Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/PVS/linear_algebra/   (Beweissystem der NASA Version 6.0.9©)  Datei vom 28.9.2014 mit Größe 289 kB image not shown  

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    (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)
    (below type-eq-decl nil naturalnumbers nil)
    (Matrix type-eq-decl nil matrices nil)
    (square? const-decl "bool" matrices nil)
    (Square type-eq-decl nil matrices nil)
    (squareMat? const-decl "bool" matrices nil)
    (SquareMat type-eq-decl nil matrices nil)
    (real_lt_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)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props 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)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (real_minus_real_is_real application-judgement "real" reals nil))
   nil))
 (ellipsoid_TCC11 0
  (tcc "tcc" 3520936757
       ("" (skosimp)
        (("" (split)
          (("1" (grind) nil nil)
           ("2" (expand "squareMat?")
            (("2" (assert)
              (("2" (typepred "M!1")
                (("2" (typepred "Q!1")
                  (("2" (expand "squareMat?")
                    (("2" (assert) (("2" (grind) nil nil)) nil)) nil))
                  nil))
                nil))
              nil))
            nil)
           ("3" (lemma "bijec_prod")
            (("3" (inst?)
              (("1" (assert)
                (("1" (split)
                  (("1" (assert)
                    (("1" (typepred "Q!1")
                      (("1" (typepred "M!1")
                        (("1" (expand "square?")
                          (("1" (assert) (("1" (grind) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (typepred "Q!1")
                    (("2" (typepred "M!1")
                      (("2" (expand "square?") (("2" (grind) nil nil))
                        nil))
                      nil))
                    nil)
                   ("3" (lemma "bijec_prod")
                    (("3" (inst?)
                      (("1" (assert)
                        (("1" (split)
                          (("1" (assert)
                            (("1" (hide 2 3)
                              (("1"
                                (typepred "M!1")
                                (("1" (grind) nil nil))
                                nil))
                              nil))
                            nil)
                           ("2" (typepred "M!1")
                            (("2" (hide 2 3) (("2" (grind) nil nil))
                              nil))
                            nil)
                           ("3" (lemma "bijec_transpose")
                            (("3" (inst?) (("3" (assertnil nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (typepred "M!1")
                        (("2" (hide 2 3) (("2" (grind) nil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (typepred "Q!1")
                (("2" (typepred "M!1")
                  (("2" (expand "square?")
                    (("2" (assert)
                      (("2" (hide 2) (("2" (grind) nil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ((transpose const-decl "Matrix" matrices nil)
        (* const-decl "Matrix" matrices nil)
        (square? const-decl "bool" matrices nil)
        (real_times_real_is_real application-judgement "real" reals
         nil)
        (int_minus_int_is_int application-judgement "int" integers nil)
        (real_le_is_total_order name-judgement "(total_order?[real])"
         real_props nil)
        (real_plus_real_is_real application-judgement "real" reals nil)
        (sigma def-decl "real" sigma "reals/")
        (boolean nonempty-type-decl nil booleans nil)
        (bool nonempty-type-eq-decl nil booleans nil)
        (NOT const-decl "[bool -> bool]" booleans nil)
        (posnat nonempty-type-eq-decl nil integers nil)
        (below type-eq-decl nil naturalnumbers nil)
        (real nonempty-type-from-decl nil reals nil)
        (Matrix type-eq-decl nil matrices nil)
        (Square type-eq-decl nil matrices 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)
        (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)
    (("" (replace -4)
      (("" (assert)
        (("" (lemma "add_cancel_neg2[n!1]")
          (("" (inst -1 "b!1" "M!1 * x!1")
            (("1" (typepred "M!1")
              (("1" (expand "squareMat?")
                (("1" (replace -2 :actuals? t :hide? t)
                  (("1" (hide -1)
                    (("1" (replace -1 :hide? t)
                      (("1" (lemma "left_distributive_vect")
                        (("1" (inst -1 "M!1" "x!1" "c!1")
                          (("1" (typepred "M!1")
                            (("1" (expand "squareMat?")
