replace "n!1=0")
                        (("1" (assert)
                          (("1" (expand "abs") (("1" (assert) nil nil))
                            nil))
                          nil)
                         ("2" (assert)
                          (("2" (case-replace "j!1=0")
                            (("1" (assert)
                              (("1"
                                (expand "abs" 2 3)
                                (("1" (assert) nil nil))
                                nil))
                              nil)
                             ("2" (assert)
                              (("2"
                                (rewrite "abs_mult" 3 :dir rl)
                                (("2"
                                  (rewrite "div_times" 3)
                                  (("2"
                                    (rewrite "expt_plus" 3 :dir rl)
                                    (("2"
                                      (rewrite "abs_div" 3)
                                      (("2"
                                        (rewrite "abs_div" 3)
                                        (("2"
                                          (expand "abs" 3 4)
                                          (("2"
                                            (expand "abs" 3 2)
                                            (("2"
                                              (rewrite
                                               "abs_expt"
                                               3
                                               :dir
                                               rl)
                                              (("2"
                                                (case "abs(a!1)>0")
                                                (("1"
                                                  (lemma
                                                   "expt_pos"
                                                   ("px"
                                                    "abs(a!1)"
                                                    "i"
                                                    "(j!1+1)*n!1"))
                                                  (("1"
                                                    (lemma
                                                     "posreal_times_posreal_is_posreal"
                                                     ("px"
                                                      "factorial(n!1)"
                                                      "py"
                                                      "factorial(j!1 * n!1)"))
                                                    (("1"
                                                      (lemma
                                                       "both_sides_div_pos_lt2"
                                                       ("px"
                                                        "factorial(j!1 * n!1 + n!1)"
                                                        "pz"
                                                        "abs(a!1) ^ (j!1 * n!1 + n!1)"
                                                        "py"
                                                        "factorial(n!1) * factorial(j!1 * n!1)"))
                                                      (("1"
                                                        (assert)
                                                        (("1"
                                                          (hide
                                                           4
                                                           5
                                                           -2
                                                           -3
                                                           6)
                                                          (("1"
                                                            (case
                                                             "forall (pn,pm:posnat): factorial(pm)*factorial(pn) < factorial(pn+pm)")
                                                            (("1"
                                                              (inst
                                                               -
                                                               "n!1"
                                                               "j!1*n!1")
                                                              (("1"
                                                                (assert)
                                                                nil
                                                                nil))
                                                              nil)
                                                             ("2"
                                                              (hide-all-but
                                                               1)
                                                              (("2"
                                                                (induct
                                                                 "pn")
                                                                (("1"
                                                                  (assert)
                                                                  nil
                                                                  nil)
                                                                 ("2"
                                                                  (assert)
                                                                  nil
                                                                  nil)
                                                                 ("3"
                                                                  (skosimp*)
                                                                  (("3"
                                                                    (case-replace
                                                                     "j!2=0")
                                                                    (("1"
                                                                      (expand
                                                                       "factorial"
                                                                       1
                                                                       2)
                                                                      (("1"
                                                                        (expand
                                                                         "factorial"
                                                                         1
                                                                         3)
                                                                        (("1"
                                                                          (expand
                                                                           "factorial"
                                                                           1
                                                                           1)
                                                                          (("1"
                                                                            (assert)
                                                                            nil
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil)
                                                                     ("2"
                                                                      (assert)
                                                                      (("2"
                                                                        (inst
                                                                         -
                                                                         "pm!1")
                                                                        (("2"
                                                                          (expand
                                                                           "factorial"
                                                                           2
                                                                           3)
                                                                          (("2"
                                                                            (expand
                                                                             "factorial"
                                                                             2
                                                                             2)
                                                                            (("2"
                                                                              (lemma
                                                                               "lt_times_lt_pos1"
                                                                               ("px"
                                                                                "factorial(pm!1) * factorial(j!2)"
                                                                                "y"
                                                                                "factorial(pm!1 + j!2)"
