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products/Sources/formale Sprachen/PVS/linear_algebra/ (Beweissystem der NASA Version 6.0.9^©) Datei vom 28.9.2014 mit Größe 289 kB

Quelle ellipsoid.prf Sprache: Lisp

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        (number_field nonempty-type-from-decl nil number_fields nil)
        (real_pred const-decl "[number_field -> boolean]" 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)
        (> const-decl "bool" reals nil)
        (SquareMat type-eq-decl nil matrices nil)
        (squareMat? const-decl "bool" matrices nil)
        (M!1 skolem-const-decl "SquareMat(n!1)" ellipsoid nil)
        (Q!1 skolem-const-decl "SquareMat(n!1)" ellipsoid nil)
        (n!1 skolem-const-decl "posnat" ellipsoid nil)
        (= const-decl "[T, T -> boolean]" equalities nil)
        (bijec_transpose formula-decl nil matrix_operator nil)
        (bijec_prod formula-decl nil matrix_operator nil))
       shostak)
  (ellipsoid_TCC11-1 nil 3520826443 ("" (subtype-tcc) nil nil)
   ((transpose const-decl "Matrix" matrices nil)
    (square? const-decl "bool" matrices nil)
    (sigma def-decl "real" sigma "reals/")
    (Matrix type-eq-decl nil matrices nil)
    (Square type-eq-decl nil matrices nil)
    (SquareMat type-eq-decl nil matrices nil)
    (squareMat? const-decl "bool" matrices nil)
    (bijec_transpose formula-decl nil matrix_operator nil)
    (bijec_prod formula-decl nil matrix_operator nil))
   nil))
(ellipsoid_TCC12 0
  (ellipsoid_TCC12-1 nil 3520826443 ("" (subtype-tcc) nil nil)
   ((T const-decl "[Mat(m, n) -> Map_linear(n, m)]" matrix_operator
     nil)
    (injective? const-decl "bool" functions nil)
    (surjective? const-decl "bool" functions nil)
    (bijective? const-decl "bool" functions nil)
    (bijective? const-decl "bool" linear_map nil)
    (inverse const-decl "Maping" linear_map nil)
    (inverse const-decl "D" function_inverse nil)
    (L const-decl "[Map_linear(n, m) -> Mat(m, n)]" matrix_operator
     nil)
    (inv const-decl "[Matrix_inv(n) -> Matrix_inv(n)]" matrix_operator
     nil)
    (- const-decl "real" vectors "vectors/")
    (* const-decl "Vector[M`rows]" matrices nil)
    (* const-decl "real" vectors "vectors/")
    (transpose const-decl "Matrix" matrices nil)
    (* const-decl "Matrix" matrices nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (int_minus_int_is_int application-judgement "int" integers nil))
   nil))
(ellipsoid_TCC13 0
  (ellipsoid_TCC13-1 nil 3520826443 ("" (subtype-tcc) nil nil)
   ((T const-decl "[Mat(m, n) -> Map_linear(n, m)]" matrix_operator
     nil)
    (injective? const-decl "bool" functions nil)
    (surjective? const-decl "bool" functions nil)
    (bijective? const-decl "bool" functions nil)
    (bijective? const-decl "bool" linear_map nil)
    (inverse const-decl "Maping" linear_map nil)
    (inverse const-decl "D" function_inverse nil)
    (L const-decl "[Map_linear(n, m) -> Mat(m, n)]" matrix_operator
     nil)
    (inv const-decl "[Matrix_inv(n) -> Matrix_inv(n)]" matrix_operator
     nil)
    (- const-decl "real" vectors "vectors/")
    (* const-decl "Vector[M`rows]" matrices nil)
    (* const-decl "real" vectors "vectors/")
    (transpose const-decl "Matrix" matrices nil)
    (* const-decl "Matrix" matrices nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (int_minus_int_is_int application-judgement "int" integers nil))
   nil))
(ellipsoid 0
  (thm "thm" 3521305299
   ("" (skosimp)
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                                         -1
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                                         "Q!1*transpose(M!1)"
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                                        (("1"
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                                                  (hide -1 -2 -3)
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                                                      (lemma
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                                                      (("1"
                                                        (inst
                                                         -1
                                                         "inv(n!1)(Q!1*transpose(M!1))"
                                                         "inv(n!1)(M!1)"
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                                                        (("1"
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                                                          (("1"
                                                            (expand
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                                                            (("1"
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                                                               -9)
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                                                                (replace
                                                                 -2
                                                                 :dir
                                                                 rl
                                                                 :actuals?
                                                                 t
                                                                 :hide?
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                                                                (("1"
                                                                  (replace
                                                                   -1
                                                                   :actuals?
                                                                   t
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                                                                  (("1"
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                                                                     -1)
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                                                                        (("1"
                                                                          (replace
                                                                           -1)
                                                                          (("1"
                                                                            (lemma
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                                                                            (("1"
                                                                              (inst
                                                                               -1
                                                                               "Q!1"
                                                                               "transpose(M!1)"
                                                                               "n!1")
                                                                              (("1"
                                                                                (assert)
                                                                                (("1"
                                                                                  (case
                                                                                   "square?(transpose(M!1)) AND
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                                                                                  (("1"
                                                                                    (flatten)
                                                                                    (("1"
                                                                                      (assert)
                                                                                      (("1"
                                                                                        (hide
                                                                                         -1
                                                                                         -2
                                                                                         -3)
                                                                                        (("1"
                                                                                          (replace
                                                                                           -1)
                                                                                          (("1"
                                                                                            (lemma
                                                                                             "mult_assoc_vect")
                                                                                            (("1"
                                                                                              (inst
                                                                                               -1
                                                                                               "inv(n!1)(transpose(M!1))"
                                                                                               "inv(n!1)(Q!1)"
                                                                                               "x!1-c!1")
                                                                                              (("1"
                                                                                                (replace
                                                                                                 -1)
                                                                                                (("1"
                                                                                                  (lemma
                                                                                                   "trans_mat_scal")
                                                                                                  (("1"
                                                                                                    (inst
                                                                                                     -1
                                                                                                     "inv(n!1)(transpose(M!1))"
                                                                                                     "M!1*(x!1-c!1)"
                                                                                                     "inv(n!1)(Q!1)*(x!1-c!1)")
