Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/PVS/analysis/   (Beweissystem der NASA Version 6.0.9©)  Datei vom 28.9.2014 mit Größe 490 kB image not shown  

Quelle  weierstrass_approximation.prf

  Sprache: Lisp
 

(weierstrass_approximation
 (IMP_real_fun_on_compact_sets_TCC1 0
  (IMP_real_fun_on_compact_sets_TCC1-1 nil 3479658253
   ("" (lemma "real_metric_space") (("" (inst?) nil nil)) nil)
   ((fullset const-decl "set" sets nil) (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (real_metric_space formula-decl nil real_metric_space nil))
   nil))
 (Weierstrass_approx_combin1_TCC1 0
  (Weierstrass_approx_combin1_TCC1-1 nil 3479744181
   ("" (subtype-tcc) nil nilnil nil))
 (Weierstrass_approx_combin1_TCC2 0
  (Weierstrass_approx_combin1_TCC2-1 nil 3479744181
   ("" (subtype-tcc) nil nil) ((/= const-decl "boolean" notequal nil))
   nil))
 (Weierstrass_approx_combin1_TCC3 0
  (Weierstrass_approx_combin1_TCC3-1 nil 3479744181
   ("" (subtype-tcc) nil nil)
   ((boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (> const-decl "bool" 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)
    (>= const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   nil))
 (Weierstrass_approx_combin1 0
  (Weierstrass_approx_combin1-1 nil 3479744182
   ("" (skeep)
    ((""
      (case "FORALL (pp: nat):      FORALL (kz: nat): pp<=2 AND kz+1 > pp IMPLIES
                                        sigma(0,kz+1,LAMBDA (i:nat): IF i > kz+1 THEN 0 ELSE i^pp*C(kz+1,i)*(-1)^(kz+1-i) ENDIF) = 0")
      (("1" (inst - "p")
        (("1" (inst - "k-1") (("1" (assertnil nil)) nil)) nil)
       ("2" (hide 2)
        (("2" (hide -1)
          (("2" (hide -1)
            (("2" (induct "pp")
              (("1" (skosimp*)
                (("1" (assert)
                  (("1" (lemma "binomial_theorem")
                    (("1" (inst - "kz!1+1" "1" "-1")
                      (("1" (assert)
                        (("1" (expand "^" -1 1)
                          (("1" (expand "expt" -1 1)
                            (("1" (lemma "sigma_restrict_eq")
                              (("1"
                                (inst
                                 -
                                 "LAMBDA (i: nat):
                                 IF i > kz!1+1 THEN 0 ELSE C(kz!1+1, i) * 1 ^ i * (-1) ^ (kz!1+1 - i) ENDIF"
                                 "LAMBDA (i: nat):
                                IF i > kz!1+1 THEN 0 ELSE C(kz!1+1, i) * i ^ 0 * (-1) ^ (kz!1+1 - i) ENDIF"
                                 "kz!1+1"
                                 "0")
                                (("1"
                                  (assert)
                                  (("1"
                                    (hide 2)
                                    (("1"
                                      (decompose-equality)
                                      (("1"
                                        (expand "restrict")
                                        (("1"
                                          (lift-if)
                                          (("1"
                                            (ground)
                                            (("1"
                                              (rewrite "expt_1i")
                                              (("1"
                                                (expand "^" +)
                                                (("1"
                                                  (expand "expt" +)
                                                  (("1"
                                                    (propax)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (skosimp*)
                                        (("2" (assertnil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (skosimp*)
                                  (("2" (assertnil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (skosimp*)
                (("2" (lemma "sigma_split")
                  (("2"
                    (inst - "LAMBDA (i: nat):
                          IF i > kz!1 + 1 THEN 0
                          ELSE (i ^ (j!1 + 1)) * C(kz!1 + 1, i) * (-1) ^ (kz!1 + 1 - i)
                          ENDIF" "kz!1 + 1" "0" "0")
                    (("1" (assert)
                      (("1" (replace -1)
                        (("1" (hide -1)
                          (("1" (expand "sigma" 1 1)
                            (("1" (expand "sigma" 1 1)
                              (("1"
                                (case "0^(1+j!1) = 0")
                                (("1"
                                  (replace -1)
                                  (("1"
                                    (hide -1)
                                    (("1"
                                      (assert)
                                      (("1"
                                        (case
                                         "NOT (LAMBDA (i: nat):
                                            IF i > 1 + kz!1 THEN 0
                                            ELSE C(1 + kz!1, i) * (i ^ (1 + j!1)) * (-1) ^ (1 - i + kz!1)
                                            ENDIF) = (LAMBDA (i: nat):
                                            -(1+kz!1)*IF i > 1 + kz!1 or i = 0 THEN 0
                                            ELSE C(kz!1, i-1) * (i ^ j!1) * (-1) ^ (kz!1-i)
                                            ENDIF)")
                                        (("1"
                                          (hide 2)
                                          (("1"
                                            (hide -1)
                                            (("1"
                                              (decompose-equality)
                                              (("1"
                                                (lift-if)
                                                (("1"
                                                  (lift-if)
                                                  (("1"
                                                    (lift-if)
                                                    (("1"
                                                      (assert)
                                                      (("1"
                                                        (ground)
                                                        (("1"
                                                          (replace -1)
                                                          (("1"
                                                            (case
                                                             "0^(1+j!1) = 0")
                                                            (("1"
                                                              (assert)
                                                              nil
                                                              nil)
                                                             ("2"
                                                              (expand
                                                               "^"
                                                               1)
                                                              (("2"
                                                                (expand
                                                                 "expt"
                                                                 1)
                                                                (("2"
                                                                  (assert)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (expand "C")
                                                          (("2"
                                                            (case
                                                             "(-1) ^ (1 - x!1 + kz!1) = -(-1)^(kz!1-x!1)")
                                                            (("1"
                                                              (replace
                                                               -1)
                                                              (("1"
                                                                (hide
                                                                 -1)
                                                                (("1"
                                                                  (expand
                                                                   "factorial"
                                                                   2
                                                                   1)
                                                                  (("1"
                                                                    (expand
                                                                     "factorial"
                                                                     2
                                                                     3)
                                                                    (("1"
                                                                      (case
                                                                       "x!1^(1+j!1) = x!1*x!1^j!1")
                                                                      (("1"
                                                                        (replace
                                                                         -1)
                                                                        (("1"
                                                                          (hide
                                                                           -1)
                                                                          (("1"
                                                                            (assert)
                                                                            nil
                                                                            nil))
                                                                          nil))
                                                                        nil)
                                                                       ("2"
                                                                        (expand
                                                                         "^"
                                                                         1)
                                                                        (("2"
                                                                          (expand
                                                                           "expt"
                                                                           1
                                                                           1)
                                                                          (("2"
                                                                            (propax)
                                                                            nil
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil)
                                                             ("2"
                                                              (case
                                                               "kz!1 = x!1")
                                                              (("1"
                                                                (replace
                                                                 -1)
                                                                (("1"
                                                                  (assert)
                                                                  (("1"
                                                                    (expand
                                                                     "^"
                                                                     1)
                                                                    (("1"
                                                                      (expand
                                                                       "expt")
                                                                      (("1"
                                                                        (expand
                                                                         "expt")
                                                                        (("1"
                                                                          (assert)
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil)
                                                               ("2"
                                                                (case
                                                                 "kz!1 = x!1 -1")
                                                                (("1"
                                                                  (replace
                                                                   -1)
                                                                  (("1"
                                                                    (assert)
                                                                    (("1"
                                                                      (expand
                                                                       "^"
                                                                       2)
                                                                      (("1"
                                                                        (expand
                                                                         "expt"
                                                                         2)
                                                                        (("1"
                                                                          (expand
                                                                           "expt"
                                                                           2)
                                                                          (("1"
                                                                            (propax)
                                                                            nil
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil)
                                                                 ("2"
                                                                  (expand
                                                                   "^"
                                                                   3)
                                                                  (("2"
                                                                    (assert)
                                                                    (("2"
                                                                      (expand
                                                                       "expt"
                                                                       3
                                                                       1)
                                                                      (("2"
                                                                        (assert)
                                                                        nil
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (skosimp*)
                                                (("2"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil)
                                               ("3"
                                                (skosimp*)
                                                (("3"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil)
                                         ("2"
                                          (replace -1)
                                          (("2"
                                            (hide -1)
                                            (("2"
                                              (lemma "sigma_shift_T2")
                                              (("2"
                                                (inst
                                                 -
                                                 "(LAMBDA (i: nat):
                                       -(1 + kz!1) *
                                        IF i > 1 + kz!1 OR i = 0 THEN 0
                                        ELSE C(kz!1, i - 1) * (i ^ j!1) * (-1) ^ (kz!1 - i)
                                        ENDIF)"
                                                 "kz!1"
                                                 "0"
                                                 "1")
                                                (("1"
                                                  (assert)
                                                  (("1"
                                                    (replace -1)
                                                    (("1"
                                                      (hide -1)
                                                      (("1"
                                                        (lemma
                                                         "sigma_scal")
                                                        (("1"
                                                          (inst
                                                           -
                                                           "LAMBDA (i_1: nat):
                                           IF 1 + i_1 > 1 + kz!1 THEN 0
                                           ELSE (C(kz!1, i_1) * (-1) ^ (- i_1 + kz!1)) *
                                                 ((1 + i_1) ^ j!1)
                                           ENDIF"
                                                           "(1+kz!1)"
                                                           "kz!1"
                                                           "0")
                                                          (("1"
                                                            (lemma
                                                             "sigma_restrict_eq")
                                                            (("1"
                                                              (inst
                                                               -
                                                               "LAMBDA (i: nat):
                              (1 + kz!1) *
                               IF 1 + i > 1 + kz!1 THEN 0
                               ELSE (C(kz!1, i) * (-1) ^ (-i + kz!1)) * ((1 + i) ^ j!1)
                               ENDIF"
                                                               "LAMBDA (i_1: nat):
                              -(1 + kz!1) *
                               IF 1 + i_1 > 1 + kz!1 THEN 0
                               ELSE (C(kz!1, i_1) * (-1) ^ (-1 - i_1 + kz!1)) *
                                     ((1 + i_1) ^ j!1)
                               ENDIF"
                                                               "kz!1"
                                                               "0")
                                                              (("1"
                                                                (split
                                                                 -1)
                                                                (("1"
                                                                  (replace
                                                                   -1)
                                                                  (("1"
                                                                    (hide
                                                                     -1)
                                                                    (("1"
                                                                      (replace
                                                                       -1)
                                                                      (("1"
                                                                        (hide
                                                                         -1)
                                                                        (("1"
                                                                          (inst
                                                                           -
                                                                           "kz!1-1")
                                                                          (("1"
                                                                            (name
                                                                             "AA"
                                                                             "LAMBDA (i_1: nat):
                                        IF 1 + i_1 > 1 + kz!1 THEN 0
                                        ELSE (C(kz!1, i_1) * (-1) ^ (-i_1 + kz!1)) *
                                              ((1 + i_1) ^ j!1)
                                        ENDIF")
                                                                            (("1"
                                                                              (replace
                                                                               -1)
                                                                              (("1"
                                                                                (case
                                                                                 "sigma(0,kz!1,AA) = 0")
                                                                                (("1"
                                                                                  (assert)
                                                                                  nil
                                                                                  nil)
                                                                                 ("2"
                                                                                  (hide
                                                                                   2)
                                                                                  (("2"
                                                                                    (assert)
                                                                                    (("2"
                                                                                      (name
                                                                                       "BB"
                                                                                       "LAMBDA (i: nat):
                                           IF i > kz!1 THEN 0
                                           ELSE C(kz!1, i) * i ^ j!1 * (-1) ^ (kz!1 - i)
                                           ENDIF")
                                                                                      (("1"
                                                                                        (replace
                                                                                         -1)
                                                                                        (("1"
                                                                                          (case
                                                                                           "j!1 = 0")
                                                                                          (("1"
                                                                                            (replace
                                                                                             -1)
                                                                                            (("1"
                                                                                              (assert)
                                                                                              (("1"
                                                                                                (case
                                                                                                 "AA = BB")
                                                                                                (("1"
                                                                                                  (assert)
                                                                                                  nil
                                                                                                  nil)
                                                                                                 ("2"
                                                                                                  (hide
                                                                                                   2)
                                                                                                  (("2"
                                                                                                    (decompose-equality)
                                                                                                    (("2"
                                                                                                      (expand
                                                                                                       "AA"
                                                                                                       +)
                                                                                                      (("2"
                                                                                                        (expand
                                                                                                         "BB"
                                                                                                         +)
                                                                                                        (("2"
                                                                                                          (lift-if)
                                                                                                          (("2"
                                                                                                            (lift-if)
                                                                                                            (("2"
                                                                                                              (assert)
                                                                                                              (("2"
                                                                                                                (ground)
                                                                                                                (("2"
                                                                                                                  (replace
                                                                                                                   -1)
                                                                                                                  (("2"
                                                                                                                    (expand
                                                                                                                     "^"
                                                                                                                     2)
                                                                                                                    (("2"
                                                                                                                      (expand
                                                                                                                       "expt"
                                                                                                                       2
                                                                                                                       2)
                                                                                                                      (("2"
                                                                                                                        (expand
                                                                                                                         "expt"
                                                                                                                         2
                                                                                                                         3)
                                                                                                                        (("2"
                                                                                                                          (assert)
                                                                                                                          nil
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil)
                                                                                           ("2"
                                                                                            (case
                                                                                             "j!1 = 1")
                                                                                            (("1"
                                                                                              (replace
                                                                                               -1)
                                                                                              (("1"
                                                                                                (hide
                                                                                                 1)
                                                                                                (("1"
                                                                                                  (hide
                                                                                                   -1)
                                                                                                  (("1"
                                                                                                    (expand
                                                                                                     "^"
                                                                                                     -1
                                                                                                     1)
                                                                                                    (("1"
                                                                                                      (expand
                                                                                                       "expt"
                                                                                                       -1)
                                                                                                      (("1"
                                                                                                        (expand
                                                                                                         "expt"
                                                                                                         -1)
                                                                                                        (("1"
                                                                                                          (expand
                                                                                                           " ^"
                                                                                                           -2
                                                                                                           2)
                                                                                                          (("1"
                                                                                                            (expand
                                                                                                             "expt"
                                                                                                             -2)
                                                                                                            (("1"
                                                                                                              (expand
                                                                                                               "expt"
                                                                                                               -2)
                                                                                                              (("1"
                                                                                                                (name
                                                                                                                 "CC"
                                                                                                                 "(LAMBDA (i_1: nat):
                                            IF 1 + i_1 > 1 + kz!1 THEN 0
                                            ELSE C(kz!1, i_1) * (-1) ^ (-i_1 + kz!1)
                                            ENDIF)")
                                                                                                                (("1"
                                                                                                                  (case
                                                                                                                   "AA = BB + CC")
                                                                                                                  (("1"
                                                                                                                    (replace
                                                                                                                     -1)
                                                                                                                    (("1"
                                                                                                                      (hide
                                                                                                                       -1)
                                                                                                                      (("1"
                                                                                                                        (lemma
                                                                                                                         "sigma_sum")
                                                                                                                        (("1"
                                                                                                                          (inst
                                                                                                                           -
                                                                                                                           "BB"
                                                                                                                           "CC"
                                                                                                                           "kz!1"
                                                                                                                           "0")
                                                                                                                          (("1"
                                                                                                                            (expand
                                                                                                                             "+"
                                                                                                                             +)
                                                                                                                            (("1"
                                                                                                                              (replace
                                                                                                                               -1
                                                                                                                               :dir
                                                                                                                               rl)
                                                                                                                              (("1"
                                                                                                                                (hide
                                                                                                                                 -1)
                                                                                                                                (("1"
                                                                                                                                  (case
                                                                                                                                   "sigma(0,kz!1,CC) = 0")
                                                                                                                                  (("1"
                                                                                                                                    (assert)
                                                                                                                                    nil
                                                                                                                                    nil)
                                                                                                                                   ("2"
