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Datei: vect_chain_rule.prf   Sprache: Unknown

(vect_chain_rule
 (composition_derivable_vfun_TCC1 0
  (composition_derivable_vfun_TCC1-1 nil 3472221152
   ("" (lemma "deriv_domain1")
    (("" (expand "deriv_domain?") (("" (propax) nil nil)) nil)) nil)
   ((deriv_domain1 formula-decl nil vect_chain_rule nil)) nil))
 (composition_derivable_vfun_TCC2 0
  (composition_derivable_vfun_TCC2-1 nil 3472221152
   ("" (lemma "not_one_element1") (("" (propax) nil nil)) nil)
   ((not_one_element1 formula-decl nil vect_chain_rule nil)) nil))
 (composition_derivable_vfun_TCC3 0
  (composition_derivable_vfun_TCC3-1 nil 3472221152
   ("" (lemma "deriv_domain2") (("" (skosimp*) nil nil)) nil)
   ((deriv_domain2 formula-decl nil vect_chain_rule nil)) nil))
 (composition_derivable_vfun_TCC4 0
  (composition_derivable_vfun_TCC4-1 nil 3472221152
   ("" (skosimp*)
    (("" (lemma "not_one_element2")
      (("" (expand "not_one_element?") (("" (inst?) nil nil)) nil))
      nil))
    nil)
   ((not_one_element2 formula-decl nil vect_chain_rule nil)) nil))
 (composition_derivable_vfun 0
  (composition_derivable_vfun-2 nil 3472381455
   ("" (skosimp*)
    (("" (expand "derivable_rv?")
      (("" (flatten)
        (("" (split +)
          (("1" (hide -3)
            (("1" (lemma "composition_derivable[T1,T2]")
              (("1" (expand "derivable?" 1)
                (("1" (skosimp*)
                  (("1"
                    (inst - "f!1" "(LAMBDA (a: T2): g!1(a)`x)" "x!1")
                    (("1" (expand "o ")
                      (("1" (assert)
                        (("1" (hide 2)
                          (("1" (expand derivable? -)
                            (("1" (inst?)
                              (("1"
                                (inst?)
                                (("1" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (lemma "not_one_element2") (("2" (propax) nil nil))
                nil)
               ("3" (lemma "deriv_domain2")
                (("3" (expand "deriv_domain?") (("3" (propax) nil nil))
                  nil))
                nil)
               ("4" (lemma "not_one_element1") (("4" (propax) nil nil))
                nil)
               ("5" (lemma "deriv_domain1")
                (("5" (expand "deriv_domain?") (("5" (propax) nil nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (hide -2)
            (("2" (lemma "composition_derivable[T1,T2]")
              (("1" (expand "derivable?" 1)
                (("1" (skosimp*)
                  (("1"
                    (inst - "f!1" "(LAMBDA (a: T2): g!1(a)`y)" "x!1")
                    (("1" (expand "o ")
                      (("1" (assert)
                        (("1" (hide 2)
                          (("1" (expand derivable? -)
                            (("1" (inst?)
                              (("1"
                                (inst?)
                                (("1" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (lemma "not_one_element2") (("2" (propax) nil nil))
                nil)
               ("3" (lemma "deriv_domain2")
                (("3" (expand "deriv_domain?") (("3" (propax) nil nil))
                  nil))
                nil)
               ("4" (lemma "not_one_element1") (("4" (propax) nil nil))
                nil)
               ("5" (lemma "deriv_domain1")
                (("5" (expand "deriv_domain?") (("5" (propax) nil nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((derivable_rv? const-decl "bool" deriv_real_vect2 nil)
    (T2 formal-subtype-decl nil vect_chain_rule nil)
    (T2_pred const-decl "[real -> boolean]" vect_chain_rule nil)
    (T1 formal-subtype-decl nil vect_chain_rule nil)
    (T1_pred const-decl "[real -> boolean]" vect_chain_rule 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)
    (composition_derivable formula-decl nil chain_rule "analysis/")
    (deriv_domain? const-decl "bool" deriv_domain_def "analysis/")
    (not_one_element? const-decl "bool" deriv_domain_def "analysis/")
    (bool nonempty-type-eq-decl nil booleans nil)
    (O const-decl "T3" function_props nil)
    (Vect2 type-eq-decl nil vectors_2D_def "vectors/")
    (derivable? const-decl "bool" derivatives "analysis/")
    (not_one_element2 formula-decl nil vect_chain_rule nil)
    (deriv_domain2 formula-decl nil vect_chain_rule nil)
    (not_one_element1 formula-decl nil vect_chain_rule nil)
    (deriv_domain1 formula-decl nil vect_chain_rule nil))
   nil)
  (composition_derivable_vfun-1 nil 3472221158
   ("" (skosimp*)
    (("" (expand "differentiable_rv?")
