SSL derivatives_def.prf Sprache: Lisp

(derivatives_def
(derivable_continuous 0
  (derivable_continuous-1 nil 3253536989
   ("" (expand "derivable?")
    (("" (expand "convergent?")
      (("" (skosimp*) (("" (forward-chain "deriv_continuous") nil nil))
        nil))
      nil))
    nil)
   nil nil))
(sum_derivable 0
  (sum_derivable-1 nil 3253536989
   ("" (auto-rewrite "derivable?" "sum_NQ" "deriv_TCC")
    (("" (skosimp)
      (("" (assert)
        (("" (use "sum_fun_convergent[(A(x!1))]")
          (("" (assert) nil nil)) nil))
        nil))
      nil))
    nil)
   nil nil))
(neg_derivable 0
  (neg_derivable-1 nil 3442586144
   ("" (auto-rewrite "derivable?" "neg_NQ" "deriv_TCC")
    (("" (skosimp)
      (("" (assert)
        (("" (use "neg_fun_convergent[(A(x!1))]")
          (("" (assert) nil nil)) nil))
        nil))
      nil))
    nil)
   nil nil))
(diff_derivable 0
  (diff_derivable-1 nil 3253536989
   ("" (auto-rewrite "derivable?" "diff_NQ" "deriv_TCC")
    (("" (skosimp)
      (("" (assert)
        (("" (use "diff_fun_convergent[(A(x!1))]")
          (("" (assert) nil nil)) nil))
        nil))
      nil))
    nil)
   nil nil))
(prod_derivable 0
  (prod_derivable-1 nil 3253536989
   ("" (expand "derivable?")
    (("" (expand "convergent?")
      (("" (skosimp*)
        (("" (use "cnv_seq_prod_NQ")
          (("" (assert) (("" (inst?) nil nil)) nil)) nil))
        nil))
      nil))
    nil)
   nil nil))
(scal_derivable 0
  (scal_derivable-1 nil 3253536989
   ("" (auto-rewrite "derivable?" "scal_NQ" "deriv_TCC")
    (("" (skosimp)
      (("" (assert)
        (("" (use "scal_fun_convergent[(A(x!1))]")
          (("" (assert) nil nil)) nil))
        nil))
      nil))
    nil)
   nil nil))
(const_derivable 0
  (const_derivable-1 nil 3253536989
   ("" (skolem!)
    (("" (auto-rewrite "derivable?" "const_NQ" "deriv_TCC")
      (("" (assert)
        (("" (lemma "const_fun_convergent[(A(x!1))]")
          (("" (inst - "0" "0") nil nil)) nil))
        nil))
      nil))
    nil)
   ((const_fun_continuous application-judgement "continuous_fun"
     continuous_functions nil))
   nil))
(inv_derivable 0
  (inv_derivable-1 nil 3253536989
   ("" (expand "derivable?")
    (("" (expand "convergent?")
      (("" (skosimp*)
        (("" (use "cnv_seq_inv_NQ")
          (("" (assert) (("" (inst?) nil nil)) nil)) nil))
        nil))
      nil))
    nil)
   nil nil))
(div_derivable 0
  (div_derivable-1 nil 3253536989
   ("" (skosimp)
    (("" (rewrite "div_function[T]")
      (("" (rewrite "prod_derivable")
        (("" (rewrite "inv_derivable") nil nil)) nil))
      nil))
    nil)
   ((div_function formula-decl nil real_fun_ops "reals/")
    (/= const-decl "boolean" notequal nil)
    (nzreal nonempty-type-eq-decl nil reals nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (T_pred const-decl "[real -> boolean]" derivatives_def nil)
    (T formal-subtype-decl nil derivatives_def nil)
    (inv_derivable formula-decl nil derivatives_def nil)
    (/ const-decl "[T -> real]" real_fun_ops "reals/")
    (prod_derivable formula-decl nil derivatives_def nil))
   nil))
(identity_derivable 0
  (identity_derivable-1 nil 3253536989
   ("" (skolem!)
