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Datei: holder_scaf.pvs   Sprache: Lisp

Original von: PVS©

(holder_scaf
 (mu_TCC1 0
  (mu_TCC1-1 nil 3477456171
   ("" (typepred "S")
    (("" (expand "sigma_algebra?")
      (("" (flatten)
        (("" (expand "subset_algebra_empty?") (("" (assertnil nil))
          nil))
        nil))
      nil))
    nil)
   ((subset_algebra_empty? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (finite_emptyset name-judgement "finite_set" finite_sets nil)
    (finite_emptyset name-judgement "finite_set[T]" countable_props
     "sets_aux/")
    (finite_emptyset name-judgement "finite_set[T]" countable_setofsets
     "sets_aux/")
    (member const-decl "bool" sets nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (T formal-type-decl nil holder_scaf nil)
    (setof type-eq-decl nil defined_types nil)
    (setofsets type-eq-decl nil sets nil)
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
     "measure_integration/")
    (S formal-const-decl "sigma_algebra[T]" holder_scaf nil))
   nil))
 (q_TCC1 0 (q_TCC1-1 nil 3477456171 ("" (assertnil nilnil nil))
 (f_TCC1 0
  (f_TCC1-1 nil 3477456171 ("" (assertnil nil)
   ((real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   nil))
 (g_TCC1 0
  (g_TCC1-1 nil 3477456171
   ("" (typepred "q")
    (("" (typepred "p")
      (("" (lemma "div_cancel3" ("x" "1" "n0z" "q" "y" "1-1/p"))
        (("" (assert)
          (("" (flatten)
            (("" (hide -2)
              (("" (split)
                (("1" (replace -1 1)
                  (("1"
                    (lemma "both_sides_times_pos_ge1"
                     ("pz" "q" "x" "1" "y" "1-1/p"))
                    (("1" (assert)
                      (("1" (hide 2)
                        (("1"
                          (lemma "posreal_div_posreal_is_posreal"
                           ("px" "1" "py" "p"))
                          (("1" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (assertnil nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nzreal_times_nzreal_is_nzreal application-judgement "nzreal"
     real_types nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (nzreal_div_nzreal_is_nzreal application-judgement "nzreal"
     real_types 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)
    (posreal_times_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (posreal_div_posreal_is_posreal judgement-tcc nil real_types nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (both_sides_times_pos_ge1 formula-decl nil real_props nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (div_cancel3 formula-decl nil real_props nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields 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 "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)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (+ const-decl "[numfield, numfield -> numfield]" 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)
    (p formal-const-decl "{a: real | a > 1}" holder_scaf nil)
    (q formal-const-decl "{a: posreal | 1 / p + 1 / a = 1}" holder_scaf
     nil))
   nil))
 (holder_aux 0
  (holder_aux-1 nil 3477461478
   ("" (skosimp)
    (("" (typepred "f!1")
      (("" (typepred "g!1")
        (("" (case "p>1")
          (("1" (case "q>1")
            (("1" (typepred "norm[T, S, mu, q](g!1)")
              (("1" (typepred "norm[T, S, mu, p](f!1)")
                (("1" (expand ">=")
                  (("1" (expand "<=" -1)
                    (("1" (split)
                      (("1" (expand "<=" -2)
                        (("1" (split)
                          (("1" (expand "p_integrable?")