                              (("1"
                                (replace -2 :actuals? t :hide? t)
                                (("1"
                                  (hide -1)
                                  (("1"
                                    (replace -1)
                                    (("1"
                                      (lemma "prod_inv_oper")
                                      (("1"
                                        (inst
                                         -1
                                         "M!1"
                                         "Q!1*transpose(M!1)"
                                         "n!1")
                                        (("1"
                                          (assert)
                                          (("1"
                                            (case
                                             "square?(Q!1 * transpose(M!1)) AND
                         squareMat?(n!1)(Q!1 * transpose(M!1)) AND
                          bijective?(n!1)(T(n!1, n!1)(Q!1 * transpose(M!1)))")
                                            (("1"
                                              (flatten)
                                              (("1"
                                                (assert)
                                                (("1"
                                                  (hide -1 -2 -3)
                                                  (("1"
                                                    (replace -1)
                                                    (("1"
                                                      (lemma
                                                       "mult_assoc_vect")
                                                      (("1"
                                                        (inst
                                                         -1
                                                         "inv(n!1)(Q!1*transpose(M!1))"
                                                         "inv(n!1)(M!1)"
                                                         "M!1*(x!1-c!1)")
                                                        (("1"
                                                          (reveal -8)
                                                          (("1"
                                                            (expand
                                                             "square?")
                                                            (("1"
                                                              (reveal
                                                               -9)
                                                              (("1"
                                                                (replace
                                                                 -2
                                                                 :dir
                                                                 rl
                                                                 :actuals?
                                                                 t
                                                                 :hide?
                                                                 t)
                                                                (("1"
                                                                  (replace
                                                                   -1
                                                                   :actuals?
                                                                   t
                                                                   :hide?
                                                                   t)
                                                                  (("1"
                                                                    (replace
                                                                     -1)
                                                                    (("1"
                                                                      (assert)
                                                                      (("1"
                                                                        (case
                                                                         "inv(n!1)(M!1) * (M!1 * (x!1 - c!1))=x!1-c!1")
                                                                        (("1"
                                                                          (replace
                                                                           -1)
                                                                          (("1"
                                                                            (lemma
                                                                             "prod_inv_oper")
                                                                            (("1"
                                                                              (inst
                                                                               -1
                                                                               "Q!1"
                                                                               "transpose(M!1)"
                                                                               "n!1")
                                                                              (("1"
                                                                                (assert)
                                                                                (("1"
                                                                                  (case
                                                                                   "square?(transpose(M!1)) AND
                                     squareMat?(n!1)(transpose(M!1)) AND
                                      bijective?(n!1)(T(n!1, n!1)(transpose(M!1)))")
                                                                                  (("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"
                                                                                                            (case
                                                                                                             "(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"
                                                                                                                    (case
                                                                                                                     "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)
                                                                                                          nil
                                                                                                          nil))
                                                                                                        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"
                                                                                          (typepred
                                                                                           "M!1")
                                                                                          (("1"
                                                                                            (grind)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil)
                                                                                       ("2"
                                                                                        (hide-all-but
                                                                                         1)
                                                                                        (("2"
                                                                                          (typepred
                                                                                           "M!1")
                                                                                          (("2"
                                                                                            (grind)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil)
                                                                                       ("3"
                                                                                        (lemma
                                                                                         "bijec_transpose")
                                                                                        (("3"
                                                                                          (inst
                                                                                           -1
                                                                                           "M!1"
                                                                                           "n!1")
                                                                                          (("3"
                                                                                            (assert)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil)
                                                                                   ("3"
                                                                                    (flatten)
                                                                                    (("3"
                                                                                      (hide
                                                                                       2)
                                                                                      (("3"
                                                                                        (grind)
                                                                                        nil
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil)
                                                                         ("2"
                                                                          (hide
                                                                           2)
                                                                          (("2"
                                                                            (assert)