    (("" (auto-rewrite "derivable?" "identity_NQ" "deriv_TCC")
      (("" (assert)
        (("" (use "const_fun_convergent[(A(x!1))]" ("b" "1" "c" "0"))
          nil nil))
        nil))
      nil))
    nil)
   ((id_fun_continuous name-judgement "continuous_fun"
     continuous_functions nil)
    (const_fun_continuous application-judgement "continuous_fun"
     continuous_functions nil))
   nil))
(deriv_sum_TCC1 0
  (deriv_sum_TCC1-1 nil 3253536989
   ("" (skosimp) (("" (rewrite "sum_derivable") nil nil)) nil)
   ((sum_derivable formula-decl nil derivatives_def nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (T_pred const-decl "[real -> boolean]" derivatives_def nil)
    (T formal-subtype-decl nil derivatives_def nil))
   nil))
(deriv_sum 0
  (deriv_sum-1 nil 3253536989
   ("" (skosimp)
    ((""
      (auto-rewrite "deriv_TCC" "sum_derivable" "derivable?" "deriv"
                    "sum_NQ" "lim_sum_fun[(A(x!1))]")
      (("" (assert) nil nil)) nil))
    nil)
   nil nil))
(deriv_neg_TCC1 0
  (deriv_neg_TCC1-1 nil 3442582979
   ("" (lemma "neg_derivable") (("" (propax) nil nil)) nil)
   ((neg_derivable formula-decl nil derivatives_def nil)) shostak))
(deriv_neg 0
  (deriv_neg-2 nil 3442586421
   ("" (skosimp)
    ((""
      (auto-rewrite "deriv_TCC" "neg_derivable" "derivable?" "deriv"
                    "neg_NQ" "lim_neg_fun[(A(x!1))]")
      (("" (assert) nil nil)) nil))
    nil)
   nil nil)
  (deriv_neg-1 nil 3442586370
   (";;; Proof for formula derivatives_def.deriv_opposite" (skosimp)
    ((";;; Proof for formula derivatives_def.deriv_opposite"
      (auto-rewrite "deriv_TCC" "opposite_derivable" "derivable?"
                    "deriv" "opposite_NQ" "lim_opposite_fun[(A(x!1))]")
      ((";;; Proof for formula derivatives_def.deriv_opposite" (assert)
        nil))))
    "")
   nil nil))
(deriv_diff_TCC1 0
  (deriv_diff_TCC1-1 nil 3253536989
   ("" (skosimp) (("" (rewrite "diff_derivable") nil nil)) nil)
   ((diff_derivable formula-decl nil derivatives_def nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (T_pred const-decl "[real -> boolean]" derivatives_def nil)
    (T formal-subtype-decl nil derivatives_def nil))
   nil))
(deriv_diff 0
  (deriv_diff-1 nil 3253536989
   ("" (skosimp)
    ((""
      (auto-rewrite "deriv_TCC" "diff_derivable" "derivable?" "deriv"
                    "diff_NQ" "lim_diff_fun[(A(x!1))]")
      (("" (assert) nil nil)) nil))
    nil)
   nil nil))
(deriv_prod_TCC1 0
  (deriv_prod_TCC1-1 nil 3253536989
   ("" (skosimp) (("" (rewrite "prod_derivable") nil nil)) nil)
   ((prod_derivable formula-decl nil derivatives_def nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (T_pred const-decl "[real -> boolean]" derivatives_def nil)
    (T formal-subtype-decl nil derivatives_def nil))
   nil))
(deriv_prod 0
  (deriv_prod-1 nil 3253536989
   ("" (skosimp)
    (("" (use "prod_derivable")
      (("" (assert)
        ((""
          (auto-rewrite "deriv" "deriv_TCC" "derivable?"