      (("" (flatten)
        (("" (split +)
          (("1" (hide -3)
            (("1" (lemma "composition_derivable[T1,T2]")
              (("1" (expand "derivable?" 1)
                (("1" (skosimp*)
                  (("1"
                    (inst - "f!1" "(LAMBDA (a: T2): g!1(a)`x)" "x!1")
                    (("1" (expand "o ")
                      (("1" (assert)
                        (("1" (hide 2)
                          (("1" (expand derivable? -)
                            (("1" (inst?)
                              (("1"
                                (inst?)
                                (("1" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (lemma "not_one_element2") (("2" (propax) nil nil))
                nil)
               ("3" (lemma "deriv_domain2") (("3" (propax) nil nil))
                nil)
               ("4" (lemma "not_one_element1") (("4" (propax) nil nil))
                nil)
               ("5" (lemma "deriv_domain1") (("5" (propax) nil nil))
                nil))
              nil))
            nil)
           ("2" (hide -2)
            (("2" (lemma "composition_derivable[T1,T2]")
              (("1" (expand "derivable?" 1)
                (("1" (skosimp*)
                  (("1"
                    (inst - "f!1" "(LAMBDA (a: T2): g!1(a)`y)" "x!1")
                    (("1" (expand "o ")
                      (("1" (assert)
                        (("1" (hide 2)
                          (("1" (expand derivable? -)
                            (("1" (inst?)
                              (("1"
                                (inst?)
                                (("1" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (lemma "not_one_element2") (("2" (propax) nil nil))
                nil)
               ("3" (lemma "deriv_domain2") (("3" (propax) nil nil))
                nil)
               ("4" (lemma "not_one_element1") (("4" (propax) nil nil))
                nil)
               ("5" (lemma "deriv_domain1") (("5" (propax) nil nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((composition_derivable formula-decl nil chain_rule "analysis/")
    (Vect2 type-eq-decl nil vectors_2D_def "vectors/")
    (derivable? const-decl "bool" derivatives "analysis/"))
   nil))
 (gg_TCC1 0
  (gg_TCC1-1 nil 3313840193
   ("" (lemma "deriv_domain2") (("" (propax) nil nil)) nil)
   ((deriv_domain2 formula-decl nil vect_chain_rule nil)) nil))
 (gg_TCC2 0
  (gg_TCC2-1 nil 3313840193
   ("" (lemma "not_one_element2") (("" (propax) nil nil)) nil)
   ((not_one_element2 formula-decl nil vect_chain_rule nil)) nil))
 (deriv_comp_vfun_TCC1 0
  (deriv_comp_vfun_TCC1-1 nil 3472221178
   ("" (skosimp*)
    (("" (lemma "composition_derivable_vfun")
      (("" (inst?) (("" (assertnil nil)) nil)) nil))
    nil)
   ((composition_derivable_vfun formula-decl nil vect_chain_rule nil)
    (derivable_rv? const-decl "bool" deriv_real_vect2 nil)
    (Vect2 type-eq-decl nil vectors_2D_def "vectors/")
    (derivable? const-decl "bool" derivatives "analysis/")
    (bool nonempty-type-eq-decl nil booleans nil)
    (T2 formal-subtype-decl nil vect_chain_rule nil)
    (T2_pred const-decl "[real -> boolean]" vect_chain_rule nil)
    (T1 formal-subtype-decl nil vect_chain_rule nil)
    (T1_pred const-decl "[real -> boolean]" vect_chain_rule 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))
   nil))
 (deriv_comp_vfun 0
  (deriv_comp_vfun-2 nil 3472381511
   ("" (skolem-typepred)
    (("" (expand "*" 1)
      (("" (lemma "composition_derivable_vfun")
        (("" (inst?)
