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Datei: Sprache: PVS

Original von: PVS^©

(factorial_props
(factorial_2n_lb_TCC1 0
  (factorial_2n_lb_TCC1-1 nil 3322393244
   ("" (skosimp*) (("" (assert) nil nil)) nil)
   ((mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   nil))
(factorial_2n_lb 0
  (factorial_2n_lb-1 nil 3322393266
   ("" (case "FORALL (n: nat): n^n <= factorial(2 * n)")
    (("1" (skosimp)
      (("1" (inst - "n!1")
        (("1" (rewrite "expt_times")
          (("1" (expand "^" 1 1)
            (("1" (expand "expt" 1)
              (("1" (expand "expt" 1)
                (("1" (expand "expt" 1)
                  (("1" (rewrite "sq_rew") nil nil)) nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil)
     ("2" (hide 2)
      (("2"
        (case "FORALL (n, N: nat): n <= N => N ^ n <= factorial(n + N)")
        (("1" (skosimp)
          (("1" (inst - "n!1" "n!1") (("1" (assert) nil nil)) nil))
          nil)
         ("2" (hide 2)
          (("2" (skolem 1 ("_" "N"))
            (("2" (induct "n")
              (("1" (assert)
                (("1" (expand "^")
                  (("1" (expand "expt") (("1" (assert) nil nil)) nil))
                  nil))
                nil)
               ("2" (skosimp*)
                (("2" (assert)
                  (("2" (rewrite "expt_plus")
                    (("2" (rewrite "expt_x1")
                      (("2" (expand "factorial" 1)
                        (("2" (lemma "le_times_le_pos")
                          (("2"
                            (inst - "N^j!1" "N" "N+j!1+1"
                             "factorial(N+j!1)")
                            (("2" (assert) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((pred type-eq-decl nil defined_types nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (expt_x1 formula-decl nil exponentiation nil)
    (le_times_le_pos formula-decl nil real_props nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (expt_plus formula-decl nil exponentiation nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (sqrt const-decl "{nnz: nnreal | nnz * nnz = nnx}" sqrt nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (nnreal type-eq-decl nil real_types nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (nzreal nonempty-type-eq-decl nil reals nil)
    (expt_times formula-decl nil exponentiation nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nat_exp application-judgement "nat" exponentiation nil)
    (nnreal_exp application-judgement "nnreal" exponentiation nil)
    (expt def-decl "real" exponentiation nil)
    (sq_rew formula-decl nil sq nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (sq_sqrt formula-decl nil sqrt nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers 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)
    (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)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (<= const-decl "bool" reals nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (/= const-decl "boolean" notequal nil)
    (^ const-decl "real" exponentiation nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (factorial def-decl "posnat" factorial "ints/")
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields
       nil))
   nil))
(factorial_2np1_lb_TCC1 0
  (factorial_2np1_lb_TCC1-1 nil 3322393244
   ("" (skosimp*) (("" (assert) nil nil)) nil)
   ((mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (even_plus_odd_is_odd application-judgement "odd_int" integers nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   nil))
(factorial_2np1_lb 0
  (factorial_2np1_lb-2 nil 3322393339
   ("" (skosimp)
    (("" (lemma "factorial_2n_lb" ("n" "n!1"))
      (("" (case-replace "n!1=0")
        (("1" (expand "factorial")
          (("1" (expand "factorial")
            (("1" (rewrite "sqrt_0")
              (("1" (expand "^")
                (("1" (expand "expt") (("1" (assert) nil nil)) nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (rewrite "expt_plus" 2)
          (("2" (expand "factorial" 2)
            (("2" (assert)
              (("2" (rewrite "expt_x1")
                (("2"
                  (lemma "le_times_le_pos"
                   ("nnx" "sqrt(n!1) ^ (2 * n!1)" "y"
                    "factorial(2 * n!1)" "nnz" "sqrt(n!1)" "w"
                    "1+2*n!1"))
                  (("2" (assert)
                    (("2" (hide-all-but (1 2))
                      (("2"
                        (lemma "sq_le"
                         ("nna" "sqrt(n!1)" "nnb" "1 + 2 * n!1"))
                        (("2" (rewrite "sq_sqrt")
                          (("2" (replace -1 1 rl)
                            (("2" (hide -1)
                              (("2"
                                (expand "sq")
                                (("2" (assert) nil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    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)
    (factorial_2n_lb formula-decl nil factorial_props nil)
    (expt_plus formula-decl nil exponentiation nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (/= const-decl "boolean" notequal nil)
    (nzreal nonempty-type-eq-decl nil reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (nnreal type-eq-decl nil real_types nil)
    (sqrt const-decl "{nnz: nnreal | nnz * nnz = nnx}" sqrt nil)
    (nnreal_exp application-judgement "nnreal" exponentiation nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (odd_plus_even_is_odd application-judgement "odd_int" integers nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (le_times_le_pos formula-decl nil real_props nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (sq_sqrt formula-decl nil sqrt nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (even_plus_even_is_even application-judgement "even_int" integers
     nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (sq const-decl "nonneg_real" sq nil)
    (sq_le formula-decl nil sq nil)
    (expt_x1 formula-decl nil exponentiation nil)
    (factorial def-decl "posnat" factorial "ints/")
    (sqrt_0 formula-decl nil sqrt nil)
    (expt def-decl "real" exponentiation nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nat_expt application-judgement "nat" exponentiation nil)
    (^ const-decl "real" exponentiation nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers 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)
    (= const-decl "[T, T -> boolean]" equalities nil))
   nil)
  (factorial_2np1_lb-1 nil 3322393290
   ("" (skosimp)
    (("" (lemma "exp_series_scaf2" ("n" "n!1"))
      (("" (case-replace "n!1=0")
        (("1" (expand "factorial")
          (("1" (expand "factorial")
            (("1" (rewrite "sqrt_0")
              (("1" (expand "^")
                (("1" (expand "expt") (("1" (assert) nil)))))))))))
         ("2" (rewrite "expt_plus" 2)
          (("2" (expand "factorial" 2)
            (("2" (assert)
              (("2" (rewrite "expt_x1")
                (("2"
                  (lemma "le_times_le_pos"
                   ("nnx" "sqrt(n!1) ^ (2 * n!1)" "y"
                    "factorial(2 * n!1)" "nnz" "sqrt(n!1)" "w"
                    "1+2*n!1"))
                  (("2" (assert)
                    (("2" (hide-all-but (1 2))
                      (("2"
                        (lemma "sq_le"
                         ("nna" "sqrt(n!1)" "nnb" "1 + 2 * n!1"))
                        (("2" (rewrite "sq_sqrt")
                          (("2" (replace -1 1 rl)
                            (("2" (hide -1)
                              (("2"
                                (expand "sq")
                                (("2"
                                  (assert)
                                  nil))))))))))))))))))))))))))))))
    nil)
   nil nil)))

¤ Dauer der Verarbeitung: 0.288 Sekunden (vorverarbeitet) ¤

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