                                                                                                                                    (hide
                                                                                                                                     2)
                                                                                                                                    (("2"
                                                                                                                                      (replace
                                                                                                                                       -1
                                                                                                                                       +
                                                                                                                                       :dir
                                                                                                                                       rl)
                                                                                                                                      (("2"
                                                                                                                                        (hide
                                                                                                                                         -1)
                                                                                                                                        (("2"
                                                                                                                                          (hide
                                                                                                                                           -1)
                                                                                                                                          (("2"
                                                                                                                                            (hide
                                                                                                                                             -1)
                                                                                                                                            (("2"
                                                                                                                                              (lemma
                                                                                                                                               "binomial_theorem")
                                                                                                                                              (("2"
                                                                                                                                                (inst
                                                                                                                                                 -
                                                                                                                                                 "kz!1"
                                                                                                                                                 "1"
                                                                                                                                                 "-1")
                                                                                                                                                (("2"
                                                                                                                                                  (assert)
                                                                                                                                                  (("2"
                                                                                                                                                    (expand
                                                                                                                                                     "^"
                                                                                                                                                     -1
                                                                                                                                                     1)
                                                                                                                                                    (("2"
                                                                                                                                                      (expand
                                                                                                                                                       "expt"
                                                                                                                                                       -1)
                                                                                                                                                      (("2"
                                                                                                                                                        (case
                                                                                                                                                         "(LAMBDA (i: nat):
                                                 IF i > kz!1 THEN 0
                                                 ELSE C(kz!1, i) * 1 ^ i * (-1) ^ (kz!1 - i)
                                                 ENDIF) = (LAMBDA (i_1: nat):
                                                 IF 1 + i_1 > 1 + kz!1 THEN 0
                                                 ELSE C(kz!1, i_1) * (-1) ^ (-i_1 + kz!1)
                                                 ENDIF)")
                                                                                                                                                        (("1"
                                                                                                                                                          (assert)
                                                                                                                                                          nil
                                                                                                                                                          nil)
                                                                                                                                                         ("2"
                                                                                                                                                          (hide
                                                                                                                                                           2)
                                                                                                                                                          (("2"
                                                                                                                                                            (decompose-equality)
                                                                                                                                                            (("1"
                                                                                                                                                              (hide
                                                                                                                                                               -1)
                                                                                                                                                              (("1"
                                                                                                                                                                (lift-if)
                                                                                                                                                                (("1"
                                                                                                                                                                  (lift-if)
                                                                                                                                                                  (("1"
                                                                                                                                                                    (assert)
                                                                                                                                                                    (("1"
                                                                                                                                                                      (ground)
                                                                                                                                                                      (("1"
                                                                                                                                                                        (case
                                                                                                                                                                         "1^x!1 = 1")
                                                                                                                                                                        (("1"
                                                                                                                                                                          (assert)
                                                                                                                                                                          nil
                                                                                                                                                                          nil)
                                                                                                                                                                         ("2"
                                                                                                                                                                          (rewrite
                                                                                                                                                                           "expt_1i")
                                                                                                                                                                          nil
                                                                                                                                                                          nil))
                                                                                                                                                                        nil))
                                                                                                                                                                      nil))
                                                                                                                                                                    nil))
                                                                                                                                                                  nil))
                                                                                                                                                                nil))
                                                                                                                                                              nil)
                                                                                                                                                             ("2"
                                                                                                                                                              (skosimp*)
                                                                                                                                                              (("2"
                                                                                                                                                                (assert)
                                                                                                                                                                nil
                                                                                                                                                                nil))
                                                                                                                                                              nil)
                                                                                                                                                             ("3"
                                                                                                                                                              (skosimp*)
                                                                                                                                                              (("3"
                                                                                                                                                                (assert)
                                                                                                                                                                nil
                                                                                                                                                                nil))
                                                                                                                                                              nil))
                                                                                                                                                            nil))
                                                                                                                                                          nil))
                                                                                                                                                        nil))
                                                                                                                                                      nil))
                                                                                                                                                    nil))
                                                                                                                                                  nil))
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil)
                                                                                                                   ("2"
                                                                                                                    (replace
                                                                                                                     -1
                                                                                                                     +
                                                                                                                     :dir
                                                                                                                     rl)
                                                                                                                    (("2"
                                                                                                                      (hide
                                                                                                                       -1)
                                                                                                                      (("2"
                                                                                                                        (replace
                                                                                                                         -1
                                                                                                                         +
                                                                                                                         :dir
                                                                                                                         rl)
                                                                                                                        (("2"
                                                                                                                          (hide
                                                                                                                           -1)
                                                                                                                          (("2"
                                                                                                                            (replace
                                                                                                                             -1
                                                                                                                             +
                                                                                                                             :dir
                                                                                                                             rl)
                                                                                                                            (("2"
                                                                                                                              (hide
                                                                                                                               -1)
                                                                                                                              (("2"
                                                                                                                                (expand
                                                                                                                                 "+"
                                                                                                                                 1)
                                                                                                                                (("2"
                                                                                                                                  (decompose-equality)
                                                                                                                                  (("1"
                                                                                                                                    (hide
                                                                                                                                     2)
                                                                                                                                    (("1"
                                                                                                                                      (lift-if)
                                                                                                                                      (("1"
                                                                                                                                        (lift-if)
                                                                                                                                        (("1"
                                                                                                                                          (assert)
                                                                                                                                          nil
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil)
                                                                                                                                   ("2"
                                                                                                                                    (skosimp*)
                                                                                                                                    (("2"
                                                                                                                                      (assert)
                                                                                                                                      nil
                                                                                                                                      nil))
                                                                                                                                    nil)
                                                                                                                                   ("3"
                                                                                                                                    (skosimp*)
                                                                                                                                    (("3"
                                                                                                                                      (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)
                                                                                              nil
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil)
                                                                                       ("2"
                                                                                        (skosimp*)
                                                                                        (("2"
                                                                                          (assert)
                                                                                          nil
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil)
                                                                             ("2"
                                                                              (skosimp*)
                                                                              (("2"
                                                                                (assert)
                                                                                nil
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil)
                                                                 ("2"
                                                                  (hide
                                                                   2)
                                                                  (("2"
                                                                    (decompose-equality)
                                                                    (("1"
                                                                      (expand
                                                                       "restrict")
                                                                      (("1"
                                                                        (lift-if)
                                                                        (("1"
                                                                          (ground)
                                                                          (("1"
                                                                            (case
                                                                             "(-1) ^ (-1 - x!1 + kz!1) = -(-1) ^ (- x!1 + kz!1)")
                                                                            (("1"
                                                                              (assert)
                                                                              nil
                                                                              nil)
                                                                             ("2"
                                                                              (hide
                                                                               3)
                                                                              (("2"
                                                                                (case
                                                                                 "x!1 = kz!1")
                                                                                (("1"
                                                                                  (replace
                                                                                   -1)
                                                                                  (("1"
                                                                                    (assert)
                                                                                    (("1"
                                                                                      (expand
                                                                                       "^"
                                                                                       1)
                                                                                      (("1"
                                                                                        (expand
                                                                                         "expt"
                                                                                         1)
                                                                                        (("1"
                                                                                          (expand
                                                                                           "expt"
                                                                                           1)
                                                                                          (("1"
                                                                                            (propax)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil)
                                                                                 ("2"
                                                                                  (case
                                                                                   "-1 - x!1 + kz!1  >= 0")
                                                                                  (("1"
                                                                                    (expand
                                                                                     "^"
                                                                                     2)
                                                                                    (("1"
                                                                                      (assert)
                                                                                      (("1"
                                                                                        (expand
                                                                                         "expt"
                                                                                         2
                                                                                         2)
                                                                                        (("1"
                                                                                          (assert)
                                                                                          nil
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil)
                                                                                   ("2"
                                                                                    (assert)
                                                                                    nil
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil)
                                                                     ("2"
                                                                      (skosimp*)
                                                                      (("2"
                                                                        (assert)
                                                                        nil
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil)
                                                               ("2"
                                                                (skosimp*)
                                                                (("2"
                                                                  (assert)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil)
                                                           ("2"
                                                            (skosimp*)
                                                            (("2"
                                                              (assert)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (skosimp*)
                                                  (("2"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil)
                                         ("3"
                                          (skosimp*)
                                          (("3" (assertnil nil))
                                          nil)
                                         ("4"
                                          (skosimp*)
                                          (("4" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (expand "^" 1)
                                  (("2"
                                    (expand "expt" 1)
                                    (("2" (assertnil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (skosimp*) (("2" (assertnil nil)) nil))
                    nil))
                  nil))
                nil)
               ("3" (skosimp*) (("3" (assertnil nil)) nil))
              nil))
            nil))
          nil))
        nil)
       ("3" (skosimp*) (("3" (assertnil nil)) nil))
      nil))
    nil)
   ((- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (C const-decl "posnat" binomial "reals/")
    (posnat nonempty-type-eq-decl nil integers nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (^ const-decl "real" exponentiation nil)
    (/= const-decl "boolean" notequal nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (sigma def-decl "real" sigma "reals/")
    (T_high type-eq-decl nil sigma "reals/")
    (T_low type-eq-decl nil sigma "reals/")
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (> const-decl "bool" reals nil) (<= const-decl "bool" reals nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans 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)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (rat_times_rat_is_rat application-judgement "rat" rationals nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (int_plus_int_is_int application-judgement "int" integers 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)
    (sigma_restrict_eq formula-decl nil sigma "reals/")
    (nat_expt application-judgement "nat" exponentiation 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)
    (restrict const-decl "[T -> real]" sigma "reals/")
    (expt_1i formula-decl nil exponentiation nil)
    (int_expt application-judgement "int" exponentiation nil)
    (nzreal_expt application-judgement "nzreal" exponentiation nil)
    (nat_exp application-judgement "nat" exponentiation nil)
    (nzreal_exp application-judgement "nzreal" exponentiation nil)
    (rat_exp application-judgement "rat" exponentiation nil)
    (posint_exp application-judgement "posint" exponentiation nil)
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (kz!1 skolem-const-decl "nat" weierstrass_approximation nil)
    (expt def-decl "real" exponentiation nil)
    (int_exp application-judgement "int" exponentiation nil)
    (odd_plus_odd_is_even application-judgement "even_int" integers
     nil)
    (nzrat_times_nzrat_is_nzrat application-judgement "nzrat" rationals
     nil)
    (binomial_theorem formula-decl nil polynomials "reals/")
    (sigma_split formula-decl nil sigma "reals/")
    (even_minus_odd_is_odd application-judgement "odd_int" integers
     nil)
    (int_times_even_is_even application-judgement "even_int" integers
     nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (sigma_shift_T2 formula-decl nil sigma "reals/")
    (even_plus_odd_is_odd application-judgement "odd_int" integers nil)
    (rat nonempty-type-eq-decl nil rationals nil)
    (odd_plus_even_is_odd application-judgement "odd_int" integers nil)
    (AA skolem-const-decl "[nat -> rational]" weierstrass_approximation
     nil)
    (BB skolem-const-decl "[nat -> rat]" weierstrass_approximation nil)
    (+ const-decl "[T -> real]" real_fun_ops "reals/")
    (rat_plus_rat_is_rat application-judgement "rat" rationals nil)
    (sigma_sum formula-decl nil sigma "reals/")
    (posnat_expt application-judgement "posnat" exponentiation nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (sigma_scal formula-decl nil sigma "reals/")
    (minus_nzint_is_nzint application-judgement "nzint" integers nil)
    (factorial_1 formula-decl nil factorial "ints/")
    (nzint_times_nzint_is_nzint application-judgement "nzint" integers
     nil)
    (factorial_0 formula-decl nil factorial "ints/")
    (minus_int_is_int application-judgement "int" integers nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (factorial def-decl "posnat" factorial "ints/")
    (nzrat_div_nzrat_is_nzrat application-judgement "nzrat" rationals
     nil)
    (minus_nzrat_is_nzrat application-judgement "nzrat" rationals nil)
    (odd_minus_even_is_odd application-judgement "odd_int" integers
     nil)
    (kz!1 skolem-const-decl "nat" weierstrass_approximation nil))
   shostak))
 (Weierstrass_approximation_0_1 0
  (Weierstrass_approximation_0_1-1 nil 3479658386
   ("" (skeep)
    (("" (lemma "closed_intervals_compact")
      (("" (inst?)
        (("" (label "intcompact" -1)
          (("" (label "fcont" -2)
            (("" (label "approx" 1)
              (("" (case "empty?(closed_intv(0,1))")
                (("1" (expand "empty?" -1)
                  (("1" (inst - "1/2")
                    (("1" (expand "member") (("1" (propax) nil nil))
                      nil))
                    nil))
                  nil)
                 ("2" (label "nonempty" 1)
                  (("2"
                    (case "NOT EXISTS (M: posreal): FORALL (x: (closed_intv(0,1))): abs(f(x)) < M")
                    (("1" (hide "approx")
                      (("1" (lemma "cont_on_compact_max")
                        (("1" (inst?)
                          (("1" (assert)
                            (("1" (lemma "cont_on_compact_min")
                              (("1"
                                (inst?)
                                (("1"
                                  (assert)
                                  (("1"
                                    (skosimp*)
                                    (("1"
                                      (name
                                       "Mz"
                                       "max(abs(f(s!1))+1,abs(f(s!2))+1)")
                                      (("1"
                                        (inst + "Mz")
                                        (("1"
                                          (skosimp*)
                                          (("1"
                                            (expand "abs" +)
                                            (("1"
                                              (lift-if)
                                              (("1"
                                                (ground)
                                                (("1"
                                                  (inst - "x!1")
                                                  (("1"
                                                    (expand "abs" -2 1)
                                                    (("1"
                                                      (assert)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (inst - "x!1")
                                                  (("2"
                                                    (inst - "x!1")
                                                    (("2"
                                                      (expand
                                                       "abs"
                                                       -1
                                                       2)
                                                      (("2"
                                                        (assert)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (lemma "Heine")
                      (("2" (inst - "closed_intv(0,1)" "f")
                        (("2" (assert)
                          (("2" (label "Mdef" -2)
                            (("2" (label "funif" -1)
                              (("2"
                                (copy "funif")
                                (("2"
                                  (label "funif2" -1)
                                  (("2"
                                    (expand
                                     "uniformly_continuous?"