                                                                                                    (("1"
                                                                                                      (case
                                                                                                       "inv(n!1)(transpose(M!1))`rows = n!1")
                                                                                                      (("1"
                                                                                                        (replace
                                                                                                         -1
                                                                                                         :actuals?
                                                                                                         t
                                                                                                         :hide?
                                                                                                         t)
                                                                                                        (("1"
                                                                                                          (replace
                                                                                                           -1)
                                                                                                          (("1"
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                                                                                                             "(transpose(inv(n!1)(transpose(M!1))) * (M!1 * (x!1 - c!1)))=x!1-c!1")
                                                                                                            (("1"
                                                                                                              (replace
                                                                                                               -1)
                                                                                                              (("1"
                                                                                                                (assert)
                                                                                                                (("1"
                                                                                                                  (hide-all-but
                                                                                                                   (-11
                                                                                                                    1))
                                                                                                                  (("1"
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                                                                                                                     "transpose(inv(n!1)(transpose(M!1)))`rows = n!1")
                                                                                                                    (("1"
                                                                                                                      (replace
                                                                                                                       -1
                                                                                                                       :actuals?
                                                                                                                       t
                                                                                                                       :hide?
                                                                                                                       t)
                                                                                                                      (("1"
                                                                                                                        (propax)
                                                                                                                        nil
                                                                                                                        nil))
                                                                                                                      nil)
                                                                                                                     ("2"
                                                                                                                      (assert)
                                                                                                                      (("2"
                                                                                                                        (hide-all-but
                                                                                                                         1)
                                                                                                                        (("2"
                                                                                                                          (grind)
                                                                                                                          nil
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil)
                                                                                                             ("2"
                                                                                                              (hide
                                                                                                               -1
                                                                                                               -2
                                                                                                               -3)
                                                                                                              (("2"
                                                                                                                (hide
                                                                                                                 2)
                                                                                                                (("2"
                                                                                                                  (lemma
                                                                                                                   "tran_inv_oper")
                                                                                                                  (("2"
                                                                                                                    (inst
                                                                                                                     -1
                                                                                                                     "M!1"
                                                                                                                     "n!1")
                                                                                                                    (("2"
                                                                                                                      (assert)
                                                                                                                      (("2"
                                                                                                                        (replace
                                                                                                                         -1)
                                                                                                                        (("2"
                                                                                                                          (lemma
                                                                                                                           "transpose2")
                                                                                                                          (("2"
                                                                                                                            (inst
                                                                                                                             -1
                                                                                                                             "inv(n!1)(M!1)")
                                                                                                                            (("2"
                                                                                                                              (replace
                                                                                                                               -1)
                                                                                                                              (("2"
                                                                                                                                (propax)
                                                                                                                                nil
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil)
                                                                                                             ("3"
                                                                                                              (hide-all-but
                                                                                                               1)
                                                                                                              (("3"
                                                                                                                (grind)
                                                                                                                nil
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil)
                                                                                                       ("2"