                            (("1" (flatten)
                              (("1"
                                (rewrite
                                 "prod_complex_measurable[T,S]")
                                (("1"
                                  (name
                                   "ALPHA"
                                   "norm[T, S, mu, p](f!1)")
                                  (("1"
                                    (replace -1)
                                    (("1"
                                      (name
                                       "BETA"
                                       "norm[T, S, mu, q](g!1)")
                                      (("1"
                                        (replace -1)
                                        (("1"
                                          (lemma
                                           "posreal_times_posreal_is_posreal"
                                           ("px" "ALPHA" "py" "BETA"))
                                          (("1"
                                            (lemma
                                             "real_expt_pos"
                                             ("px" "ALPHA" "a" "p"))
                                            (("1"
                                              (lemma
                                               "real_expt_pos"
                                               ("px" "BETA" "a" "q"))
                                              (("1"
                                                (case
                                                 "FORALL (x: T):
        abs(f!1(x) * g!1(x)) / (ALPHA * BETA) <=
         (1 / p) * (abs(f!1(x)) ^ p / (ALPHA ^ p)) +
          (1 / q) * (abs(g!1(x)) ^ q / (BETA ^ q))")
                                                (("1"
                                                  (case-replace
                                                   "abs(f!1 * g!1) ^ 1=abs(f!1 * g!1)")
                                                  (("1"
                                                    (hide -1)
                                                    (("1"
                                                      (expand "norm")
                                                      (("1"
                                                        (case-replace
                                                         "abs(f!1 * g!1) ^ 1=abs(f!1 * g!1)")
                                                        (("1"
                                                          (hide -1)
                                                          (("1"
                                                            (hide
                                                             -11
                                                             -13)
                                                            (("1"
                                                              (lemma
                                                               "both_sides_real_expt"
                                                               ("x"
                                                                "integral.integral(abs(g!1) ^ q) ^ (1 / q)"
                                                                "y"
                                                                "BETA"
                                                                "pa"
                                                                "q"))
                                                              (("1"
                                                                (rewrite
                                                                 "real_expt_times"
                                                                 -1
                                                                 :dir
                                                                 rl)
                                                                (("1"
                                                                  (assert)
                                                                  (("1"
                                                                    (lemma
                                                                     "both_sides_real_expt"
                                                                     ("x"
                                                                      "integral.integral(abs(f!1)^p)^(1/p)"
                                                                      "y"
                                                                      "ALPHA"
                                                                      "pa"
                                                                      "p"))
                                                                    (("1"
                                                                      (rewrite
                                                                       "real_expt_times"
                                                                       -1
                                                                       :dir
                                                                       rl)
                                                                      (("1"
                                                                        (assert)
                                                                        (("1"
                                                                          (rewrite
                                                                           "real_expt_x1"
                                                                           -1)
                                                                          (("1"
                                                                            (rewrite
                                                                             "real_expt_x1"
                                                                             -2)
                                                                            (("1"
                                                                              (lemma
                                                                               "integral.integrable_is_measurable")
                                                                              (("1"
                                                                                (inst-cp
                                                                                 -
                                                                                 "abs(f!1) ^ p")
                                                                                (("1"
                                                                                  (inst
                                                                                   -
                                                                                   "abs(g!1) ^ q")
                                                                                  (("1"
                                                                                    (lemma
                                                                                     "expt_measurable"
                                                                                     ("g"
                                                                                      "abs(f!1) ^ p"
                                                                                      "a"
                                                                                      "1/p"))
                                                                                    (("1"
                                                                                      (lemma
                                                                                       "expt_measurable"
                                                                                       ("g"
                                                                                        "abs(g!1) ^ q"
                                                                                        "a"
                                                                                        "1/q"))
                                                                                      (("1"
                                                                                        (case-replace
                                                                                         "(^[T](abs(g!1) ^ q, 1 / q))=abs(g!1)")
                                                                                        (("1"
                                                                                          (case-replace
                                                                                           "(^[T](abs(f!1)^p, 1 / p))=abs(f!1)")
                                                                                          (("1"
                                                                                            (hide
                                                                                             -1
                                                                                             -2
                                                                                             -5
                                                                                             -6)
                                                                                            (("1"
                                                                                              (case
                                                                                               "measurable_function?[T,S](abs(f!1*g!1))")
                                                                                              (("1"
                                                                                                (lemma
                                                                                                 "scal_measurable"
                                                                                                 ("c"
                                                                                                  "1/(ALPHA * BETA)"
                                                                                                  "g"
                                                                                                  "abs(f!1 * g!1)"))
                                                                                                (("1"
                                                                                                  (hide
                                                                                                   -2
                                                                                                   -3
                                                                                                   -4)
                                                                                                  (("1"
                                                                                                    (lemma
                                                                                                     "integral.integrable_scal"
                                                                                                     ("c"
                                                                                                      "(1/p)*(1/ALPHA ^ p)"
                                                                                                      "f"
                                                                                                      "abs(f!1) ^ p"))
                                                                                                    (("1"
                                                                                                      (lemma
                                                                                                       "integral.integrable_scal"
                                                                                                       ("c"
                                                                                                        "(1/q)*(1/BETA ^ q)"
                                                                                                        "f"
                                                                                                        "abs(g!1) ^ q"))
                                                                                                      (("1"
                                                                                                        (lemma
                                                                                                         "integral.integrable_add"
                                                                                                         ("f1"
                                                                                                          "*[T]((1 / p) * (1 / ALPHA ^ p), abs(f!1) ^ p)"
                                                                                                          "f2"
                                                                                                          "*[T]((1 / q) * (1 / BETA ^ q), abs(g!1) ^ q)"))
                                                                                                        (("1"
                                                                                                          (hide
                                                                                                           -2
                                                                                                           -3)
                                                                                                          (("1"
                                                                                                            (lemma
                                                                                                             "integral_ae_abs"
                                                                                                             ("h"
                                                                                                              "*[T](1 / (ALPHA * BETA), abs(f!1 * g!1))"
                                                                                                              "f"
                                                                                                              "(+[T])