                                     "funif2")
                                    (("2"
                                      (skosimp*)
                                      (("2"
                                        (inst - "epsilon!1/2")
                                        (("2"
                                          (skosimp*)
                                          (("2"
                                            (expand
                                             "real_dist"
                                             "funif2")
                                            (("2"
                                              (case
                                               "EXISTS (N:above(5)): N>(M!1/(delta!1^2*epsilon!1))")
                                              (("1"
                                                (skosimp*)
                                                (("1"
                                                  (label "Ndef" -1)
                                                  (("1"
                                                    (name
                                                     "fbseq"
                                                     "(# index:= N!1,bern_seq := (LAMBDA (pj: upto(N!1)): f(pj/N!1)) #)")
                                                    (("1"
                                                      (label
                                                       "fbsname"
                                                       -1)
                                                      (("1"
                                                        (name
                                                         "BNf"
                                                         "Bern_poly(fbseq)")
                                                        (("1"
                                                          (label
                                                           "BNfname"
                                                           -1)
                                                          (("1"
                                                            (case
                                                             "FORALL (x:(closed_intv(0,1))): abs(BNf(x)-f(x)) <= (epsilon!1/2) + (M!1/(2*sq(delta!1)*N!1))")
                                                            (("1"
                                                              (lemma
                                                               "Bern_poly_inverse_def")
                                                              (("1"
                                                                (inst
                                                                 -
                                                                 "fbseq")
                                                                (("1"
                                                                  (inst
                                                                   +
                                                                   "Bern_poly_inverse(fbseq)"
                                                                   "fbseq`index")
                                                                  (("1"
                                                                    (skosimp*)
                                                                    (("1"
                                                                      (inst
                                                                       -
                                                                       "x!1")
                                                                      (("1"
                                                                        (case
                                                                         "(epsilon!1 / 2) + (M!1 / (2 * sq(delta!1) * N!1)) < epsilon!1")
                                                                        (("1"
                                                                          (assert)
                                                                          nil
                                                                          nil)
                                                                         ("2"
                                                                          (hide
                                                                           "approx")
                                                                          (("2"
                                                                            (hide
                                                                             -1)
                                                                            (("2"
                                                                              (hide
                                                                               -1)
                                                                              (("2"
                                                                                (case
                                                                                 "(M!1 / (2 * sq(delta!1) * N!1)) < epsilon!1/2")
                                                                                (("1"
                                                                                  (assert)
                                                                                  nil
                                                                                  nil)
                                                                                 ("2"
                                                                                  (cross-mult
                                                                                   1)
                                                                                  (("2"
                                                                                    (copy
                                                                                     "Ndef")
                                                                                    (("2"
                                                                                      (cross-mult
                                                                                       -1)
                                                                                      (("2"
                                                                                        (expand
                                                                                         "^")
                                                                                        (("2"
                                                                                          (expand
                                                                                           "expt")
                                                                                          (("2"
                                                                                            (expand
                                                                                             "expt")
                                                                                            (("2"
                                                                                              (expand
                                                                                               "expt")
                                                                                              (("2"
                                                                                                (mult-by
                                                                                                 -1
                                                                                                 "2")
                                                                                                (("2"
                                                                                                  (expand
                                                                                                   "sq")
                                                                                                  (("2"
                                                                                                    (assert)
                                                                                                    nil
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil)
                                                             ("2"
                                                              (hide
                                                               "approx")
                                                              (("2"
                                                                (label
                                                                 "bernapprox"
                                                                 1)
                                                                (("2"
                                                                  (case
                                                                   "FORALL (x,y:(closed_intv(0,1))): abs(BNf(x)-f(y)) <= (epsilon!1/2) + ((2*M!1)/sq(delta!1))*sq(x-y) + ((2*M!1)/sq(delta!1))*(1/N!1)*(x-sq(x))")
                                                                  (("1"
                                                                    (skosimp*)
                                                                    (("1"
                                                                      (inst
                                                                       -
                                                                       "x!1"
                                                                       "x!1")
                                                                      (("1"
                                                                        (assert)
                                                                        (("1"
                                                                          (case
                                                                           "x!1 - sq(x!1) <= 1/4")
                                                                          (("1"
                                                                            (mult-by
                                                                             -1
                                                                             "((2 * M!1) / sq(delta!1)) * sq(x!1 - x!1) +
                                                                                                                        ((2 * M!1) / sq(delta!1)) * (1 / N!1)")
                                                                            (("1"
                                                                              (assert)
                                                                              nil
                                                                              nil))
                                                                            nil)
                                                                           ("2"
                                                                            (hide-all-but
                                                                             1)
                                                                            (("2"
                                                                              (cross-mult
                                                                               1)
                                                                              (("2"
                                                                                (typepred
                                                                                 "sq(2*x!1-1)")
                                                                                (("2"
                                                                                  (expand
                                                                                   "sq")
                                                                                  (("2"
                                                                                    (assert)
                                                                                    nil
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil)
                                                                   ("2"
                                                                    (hide
                                                                     "bernapprox")
                                                                    (("2"
                                                                      (label
                                                                       "bapprox2"
                                                                       1)
                                                                      (("2"
                                                                        (skosimp*)
                                                                        (("2"
                                                                          (name
                                                                           "sqdiffseq"
                                                                           "(# index:= N!1, bern_seq := (LAMBDA (pj:upto(N!1)): sq((pj/N!1)-y!1)) #)")
                                                                          (("2"
                                                                            (label
                                                                             "sqdiffseqdef"
                                                                             -1)
                                                                            (("2"
                                                                              (name
                                                                               "BNdiff"
                                                                               "Bern_poly(sqdiffseq)")
                                                                              (("1"
                                                                                (label
                                                                                 "BNdiffname"
                                                                                 -1)
                                                                                (("1"
                                                                                  (case
                                                                                   "abs(BNf(x!1)-f(y!1)) <= ((2*M!1)/sq(delta!1))*BNdiff(x!1) + epsilon!1/2")
                                                                                  (("1"
                                                                                    (case
                                                                                     "BNdiff(x!1) = sq(x!1)+(1/N!1)*(x!1-sq(x!1))-2*y!1*x!1+sq(y!1)")
                                                                                    (("1"
                                                                                      (replace
                                                                                       -1
                                                                                       -2)
                                                                                      (("1"
                                                                                        (hide
                                                                                         -1)
                                                                                        (("1"
                                                                                          (case
                                                                                           "((2 * M!1) / sq(delta!1)) *
                                                                                                                                                                                       (sq(x!1) + (1 / N!1) * (x!1 - sq(x!1)) - 2 * y!1 * x!1 + sq(y!1))
                                                                                                                                                                                       + epsilon!1 / 2 = (epsilon!1 / 2) + ((2 * M!1) / sq(delta!1)) * sq(x!1 - y!1) +
                                                                                                                                                                                       ((2 * M!1) / sq(delta!1)) * (1 / N!1) * (x!1 - sq(x!1))")
                                                                                          (("1"
                                                                                            (assert)
                                                                                            nil
                                                                                            nil)
                                                                                           ("2"
                                                                                            (hide
                                                                                             "bapprox2")
                                                                                            (("2"
                                                                                              (expand
                                                                                               "sq"
                                                                                               1)
                                                                                              (("2"
                                                                                                (field
                                                                                                 1)
                                                                                                nil
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil)
                                                                                     ("2"
                                                                                      (lemma
                                                                                       "Bern_poly_inverse_def")
                                                                                      (("2"
                                                                                        (inst
                                                                                         -
                                                                                         "sqdiffseq")
                                                                                        (("2"
                                                                                          (hide
                                                                                           "bapprox2")
                                                                                          (("2"
                                                                                            (hide
                                                                                             "nonempty")
                                                                                            (("2"
                                                                                              (hide
                                                                                               "fcont")
                                                                                              (("2"
                                                                                                (hide
                                                                                                 "intcompact")
                                                                                                (("2"
                                                                                                  (hide
                                                                                                   "Mdef")
                                                                                                  (("2"
                                                                                                    (hide
                                                                                                     "funif")
                                                                                                    (("2"
                                                                                                      (hide
                                                                                                       "funif2")
                                                                                                      (("2"
                                                                                                        (hide
                                                                                                         "Ndef")
                                                                                                        (("2"
                                                                                                          (hide
                                                                                                           "fbsname")
                                                                                                          (("2"
                                                                                                            (hide
                                                                                                             "BNfname")
                                                                                                            (("2"
                                                                                                              (replace
                                                                                                               -1
                                                                                                               :dir
                                                                                                               rl)
                                                                                                              (("2"
                                                                                                                (replace
                                                                                                                 "BNdiffname"
                                                                                                                 :dir
                                                                                                                 rl)
                                                                                                                (("2"
                                                                                                                  (hide
                                                                                                                   -1)
                                                                                                                  (("2"
                                                                                                                    (hide
                                                                                                                     -1)
                                                                                                                    (("2"
                                                                                                                      (hide
                                                                                                                       -1)
                                                                                                                      (("2"
                                                                                                                        (expand
                                                                                                                         "Bern_poly_inverse")
                                                                                                                        (("2"
                                                                                                                          (expand
                                                                                                                           "polynomial")
                                                                                                                          (("2"
                                                                                                                            (replace
                                                                                                                             "sqdiffseqdef"
                                                                                                                             :dir
                                                                                                                             rl)
                                                                                                                            (("2"
                                                                                                                              (assert)
                                                                                                                              (("2"
                                                                                                                                (case
                                                                                                                                 "NOT (LAMBDA (i_1: nat):
                                                                                                                                                                                                 IF i_1 > N!1 THEN 0
                                                                                                                                                                                                 ELSE sigma(0, i_1,
                                                                                                                                                                                                            LAMBDA (i: nat):
                                                                                                                                                                                                              IF i > i_1 THEN 0
                                                                                                                                                                                                              ELSE sq((i / N!1) - y!1) * C(i_1, i) *
                                                                                                                                                                                                                    C(N!1, i_1)
                                                                                                                                                                                                                    * (-1) ^ (i_1 - i)
                                                                                                                                                                                                              ENDIF)
                                                                                                                                                                                                 ENDIF
                                                                                                                                                                                                  * (IF i_1 = 0 THEN 1 ELSE x!1 ^ i_1 ENDIF)) = (LAMBDA (i_1: nat):
                                                                                                                                                                                                 IF i_1 > 3 THEN 0
                                                                                                                                                                                                 ELSE sigma(0, i_1,
                                                                                                                                                                                                            LAMBDA (i: nat):
                                                                                                                                                                                                              IF i > i_1 THEN 0
                                                                                                                                                                                                              ELSE sq((i / N!1) - y!1) * C(i_1, i) *
                                                                                                                                                                                                                    C(N!1, i_1)
                                                                                                                                                                                                                    * (-1) ^ (i_1 - i)
                                                                                                                                                                                                              ENDIF)
                                                                                                                                                                                                 ENDIF
                                                                                                                                                                                                  * (IF i_1 = 0 THEN 1 ELSE x!1 ^ i_1 ENDIF))")
                                                                                                                                (("1"
                                                                                                                                  (hide
                                                                                                                                   2)
                                                                                                                                  (("1"
                                                                                                                                    (decompose-equality)
                                                                                                                                    (("1"
                                                                                                                                      (lift-if)
                                                                                                                                      (("1"
                                                                                                                                        (lift-if)
                                                                                                                                        (("1"
                                                                                                                                          (lift-if)
                                                                                                                                          (("1"
                                                                                                                                            (case
                                                                                                                                             "x!2 > 3 ")
                                                                                                                                            (("1"
                                                                                                                                              (assert)
                                                                                                                                              (("1"
                                                                                                                                                (ground)
                                                                                                                                                (("1"
                                                                                                                                                  (lemma
                                                                                                                                                   "Weierstrass_approx_combin1")
                                                                                                                                                  (("1"
                                                                                                                                                    (name
                                                                                                                                                     "AA"
                                                                                                                                                     "LAMBDA (i: nat):
                                                                                                                                                                                                                                                              IF i > x!2 THEN 0
                                                                                                                                                                                                                                                              ELSE sq((i / N!1) - y!1) * C(N!1, x!2) * C(x!2, i) *
                                                                                                                                                                                                                                                                    (-1) ^ (x!2 - i)
                                                                                                                                                                                                                                                              ENDIF")
                                                                                                                                                    (("1"
                                                                                                                                                      (replace
                                                                                                                                                       -1)
                                                                                                                                                      (("1"
                                                                                                                                                        (name
                                                                                                                                                         "BB"
                                                                                                                                                         "LAMBDA (i: nat):
                                                                                                                                                                                                                                                                              IF i > x!2 THEN 0
                                                                                                                                                                                                                                                                              ELSE sq(i / N!1) * C(N!1, x!2) * C(x!2, i) *
                                                                                                                                                                                                                                                                                    (-1) ^ (x!2 - i)
                                                                                                                                                                                                                                                                              ENDIF")
                                                                                                                                                        (("1"
                                                                                                                                                          (name
                                                                                                                                                           "CC"
                                                                                                                                                           "LAMBDA (i: nat):
                                                                                                                                                                                                                                                                              IF i > x!2 THEN 0
                                                                                                                                                                                                                                                                              ELSE -2*(i / N!1)*y!1 * C(N!1, x!2) * C(x!2, i) *
                                                                                                                                                                                                                                                                                    (-1) ^ (x!2 - i)
                                                                                                                                                                                                                                                                              ENDIF")
                                                                                                                                                          (("1"
                                                                                                                                                            (name
                                                                                                                                                             "DD"
                                                                                                                                                             "LAMBDA (i: nat):
                                                                                                                                                                                                                                                                              IF i > x!2 THEN 0
                                                                                                                                                                                                                                                                              ELSE sq(y!1) * C(N!1, x!2) * C(x!2, i) *
                                                                                                                                                                                                                                                                                    (-1) ^ (x!2 - i)
                                                                                                                                                                                                                                                                              ENDIF")
                                                                                                                                                            (("1"
                                                                                                                                                              (case
                                                                                                                                                               "NOT AA = BB+CC+DD")
                                                                                                                                                              (("1"
                                                                                                                                                                (decompose-equality)
                                                                                                                                                                (("1"
                                                                                                                                                                  (expand
                                                                                                                                                                   "+"
                                                                                                                                                                   1)
                                                                                                                                                                  (("1"
                                                                                                                                                                    (replace
                                                                                                                                                                     -1
                                                                                                                                                                     :dir
                                                                                                                                                                     rl)
                                                                                                                                                                    (("1"
                                                                                                                                                                      (hide
                                                                                                                                                                       -1)
                                                                                                                                                                      (("1"
                                                                                                                                                                        (replace
                                                                                                                                                                         -1
                                                                                                                                                                         :dir
                                                                                                                                                                         rl)
                                                                                                                                                                        (("1"
                                                                                                                                                                          (hide
                                                                                                                                                                           -1)
                                                                                                                                                                          (("1"
                                                                                                                                                                            (replace
                                                                                                                                                                             -1
                                                                                                                                                                             :dir
                                                                                                                                                                             rl)
                                                                                                                                                                            (("1"
                                                                                                                                                                              (hide
                                                                                                                                                                               -1)
                                                                                                                                                                              (("1"
                                                                                                                                                                                (replace
                                                                                                                                                                                 -1
                                                                                                                                                                                 :dir
                                                                                                                                                                                 rl)
                                                                                                                                                                                (("1"
                                                                                                                                                                                  (hide
                                                                                                                                                                                   -1)
                                                                                                                                                                                  (("1"
                                                                                                                                                                                    (assert)
                                                                                                                                                                                    (("1"
                                                                                                                                                                                      (lift-if)
                                                                                                                                                                                      (("1"
                                                                                                                                                                                        (assert)
                                                                                                                                                                                        (("1"
                                                                                                                                                                                          (ground)
                                                                                                                                                                                          (("1"
                                                                                                                                                                                            (hide
                                                                                                                                                                                             4)
                                                                                                                                                                                            (("1"
                                                                                                                                                                                              (grind
                                                                                                                                                                                               :exclude
                                                                                                                                                                                               "C")
                                                                                                                                                                                              nil
                                                                                                                                                                                              nil))
                                                                                                                                                                                            nil))
                                                                                                                                                                                          nil))
                                                                                                                                                                                        nil))
                                                                                                                                                                                      nil))
                                                                                                                                                                                    nil))
                                                                                                                                                                                  nil))
                                                                                                                                                                                nil))
                                                                                                                                                                              nil))
                                                                                                                                                                            nil))
                                                                                                                                                                          nil))
                                                                                                                                                                        nil))
                                                                                                                                                                      nil))
                                                                                                                                                                    nil))
                                                                                                                                                                  nil))
                                                                                                                                                                nil)
                                                                                                                                                               ("2"
                                                                                                                                                                (replace
                                                                                                                                                                 -1)
                                                                                                                                                                (("2"
                                                                                                                                                                  (lemma
                                                                                                                                                                   "sigma_sum")
                                                                                                                                                                  (("2"
                                                                                                                                                                    (inst
                                                                                                                                                                     -
                                                                                                                                                                     "BB+CC"
                                                                                                                                                                     "DD"
                                                                                                                                                                     "x!2"
                                                                                                                                                                     "0")
                                                                                                                                                                    (("2"
                                                                                                                                                                      (expand