                                                                            (("2"
                                                                              (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))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil)
                                                                                 ("2"
                                                                                  (hide
                                                                                   2)
                                                                                  (("2"
                                                                                    (hide-all-but
                                                                                     1)
                                                                                    (("2"
                                                                                      (typepred
                                                                                       "M!1")
                                                                                      (("2"
                                                                                        (grind)
                                                                                        nil
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil)
                                                                         ("3"
                                                                          (hide-all-but
                                                                           1)
                                                                          (("3"
                                                                            (typepred
                                                                             "M!1")
                                                                            (("3"
                                                                              (grind)
                                                                              nil
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (hide-all-but
                                                           1)
                                                          (("2"
                                                            (typepred
                                                             "M!1")
                                                            (("2"
                                                              (grind)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("3"
                                                          (hide-all-but
                                                           1)
                                                          (("3"
                                                            (typepred
                                                             "M!1")
                                                            (("3"
                                                              (grind)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("2"
                                              (split)
                                              (("1"
                                                (hide-all-but 1)
                                                (("1"
                                                  (grind)
                                                  (("1"
                                                    (typepred "M!1")
                                                    (("1"
                                                      (typepred "Q!1")
                                                      (("1"
                                                        (grind)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (hide-all-but 1)
                                                (("2"
                                                  (typepred "M!1")
                                                  (("2"
                                                    (typepred "Q!1")
                                                    (("2"
                                                      (grind)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("3"
                                                (lemma "bijec_prod")
                                                (("3"
                                                  (inst
                                                   -1
                                                   "Q!1"
                                                   "transpose(M!1)"
                                                   "n!1")
                                                  (("1"
                                                    (assert)
                                                    (("1"
                                                      (split)
                                                      (("1"
                                                        (hide-all-but
                                                         1)
                                                        (("1"
                                                          (typepred
                                                           "M!1")
                                                          (("1"
                                                            (grind)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (hide-all-but
                                                         1)
                                                        (("2"
                                                          (typepred
                                                           "M!1")
                                                          (("2"
                                                            (grind)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("3"
                                                        (lemma
                                                         "bijec_transpose")
                                                        (("3"
                                                          (inst
                                                           -1
                                                           "M!1"
                                                           "n!1")
                                                          (("3"
                                                            (assert)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (hide-all-but 1)
                                                    (("2"
                                                      (typepred "M!1")
                                                      (("2"
                                                        (grind)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("3"
                                              (flatten)
                                              (("3"
                                                (hide 2)
                                                (("3"
                                                  (hide-all-but 1)
                                                  (("3"
                                                    (typepred "M!1")
                                                    (("3"
                                                      (typepred "Q!1")
                                                      (("3"
                                                        (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" (hide-all-but 1)
              (("2" (typepred "M!1") (("2" (grind) nil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((posnat 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)
    (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)
    (add_cancel_neg2 formula-decl nil vectors "vectors/")
    (NOT const-decl "[bool -> bool]" booleans nil)
    (prod_inv_oper formula-decl nil matrix_operator nil)
    (bijec_prod formula-decl nil matrix_operator nil)
    (mult_assoc_vect formula-decl nil matrices nil)
    (sigma def-decl "real" sigma "reals/")
    (iso_inv formula-decl nil matrix_operator nil)
    (ident_vect formula-decl nil matrices nil)
    (identity? const-decl "bool" matrices nil)
    (I const-decl "(identity?)" matrices nil)
    (trans_mat_scal formula-decl nil matrices nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (tran_inv_oper formula-decl nil matrix_operator nil)
    (transpose2 formula-decl nil matrices 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)
    (real_times_real_is_real application-judgement "real" reals nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (bijec_transpose formula-decl nil matrix_operator nil)
    (- const-decl "real" vectors "vectors/")
    (inv const-decl "[Matrix_inv(n) -> Matrix_inv(n)]" matrix_operator
     nil)
    (Matrix_inv type-eq-decl nil matrix_operator nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (T const-decl "[Mat(m, n) -> Map_linear(n, m)]" matrix_operator
     nil)
    (Map_linear type-eq-decl nil linear_map_def nil)
    (linear_map_e? const-decl "bool" linear_map_def nil)
    (Mat type-eq-decl nil matrices nil)
    (bijective? const-decl "bool" linear_map nil)
    (Map type-eq-decl nil linear_map nil)
    (Maping type-eq-decl nil linear_map nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (* const-decl "Matrix" matrices nil)
    (transpose const-decl "Matrix" matrices nil)
    (left_distributive_vect formula-decl nil matrices nil)
    (* const-decl "Vector[M`rows]" matrices nil)
    (Vector type-eq-decl nil vectors "vectors/")
    (Index type-eq-decl nil vectors "vectors/")
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (IFF const-decl "[bool, bool -> bool]" booleans nil)
    (< const-decl "bool" reals nil)
    (below type-eq-decl nil naturalnumbers nil)
    (Matrix type-eq-decl nil matrices nil)
    (square? const-decl "bool" matrices nil)
    (Square type-eq-decl nil matrices nil)
    (squareMat? const-decl "bool" matrices nil)
    (n!1 skolem-const-decl "posnat" ellipsoid nil)
    (SquareMat type-eq-decl nil matrices nil)
    (M!1 skolem-const-decl "SquareMat(n!1)" ellipsoid nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   shostak)
  (ellipsoid-3 "thm" 3521305283
   ("" (skosimp)
    (("" (replace -4)
      (("" (assert)
        (("" (lemma "add_cancel_neg2[n!1]")
          (("" (inst -1 "b!1" "M!1 * x!1")
            (("1" (typepred "M!1")
              (("1" (expand "squareMat?")