          (("" (assert)
            (("" (expand "deriv")
              (("" (apply-extensionality 1 :hide? t)
                (("1" (expand "deriv_rv")
                  (("1" (expand "o ")
                    (("1" (expand "*")
                      (("1" (split +)
                        (("1" (lemma "deriv_composition[T1,T2]")
                          (("1" (inst?)
                            (("1" (assert)
                              (("1"
                                (split -1)
                                (("1"
                                  (expand "o ")
                                  (("1"
                                    (replace -1)
                                    (("1"
                                      (hide -1)
                                      (("1" (assertnil nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (hide 2)
                                  (("2"
                                    (expand "derivable?" -2)
                                    (("2" (inst?) nil nil))
                                    nil))
                                  nil)
                                 ("3"
                                  (hide 2)
                                  (("3"
                                    (expand "derivable_rv?")
                                    (("3"
                                      (flatten)
                                      (("3"
                                        (expand "derivable?" -)
                                        (("3"
                                          (inst -4 "ff!1(x!1)")
                                          nil
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (lemma "not_one_element2")
                            (("2" (propax) nil nil)) nil)
                           ("3" (lemma "deriv_domain2")
                            (("3" (propax) nil nil)) nil)
                           ("4" (lemma "not_one_element1")
                            (("4" (propax) nil nil)) nil)
                           ("5" (lemma "deriv_domain1")
                            (("5" (propax) nil nil)) nil))
                          nil)
                         ("2" (lemma "deriv_composition[T1,T2]")
                          (("1" (inst?)
                            (("1" (assert)
                              (("1"
                                (split -1)
                                (("1"
                                  (expand "o ")
                                  (("1"
                                    (replace -1)
                                    (("1"
                                      (hide -1)
                                      (("1" (assertnil nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (hide 2)
                                  (("2"
                                    (expand "derivable?" -2)
                                    (("2" (inst?) nil nil))
                                    nil))
                                  nil)
                                 ("3"
                                  (hide 2)
                                  (("3"
                                    (expand "derivable_rv?")
                                    (("3"
                                      (flatten)
                                      (("3"
                                        (expand "derivable?" -)
                                        (("3"
                                          (inst -5 "ff!1(x!1)")
                                          nil
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (lemma "not_one_element2")
                            (("2" (propax) nil nil)) nil)
                           ("3" (lemma "deriv_domain2")
                            (("3" (propax) nil nil)) nil)
                           ("4" (lemma "not_one_element1")
                            (("4" (propax) nil nil)) nil)
                           ("5" (lemma "deriv_domain1")
                            (("5" (propax) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (lemma "not_one_element1")
                  (("2" (skosimp*) nil nil)) nil)
                 ("3" (skosimp*)
                  (("3" (lemma "deriv_domain1")
                    (("3" (expand "deriv_domain?")
                      (("3" (inst?) nil nil)) nil))
                    nil))
                  nil)
                 ("4" (expand "derivable?" -2) (("4" (propax) nil nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((* const-decl "[T -> Vect2]" vect_fun_ops_rv nil)
    (deriv const-decl "[T -> real]" derivatives "analysis/")
    (deriv const-decl "[T -> Vect2]" deriv_real_vect2 nil)
    (deriv_composition formula-decl nil chain_rule "analysis/")
    (real_times_real_is_real application-judgement "real" reals nil)
    (not_one_element2 formula-decl nil vect_chain_rule nil)
    (deriv_domain2 formula-decl nil vect_chain_rule nil)
    (not_one_element1 formula-decl nil vect_chain_rule nil)
    (deriv_domain1 formula-decl nil vect_chain_rule nil)
    (derivable? const-decl "bool" derivatives_def "analysis/")
    (ff!1 skolem-const-decl "{f | derivable?(f)}" vect_chain_rule nil)
    (deriv_domain? const-decl "bool" deriv_domain_def "analysis/")
    (not_one_element? const-decl "bool" deriv_domain_def "analysis/")
    (deriv_rv const-decl "Vect2" deriv_real_vect2 nil)
    (O const-decl "T3" function_props nil)
    (Vector type-eq-decl nil vectors_2D "vectors/")
    (* const-decl "Vector" vectors_2D "vectors/")
    (deriv const-decl "real" derivatives_def "analysis/")
    (composition_derivable_vfun formula-decl nil vect_chain_rule nil)
    (derivable_rv? const-decl "bool" deriv_real_vect2 nil)
    (Vect2 type-eq-decl nil vectors_2D_def "vectors/")
    (T2 formal-subtype-decl nil vect_chain_rule nil)
    (T2_pred const-decl "[real -> boolean]" vect_chain_rule nil)
    (derivable? const-decl "bool" derivatives "analysis/")
    (T1 formal-subtype-decl nil vect_chain_rule nil)
    (T1_pred const-decl "[real -> boolean]" vect_chain_rule 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)
    (number nonempty-type-decl nil numbers nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil))
   nil)
  (deriv_comp_vfun-1 nil 3472221184
   ("" (skolem-typepred)
    (("" (expand "*" 1)
      (("" (lemma "composition_derivable_vfun")
        (("" (inst?)