                                                                                                                                                                       "+"
                                                                                                                                                                       +)
                                                                                                                                                                      (("2"
                                                                                                                                                                        (expand
                                                                                                                                                                         "+"
                                                                                                                                                                         -1)
                                                                                                                                                                        (("2"
                                                                                                                                                                          (replace
                                                                                                                                                                           -1
                                                                                                                                                                           +
                                                                                                                                                                           :dir
                                                                                                                                                                           rl)
                                                                                                                                                                          (("2"
                                                                                                                                                                            (hide
                                                                                                                                                                             -1)
                                                                                                                                                                            (("2"
                                                                                                                                                                              (lemma
                                                                                                                                                                               "sigma_sum")
                                                                                                                                                                              (("2"
                                                                                                                                                                                (inst
                                                                                                                                                                                 -
                                                                                                                                                                                 "BB"
                                                                                                                                                                                 "CC"
                                                                                                                                                                                 "x!2"
                                                                                                                                                                                 "0")
                                                                                                                                                                                (("2"
                                                                                                                                                                                  (replace
                                                                                                                                                                                   -1
                                                                                                                                                                                   :dir
                                                                                                                                                                                   rl)
                                                                                                                                                                                  (("2"
                                                                                                                                                                                    (hide
                                                                                                                                                                                     -1)
                                                                                                                                                                                    (("2"
                                                                                                                                                                                      (case
                                                                                                                                                                                       "sigma(0,x!2,DD) = 0 AND sigma(0,x!2,BB) = 0 AND sigma(0,x!2,CC) = 0")
                                                                                                                                                                                      (("1"
                                                                                                                                                                                        (flatten)
                                                                                                                                                                                        (("1"
                                                                                                                                                                                          (assert)
                                                                                                                                                                                          nil
                                                                                                                                                                                          nil))
                                                                                                                                                                                        nil)
                                                                                                                                                                                       ("2"
                                                                                                                                                                                        (hide
                                                                                                                                                                                         3)
                                                                                                                                                                                        (("2"
                                                                                                                                                                                          (hide
                                                                                                                                                                                           -1)
                                                                                                                                                                                          (("2"
                                                                                                                                                                                            (split
                                                                                                                                                                                             +)
                                                                                                                                                                                            (("1"
                                                                                                                                                                                              (replace
                                                                                                                                                                                               -1
                                                                                                                                                                                               :dir
                                                                                                                                                                                               rl)
                                                                                                                                                                                              (("1"
                                                                                                                                                                                                (hide
                                                                                                                                                                                                 -1)
                                                                                                                                                                                                (("1"
                                                                                                                                                                                                  (hide
                                                                                                                                                                                                   -1)
                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                    (hide
                                                                                                                                                                                                     -1)
                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                      (hide
                                                                                                                                                                                                       -1)
                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                        (inst
                                                                                                                                                                                                         -
                                                                                                                                                                                                         "x!2"
                                                                                                                                                                                                         "0")
                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                          (assert)
                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                            (mult-by
                                                                                                                                                                                                             -1
                                                                                                                                                                                                             "sq(y!1)*C(N!1,x!2)")
                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                              (lemma
                                                                                                                                                                                                               "sigma_scal")
                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                (inst
                                                                                                                                                                                                                 -
                                                                                                                                                                                                                 "LAMBDA (i: nat):
                                                                                                                                                                                                                            IF i > x!2 THEN 0
                                                                                                                                                                                                                            ELSE C(x!2, i) * i ^ 0 * (-1) ^ (x!2 - i)
                                                                                                                                                                                                                            ENDIF"
                                                                                                                                                                                                                 "sq(y!1)*C(N!1,x!2)"
                                                                                                                                                                                                                 "x!2"
                                                                                                                                                                                                                 "0")
                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                  (replace
                                                                                                                                                                                                                   -1
                                                                                                                                                                                                                   :dir
                                                                                                                                                                                                                   rl)
                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                    (hide
                                                                                                                                                                                                                     -1)
                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                      (case
                                                                                                                                                                                                                       "(LAMBDA (i_1: nat):
                                                                                                                                                                                                                                        sq(y!1) * C(N!1, x!2) *
                                                                                                                                                                                                                                         IF i_1 > x!2 THEN 0
                                                                                                                                                                                                                                         ELSE C(x!2, i_1) * i_1 ^ 0 * (-1) ^ (x!2 - i_1)
                                                                                                                                                                                                                                         ENDIF) = (LAMBDA (i: nat):
                                                                                                                                                                                                                                        IF i > x!2 THEN 0
                                                                                                                                                                                                                                        ELSE sq(y!1) * C(N!1, x!2) * C(x!2, i) * (-1) ^ (x!2 - i)
                                                                                                                                                                                                                                        ENDIF)")
                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                        (assert)
                                                                                                                                                                                                                        nil
                                                                                                                                                                                                                        nil)
                                                                                                                                                                                                                       ("2"
                                                                                                                                                                                                                        (hide
                                                                                                                                                                                                                         2)
                                                                                                                                                                                                                        (("2"
                                                                                                                                                                                                                          (decompose-equality)
                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                            (lift-if)
                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                              (ground)
                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                (case
                                                                                                                                                                                                                                 "x!3^0 = 1")
                                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                                  (assert)
                                                                                                                                                                                                                                  nil
                                                                                                                                                                                                                                  nil)
                                                                                                                                                                                                                                 ("2"
                                                                                                                                                                                                                                  (expand
                                                                                                                                                                                                                                   "^"
                                                                                                                                                                                                                                   1)
                                                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                                                    (expand
                                                                                                                                                                                                                                     "expt")
                                                                                                                                                                                                                                    (("2"
                                                                                                                                                                                                                                      (propax)
                                                                                                                                                                                                                                      nil
                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                            nil)
                                                                                                                                                                                                                           ("2"
                                                                                                                                                                                                                            (skosimp*)
                                                                                                                                                                                                                            (("2"
                                                                                                                                                                                                                              (assert)
                                                                                                                                                                                                                              nil
                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                  nil)
                                                                                                                                                                                                                 ("2"
                                                                                                                                                                                                                  (skosimp*)
                                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                                    (assert)
                                                                                                                                                                                                                    nil
                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                nil))
                                                                                                                                                                                                              nil)
                                                                                                                                                                                                             ("2"
                                                                                                                                                                                                              (skosimp*)
                                                                                                                                                                                                              (("2"
                                                                                                                                                                                                                (assert)
                                                                                                                                                                                                                nil
                                                                                                                                                                                                                nil))
                                                                                                                                                                                                              nil))
                                                                                                                                                                                                            nil))
                                                                                                                                                                                                          nil))
                                                                                                                                                                                                        nil))
                                                                                                                                                                                                      nil))
                                                                                                                                                                                                    nil))
                                                                                                                                                                                                  nil))
                                                                                                                                                                                                nil))
                                                                                                                                                                                              nil)
                                                                                                                                                                                             ("2"
                                                                                                                                                                                              (replace
                                                                                                                                                                                               -3
                                                                                                                                                                                               :dir
                                                                                                                                                                                               rl)
                                                                                                                                                                                              (("2"
                                                                                                                                                                                                (hide
                                                                                                                                                                                                 -1)
                                                                                                                                                                                                (("2"
                                                                                                                                                                                                  (hide
                                                                                                                                                                                                   -1)
                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                    (hide
                                                                                                                                                                                                     -1)
                                                                                                                                                                                                    (("2"
                                                                                                                                                                                                      (hide
                                                                                                                                                                                                       -1)
                                                                                                                                                                                                      (("2"
                                                                                                                                                                                                        (inst
                                                                                                                                                                                                         -
                                                                                                                                                                                                         "x!2"
                                                                                                                                                                                                         "2")
                                                                                                                                                                                                        (("2"
                                                                                                                                                                                                          (assert)
                                                                                                                                                                                                          (("2"
                                                                                                                                                                                                            (mult-by
                                                                                                                                                                                                             -1
                                                                                                                                                                                                             "(1/sq(N!1))*C(N!1,x!2)")
                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                              (lemma
                                                                                                                                                                                                               "sigma_scal")
                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                (inst
                                                                                                                                                                                                                 -
                                                                                                                                                                                                                 "LAMBDA (i: nat):
                                                                                                                                                                                                                            IF i > x!2 THEN 0
                                                                                                                                                                                                                            ELSE C(x!2, i) * i ^ 2 * (-1) ^ (x!2 - i)
                                                                                                                                                                                                                            ENDIF"
                                                                                                                                                                                                                 "((1 / sq(N!1)) * C(N!1, x!2))"
                                                                                                                                                                                                                 "x!2"
                                                                                                                                                                                                                 "0")
                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                  (replace
                                                                                                                                                                                                                   -1
                                                                                                                                                                                                                   :dir
                                                                                                                                                                                                                   rl)
                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                    (hide
                                                                                                                                                                                                                     -1)
                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                      (case
                                                                                                                                                                                                                       "(LAMBDA (i_1: nat):
                                                                                                                                                                                                                                        ((1 / sq(N!1)) * C(N!1, x!2)) *
                                                                                                                                                                                                                                         IF i_1 > x!2 THEN 0
                                                                                                                                                                                                                                         ELSE C(x!2, i_1) * i_1 ^ 2 * (-1) ^ (x!2 - i_1)
                                                                                                                                                                                                                                         ENDIF) = (LAMBDA (i: nat):
                                                                                                                                                                                                                                        IF i > x!2 THEN 0
                                                                                                                                                                                                                                        ELSE sq(i / N!1) * C(N!1, x!2) * C(x!2, i) * (-1) ^ (x!2 - i)
                                                                                                                                                                                                                                        ENDIF)")
                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                        (assert)
                                                                                                                                                                                                                        nil
                                                                                                                                                                                                                        nil)
                                                                                                                                                                                                                       ("2"
                                                                                                                                                                                                                        (hide
                                                                                                                                                                                                                         2)
                                                                                                                                                                                                                        (("2"
                                                                                                                                                                                                                          (decompose-equality)
                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                            (lift-if)
                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                              (ground)
                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                (case
                                                                                                                                                                                                                                 "x!3^2 = sq(x!3)")
                                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                                  (replace
                                                                                                                                                                                                                                   -1)
                                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                                    (hide
                                                                                                                                                                                                                                     -1)
                                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                                      (hide
                                                                                                                                                                                                                                       -1)
                                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                                        (grind
                                                                                                                                                                                                                                         :exclude
                                                                                                                                                                                                                                         ("C"
                                                                                                                                                                                                                                          "^"))
                                                                                                                                                                                                                                        nil
                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                  nil)
                                                                                                                                                                                                                                 ("2"
                                                                                                                                                                                                                                  (hide-all-but
                                                                                                                                                                                                                                   1)
                                                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                                                    (grind)
                                                                                                                                                                                                                                    nil
                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                            nil)
                                                                                                                                                                                                                           ("2"
                                                                                                                                                                                                                            (skosimp*)
                                                                                                                                                                                                                            (("2"
                                                                                                                                                                                                                              (assert)
                                                                                                                                                                                                                              nil
                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                  nil)
                                                                                                                                                                                                                 ("2"
                                                                                                                                                                                                                  (skosimp*)
                                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                                    (assert)
                                                                                                                                                                                                                    nil
                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                nil))
                                                                                                                                                                                                              nil)
                                                                                                                                                                                                             ("2"
                                                                                                                                                                                                              (skosimp*)
                                                                                                                                                                                                              (("2"
                                                                                                                                                                                                                (assert)
                                                                                                                                                                                                                nil
                                                                                                                                                                                                                nil))
                                                                                                                                                                                                              nil))
                                                                                                                                                                                                            nil))
                                                                                                                                                                                                          nil))
                                                                                                                                                                                                        nil))
                                                                                                                                                                                                      nil))
                                                                                                                                                                                                    nil))
                                                                                                                                                                                                  nil))
                                                                                                                                                                                                nil))
                                                                                                                                                                                              nil)
                                                                                                                                                                                             ("3"
                                                                                                                                                                                              (replace
                                                                                                                                                                                               -2
                                                                                                                                                                                               :dir
                                                                                                                                                                                               rl)
                                                                                                                                                                                              (("3"
                                                                                                                                                                                                (hide
                                                                                                                                                                                                 -1)
                                                                                                                                                                                                (("3"
                                                                                                                                                                                                  (hide
                                                                                                                                                                                                   -1)
                                                                                                                                                                                                  (("3"
                                                                                                                                                                                                    (hide
                                                                                                                                                                                                     -1)
                                                                                                                                                                                                    (("3"
                                                                                                                                                                                                      (hide
                                                                                                                                                                                                       -1)
                                                                                                                                                                                                      (("3"
                                                                                                                                                                                                        (inst
                                                                                                                                                                                                         -
                                                                                                                                                                                                         "x!2"
                                                                                                                                                                                                         "1")
                                                                                                                                                                                                        (("3"
                                                                                                                                                                                                          (assert)
                                                                                                                                                                                                          (("3"
                                                                                                                                                                                                            (mult-by
                                                                                                                                                                                                             -1
                                                                                                                                                                                                             "(-2*y!1/N!1)*C(N!1,x!2)")
                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                              (lemma
                                                                                                                                                                                                               "sigma_scal")
                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                (inst
                                                                                                                                                                                                                 -
                                                                                                                                                                                                                 "(LAMBDA (i: nat):
                                                                                                                                                                                                                            IF i > x!2 THEN 0