                      (*[T]((1 / p) * (1 / ALPHA ^ p), abs(f!1) ^ p),
                       *[T]((1 / q) * (1 / BETA ^ q), abs(g!1) ^ q))"))
                                                                                                            (("1"
                                                                                                              (split
                                                                                                               -1)
                                                                                                              (("1"
                                                                                                                (flatten
                                                                                                                 -1)
                                                                                                                (("1"
                                                                                                                  (lemma
                                                                                                                   "integral.integrable_scal"
                                                                                                                   ("c"
                                                                                                                    "ALPHA*BETA"
                                                                                                                    "f"
                                                                                                                    "*[T](1 / (ALPHA * BETA), abs(f!1 * g!1))"))
                                                                                                                  (("1"
                                                                                                                    (case-replace
                                                                                                                     "(*[T]
                      (ALPHA * BETA,
                       *[T](1 / (ALPHA * BETA), abs(f!1 * g!1))))=abs(f!1 * g!1)")
                                                                                                                    (("1"
                                                                                                                      (rewrite
                                                                                                                       "real_expt_x1")
                                                                                                                      (("1"
                                                                                                                        (lemma
                                                                                                                         "integral.integral_scal"
                                                                                                                         ("c"
                                                                                                                          "1/(ALPHA*BETA)"
                                                                                                                          "f"
                                                                                                                          "abs(f!1*g!1)"))
                                                                                                                        (("1"
                                                                                                                          (replace
                                                                                                                           -1)
                                                                                                                          (("1"
                                                                                                                            (lemma
                                                                                                                             "integral_nonneg"
                                                                                                                             ("f"
                                                                                                                              "abs(f!1*g!1)"))
                                                                                                                            (("1"
                                                                                                                              (split
                                                                                                                               -1)
                                                                                                                              (("1"
                                                                                                                                (rewrite
                                                                                                                                 "real_props.abs_mult"
                                                                                                                                 -6)
                                                                                                                                (("1"
                                                                                                                                  (lemma
                                                                                                                                   "posreal_div_posreal_is_posreal"
                                                                                                                                   ("px"
                                                                                                                                    "1"
                                                                                                                                    "py"
                                                                                                                                    "ALPHA*BETA"))
                                                                                                                                  (("1"
                                                                                                                                    (expand
                                                                                                                                     "abs"
                                                                                                                                     -7
                                                                                                                                     1)
                                                                                                                                    (("1"
                                                                                                                                      (assert)
                                                                                                                                      (("1"
                                                                                                                                        (expand
                                                                                                                                         "abs"
                                                                                                                                         -7
                                                                                                                                         1)
                                                                                                                                        (("1"
                                                                                                                                          (case
                                                                                                                                           "integral(abs((+[T])
                        (*[T]((1 / p) * (1 / ALPHA ^ p), abs(f!1) ^ p),
                         *[T]((1 / q) * (1 / BETA ^ q), abs(g!1) ^ q)))) <=1")
                                                                                                                                          (("1"
                                                                                                                                            (assert)
                                                                                                                                            (("1"
                                                                                                                                              (lemma
                                                                                                                                               "div_mult_pos_le1"
                                                                                                                                               ("z"
                                                                                                                                                "integral.integral(abs(f!1 * g!1))"
                                                                                                                                                "py"
                                                                                                                                                "ALPHA * BETA"
                                                                                                                                                "x"
                                                                                                                                                "1"))
                                                                                                                                              (("1"
                                                                                                                                                (assert)
                                                                                                                                                nil
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil)
                                                                                                                                           ("2"
                                                                                                                                            (hide
                                                                                                                                             2
                                                                                                                                             -1
                                                                                                                                             -2
                                                                                                                                             -3
                                                                                                                                             -4
                                                                                                                                             -5
                                                                                                                                             -6
                                                                                                                                             -7)
                                                                                                                                            (("2"
                                                                                                                                              (case-replace
                                                                                                                                               "abs((+[T])
                       (*[T]((1 / p) * (1 / ALPHA ^ p), abs(f!1) ^ p),
                        *[T]((1 / q) * (1 / BETA ^ q), abs(g!1) ^ q)))= ((1/p)*1/(ALPHA^p))* abs(f!1)^p + ((1/q)*1/(BETA^q))* abs(g!1)^q")
                                                                                                                                              (("1"
                                                                                                                                                (hide
                                                                                                                                                 -1)
                                                                                                                                                (("1"
                                                                                                                                                  (lemma
                                                                                                                                                   "posreal_times_posreal_is_posreal"
                                                                                                                                                   ("px"
                                                                                                                                                    "1/p"
                                                                                                                                                    "py"
                                                                                                                                                    "1/(ALPHA^p)"))
                                                                                                                                                  (("1"
                                                                                                                                                    (lemma