                                                                                                        (hide-all-but
                                                                                                         1)
                                                                                                        (("2"
                                                                                                          (grind)
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                                                                                                        nil))
                                                                                                      nil)
                                                                                                     ("2"
                                                                                                      (hide-all-but
                                                                                                       1)
                                                                                                      (("2"
                                                                                                        (grind)
                                                                                                        (("1"
                                                                                                          (reveal
                                                                                                           -24)
                                                                                                          (("1"
                                                                                                            (assert)
                                                                                                            nil
                                                                                                            nil))
                                                                                                          nil)
                                                                                                         ("2"
                                                                                                          (reveal
                                                                                                           -24)
                                                                                                          (("2"
                                                                                                            (assert)
                                                                                                            nil
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil)
                                                                                   ("2"
                                                                                    (hide
                                                                                     2)
                                                                                    (("2"
                                                                                      (split)
                                                                                      (("1"
                                                                                        (hide-all-but
                                                                                         1)
                                                                                        (("1"
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                                                                                           "M!1")
                                                                                          (("1"
                                                                                            (grind)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil)
                                                                                       ("2"
                                                                                        (hide-all-but
                                                                                         1)
                                                                                        (("2"
                                                                                          (typepred
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                                                                                            (grind)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil)
                                                                                       ("3"
                                                                                        (lemma
                                                                                         "bijec_transpose")
                                                                                        (("3"
                                                                                          (inst
                                                                                           -1
                                                                                           "M!1"
                                                                                           "n!1")
                                                                                          (("3"
                                                                                            (assert)
                                                                                            nil
                                                                                            nil))
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                                                                                   ("3"
                                                                                    (flatten)
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                                                                              nil))
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                                                                          nil)
                                                                         ("2"
                                                                          (hide
                                                                           2)
                                                                          (("2"
                                                                            (assert)
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                                                                              (lemma
                                                                               "mult_assoc_vect")
                                                                              (("2"
                                                                                (inst
                                                                                 -1
                                                                                 "inv(n!1)(M!1)"
                                                                                 "M!1"
                                                                                 "x!1-c!1")
                                                                                (("1"
                                                                                  (replace
                                                                                   -1
                                                                                   :dir
                                                                                   rl)
                                                                                  (("1"
                                                                                    (assert)
                                                                                    (("1"
                                                                                      (lemma
                                                                                       "iso_inv")
                                                                                      (("1"
                                                                                        (inst
                                                                                         -1
                                                                                         "M!1"
                                                                                         "n!1")
                                                                                        (("1"
                                                                                          (assert)
                                                                                          (("1"
                                                                                            (flatten)
                                                                                            (("1"
                                                                                              (replace
                                                                                               -1)
                                                                                              (("1"
                                                                                                (assert)
                                                                                                (("1"
                                                                                                  (lemma
                                                                                                   "ident_vect")
                                                                                                  (("1"
                                                                                                    (inst
                                                                                                     -1
                                                                                                     "I(n!1)"
                                                                                                     "x!1-c!1")
                                                                                                    nil
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
--> --------------------

--> maximum size reached

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

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