                                                                                                                                                                                                                            ELSE C(x!2, i) * i ^ 1 * (-1) ^ (x!2 - i)
                                                                                                                                                                                                                            ENDIF)"
                                                                                                                                                                                                                 "((-2 * y!1 / N!1) * C(N!1, x!2))"
                                                                                                                                                                                                                 "x!2"
                                                                                                                                                                                                                 "0")
                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                  (replace
                                                                                                                                                                                                                   -1
                                                                                                                                                                                                                   :dir
                                                                                                                                                                                                                   rl)
                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                    (hide
                                                                                                                                                                                                                     -1)
                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                      (case
                                                                                                                                                                                                                       "(LAMBDA (i_1: nat):
                                                                                                                                                                                                                                        ((-2 * y!1 / N!1) * C(N!1, x!2)) *
                                                                                                                                                                                                                                         IF i_1 > x!2 THEN 0
                                                                                                                                                                                                                                         ELSE C(x!2, i_1) * i_1 ^ 1 * (-1) ^ (x!2 - i_1)
                                                                                                                                                                                                                                         ENDIF) = (LAMBDA (i: nat):
                                                                                                                                                                                                                                        IF i > x!2 THEN 0
                                                                                                                                                                                                                                        ELSE -2 *
                                                                                                                                                                                                                                              (C(N!1, x!2) * C(x!2, i) * (-1) ^ (x!2 - i) * (i / N!1)
                                                                                                                                                                                                                                                * y!1)
                                                                                                                                                                                                                                        ENDIF)")
                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                        (assert)
                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                          (grind-reals)
                                                                                                                                                                                                                          nil
                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                        nil)
                                                                                                                                                                                                                       ("2"
                                                                                                                                                                                                                        (hide
                                                                                                                                                                                                                         2)
                                                                                                                                                                                                                        (("2"
                                                                                                                                                                                                                          (decompose-equality)
                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                            (lift-if)
                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                              (ground)
                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                (case
                                                                                                                                                                                                                                 "x!3^1 = x!3")
                                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                                  (replace
                                                                                                                                                                                                                                   -1)
                                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                                    (hide
                                                                                                                                                                                                                                     -1)
                                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                                      (assert)
                                                                                                                                                                                                                                      nil
                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                  nil)
                                                                                                                                                                                                                                 ("2"
                                                                                                                                                                                                                                  (expand
                                                                                                                                                                                                                                   "^"
                                                                                                                                                                                                                                   1)
                                                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                                                    (expand
                                                                                                                                                                                                                                     "expt")
                                                                                                                                                                                                                                    (("2"
                                                                                                                                                                                                                                      (expand
                                                                                                                                                                                                                                       "expt")
                                                                                                                                                                                                                                      (("2"
                                                                                                                                                                                                                                        (propax)
                                                                                                                                                                                                                                        nil
                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                            nil)
                                                                                                                                                                                                                           ("2"
                                                                                                                                                                                                                            (skosimp*)
                                                                                                                                                                                                                            (("2"
                                                                                                                                                                                                                              (assert)
                                                                                                                                                                                                                              nil
                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                  nil)
                                                                                                                                                                                                                 ("2"
                                                                                                                                                                                                                  (skosimp*)
                                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                                    (assert)
                                                                                                                                                                                                                    nil
                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                nil))
                                                                                                                                                                                                              nil)
                                                                                                                                                                                                             ("2"
                                                                                                                                                                                                              (skosimp*)
                                                                                                                                                                                                              (("2"
                                                                                                                                                                                                                (assert)
                                                                                                                                                                                                                nil
                                                                                                                                                                                                                nil))
                                                                                                                                                                                                              nil))
                                                                                                                                                                                                            nil))
                                                                                                                                                                                                          nil))
                                                                                                                                                                                                        nil))
                                                                                                                                                                                                      nil))
                                                                                                                                                                                                    nil))
                                                                                                                                                                                                  nil))
                                                                                                                                                                                                nil))
                                                                                                                                                                                              nil))
                                                                                                                                                                                            nil))
                                                                                                                                                                                          nil))
                                                                                                                                                                                        nil))
                                                                                                                                                                                      nil))
                                                                                                                                                                                    nil))
                                                                                                                                                                                  nil))
                                                                                                                                                                                nil))
                                                                                                                                                                              nil))
                                                                                                                                                                            nil))
                                                                                                                                                                          nil))
                                                                                                                                                                        nil))
                                                                                                                                                                      nil))
                                                                                                                                                                    nil))
                                                                                                                                                                  nil))
                                                                                                                                                                nil))
                                                                                                                                                              nil))
                                                                                                                                                            nil))
                                                                                                                                                          nil))
                                                                                                                                                        nil))
                                                                                                                                                      nil)
                                                                                                                                                     ("2"
                                                                                                                                                      (skosimp*)
                                                                                                                                                      (("2"
                                                                                                                                                        (assert)
                                                                                                                                                        nil
                                                                                                                                                        nil))
                                                                                                                                                      nil))
                                                                                                                                                    nil))
                                                                                                                                                  nil))
                                                                                                                                                nil))
                                                                                                                                              nil)
                                                                                                                                             ("2"
                                                                                                                                              (assert)
                                                                                                                                              nil
                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil)
                                                                                                                                     ("2"
                                                                                                                                      (skosimp*)
                                                                                                                                      (("2"
                                                                                                                                        (assert)
                                                                                                                                        nil
                                                                                                                                        nil))
                                                                                                                                      nil)
                                                                                                                                     ("3"
                                                                                                                                      (skosimp*)
                                                                                                                                      (("3"
                                                                                                                                        (assert)
                                                                                                                                        nil
                                                                                                                                        nil))
                                                                                                                                      nil)
                                                                                                                                     ("4"
                                                                                                                                      (skosimp*)
                                                                                                                                      (("4"
                                                                                                                                        (assert)
                                                                                                                                        nil
                                                                                                                                        nil))
                                                                                                                                      nil)
                                                                                                                                     ("5"
                                                                                                                                      (skosimp*)
                                                                                                                                      (("5"
                                                                                                                                        (assert)
                                                                                                                                        nil
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil))
                                                                                                                                  nil)
                                                                                                                                 ("2"
                                                                                                                                  (replace
                                                                                                                                   -1)
                                                                                                                                  (("2"
                                                                                                                                    (hide
                                                                                                                                     -1)
                                                                                                                                    (("2"
                                                                                                                                      (hide
                                                                                                                                       "sqdiffseqdef")
                                                                                                                                      (("2"
                                                                                                                                        (name
                                                                                                                                         "AA"
                                                                                                                                         "(LAMBDA (i_1: nat):
                                                                                                                           IF i_1 > 3 THEN 0
                                                                                                                           ELSE sigma(0, i_1,
                                                                                                                                      LAMBDA (i: nat):
                                                                                                                                        IF i > i_1 THEN 0
                                                                                                                                        ELSE sq((i / N!1) - y!1) * C(i_1, i) *
                                                                                                                                              C(N!1, i_1)
                                                                                                                                              * (-1) ^ (i_1 - i)
                                                                                                                                        ENDIF)
                                                                                                                           ENDIF
                                                                                                                            * (IF i_1 = 0 THEN 1 ELSE x!1 ^ i_1 ENDIF))")
                                                                                                                                        (("2"
                                                                                                                                          (replace
                                                                                                                                           -1)
                                                                                                                                          (("2"
                                                                                                                                            (lemma
                                                                                                                                             "sigma_split")
                                                                                                                                            (("2"
                                                                                                                                              (inst
                                                                                                                                               -
                                                                                                                                               "AA"
                                                                                                                                               "N!1"
                                                                                                                                               "0"
                                                                                                                                               "3")
                                                                                                                                              (("2"
                                                                                                                                                (assert)
                                                                                                                                                (("2"
                                                                                                                                                  (replace
                                                                                                                                                   -1)
                                                                                                                                                  (("2"
                                                                                                                                                    (hide
                                                                                                                                                     -1)
                                                                                                                                                    (("2"
                                                                                                                                                      (case
                                                                                                                                                       "sigma(4,N!1,AA) = 0")
                                                                                                                                                      (("1"
                                                                                                                                                        (replace
                                                                                                                                                         -1)
                                                                                                                                                        (("1"
                                                                                                                                                          (assert)
                                                                                                                                                          (("1"
                                                                                                                                                            (hide
                                                                                                                                                             -1)
                                                                                                                                                            (("1"
                                                                                                                                                              (replace
                                                                                                                                                               -1
                                                                                                                                                               :dir
                                                                                                                                                               rl)
                                                                                                                                                              (("1"
                                                                                                                                                                (hide
                                                                                                                                                                 -1)
                                                                                                                                                                (("1"
                                                                                                                                                                  (expand
                                                                                                                                                                   "sigma")
                                                                                                                                                                  (("1"
                                                                                                                                                                    (expand
                                                                                                                                                                     "sigma")
                                                                                                                                                                    (("1"
                                                                                                                                                                      (expand
                                                                                                                                                                       "sigma")
                                                                                                                                                                      (("1"
                                                                                                                                                                        (expand
                                                                                                                                                                         "sigma")
                                                                                                                                                                        (("1"
                                                                                                                                                                          (assert)
                                                                                                                                                                          (("1"
                                                                                                                                                                            (case
                                                                                                                                                                             "C(N!1,0) = 1 AND C(N!1,1) = N!1 AND C(N!1,2) = N!1*(N!1-1)/2 AND C(N!1,3) = N!1*(N!1-1)*(N!1-2)/6")
                                                                                                                                                                            (("1"
                                                                                                                                                                              (flatten)
                                                                                                                                                                              (("1"
                                                                                                                                                                                (replace
                                                                                                                                                                                 -1)
                                                                                                                                                                                (("1"
                                                                                                                                                                                  (hide
                                                                                                                                                                                   -1)
                                                                                                                                                                                  (("1"
                                                                                                                                                                                    (replace
                                                                                                                                                                                     -1)
                                                                                                                                                                                    (("1"
                                                                                                                                                                                      (hide
                                                                                                                                                                                       -1)
                                                                                                                                                                                      (("1"
                                                                                                                                                                                        (replace
                                                                                                                                                                                         -1)
                                                                                                                                                                                        (("1"
                                                                                                                                                                                          (hide
                                                                                                                                                                                           -1)
                                                                                                                                                                                          (("1"
                                                                                                                                                                                            (replace
                                                                                                                                                                                             -1)
                                                                                                                                                                                            (("1"
                                                                                                                                                                                              (hide
                                                                                                                                                                                               -1)
                                                                                                                                                                                              (("1"
                                                                                                                                                                                                (mult-by
                                                                                                                                                                                                 1
                                                                                                                                                                                                 "sq(N!1)")
                                                                                                                                                                                                (("1"
                                                                                                                                                                                                  (grind)
                                                                                                                                                                                                  nil
                                                                                                                                                                                                  nil))
                                                                                                                                                                                                nil))
                                                                                                                                                                                              nil))
                                                                                                                                                                                            nil))
                                                                                                                                                                                          nil))
                                                                                                                                                                                        nil))
                                                                                                                                                                                      nil))
                                                                                                                                                                                    nil))
                                                                                                                                                                                  nil))
                                                                                                                                                                                nil))
                                                                                                                                                                              nil)
                                                                                                                                                                             ("2"
                                                                                                                                                                              (hide
                                                                                                                                                                               2)
                                                                                                                                                                              (("2"
                                                                                                                                                                                (split
                                                                                                                                                                                 +)
                                                                                                                                                                                (("1"
                                                                                                                                                                                  (expand
                                                                                                                                                                                   "C")
                                                                                                                                                                                  (("1"
                                                                                                                                                                                    (assert)
                                                                                                                                                                                    nil
                                                                                                                                                                                    nil))
                                                                                                                                                                                  nil)
                                                                                                                                                                                 ("2"
                                                                                                                                                                                  (expand
                                                                                                                                                                                   "C")
                                                                                                                                                                                  (("2"
                                                                                                                                                                                    (expand
                                                                                                                                                                                     "factorial"
                                                                                                                                                                                     1
                                                                                                                                                                                     1)
                                                                                                                                                                                    (("2"
                                                                                                                                                                                      (assert)
                                                                                                                                                                                      nil
                                                                                                                                                                                      nil))
                                                                                                                                                                                    nil))
                                                                                                                                                                                  nil)
                                                                                                                                                                                 ("3"
                                                                                                                                                                                  (expand
                                                                                                                                                                                   "C")
                                                                                                                                                                                  (("3"
                                                                                                                                                                                    (expand
                                                                                                                                                                                     "factorial"
                                                                                                                                                                                     1
                                                                                                                                                                                     1)
                                                                                                                                                                                    (("3"
                                                                                                                                                                                      (expand
                                                                                                                                                                                       "factorial"
                                                                                                                                                                                       1
                                                                                                                                                                                       1)
                                                                                                                                                                                      (("3"
                                                                                                                                                                                        (expand
                                                                                                                                                                                         "factorial"
                                                                                                                                                                                         1
                                                                                                                                                                                         3)
                                                                                                                                                                                        (("3"
                                                                                                                                                                                          (assert)
                                                                                                                                                                                          nil
                                                                                                                                                                                          nil))
                                                                                                                                                                                        nil))
                                                                                                                                                                                      nil))
                                                                                                                                                                                    nil))
                                                                                                                                                                                  nil)
                                                                                                                                                                                 ("4"
                                                                                                                                                                                  (expand
                                                                                                                                                                                   "C")
                                                                                                                                                                                  (("4"
                                                                                                                                                                                    (expand