                                                                                "nnz"
                                                                                "1+j!2"
                                                                                "w"
                                                                                "1+j!2+pm!1"))
                                                                              (("2"
                                                                                (assert)
                                                                                nil
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (assert)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (propax)
                                                    nil
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (expand "abs")
                                                  (("2"
                                                    (case-replace
                                                     "a!1<0")
                                                    (("1"
                                                      (assert)
                                                      nil
                                                      nil)
                                                     ("2"
                                                      (assert)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (hide-all-but (1 2))
              (("2" (expand "exp_term")
                (("2" (case-replace "n!1=0")
                  (("1" (expand "abs") (("1" (assert) nil nil)) nil)
                   ("2" (assert)
                    (("2" (rewrite "abs_div")
                      (("2" (expand "abs" -1 2)
                        (("2" (rewrite "abs_expt" -1 :dir rl)
                          (("2"
                            (lemma "expt_nonzero"
                             ("n0x" "abs(a!1)" "i" "n!1"))
                            (("1" (assert) nil nil)
                             ("2" (expand "abs")
                              (("2"
                                (case-replace "a!1<0")
                                (("1" (assert) nil nil)
                                 ("2" (assert) nil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nnreal_exp application-judgement "nnreal" exponentiation nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (expt_nonzero formula-decl nil exponentiation nil)
    (nzreal nonempty-type-eq-decl nil reals nil)
    (expt_plus formula-decl nil exponentiation nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (both_sides_times_pos_le1 formula-decl nil real_props nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (div_times formula-decl nil real_props nil)
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (abs_div formula-decl nil real_props nil)
    (nnreal_div_posreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (abs_expt formula-decl nil exponentiation nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (expt_pos formula-decl nil exponentiation nil)
    (both_sides_div_pos_lt2 formula-decl nil real_props nil)
    (lt_times_lt_pos1 formula-decl nil real_props nil)
    (even_plus_odd_is_odd application-judgement "odd_int" integers nil)
    (factorial_0 formula-decl nil factorial "ints/")
    (< const-decl "bool" reals nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (posreal_times_posreal_is_posreal judgement-tcc nil real_types nil)
    (abs_mult formula-decl nil real_props nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (factorial def-decl "posnat" factorial "ints/")
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (int_times_even_is_even application-judgement "even_int" integers
     nil)
    (even_plus_even_is_even application-judgement "even_int" integers
     nil)
    (nzrat_abs_is_pos application-judgement "{r: posrat | r >= q}"
     real_defs nil)
    (int_abs_is_nonneg application-judgement "{j: nonneg_int | j >= i}"
     real_defs nil)
    (nzint_abs_is_pos application-judgement "{j: posint | j >= i}"
     real_defs nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (expt_x1 formula-decl nil exponentiation nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (expt_x0 formula-decl nil exponentiation nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (^ const-decl "real" exponentiation nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (exp_term const-decl "real" exp_term nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (<= const-decl "bool" reals nil)
    (/= const-decl "boolean" notequal nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (pred type-eq-decl nil defined_types nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil))
   shostak))
 (exp_term_decr 0
  (exp_term_decr-1 nil 3320472607
   ("" (skosimp)
    (("" (expand "exp_term")
      (("" (case-replace "n!1=0")
        (("1" (expand "factorial")
          (("1" (expand "factorial")
            (("1" (rewrite "expt_x1")
              (("1" (expand "abs" 2 2) (("1" (assert) nil nil)) nil))
              nil))
            nil))
          nil)
         ("2" (assert)
          (("2" (rewrite "abs_div")
            (("2" (rewrite "abs_div")
              (("2" (expand "abs" 3 4)
                (("2" (expand "abs" 3 2)
                  (("2" (rewrite "abs_expt" 3 :dir rl)
                    (("2" (rewrite "abs_expt" 3 :dir rl)
                      (("2" (case "0 < abs(a!1)")
                        (("1" (rewrite "expt_plus" 3)
                          (("1" (rewrite "expt_x1")
                            (("1" (expand "factorial" 3 1)
                              (("1"
                                (lemma
                                 "both_sides_times_pos_lt1"
                                 ("x"
                                  "abs(a!1)/(1+n!1)"
                                  "y"
                                  "1"
                                  "pz"
                                  "abs(a!1) ^ n!1 / factorial(n!1)"))
                                (("1"
                                  (rewrite "div_times" -1)
                                  (("1"
                                    (assert)
                                    (("1"
                                      (hide 4)
                                      (("1"
                                        (rewrite "div_mult_pos_lt1" 1)
                                        nil
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (lemma
                                   "expt_pos"
                                   ("px" "abs(a!1)" "i" "n!1"))
                                  (("1"
                                    (rewrite "div_mult_pos_gt1" 1)
                                    (("1" (assert) nil nil))
                                    nil)
                                   ("2" (assert) nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (expand "abs")
                          (("2" (case "a!1<0")
                            (("1" (assert) nil nil)
                             ("2" (assert) nil nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((real_div_nzreal_is_real application-judgement "real" reals nil)
    (exp_term const-decl "real" exp_term nil)
    (nzint_abs_is_pos application-judgement "{j: posint | j >= i}"
     real_defs nil)
    (int_abs_is_nonneg application-judgement "{j: nonneg_int | j >= i}"
     real_defs nil)
    (nzrat_abs_is_pos application-judgement "{r: posrat | r >= q}"
     real_defs nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (expt_plus formula-decl nil exponentiation nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (expt_pos formula-decl nil exponentiation nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (div_mult_pos_gt1 formula-decl nil extra_real_props nil)
    (div_times formula-decl nil real_props nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (div_mult_pos_lt1 formula-decl nil real_props nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (both_sides_times_pos_lt1 formula-decl nil real_props nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (< const-decl "bool" reals nil)
    (nnreal_exp application-judgement "nnreal" exponentiation nil)
    (abs_expt formula-decl nil exponentiation nil)
    (nzreal nonempty-type-eq-decl nil reals nil)
    (nnreal_div_posreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (^ const-decl "real" exponentiation nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (/= const-decl "boolean" notequal nil)
    (abs_div formula-decl nil real_props nil)
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (factorial def-decl "posnat" factorial "ints/")
    (expt_x1 formula-decl nil exponentiation nil)
    (even_plus_odd_is_odd application-judgement "odd_int" integers nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
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    (bool nonempty-type-eq-decl nil booleans nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
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    (rational_pred const-decl "[real -> boolean]" rationals nil)
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    (number nonempty-type-decl nil numbers nil))
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                                                                                            "1"
                                                                                            "nnz"
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