                                                                                                                                                     "posreal_times_posreal_is_posreal"
                                                                                                                                                     ("px"
                                                                                                                                                      "1/q"
                                                                                                                                                      "py"
                                                                                                                                                      "1/(BETA^q)"))
                                                                                                                                                    (("1"
                                                                                                                                                      (name-replace
                                                                                                                                                       "AA"
                                                                                                                                                       "1 / p * (1 / (ALPHA ^ p))")
                                                                                                                                                      (("1"
                                                                                                                                                        (name-replace
                                                                                                                                                         "BB"
                                                                                                                                                         "1 / q * (1 / (BETA ^ q))")
                                                                                                                                                        (("1"
                                                                                                                                                          (lemma
                                                                                                                                                           "integral.integral_scal"
                                                                                                                                                           ("c"
                                                                                                                                                            "AA"
                                                                                                                                                            "f"
                                                                                                                                                            "abs(f!1)^p"))
                                                                                                                                                          (("1"
                                                                                                                                                            (lemma
                                                                                                                                                             "integral.integral_scal"
                                                                                                                                                             ("c"
                                                                                                                                                              "BB"
                                                                                                                                                              "f"
                                                                                                                                                              "abs(g!1)^q"))
                                                                                                                                                            (("1"
                                                                                                                                                              (replace
                                                                                                                                                               -7)
                                                                                                                                                              (("1"
                                                                                                                                                                (replace
                                                                                                                                                                 -8)
                                                                                                                                                                (("1"
                                                                                                                                                                  (lemma
                                                                                                                                                                   "integral.integral_add"
                                                                                                                                                                   ("f1"
                                                                                                                                                                    "AA * abs(f!1) ^ p"
                                                                                                                                                                    "f2"
                                                                                                                                                                    "BB * abs(g!1) ^ q"))
                                                                                                                                                                  (("1"
                                                                                                                                                                    (replace
                                                                                                                                                                     -2)
                                                                                                                                                                    (("1"
                                                                                                                                                                      (replace
                                                                                                                                                                       -3)
                                                                                                                                                                      (("1"
                                                                                                                                                                        (replace
                                                                                                                                                                         -1)
                                                                                                                                                                        (("1"
                                                                                                                                                                          (hide
                                                                                                                                                                           -1
                                                                                                                                                                           -2
                                                                                                                                                                           -3
                                                                                                                                                                           -20
                                                                                                                                                                           -21
                                                                                                                                                                           -8
                                                                                                                                                                           -9
                                                                                                                                                                           -6
                                                                                                                                                                           -7
                                                                                                                                                                           -14
                                                                                                                                                                           -15
                                                                                                                                                                           -10)
                                                                                                                                                                          (("1"
                                                                                                                                                                            (expand
                                                                                                                                                                             "AA")
                                                                                                                                                                            (("1"
                                                                                                                                                                              (expand
                                                                                                                                                                               "BB")
                                                                                                                                                                              (("1"
                                                                                                                                                                                (name-replace
                                                                                                                                                                                 "AP"
                                                                                                                                                                                 "ALPHA ^ p")
                                                                                                                                                                                (("1"
                                                                                                                                                                                  (rewrite
                                                                                                                                                                                   "div_cancel1")
                                                                                                                                                                                  (("1"
                                                                                                                                                                                    (typepred
                                                                                                                                                                                     "q")
                                                                                                                                                                                    (("1"
                                                                                                                                                                                      (assert)
                                                                                                                                                                                      (("1"
                                                                                                                                                                                        (assert)
                                                                                                                                                                                        (("1"
                                                                                                                                                                                          (rewrite
                                                                                                                                                                                           "one_times")
                                                                                                                                                                                          (("1"