                                                                                                                                                                                     "factorial"
                                                                                                                                                                                     1
                                                                                                                                                                                     1)
                                                                                                                                                                                    (("4"
                                                                                                                                                                                      (expand
                                                                                                                                                                                       "factorial"
                                                                                                                                                                                       1
                                                                                                                                                                                       1)
                                                                                                                                                                                      (("4"
                                                                                                                                                                                        (expand
                                                                                                                                                                                         "factorial"
                                                                                                                                                                                         1
                                                                                                                                                                                         1)
                                                                                                                                                                                        (("4"
                                                                                                                                                                                          (expand
                                                                                                                                                                                           "factorial"
                                                                                                                                                                                           1
                                                                                                                                                                                           2)
                                                                                                                                                                                          (("4"
                                                                                                                                                                                            (assert)
                                                                                                                                                                                            (("4"
                                                                                                                                                                                              (case
                                                                                                                                                                                               "factorial(3) = 6")
                                                                                                                                                                                              (("1"
                                                                                                                                                                                                (replaces
                                                                                                                                                                                                 -1)
                                                                                                                                                                                                (("1"
                                                                                                                                                                                                  (assert)
                                                                                                                                                                                                  nil
                                                                                                                                                                                                  nil))
                                                                                                                                                                                                nil)
                                                                                                                                                                                               ("2"
                                                                                                                                                                                                (hide
                                                                                                                                                                                                 2)
                                                                                                                                                                                                (("2"
                                                                                                                                                                                                  (grind)
                                                                                                                                                                                                  nil
                                                                                                                                                                                                  nil))
                                                                                                                                                                                                nil))
                                                                                                                                                                                              nil))
                                                                                                                                                                                            nil))
                                                                                                                                                                                          nil))
                                                                                                                                                                                        nil))
                                                                                                                                                                                      nil))
                                                                                                                                                                                    nil))
                                                                                                                                                                                  nil))
                                                                                                                                                                                nil))
                                                                                                                                                                              nil))
                                                                                                                                                                            nil))
                                                                                                                                                                          nil))
                                                                                                                                                                        nil))
                                                                                                                                                                      nil))
                                                                                                                                                                    nil))
                                                                                                                                                                  nil))
                                                                                                                                                                nil))
                                                                                                                                                              nil))
                                                                                                                                                            nil))
                                                                                                                                                          nil))
                                                                                                                                                        nil)
                                                                                                                                                       ("2"
                                                                                                                                                        (lemma
                                                                                                                                                         "sigma_restrict_eq")
                                                                                                                                                        (("2"
                                                                                                                                                          (inst
                                                                                                                                                           -
                                                                                                                                                           "AA"
                                                                                                                                                           "LAMBDA (i:nat): 0"
                                                                                                                                                           "N!1"
                                                                                                                                                           "4")
                                                                                                                                                          (("2"
                                                                                                                                                            (split
                                                                                                                                                             -1)
                                                                                                                                                            (("1"
                                                                                                                                                              (lemma
                                                                                                                                                               "sigma_zero")
                                                                                                                                                              (("1"
                                                                                                                                                                (inst
                                                                                                                                                                 -
                                                                                                                                                                 "N!1"
                                                                                                                                                                 "4")
                                                                                                                                                                (("1"
                                                                                                                                                                  (assert)
                                                                                                                                                                  nil
                                                                                                                                                                  nil))
                                                                                                                                                                nil))
                                                                                                                                                              nil)
                                                                                                                                                             ("2"
                                                                                                                                                              (hide
                                                                                                                                                               3)
                                                                                                                                                              (("2"
                                                                                                                                                                (decompose-equality)
                                                                                                                                                                (("2"
                                                                                                                                                                  (expand
                                                                                                                                                                   "restrict")
                                                                                                                                                                  (("2"
                                                                                                                                                                    (lift-if)
                                                                                                                                                                    (("2"
                                                                                                                                                                      (hide
                                                                                                                                                                       2)
                                                                                                                                                                      (("2"
                                                                                                                                                                        (replace
                                                                                                                                                                         -1
                                                                                                                                                                         +
                                                                                                                                                                         :dir
                                                                                                                                                                         rl)
                                                                                                                                                                        (("2"
                                                                                                                                                                          (hide
                                                                                                                                                                           -1)
                                                                                                                                                                          (("2"
                                                                                                                                                                            (ground)
                                                                                                                                                                            nil
                                                                                                                                                                            nil))
                                                                                                                                                                          nil))
                                                                                                                                                                        nil))
                                                                                                                                                                      nil))
                                                                                                                                                                    nil))
                                                                                                                                                                  nil))
                                                                                                                                                                nil))
                                                                                                                                                              nil))
                                                                                                                                                            nil))
                                                                                                                                                          nil))
                                                                                                                                                        nil))
                                                                                                                                                      nil))
                                                                                                                                                    nil))
                                                                                                                                                  nil))
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil))
                                                                                                                                  nil)
                                                                                                                                 ("3"
                                                                                                                                  (skosimp*)
                                                                                                                                  (("3"
                                                                                                                                    (assert)
                                                                                                                                    nil
                                                                                                                                    nil))
                                                                                                                                  nil)
                                                                                                                                 ("4"
                                                                                                                                  (skosimp*)
                                                                                                                                  (("4"
                                                                                                                                    (assert)
                                                                                                                                    nil
                                                                                                                                    nil))
                                                                                                                                  nil)
                                                                                                                                 ("5"
                                                                                                                                  (skosimp*)
                                                                                                                                  (("5"
                                                                                                                                    (assert)
                                                                                                                                    nil
                                                                                                                                    nil))
                                                                                                                                  nil)
                                                                                                                                 ("6"
                                                                                                                                  (skosimp*)
                                                                                                                                  (("6"
                                                                                                                                    (assert)
                                                                                                                                    nil
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil)
                                                                                   ("2"
                                                                                    (name
                                                                                     "sqdiffzzz"
                                                                                     "(# index := N!1,bern_seq := (LAMBDA (pj: upto(N!1)): 2*M!1*sq(((pj/N!1)-y!1)/delta!1) + epsilon!1/2) #)")
                                                                                    (("2"
                                                                                      (label
                                                                                       "sqdiffzzzname"
                                                                                       -1)
                                                                                      (("2"
                                                                                        (name
                                                                                         "BNzzz"
                                                                                         "Bern_poly(sqdiffzzz)")
                                                                                        (("1"
                                                                                          (label
                                                                                           "BNzzzname"
                                                                                           -1)
                                                                                          (("1"
                                                                                            (case
                                                                                             "abs(BNf(x!1)-f(y!1)) <= BNzzz(x!1)")
                                                                                            (("1"
                                                                                              (case
                                                                                               "BNzzz(x!1) = ((2 * M!1) / sq(delta!1)) * BNdiff(x!1) + epsilon!1 / 2")
                                                                                              (("1"
                                                                                                (assert)
                                                                                                (("1"
                                                                                                  (replace
                                                                                                   -1)
                                                                                                  (("1"
                                                                                                    (propax)
                                                                                                    nil
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil)
                                                                                               ("2"
                                                                                                (assert)
                                                                                                (("2"
                                                                                                  (hide
                                                                                                   2)
                                                                                                  (("2"
                                                                                                    (hide
                                                                                                     "bapprox2")
                                                                                                    (("2"
                                                                                                      (hide
                                                                                                       -1)
                                                                                                      (("2"
                                                                                                        (replace
                                                                                                         "BNzzzname"
                                                                                                         1
                                                                                                         :dir
                                                                                                         rl)
                                                                                                        (("2"
                                                                                                          (replace
                                                                                                           "sqdiffzzzname"
                                                                                                           1
                                                                                                           :dir
                                                                                                           rl)
                                                                                                          (("2"
                                                                                                            (hide
                                                                                                             -1)
                                                                                                            (("2"
                                                                                                              (hide
                                                                                                               -1)
                                                                                                              (("2"
                                                                                                                (replace
                                                                                                                 "BNdiffname"
                                                                                                                 :dir
                                                                                                                 rl)
                                                                                                                (("2"
                                                                                                                  (replace
                                                                                                                   "sqdiffseqdef"
                                                                                                                   1
                                                                                                                   :dir
                                                                                                                   rl)
                                                                                                                  (("2"
                                                                                                                    (hide
                                                                                                                     "BNdiffname")
                                                                                                                    (("2"
                                                                                                                      (hide
                                                                                                                       "sqdiffseqdef")
                                                                                                                      (("2"
                                                                                                                        (hide
                                                                                                                         "BNfname")
                                                                                                                        (("2"
                                                                                                                          (hide
                                                                                                                           "fbsname")
                                                                                                                          (("2"
                                                                                                                            (hide
                                                                                                                             "fcont")
                                                                                                                            (("2"
                                                                                                                              (hide
                                                                                                                               "intcompact")
                                                                                                                              (("2"
                                                                                                                                (hide
                                                                                                                                 "funif")
                                                                                                                                (("2"
                                                                                                                                  (hide
                                                                                                                                   "nonempty")
                                                                                                                                  (("2"
                                                                                                                                    (expand
                                                                                                                                     "Bern_poly")
                                                                                                                                    (("2"
                                                                                                                                      (lemma
                                                                                                                                       "sigma_scal")
                                                                                                                                      (("2"
                                                                                                                                        (inst
                                                                                                                                         -
                                                                                                                                         "LAMBDA (i: nat):
                                                                 IF i > N!1 THEN 0
                                                                 ELSE sq((i / N!1) - y!1) * Bern(i, N!1)(x!1)
                                                                 ENDIF"
                                                                                                                                         "((2 * M!1) / sq(delta!1))"
                                                                                                                                         "N!1"
                                                                                                                                         "0")
                                                                                                                                        (("1"
                                                                                                                                          (replace
                                                                                                                                           -1
                                                                                                                                           :dir
                                                                                                                                           rl)
                                                                                                                                          (("1"
                                                                                                                                            (hide
                                                                                                                                             -1)
                                                                                                                                            (("1"
                                                                                                                                              (lemma
                                                                                                                                               "Bernstein_partition_of_unity")
                                                                                                                                              (("1"
                                                                                                                                                (inst
                                                                                                                                                 -
                                                                                                                                                 "N!1"
                                                                                                                                                 "x!1")
                                                                                                                                                (("1"
                                                                                                                                                  (mult-by
                                                                                                                                                   -1
                                                                                                                                                   "epsilon!1/2")
                                                                                                                                                  (("1"
                                                                                                                                                    (lemma
                                                                                                                                                     "sigma_scal")
                                                                                                                                                    (("1"
                                                                                                                                                      (inst
                                                                                                                                                       -
                                                                                                                                                       "LAMBDA (i: nat):
                                                                      IF i > N!1 THEN 0 ELSE Bern(i, N!1)(x!1) ENDIF"
                                                                                                                                                       "epsilon!1/2"
                                                                                                                                                       "N!1"
                                                                                                                                                       "0")
                                                                                                                                                      (("1"
                                                                                                                                                        (replace
                                                                                                                                                         -1
                                                                                                                                                         :dir
                                                                                                                                                         rl)
                                                                                                                                                        (("1"
                                                                                                                                                          (hide
                                                                                                                                                           -1)
                                                                                                                                                          (("1"
                                                                                                                                                            (name
                                                                                                                                                             "AA"
                                                                                                                                                             "LAMBDA (i: nat):
                                                                                    IF i > N!1 THEN 0
                                                                                    ELSE Bern(i, N!1)(x!1) * (epsilon!1 / 2) +
                                                                                          2 *
                                                                                           (Bern(i, N!1)(x!1) * sq(((i / N!1) - y!1) / delta!1) *
                                                                                             M!1)
                                                                                    ENDIF")
                                                                                                                                                            (("1"
                                                                                                                                                              (replace
                                                                                                                                                               -1)
                                                                                                                                                              (("1"
                                                                                                                                                                (name
                                                                                                                                                                 "BB"
                                                                                                                                                                 "LAMBDA (i_1: nat):
                                                                              epsilon!1 / 2 *
                                                                               IF i_1 > N!1 THEN 0 ELSE Bern(i_1, N!1)(x!1) ENDIF")
                                                                                                                                                                (("1"
                                                                                                                                                                  (replace
                                                                                                                                                                   -1)
                                                                                                                                                                  (("1"
                                                                                                                                                                    (hide
                                                                                                                                                                     -1)
                                                                                                                                                                    (("1"
                                                                                                                                                                      (name
                                                                                                                                                                       "CC"
                                                                                                                                                                       "LAMBDA (i_1: nat):
                                                                                ((2 * M!1) / sq(delta!1)) *
                                                                                 IF i_1 > N!1 THEN 0
                                                                                 ELSE sq((i_1 / N!1) - y!1) * Bern(i_1, N!1)(x!1)
                                                                                 ENDIF")
                                                                                                                                                                      (("1"
                                                                                                                                                                        (replace
                                                                                                                                                                         -1)
                                                                                                                                                                        (("1"
                                                                                                                                                                          (replace
                                                                                                                                                                           -3
                                                                                                                                                                           :dir
                                                                                                                                                                           rl)
                                                                                                                                                                          (("1"
                                                                                                                                                                            (case
                                                                                                                                                                             "AA = BB + CC")
                                                                                                                                                                            (("1"
                                                                                                                                                                              (replace
                                                                                                                                                                               -1)
                                                                                                                                                                              (("1"
                                                                                                                                                                                (hide
                                                                                                                                                                                 -1)
                                                                                                                                                                                (("1"
                                                                                                                                                                                  (expand
                                                                                                                                                                                   "+"
                                                                                                                                                                                   1
                                                                                                                                                                                   1)
                                                                                                                                                                                  (("1"
                                                                                                                                                                                    (lemma
                                                                                                                                                                                     "sigma_sum")
                                                                                                                                                                                    (("1"
                                                                                                                                                                                      (inst?)
                                                                                                                                                                                      (("1"
                                                                                                                                                                                        (assert)
                                                                                                                                                                                        nil
                                                                                                                                                                                        nil))
                                                                                                                                                                                      nil))
                                                                                                                                                                                    nil))
                                                                                                                                                                                  nil))
                                                                                                                                                                                nil))
                                                                                                                                                                              nil)
                                                                                                                                                                             ("2"
                                                                                                                                                                              (hide
                                                                                                                                                                               2)
                                                                                                                                                                              (("2"
                                                                                                                                                                                (decompose-equality)
                                                                                                                                                                                (("2"
                                                                                                                                                                                  (expand
                                                                                                                                                                                   "+"
                                                                                                                                                                                   1)