                (("1" (replace -2 :actuals? t :hide? t)
                  (("1" (hide -1)
                    (("1" (replace -1 :hide? t)
                      (("1" (lemma "left_distributive_vect")
                        (("1" (inst -1 "M!1" "x!1" "c!1")
                          (("1" (typepred "M!1")
                            (("1" (expand "squareMat?")
                              (("1"
                                (replace -2 :actuals? t :hide? t)
                                (("1"
                                  (hide -1)
                                  (("1"
                                    (replace -1)
                                    (("1"
                                      (lemma "prod_inv_oper")
                                      (("1"
                                        (inst
                                         -1
                                         "M!1"
                                         "Q!1*transpose(M!1)"
                                         "n!1")
                                        (("1"
                                          (assert)
                                          (("1"
                                            (case
                                             "square?(Q!1 * transpose(M!1)) AND
                         squareMat?(n!1)(Q!1 * transpose(M!1)) AND
                          bijective?(n!1)(T(n!1, n!1)(Q!1 * transpose(M!1)))")
                                            (("1"
                                              (flatten)
                                              (("1"
                                                (assert)
                                                (("1"
                                                  (hide -1 -2 -3)
                                                  (("1"
                                                    (replace -1)
                                                    (("1"
                                                      (lemma
                                                       "mult_assoc_vect")
                                                      (("1"
                                                        (inst
                                                         -1
                                                         "inv(n!1)(Q!1*transpose(M!1))"
                                                         "inv(n!1)(M!1)"
                                                         "M!1*(x!1-c!1)")
                                                        (("1"
                                                          (reveal -8)
                                                          (("1"
                                                            (expand
                                                             "square?")
                                                            (("1"
                                                              (reveal
                                                               -9)
                                                              (("1"
                                                                (replace
                                                                 -2
                                                                 :dir
                                                                 rl
                                                                 :actuals?
                                                                 t
                                                                 :hide?
                                                                 t)
                                                                (("1"
                                                                  (replace
                                                                   -1
                                                                   :actuals?
                                                                   t
                                                                   :hide?
                                                                   t)
                                                                  (("1"
                                                                    (replace
                                                                     -1)
                                                                    (("1"
                                                                      (assert)
                                                                      (("1"
                                                                        (case
                                                                         "inv(n!1)(M!1) * (M!1 * (x!1 - c!1))=x!1-c!1")
                                                                        (("1"
                                                                          (replace
                                                                           -1)
                                                                          (("1"
                                                                            (lemma
                                                                             "prod_inv_oper")
                                                                            (("1"
                                                                              (inst
                                                                               -1
                                                                               "Q!1"
                                                                               "transpose(M!1)"
                                                                               "n!1")
                                                                              (("1"
                                                                                (assert)
                                                                                (("1"
                                                                                  (case
                                                                                   "square?(transpose(M!1)) AND
                                     squareMat?(n!1)(transpose(M!1)) AND
                                      bijective?(n!1)(T(n!1, n!1)(transpose(M!1)))")
                                                                                  (("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"
                                                                                                            (case
                                                                                                             "(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"
                                                                                                                    (case
                                                                                                                     "transpose(inv(n!1)(transpose(M!1)))`rows = n!1")
                                                                                                                    (("1"
                                                                                                                      (replace
                                                                                                                       -1
                                                                                                                       :actuals?
                                                                                                                       t
                                                                                                                       :hide?