          (("" (assert)
            (("" (expand "deriv")
              (("" (apply-extensionality 1 :hide? t)
                (("1" (expand "deriv_rv")
                  (("1" (expand "o ")
                    (("1" (expand "*")
                      (("1" (split +)
                        (("1" (lemma "deriv_composition[T1,T2]")
                          (("1" (inst?)
                            (("1" (assert)
                              (("1"
                                (split -1)
                                (("1"
                                  (expand "o ")
                                  (("1"
                                    (replace -1)
                                    (("1"
                                      (hide -1)
                                      (("1" (assertnil nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (hide 2)
                                  (("2"
                                    (expand "derivable?" -2)
                                    (("2" (inst?) nil nil))
                                    nil))
                                  nil)
                                 ("3"
                                  (hide 2)
                                  (("3"
                                    (expand "differentiable_rv?")
                                    (("3"
                                      (flatten)
                                      (("3"
                                        (expand "derivable?" -)
                                        (("3"
                                          (inst -4 "ff!1(x!1)")
                                          nil
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (lemma "not_one_element2")
                            (("2" (propax) nil nil)) nil)
                           ("3" (lemma "deriv_domain2")
                            (("3" (propax) nil nil)) nil)
                           ("4" (lemma "not_one_element1")
                            (("4" (propax) nil nil)) nil)
                           ("5" (lemma "deriv_domain1")
                            (("5" (propax) nil nil)) nil))
                          nil)
                         ("2" (lemma "deriv_composition[T1,T2]")
                          (("1" (inst?)
                            (("1" (assert)
                              (("1"
                                (split -1)
                                (("1"
                                  (expand "o ")
                                  (("1"
                                    (replace -1)
                                    (("1"
                                      (hide -1)
                                      (("1" (assertnil nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (hide 2)
                                  (("2"
                                    (expand "derivable?" -2)
                                    (("2" (inst?) nil nil))
                                    nil))
                                  nil)
                                 ("3"
                                  (hide 2)
                                  (("3"
                                    (expand "differentiable_rv?")
                                    (("3"
                                      (flatten)
                                      (("3"
                                        (expand "derivable?" -)
                                        (("3"
                                          (inst -5 "ff!1(x!1)")
                                          nil
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (lemma "not_one_element2")
                            (("2" (propax) nil nil)) nil)
                           ("3" (lemma "deriv_domain2")
                            (("3" (propax) nil nil)) nil)
                           ("4" (lemma "not_one_element1")
                            (("4" (propax) nil nil)) nil)
                           ("5" (lemma "deriv_domain1")
                            (("5" (propax) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (lemma "not_one_element1")
                  (("2" (skosimp*) nil nil)) nil)
                 ("3" (skosimp*)
                  (("3" (lemma "deriv_domain1") (("3" (inst?) nil nil))
                    nil))
                  nil)
                 ("4" (expand "derivable?" -2) (("4" (propax) nil nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((deriv const-decl "[T -> real]" derivatives "analysis/")
    (deriv_composition formula-decl nil chain_rule "analysis/")
    (derivable? const-decl "bool" derivatives_def "analysis/")
    (Vector type-eq-decl nil vectors_2D "vectors/")
    (* const-decl "Vector" vectors_2D "vectors/")
    (deriv const-decl "real" derivatives_def "analysis/")
    (Vect2 type-eq-decl nil vectors_2D_def "vectors/")
    (derivable? const-decl "bool" derivatives "analysis/"))
   nil)))


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