                                                                                                                                                                                  (("2"
                                                                                                                                                                                    (expand
                                                                                                                                                                                     "AA"
                                                                                                                                                                                     +)
                                                                                                                                                                                    (("2"
                                                                                                                                                                                      (expand
                                                                                                                                                                                       "BB"
                                                                                                                                                                                       +)
                                                                                                                                                                                      (("2"
                                                                                                                                                                                        (expand
                                                                                                                                                                                         "CC"
                                                                                                                                                                                         +)
                                                                                                                                                                                        (("2"
                                                                                                                                                                                          (lift-if)
                                                                                                                                                                                          (("2"
                                                                                                                                                                                            (assert)
                                                                                                                                                                                            (("2"
                                                                                                                                                                                              (ground)
                                                                                                                                                                                              (("2"
                                                                                                                                                                                                (hide
                                                                                                                                                                                                 -1)
                                                                                                                                                                                                (("2"
                                                                                                                                                                                                  (hide
                                                                                                                                                                                                   -1)
                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                    (hide
                                                                                                                                                                                                     -1)
                                                                                                                                                                                                    (("2"
                                                                                                                                                                                                      (hide
                                                                                                                                                                                                       "funif2")
                                                                                                                                                                                                      (("2"
                                                                                                                                                                                                        (hide
                                                                                                                                                                                                         "Mdef")
                                                                                                                                                                                                        (("2"
                                                                                                                                                                                                          (grind
                                                                                                                                                                                                           :exclude
                                                                                                                                                                                                           "Bern")
                                                                                                                                                                                                          nil
                                                                                                                                                                                                          nil))
                                                                                                                                                                                                        nil))
                                                                                                                                                                                                      nil))
                                                                                                                                                                                                    nil))
                                                                                                                                                                                                  nil))
                                                                                                                                                                                                nil))
                                                                                                                                                                                              nil))
                                                                                                                                                                                            nil))
                                                                                                                                                                                          nil))
                                                                                                                                                                                        nil))
                                                                                                                                                                                      nil))
                                                                                                                                                                                    nil))
                                                                                                                                                                                  nil))
                                                                                                                                                                                nil))
                                                                                                                                                                              nil))
                                                                                                                                                                            nil))
                                                                                                                                                                          nil))
                                                                                                                                                                        nil))
                                                                                                                                                                      nil))
                                                                                                                                                                    nil))
                                                                                                                                                                  nil))
                                                                                                                                                                nil))
                                                                                                                                                              nil)
                                                                                                                                                             ("2"
                                                                                                                                                              (skosimp*)
                                                                                                                                                              (("2"
                                                                                                                                                                (assert)
                                                                                                                                                                nil
                                                                                                                                                                nil))
                                                                                                                                                              nil))
                                                                                                                                                            nil))
                                                                                                                                                          nil))
                                                                                                                                                        nil)
                                                                                                                                                       ("2"
                                                                                                                                                        (skosimp*)
                                                                                                                                                        (("2"
                                                                                                                                                          (assert)
                                                                                                                                                          nil
                                                                                                                                                          nil))
                                                                                                                                                        nil))
                                                                                                                                                      nil))
                                                                                                                                                    nil)
                                                                                                                                                   ("2"
                                                                                                                                                    (skosimp*)
                                                                                                                                                    (("2"
                                                                                                                                                      (assert)
                                                                                                                                                      nil
                                                                                                                                                      nil))
                                                                                                                                                    nil))
                                                                                                                                                  nil))
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil)
                                                                                                                                         ("2"
                                                                                                                                          (skosimp*)
                                                                                                                                          (("2"
                                                                                                                                            (assert)
                                                                                                                                            nil
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil)
                                                                                             ("2"
                                                                                              (hide
                                                                                               2)
                                                                                              (("2"
                                                                                                (name
                                                                                                 "BNfdiffseq"
                                                                                                 "(# index:= N!1, bern_seq := (LAMBDA (pj:upto(N!1)): f(pj/N!1) - f(y!1)) #)")
                                                                                                (("2"
                                                                                                  (label
                                                                                                   "BNfdiffseq"
                                                                                                   -1)
                                                                                                  (("2"
                                                                                                    (name
                                                                                                     "BNfdiff"
                                                                                                     "Bern_poly(BNfdiffseq)")
                                                                                                    (("1"
                                                                                                      (label
                                                                                                       "BNfdiffname"
                                                                                                       -1)
                                                                                                      (("1"
                                                                                                        (label
                                                                                                         "BNfdiffseqname"
                                                                                                         -2)
                                                                                                        (("1"
                                                                                                          (case
                                                                                                           "abs(BNf(x!1)-f(y!1)) = abs(BNfdiff(x!1))")
                                                                                                          (("1"
                                                                                                            (case
                                                                                                             "abs(BNfdiff(x!1)) <= BNzzz(x!1)")
                                                                                                            (("1"
                                                                                                              (assert)
                                                                                                              nil
                                                                                                              nil)
                                                                                                             ("2"
                                                                                                              (hide
                                                                                                               2)
                                                                                                              (("2"
                                                                                                                (hide
                                                                                                                 -1)
                                                                                                                (("2"
                                                                                                                  (replace
                                                                                                                   "BNfdiffname"
                                                                                                                   1
                                                                                                                   :dir
                                                                                                                   rl)
                                                                                                                  (("2"
                                                                                                                    (replace
                                                                                                                     "BNzzzname"
                                                                                                                     1
                                                                                                                     :dir
                                                                                                                     rl)
                                                                                                                    (("2"
                                                                                                                      (expand
                                                                                                                       "BNfdiffseq"
                                                                                                                       1)
                                                                                                                      (("2"
                                                                                                                        (expand
                                                                                                                         "sqdiffzzz"
                                                                                                                         1)
                                                                                                                        (("2"
                                                                                                                          (expand
                                                                                                                           "Bern_poly"
                                                                                                                           1)
                                                                                                                          (("2"
                                                                                                                            (hide
                                                                                                                             "bapprox2")
                                                                                                                            (("2"
                                                                                                                              (hide
                                                                                                                               "nonempty")
                                                                                                                              (("2"
                                                                                                                                (hide
                                                                                                                                 "BNfname")
                                                                                                                                (("2"
                                                                                                                                  (hide
                                                                                                                                   "fbsname")
                                                                                                                                  (("2"
                                                                                                                                    (hide
                                                                                                                                     "sqdiffseqdef")
                                                                                                                                    (("2"
                                                                                                                                      (hide
                                                                                                                                       "sqdiffzzzname")
                                                                                                                                      (("2"
                                                                                                                                        (lemma
                                                                                                                                         "sigma_abs")
                                                                                                                                        (("2"
                                                                                                                                          (inst
                                                                                                                                           -
                                                                                                                                           "(LAMBDA (i: nat):
                                                                          IF i > N!1 THEN 0
                                                                          ELSE Bern(i, N!1)(x!1) * f(i / N!1) -
                                                                                Bern(i, N!1)(x!1) * f(y!1)
                                                                          ENDIF)"
                                                                                                                                           "N!1"
                                                                                                                                           "0")
                                                                                                                                          (("1"
                                                                                                                                            (lemma
                                                                                                                                             "sigma_le")
                                                                                                                                            (("1"
                                                                                                                                              (inst
                                                                                                                                               -
                                                                                                                                               "LAMBDA (n: nat):
                                                                           abs(IF n > N!1 THEN 0
                                                                               ELSE Bern(n, N!1)(x!1) * f(n / N!1) -
                                                                                     Bern(n, N!1)(x!1) * f(y!1)
                                                                               ENDIF)"
                                                                                                                                               "LAMBDA (i: nat):
                                                                           IF i > N!1 THEN 0
                                                                           ELSE Bern(i, N!1)(x!1) * (epsilon!1 / 2) +
                                                                                 2 *
                                                                                  (Bern(i, N!1)(x!1) * sq(((i / N!1) - y!1) / delta!1)
                                                                                    * M!1)
                                                                           ENDIF"
                                                                                                                                               "N!1"
                                                                                                                                               "0")
                                                                                                                                              (("1"
                                                                                                                                                (assert)
                                                                                                                                                (("1"
                                                                                                                                                  (hide
                                                                                                                                                   2)
                                                                                                                                                  (("1"
                                                                                                                                                    (hide
                                                                                                                                                     -1)
                                                                                                                                                    (("1"
                                                                                                                                                      (skosimp*)
                                                                                                                                                      (("1"
                                                                                                                                                        (case
                                                                                                                                                         "FORALL (x1,y1: (closed_intv(0,1))): abs(f(x1)-f(y1)) <= 2*M!1*sq((x1-y1)/delta!1) + epsilon!1/2")
                                                                                                                                                        (("1"
                                                                                                                                                          (inst
                                                                                                                                                           -
                                                                                                                                                           "n!1/N!1"
                                                                                                                                                           "y!1")
                                                                                                                                                          (("1"
                                                                                                                                                            (lemma
                                                                                                                                                             "Bern_nonnegative")
                                                                                                                                                            (("1"
                                                                                                                                                              (inst
                                                                                                                                                               -
                                                                                                                                                               "x!1")
                                                                                                                                                              (("1"
                                                                                                                                                                (assert)
                                                                                                                                                                (("1"
                                                                                                                                                                  (inst
                                                                                                                                                                   -
                                                                                                                                                                   "n!1"
                                                                                                                                                                   "N!1")
                                                                                                                                                                  (("1"
                                                                                                                                                                    (mult-by
                                                                                                                                                                     -2
                                                                                                                                                                     "Bern(n!1,N!1)(x!1)")
                                                                                                                                                                    (("1"
                                                                                                                                                                      (case
                                                                                                                                                                       "abs(Bern(n!1,N!1)(x!1)) = Bern(n!1,N!1)(x!1)")
                                                                                                                                                                      (("1"
                                                                                                                                                                        (replace
                                                                                                                                                                         -1
                                                                                                                                                                         -2
                                                                                                                                                                         :dir
                                                                                                                                                                         rl)
                                                                                                                                                                        (("1"
                                                                                                                                                                          (hide
                                                                                                                                                                           -1)
                                                                                                                                                                          (("1"
                                                                                                                                                                            (rewrite
                                                                                                                                                                             "abs_mult"
                                                                                                                                                                             :dir
                                                                                                                                                                             rl)
                                                                                                                                                                            (("1"
                                                                                                                                                                              (expand
                                                                                                                                                                               "abs"
                                                                                                                                                                               -1
                                                                                                                                                                               2)
                                                                                                                                                                              (("1"
                                                                                                                                                                                (assert)
                                                                                                                                                                                nil
                                                                                                                                                                                nil))
                                                                                                                                                                              nil))
                                                                                                                                                                            nil))
                                                                                                                                                                          nil))
                                                                                                                                                                        nil)
                                                                                                                                                                       ("2"
                                                                                                                                                                        (expand
                                                                                                                                                                         "abs"
                                                                                                                                                                         1)
                                                                                                                                                                        (("2"
                                                                                                                                                                          (assert)
                                                                                                                                                                          nil
                                                                                                                                                                          nil))
                                                                                                                                                                        nil))
                                                                                                                                                                      nil))
                                                                                                                                                                    nil))
                                                                                                                                                                  nil))
                                                                                                                                                                nil))
                                                                                                                                                              nil))
                                                                                                                                                            nil)
                                                                                                                                                           ("2"
                                                                                                                                                            (assert)
                                                                                                                                                            (("2"
                                                                                                                                                              (cross-mult
                                                                                                                                                               1)
                                                                                                                                                              nil
                                                                                                                                                              nil))
                                                                                                                                                            nil))
                                                                                                                                                          nil)
                                                                                                                                                         ("2"
                                                                                                                                                          (assert)
                                                                                                                                                          (("2"
                                                                                                                                                            (hide
                                                                                                                                                             2)
                                                                                                                                                            (("2"
                                                                                                                                                              (skosimp*)
                                                                                                                                                              (("2"
                                                                                                                                                                (case
                                                                                                                                                                 "abs(x1!1-y1!1) < delta!1")
                                                                                                                                                                (("1"
                                                                                                                                                                  (inst
                                                                                                                                                                   "funif2"
                                                                                                                                                                   "x1!1"
                                                                                                                                                                   "y1!1")
                                                                                                                                                                  (("1"
                                                                                                                                                                    (assert)
                                                                                                                                                                    nil
                                                                                                                                                                    nil))
                                                                                                                                                                  nil)
                                                                                                                                                                 ("2"
                                                                                                                                                                  (case
                                                                                                                                                                   "sq((x1!1 - y1!1)/delta!1) >= 1")
                                                                                                                                                                  (("1"
                                                                                                                                                                    (mult-by
                                                                                                                                                                     -1
                                                                                                                                                                     "2*M!1")
                                                                                                                                                                    (("1"
                                                                                                                                                                      (case
                                                                                                                                                                       "abs(f(x1!1)-f(y1!1)) <= 2*M!1")
                                                                                                                                                                      (("1"
                                                                                                                                                                        (assert)
                                                                                                                                                                        nil
                                                                                                                                                                        nil)
                                                                                                                                                                       ("2"
                                                                                                                                                                        (lemma
                                                                                                                                                                         "triangle")
                                                                                                                                                                        (("2"
                                                                                                                                                                          (inst
                                                                                                                                                                           -
                                                                                                                                                                           "f(x1!1)"
                                                                                                                                                                           "-f(y1!1)")
                                                                                                                                                                          (("2"
                                                                                                                                                                            (rewrite
                                                                                                                                                                             "abs_neg")
                                                                                                                                                                            (("2"
                                                                                                                                                                              (copy
                                                                                                                                                                               "Mdef")
                                                                                                                                                                              (("2"
                                                                                                                                                                                (inst-cp
                                                                                                                                                                                 -
                                                                                                                                                                                 "x1!1")
                                                                                                                                                                                (("2"
                                                                                                                                                                                  (inst
                                                                                                                                                                                   -
                                                                                                                                                                                   "y1!1")
                                                                                                                                                                                  (("2"
                                                                                                                                                                                    (assert)
                                                                                                                                                                                    nil
                                                                                                                                                                                    nil))
                                                                                                                                                                                  nil))
                                                                                                                                                                                nil))
                                                                                                                                                                              nil))
                                                                                                                                                                            nil))
                                                                                                                                                                          nil))
                                                                                                                                                                        nil))
                                                                                                                                                                      nil))
                                                                                                                                                                    nil)
                                                                                                                                                                   ("2"
                                                                                                                                                                    (lemma