                                                                                                                                                                                            (replace
                                                                                                                                                                                             -3)
                                                                                                                                                                                            (("1"
                                                                                                                                                                                              (assert)
                                                                                                                                                                                              nil
                                                                                                                                                                                              nil))
                                                                                                                                                                                            nil))
                                                                                                                                                                                          nil))
                                                                                                                                                                                        nil))
                                                                                                                                                                                      nil))
                                                                                                                                                                                    nil))
                                                                                                                                                                                  nil))
                                                                                                                                                                                nil))
                                                                                                                                                                              nil))
                                                                                                                                                                            nil))
                                                                                                                                                                          nil))
                                                                                                                                                                        nil))
                                                                                                                                                                      nil))
                                                                                                                                                                    nil))
                                                                                                                                                                  nil))
                                                                                                                                                                nil))
                                                                                                                                                              nil))
                                                                                                                                                            nil))
                                                                                                                                                          nil))
                                                                                                                                                        nil))
                                                                                                                                                      nil)
                                                                                                                                                     ("2"
                                                                                                                                                      (assert)
                                                                                                                                                      (("2"
                                                                                                                                                        (lemma
                                                                                                                                                         "posreal_div_posreal_is_posreal"
                                                                                                                                                         ("px"
                                                                                                                                                          "1"
                                                                                                                                                          "py"
                                                                                                                                                          "BETA^q"))
                                                                                                                                                        (("2"
                                                                                                                                                          (assert)
                                                                                                                                                          nil
                                                                                                                                                          nil))
                                                                                                                                                        nil))
                                                                                                                                                      nil))
                                                                                                                                                    nil)
                                                                                                                                                   ("2"
                                                                                                                                                    (lemma
                                                                                                                                                     "posreal_div_posreal_is_posreal"
                                                                                                                                                     ("px"
                                                                                                                                                      "1"
                                                                                                                                                      "py"
                                                                                                                                                      "ALPHA^p"))
                                                                                                                                                    (("2"
                                                                                                                                                      (assert)
                                                                                                                                                      nil
                                                                                                                                                      nil))
                                                                                                                                                    nil))
                                                                                                                                                  nil))
                                                                                                                                                nil)
                                                                                                                                               ("2"
                                                                                                                                                (hide
                                                                                                                                                 2)
                                                                                                                                                (("2"
                                                                                                                                                  (apply-extensionality
                                                                                                                                                   :hide?
                                                                                                                                                   t)
                                                                                                                                                  (("1"
                                                                                                                                                    (expand
                                                                                                                                                     "abs"
                                                                                                                                                     +)
                                                                                                                                                    (("1"
                                                                                                                                                      (expand
                                                                                                                                                       "+"
                                                                                                                                                       +)
                                                                                                                                                      (("1"
                                                                                                                                                        (expand
                                                                                                                                                         "*"
                                                                                                                                                         +)
                                                                                                                                                        (("1"
                                                                                                                                                          (expand
                                                                                                                                                           "^"
                                                                                                                                                           1
                                                                                                                                                           1)
                                                                                                                                                          (("1"
                                                                                                                                                            (expand
                                                                                                                                                             "^"
                                                                                                                                                             1
                                                                                                                                                             1)
                                                                                                                                                            (("1"
                                                                                                                                                              (expand
                                                                                                                                                               "^"
                                                                                                                                                               1
                                                                                                                                                               6)
--> --------------------

--> maximum size reached

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

¤ Dauer der Verarbeitung: 0.54 Sekunden  (vorverarbeitet)  ¤





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