                                                                                                                       t)
                                                                                                                      (("1"
                                                                                                                        (propax)
                                                                                                                        nil)))
                                                                                                                     ("2"
                                                                                                                      (assert)
                                                                                                                      (("2"
                                                                                                                        (hide-all-but
                                                                                                                         1)
                                                                                                                        (("2"
                                                                                                                          (grind)
                                                                                                                          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)))))))))))))))))))
                                                                                                             ("3"
                                                                                                              (hide-all-but
                                                                                                               1)
                                                                                                              (("3"
                                                                                                                (grind)
                                                                                                                nil)))))))))
                                                                                                       ("2"
                                                                                                        (hide-all-but
                                                                                                         1)
                                                                                                        (("2"
                                                                                                          (grind)
                                                                                                          nil)))))
                                                                                                     ("2"
                                                                                                      (hide-all-but
                                                                                                       1)
                                                                                                      (("2"
                                                                                                        (grind)
                                                                                                        (("1"
                                                                                                          (reveal
                                                                                                           -24)
                                                                                                          (("1"
                                                                                                            (assert)
                                                                                                            nil)))
                                                                                                         ("2"
                                                                                                          (reveal
                                                                                                           -24)
                                                                                                          (("2"
                                                                                                            (assert)
                                                                                                            nil)))))))))))))))))))))))))
                                                                                   ("2"
                                                                                    (hide
                                                                                     2)
                                                                                    (("2"
                                                                                      (split)
                                                                                      (("1"
                                                                                        (hide-all-but
                                                                                         1)
                                                                                        (("1"
                                                                                          (typepred
                                                                                           "M!1")
                                                                                          (("1"
                                                                                            (grind)
                                                                                            nil)))))
                                                                                       ("2"
                                                                                        (hide-all-but
                                                                                         1)
                                                                                        (("2"
                                                                                          (typepred
                                                                                           "M!1")
                                                                                          (("2"
                                                                                            (grind)
                                                                                            nil)))))
                                                                                       ("3"
                                                                                        (lemma
                                                                                         "bijec_transpose")
                                                                                        (("3"
                                                                                          (inst
                                                                                           -1
                                                                                           "M!1"
                                                                                           "n!1")
                                                                                          (("3"
                                                                                            (assert)
                                                                                            nil)))))))))
                                                                                   ("3"
                                                                                    (flatten)
                                                                                    (("3"
                                                                                      (hide
                                                                                       2)
                                                                                      (("3"
                                                                                        (grind)
                                                                                        nil)))))))))))))))
                                                                         ("2"
                                                                          (hide
                                                                           2)
                                                                          (("2"
                                                                            (assert)
                                                                            (("2"
                                                                              (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)))))))))))))))))))
                                                                                 ("2"
                                                                                  (hide
                                                                                   2)
                                                                                  (("2"
                                                                                    (hide-all-but
                                                                                     1)
                                                                                    (("2"
                                                                                      (typepred
                                                                                       "M!1")
                                                                                      (("2"