                        ("lim_fun_def[(A(x!1))]"
                         "lim_fun_lemma[(A(x!1))]"))
          (("" (assert)
            ((""
              (use "cnv_seq_prod_NQ"
                   ("l1" "lim(NQ(f1!1, x!1), 0)" "l2"
                    "lim(NQ(f2!1, x!1), 0)"))
              (("" (ground) nil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   nil nil))
(deriv_const_TCC1 0
  (deriv_const_TCC1-1 nil 3253536989
   ("" (lemma "const_derivable") (("" (propax) nil nil)) nil)
   ((const_derivable formula-decl nil derivatives_def nil)) nil))
(deriv_const 0
  (deriv_const-1 nil 3253536989
   ("" (skosimp)
    ((""
      (auto-rewrite "deriv_TCC" "const_derivable" "derivable?" "deriv"
                    "const_NQ")
      (("" (assert)
        (("" (use "lim_const_fun[(A(x!1))]" ("b" "0" "c" "0")) nil
          nil))
        nil))
      nil))
    nil)
   ((const_fun_continuous application-judgement "continuous_fun"
     continuous_functions nil))
   nil))
(deriv_scal_TCC1 0
  (deriv_scal_TCC1-1 nil 3253536989
   ("" (lemma "scal_derivable") (("" (propax) nil nil)) nil)
   ((scal_derivable formula-decl nil derivatives_def nil)) nil))
(deriv_scal 0
  (deriv_scal-1 nil 3253536989
   ("" (skosimp)
    ((""
      (auto-rewrite "deriv_TCC" "scal_derivable" "derivable?" "deriv"
                    "scal_NQ" "lim_scal_fun[(A(x!1))]")
      (("" (assert) nil nil)) nil))
    nil)
   nil nil))
(deriv_inv_TCC1 0
  (deriv_inv_TCC1-1 nil 3253536989
   ("" (lemma "inv_derivable") (("" (propax) nil nil)) nil)
   ((inv_derivable formula-decl nil derivatives_def nil)) nil))
(deriv_inv 0
  (deriv_inv-1 nil 3253536989
   ("" (skosimp)
    (("" (forward-chain "inv_derivable")
      (("" (assert)
        ((""
          (auto-rewrite "deriv" "deriv_TCC" "derivable?"
                        ("lim_fun_def[(A(x!1))]"
                         "lim_fun_lemma[(A(x!1))]"))
          (("" (assert)
            (("" (use "cnv_seq_inv_NQ" ("l1" "lim(NQ(g!1, x!1), 0)"))
              (("" (assert) nil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   nil nil))
(deriv_div_TCC1 0
  (deriv_div_TCC1-1 nil 3253536989
   ("" (skosimp) (("" (rewrite "div_derivable") nil nil)) nil)
   ((div_derivable formula-decl nil derivatives_def nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (T_pred const-decl "[real -> boolean]" derivatives_def nil)
    (T formal-subtype-decl nil derivatives_def nil)
    (/= const-decl "boolean" notequal nil)
    (nzreal nonempty-type-eq-decl nil reals nil))
   nil))
(deriv_div 0
  (deriv_div-1 nil 3253536989
   (""
    (grind :defs nil :rewrites
     ("inv_derivable" "deriv_inv" "deriv_prod" "div_function[T]" "/"))
    nil nil)
   ((NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (inv_derivable formula-decl nil derivatives_def nil)
    (/ const-decl "[T -> real]" real_fun_ops "reals/")
    (deriv_inv formula-decl nil derivatives_def nil)
    (deriv_prod formula-decl nil derivatives_def nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (nzreal_div_nzreal_is_nzreal application-judgement "nzreal"
     real_types nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (T formal-subtype-decl nil derivatives_def nil)
    (T_pred const-decl "[real -> boolean]" derivatives_def 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)
    (div_function formula-decl nil real_fun_ops "reals/")
    (real_times_real_is_real application-judgement "real" reals nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (nzreal_times_nzreal_is_nzreal application-judgement "nzreal"
     real_types nil))
   nil))
(deriv_identity_TCC1 0
  (deriv_identity_TCC1-1 nil 3253536989
   ("" (lemma "identity_derivable") (("" (propax) nil nil)) nil)
   ((identity_derivable formula-decl nil derivatives_def nil)) nil))
(deriv_identity 0
  (deriv_identity-1 nil 3253536989
   ("" (skosimp)
    ((""
      (auto-rewrite "deriv_TCC" "identity_derivable" "derivable?"
                    "deriv" "identity_NQ")
      (("" (use "lim_const_fun[(A(x!1))]" ("b" "1" "c" "0"))
        (("" (assert) nil nil)) nil))
      nil))
    nil)
   ((id_fun_continuous name-judgement "continuous_fun"
     continuous_functions nil)
    (const_fun_continuous application-judgement "continuous_fun"
     continuous_functions nil))
   nil)))

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