                                                                                                                                                                     "sq_lt")
                                                                                                                                                                    (("2"
                                                                                                                                                                      (inst
                                                                                                                                                                       -
                                                                                                                                                                       "abs(x1!1-y1!1)"
                                                                                                                                                                       "delta!1")
                                                                                                                                                                      (("2"
                                                                                                                                                                        (rewrite
                                                                                                                                                                         "sq_abs")
                                                                                                                                                                        (("2"
                                                                                                                                                                          (assert)
                                                                                                                                                                          (("2"
                                                                                                                                                                            (rewrite
                                                                                                                                                                             "sq_div")
                                                                                                                                                                            (("2"
                                                                                                                                                                              (cross-mult
                                                                                                                                                                               2)
                                                                                                                                                                              nil
                                                                                                                                                                              nil))
                                                                                                                                                                            nil))
                                                                                                                                                                          nil))
                                                                                                                                                                        nil))
                                                                                                                                                                      nil))
                                                                                                                                                                    nil))
                                                                                                                                                                  nil))
                                                                                                                                                                nil))
                                                                                                                                                              nil))
                                                                                                                                                            nil))
                                                                                                                                                          nil))
                                                                                                                                                        nil))
                                                                                                                                                      nil))
                                                                                                                                                    nil))
                                                                                                                                                  nil))
                                                                                                                                                nil)
                                                                                                                                               ("2"
                                                                                                                                                (assert)
                                                                                                                                                (("2"
                                                                                                                                                  (skosimp*)
                                                                                                                                                  (("2"
                                                                                                                                                    (assert)
                                                                                                                                                    nil
                                                                                                                                                    nil))
                                                                                                                                                  nil))
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil)
                                                                                                                                           ("2"
                                                                                                                                            (skosimp*)
                                                                                                                                            (("2"
                                                                                                                                              (assert)
                                                                                                                                              nil
                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil)
                                                                                                           ("2"
                                                                                                            (case
                                                                                                             "BNf(x!1) - f(y!1) = BNfdiff(x!1)")
                                                                                                            (("1"
                                                                                                              (assert)
                                                                                                              nil
                                                                                                              nil)
                                                                                                             ("2"
                                                                                                              (hide
                                                                                                               2)
                                                                                                              (("2"
                                                                                                                (hide
                                                                                                                 2)
                                                                                                                (("2"
                                                                                                                  (hide
                                                                                                                   2)
                                                                                                                  (("2"
                                                                                                                    (hide
                                                                                                                     2)
                                                                                                                    (("2"
                                                                                                                      (replace
                                                                                                                       "BNfname"
                                                                                                                       1
                                                                                                                       :dir
                                                                                                                       rl)
                                                                                                                      (("2"
                                                                                                                        (replace
                                                                                                                         "BNfdiffname"
                                                                                                                         1
                                                                                                                         :dir
                                                                                                                         rl)
                                                                                                                        (("2"
                                                                                                                          (replace
                                                                                                                           "fbsname"
                                                                                                                           1
                                                                                                                           :dir
                                                                                                                           rl)
                                                                                                                          (("2"
                                                                                                                            (replace
                                                                                                                             "BNfdiffseqname"
                                                                                                                             1
                                                                                                                             :dir
                                                                                                                             rl)
                                                                                                                            (("2"
                                                                                                                              (expand
                                                                                                                               "Bern_poly"
                                                                                                                               1)
                                                                                                                              (("2"
                                                                                                                                (hide
                                                                                                                                 "fcont")
                                                                                                                                (("2"
                                                                                                                                  (hide
                                                                                                                                   "intcompact")
                                                                                                                                  (("2"
                                                                                                                                    (hide
                                                                                                                                     "Mdef")
                                                                                                                                    (("2"
                                                                                                                                      (hide
                                                                                                                                       "funif")
                                                                                                                                      (("2"
                                                                                                                                        (hide
                                                                                                                                         "funif2")
                                                                                                                                        (("2"
                                                                                                                                          (lemma
                                                                                                                                           "Bernstein_partition_of_unity")
                                                                                                                                          (("2"
                                                                                                                                            (inst
                                                                                                                                             -
                                                                                                                                             "N!1"
                                                                                                                                             "x!1")
                                                                                                                                            (("2"
                                                                                                                                              (mult-by
                                                                                                                                               -1
                                                                                                                                               "-f(y!1)")
                                                                                                                                              (("1"
                                                                                                                                                (lemma
                                                                                                                                                 "sigma_scal")
                                                                                                                                                (("1"
                                                                                                                                                  (inst
                                                                                                                                                   -
                                                                                                                                                   "LAMBDA (i: nat):
                                            IF i > N!1 THEN 0 ELSE Bern(i, N!1)(x!1) ENDIF"
                                                                                                                                                   "-f(y!1)"
                                                                                                                                                   "N!1"
                                                                                                                                                   "0")
                                                                                                                                                  (("1"
                                                                                                                                                    (replace
                                                                                                                                                     -1
                                                                                                                                                     :dir
                                                                                                                                                     rl)
                                                                                                                                                    (("1"
                                                                                                                                                      (hide
                                                                                                                                                       -1)
                                                                                                                                                      (("1"
                                                                                                                                                        (hide
                                                                                                                                                         "sqdiffzzzname")
                                                                                                                                                        (("1"
                                                                                                                                                          (hide
                                                                                                                                                           "BNfdiffseqname")
                                                                                                                                                          (("1"
                                                                                                                                                            (name
                                                                                                                                                             "AA"
                                                                                                                                                             "LAMBDA (i: nat):
                                               IF i > N!1 THEN 0
                                               ELSE Bern(i, N!1)(x!1) * f(i / N!1) -
                                                     Bern(i, N!1)(x!1) * f(y!1)
                                               ENDIF")
                                                                                                                                                            (("1"
                                                                                                                                                              (replace
                                                                                                                                                               -1)
                                                                                                                                                              (("1"
                                                                                                                                                                (name
                                                                                                                                                                 "BB"
                                                                                                                                                                 "LAMBDA (i: nat):
                                                IF i > N!1 THEN 0 ELSE f(i / N!1) * Bern(i, N!1)(x!1) ENDIF")
                                                                                                                                                                (("1"
                                                                                                                                                                  (replace
                                                                                                                                                                   -1)
                                                                                                                                                                  (("1"
                                                                                                                                                                    (name
                                                                                                                                                                     "CC"
                                                                                                                                                                     "LAMBDA (i_1: nat):
                                                -f(y!1) * IF i_1 > N!1 THEN 0 ELSE Bern(i_1, N!1)(x!1) ENDIF")
                                                                                                                                                                    (("1"
                                                                                                                                                                      (replace
                                                                                                                                                                       -1)
                                                                                                                                                                      (("1"
                                                                                                                                                                        (case
                                                                                                                                                                         "AA = BB + CC")
                                                                                                                                                                        (("1"
                                                                                                                                                                          (lemma
                                                                                                                                                                           "sigma_sum")
                                                                                                                                                                          (("1"
                                                                                                                                                                            (replace
                                                                                                                                                                             -2
                                                                                                                                                                             1)
                                                                                                                                                                            (("1"
                                                                                                                                                                              (expand
                                                                                                                                                                               "+"
                                                                                                                                                                               +)
                                                                                                                                                                              (("1"
                                                                                                                                                                                (inst
                                                                                                                                                                                 -
                                                                                                                                                                                 "BB"
                                                                                                                                                                                 "CC"
                                                                                                                                                                                 "N!1"
                                                                                                                                                                                 "0")
                                                                                                                                                                                (("1"
                                                                                                                                                                                  (assert)
                                                                                                                                                                                  nil
                                                                                                                                                                                  nil))
                                                                                                                                                                                nil))
                                                                                                                                                                              nil))
                                                                                                                                                                            nil))
                                                                                                                                                                          nil)
                                                                                                                                                                         ("2"
                                                                                                                                                                          (hide
                                                                                                                                                                           2)
                                                                                                                                                                          (("2"
                                                                                                                                                                            (decompose-equality)
                                                                                                                                                                            (("2"
                                                                                                                                                                              (expand
                                                                                                                                                                               "+"
                                                                                                                                                                               1)
                                                                                                                                                                              (("2"
                                                                                                                                                                                (expand
                                                                                                                                                                                 "AA"
                                                                                                                                                                                 +)
                                                                                                                                                                                (("2"
                                                                                                                                                                                  (expand
                                                                                                                                                                                   "BB"
                                                                                                                                                                                   +)
                                                                                                                                                                                  (("2"
                                                                                                                                                                                    (expand
                                                                                                                                                                                     "CC"
                                                                                                                                                                                     +)
                                                                                                                                                                                    (("2"
                                                                                                                                                                                      (lift-if)
                                                                                                                                                                                      (("2"
                                                                                                                                                                                        (ground)
                                                                                                                                                                                        nil
                                                                                                                                                                                        nil))
                                                                                                                                                                                      nil))
                                                                                                                                                                                    nil))
                                                                                                                                                                                  nil))
                                                                                                                                                                                nil))
                                                                                                                                                                              nil))
                                                                                                                                                                            nil))
                                                                                                                                                                          nil))
                                                                                                                                                                        nil))
                                                                                                                                                                      nil))
                                                                                                                                                                    nil))
                                                                                                                                                                  nil))
                                                                                                                                                                nil))
                                                                                                                                                              nil)
                                                                                                                                                             ("2"
                                                                                                                                                              (skosimp*)
                                                                                                                                                              (("2"
                                                                                                                                                                (assert)
                                                                                                                                                                nil
                                                                                                                                                                nil))
                                                                                                                                                              nil))
                                                                                                                                                            nil))
                                                                                                                                                          nil))
                                                                                                                                                        nil))
                                                                                                                                                      nil))
                                                                                                                                                    nil)
                                                                                                                                                   ("2"
                                                                                                                                                    (skosimp*)
                                                                                                                                                    (("2"
                                                                                                                                                      (assert)
                                                                                                                                                      nil
                                                                                                                                                      nil))
                                                                                                                                                    nil))
                                                                                                                                                  nil))
                                                                                                                                                nil)
                                                                                                                                               ("2"
                                                                                                                                                (skosimp*)
                                                                                                                                                (("2"
                                                                                                                                                  (assert)
                                                                                                                                                  nil
                                                                                                                                                  nil))
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil)
                                                                                                     ("2"
                                                                                                      (skosimp*)
                                                                                                      (("2"
                                                                                                        (ground)
                                                                                                        nil
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil)
                                                                                         ("2"
                                                                                          (skosimp*)
                                                                                          (("2"
                                                                                            (hide
                                                                                             2)
                                                                                            (("2"
                                                                                              (hide
                                                                                               2)
                                                                                              (("2"
                                                                                                (hide
                                                                                                 2)
                                                                                                (("2"
                                                                                                  (assert)
                                                                                                  (("2"
                                                                                                    (split
                                                                                                     +)
                                                                                                    (("1"
                                                                                                      (skosimp*)
                                                                                                      (("1"
                                                                                                        (ground)
                                                                                                        nil
                                                                                                        nil))
                                                                                                      nil)
                                                                                                     ("2"
                                                                                                      (skosimp*)
                                                                                                      (("2"
                                                                                                        (assert)
                                                                                                        nil
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil)
                                                                               ("2"
                                                                                (ground)
                                                                                (("2"
                                                                                  (skosimp*)
                                                                                  (("2"
                                                                                    (ground)
                                                                                    nil
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (lemma
                                                           "archimedean")
                                                          (("2"
                                                            (inst
                                                             -
                                                             "1/max(5,(M!1 / (delta!1 ^ 2 * epsilon!1)))")
                                                            (("2"
                                                              (skosimp*)
                                                              (("2"
                                                                (cross-mult
                                                                 -1)
                                                                (("2"
                                                                  (expand
                                                                   "max")
                                                                  (("2"
                                                                    (lift-if)
                                                                    (("2"
                                                                      (ground)
                                                                      nil
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (lemma "archimedean")
                                                (("2"
                                                  (inst
                                                   -
                                                   "1/max(5,(M!1 / (delta!1 ^ 2 * epsilon!1)))")
                                                  (("2"
                                                    (skosimp*)
                                                    (("2"
                                                      (inst + "n!1")
                                                      (("1"
                                                        (cross-mult -1)
                                                        (("1"
                                                          (expand
                                                           "max")
                                                          (("1"
                                                            (lift-if)
                                                            (("1"
                                                              (ground)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (cross-mult -1)
                                                        (("2"
                                                          (expand
                                                           "max")
                                                          (("2"
                                                            (lift-if)
                                                            (("2"
                                                              (ground)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((closed_intervals_compact formula-decl nil real_metric_space nil)
    (Heine formula-decl nil uniform_continuity nil)
    (posreal_times_posreal_is_posreal application-judgement "posreal"
     real_types 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)
    (above nonempty-type-eq-decl nil integers nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (^ const-decl "real" exponentiation nil)
    (archimedean formula-decl nil real_props nil)
    (odd_times_odd_is_odd application-judgement "odd_int" integers nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (both_sides_times_pos_le1_imp formula-decl nil extra_real_props
     nil)
    (div_mult_pos_le2 formula-decl nil real_props nil)
    (sq_0 formula-decl nil sq "reals/")
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (polynomial const-decl "[real -> real]" polynomials "reals/")
    (int_exp application-judgement "int" exponentiation nil)
    (sigma_0_neg formula-decl nil sigma_nat "reals/")
    (factorial_1 formula-decl nil factorial "ints/")
    (factorial_0 formula-decl nil factorial "ints/")
    (factorial def-decl "posnat" factorial "ints/")
    (posrat_times_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (both_sides_times1 formula-decl nil real_props nil)
    (rat_div_nzrat_is_rat application-judgement "rat" rationals nil)
    (restrict const-decl "[T -> real]" sigma "reals/")
    (sigma_zero formula-decl nil sigma "reals/")
    (sigma_nat application-judgement "nat" sigma_nat "reals/")
    (sigma_nnreal application-judgement "nnreal" sigma_nat "reals/")
    (sigma_restrict_eq formula-decl nil sigma "reals/")
    (even_minus_odd_is_odd application-judgement "odd_int" integers
     nil)
    (sigma_split formula-decl nil sigma "reals/")
    (Weierstrass_approx_combin1 formula-decl nil
     weierstrass_approximation nil)
    (+ const-decl "[T -> real]" real_fun_ops "reals/")
    (nzreal_expt application-judgement "nzreal" exponentiation nil)
    (int_expt application-judgement "int" exponentiation nil)
    (nzrat_div_nzrat_is_nzrat application-judgement "nzrat" rationals
     nil)
    (nzint_times_nzint_is_nzint application-judgement "nzint" integers
     nil)
    (nnrat_times_nnrat_is_nnrat application-judgement "nonneg_rat"
     rationals 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)
    (rat_exp application-judgement "rat" exponentiation nil)
    (nzrat_times_nzrat_is_nzrat application-judgement "nzrat" rationals
     nil)
    (rat_times_rat_is_rat application-judgement "rat" rationals nil)
    (sigma_sum formula-decl nil sigma "reals/")
    (minus_nzint_is_nzint application-judgement "nzint" integers nil)
    (minus_even_is_even application-judgement "even_int" integers nil)
    (nat_exp application-judgement "nat" exponentiation nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (times_div1 formula-decl nil real_props nil)
    (zero_times1 formula-decl nil real_props nil)
    (nat_expt application-judgement "nat" exponentiation nil)
    (sigma_scal formula-decl nil sigma "reals/")
    (x!2 skolem-const-decl "nat" weierstrass_approximation nil)
    (both_sides_times1_imp formula-decl nil extra_real_props nil)
    (N!1 skolem-const-decl "above(5)" weierstrass_approximation nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (C const-decl "posnat" binomial "reals/")
    (posnat nonempty-type-eq-decl nil integers nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (sigma def-decl "real" sigma "reals/")
    (T_high type-eq-decl nil sigma "reals/")
    (T_low type-eq-decl nil sigma "reals/")
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (BNfdiffseq skolem-const-decl
     "[# bern_seq: [upto(N!1) -> real], index: above(5) #]"
     weierstrass_approximation nil)
    (abs_neg formula-decl nil abs_lems "reals/")
    (triangle formula-decl nil real_props nil)
    (both_sides_times_pos_ge1_imp formula-decl nil extra_real_props
     nil)
    (div_mult_pos_ge1 formula-decl nil real_props nil)
    (nnreal_div_posreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (sq_div formula-decl nil sq "reals/")
    (sq_abs formula-decl nil sq "reals/")
    (sq_lt formula-decl nil sq "reals/")
    (subrange type-eq-decl nil integers nil)
    (n!1 skolem-const-decl "subrange(0, N!1)" weierstrass_approximation
     nil)
    (abs_mult formula-decl nil real_props nil)
    (x!1 skolem-const-decl "(LAMBDA (x: real): 0 <= x AND x <= 1)"
     weierstrass_approximation nil)
    (Bern_nonnegative formula-decl nil bernstein_polynomials "reals/")
    (div_mult_pos_le1 formula-decl nil real_props nil)
    (abs_nat formula-decl nil abs_lems "reals/")
    (sigma_le formula-decl nil sigma "reals/")
    (sigma_abs formula-decl nil sigma "reals/")
    (sqdiffzzz skolem-const-decl
     "[# bern_seq: [upto(N!1) -> posreal], index: above(5) #]"
     weierstrass_approximation nil)
    (BB skolem-const-decl "[nat -> numfield]" weierstrass_approximation
     nil)
    (CC skolem-const-decl "[nat -> real]" weierstrass_approximation
     nil)
    (AA skolem-const-decl "[nat -> numfield]" weierstrass_approximation
     nil)
    (Bernstein_partition_of_unity formula-decl nil
     bernstein_polynomials "reals/")
    (posreal_exp application-judgement "posreal" exponentiation nil)
    (BB skolem-const-decl "[nat -> real]" weierstrass_approximation
     nil)
    (CC skolem-const-decl "[nat -> real]" weierstrass_approximation
     nil)
    (AA skolem-const-decl "[nat -> numfield]" weierstrass_approximation
     nil)
    (Bern const-decl "real" bernstein_polynomials "reals/")
    (Bern_poly_inverse_def formula-decl nil bernstein_polynomials
     "reals/")
    (Bern_poly_inverse const-decl "sequence[real]"
     bernstein_polynomials "reals/")
    (sequence type-eq-decl nil sequences nil)
    (div_mult_pos_lt1 formula-decl nil real_props nil)
    (times_div2 formula-decl nil real_props nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (div_mult_pos_lt2 formula-decl nil real_props nil)
    (div_mult_pos_gt2 formula-decl nil extra_real_props nil)
    (expt def-decl "real" exponentiation nil)
    (both_sides_times_pos_gt1 formula-decl nil real_props nil)
    (nzreal_exp application-judgement "nzreal" exponentiation nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (sq_nz_pos application-judgement "posreal" sq "reals/")
    (sq const-decl "nonneg_real" sq "reals/")
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (posreal_plus_nnreal_is_posreal application-judgement "posreal"
     real_types nil)
    (IFF const-decl "[bool, bool -> bool]" booleans nil)
    (Bern_poly const-decl "[real -> real]" bernstein_polynomials
     "reals/")
    (Bernstein_Polynomial type-eq-decl nil bernstein_polynomials
     "reals/")
    (upto nonempty-type-eq-decl nil naturalnumbers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (nnrat_div_posrat_is_nnrat application-judgement "nonneg_rat"
     rationals nil)
    (n!1 skolem-const-decl "posnat" weierstrass_approximation nil)
    (posreal_div_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (uniformly_continuous? const-decl "bool" uniform_continuity nil)
    (cont_on_compact_min formula-decl nil real_fun_on_compact_sets nil)
    (nnreal_plus_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (posreal_max application-judgement
     "{z: posreal | z >= x AND z >= y}" real_defs nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (max const-decl "{p: real | p >= m AND p >= n}" real_defs nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_dist const-decl "nnreal" real_metric_space nil)
    (nnreal type-eq-decl nil real_types nil)
    (cont_on_compact_max formula-decl nil real_fun_on_compact_sets nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (>= const-decl "bool" reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (member const-decl "bool" sets nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (/= const-decl "boolean" notequal nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (set type-eq-decl nil sets nil) (empty? const-decl "bool" sets nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil) (< const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_minus_real_is_real application-judgement "real" reals nil))
   shostak)))


Messung V0.5 in Prozent
C=100 H=99 G=99

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