                                                                                        (grind)
                                                                                        nil)))))))))))))))
                                                                         ("3"
                                                                          (hide-all-but
                                                                           1)
                                                                          (("3"
                                                                            (typepred
                                                                             "M!1")
                                                                            (("3"
                                                                              (grind)
                                                                              nil)))))))))))))))))))))
                                                         ("2"
                                                          (hide-all-but
                                                           1)
                                                          (("2"
                                                            (typepred
                                                             "M!1")
                                                            (("2"
                                                              (grind)
                                                              nil)))))
                                                         ("3"
                                                          (hide-all-but
                                                           1)
                                                          (("3"
                                                            (typepred
                                                             "M!1")
                                                            (("3"
                                                              (grind)
                                                              nil)))))))))))))))))
                                             ("2"
                                              (split)
                                              (("1"
                                                (hide-all-but 1)
                                                (("1"
                                                  (grind)
                                                  (("1"
                                                    (typepred "M!1")
                                                    (("1"
                                                      (typepred "Q!1")
                                                      (("1"
                                                        (grind)
                                                        nil)))))))))
                                               ("2"
                                                (hide-all-but 1)
                                                (("2"
                                                  (typepred "M!1")
                                                  (("2"
                                                    (typepred "Q!1")
                                                    (("2"
                                                      (grind)
                                                      nil)))))))
                                               ("3"
                                                (lemma "bijec_prod")
                                                (("3"
                                                  (inst
                                                   -1
                                                   "Q!1"
                                                   "transpose(M!1)"
                                                   "n!1")
                                                  (("1"
                                                    (assert)
                                                    (("1"
                                                      (split)
                                                      (("1"
                                                        (hide-all-but
                                                         1)
                                                        (("1"
                                                          (typepred
                                                           "M!1")
                                                          (("1"
                                                            (grind)
                                                            nil)))))
                                                       ("2"
                                                        (hide-all-but
                                                         1)
                                                        (("2"
                                                          (typepred
                                                           "M!1")
                                                          (("2"
                                                            (grind)
                                                            nil)))))
                                                       ("3"
                                                        (lemma
                                                         "bijec_transpose")
                                                        (("3"
                                                          (inst
                                                           -1
                                                           "M!1"
                                                           "n!1")
                                                          (("3"
                                                            (assert)
                                                            nil)))))))))
                                                   ("2"
                                                    (hide-all-but 1)
                                                    (("2"
                                                      (typepred "M!1")
                                                      (("2"
                                                        (grind)
                                                        nil)))))))))))
                                             ("3"
                                              (flatten)
                                              (("3"
                                                (hide 2)
                                                (("3"
                                                  (hide-all-but 1)
                                                  (("3"
                                                    (typepred "M!1")
                                                    (("3"
                                                      (typepred "Q!1")
                                                      (("3"
                                                        (grind)
                                                        nil)))))))))))))))))))))))))))))))))))))))))))
             ("2" (hide-all-but 1)
              (("2" (typepred "M!1") (("2" (grind) nil))))))))))))))
    nil)
   ((SquareMat type-eq-decl nil matrices nil)
    (squareMat? const-decl "bool" matrices nil)
    (Square type-eq-decl nil matrices nil)
    (square? const-decl "bool" matrices nil)
    (Matrix type-eq-decl nil matrices nil)
    (Index type-eq-decl nil vectors "vectors/")
    (Vector type-eq-decl nil vectors "vectors/")
    (* const-decl "Vector[M`rows]" matrices nil)
    (left_distributive_vect formula-decl nil matrices nil)
    (transpose const-decl "Matrix" matrices nil)
    (Maping type-eq-decl nil linear_map nil)
    (Map type-eq-decl nil linear_map nil)
    (bijective? const-decl "bool" linear_map nil)
    (Mat type-eq-decl nil matrices nil)
    (Map_linear type-eq-decl nil linear_map_def nil)
    (T const-decl "[Mat(m, n) -> Map_linear(n, m)]" matrix_operator
     nil)
    (Matrix_inv type-eq-decl nil matrix_operator nil)
    (inv const-decl "[Matrix_inv(n) -> Matrix_inv(n)]" matrix_operator
     nil)
    (- const-decl "real" vectors "vectors/")
    (bijec_transpose formula-decl nil matrix_operator nil)
    (L const-decl "[Map_linear(n, m) -> Mat(m, n)]" matrix_operator
     nil)
    (inverse const-decl "Maping" linear_map nil)
    (transpose2 formula-decl nil matrices nil)
    (tran_inv_oper formula-decl nil matrix_operator nil)
    (trans_mat_scal formula-decl nil matrices nil)
    (I const-decl "(identity?)" matrices nil)
    (identity? const-decl "bool" matrices nil)
    (ident_vect formula-decl nil matrices nil)
    (iso_inv formula-decl nil matrix_operator nil)
    (sigma def-decl "real" sigma "reals/")
    (mult_assoc_vect formula-decl nil matrices nil)
    (bijec_prod formula-decl nil matrix_operator nil)
    (prod_inv_oper formula-decl nil matrix_operator nil)
    (add_cancel_neg2 formula-decl nil vectors "vectors/"))
   nil)
  (thm "thm" 3521286628
   ("" (skosimp)
    (("" (replace -4)
      (("" (assert)
        (("" (lemma "add_cancel_neg2[n!1]")
          (("" (inst -1 "b!1" "M!1 * x!1")
            (("1" (typepred "M!1")
              (("1" (expand "squareMat?")
                (("1" (replace -2 :actuals? t :hide? t)
                  (("1" (hide -1)
                    (("1" (replace -1 :hide? t)
                      (("1" (lemma "left_distributive_vect")
                        (("1" (inst -1 "M!1" "x!1" "c!1")
                          (("1" (typepred "M!1")
                            (("1" (expand "squareMat?")
                              (("1"
                                (replace -2 :actuals? t :hide? t)
                                (("1"
                                  (hide -1)
                                  (("1"
                                    (replace -1)
                                    (("1"
                                      (lemma "prod_inv_oper")
                                      (("1"
                                        (inst
                                         -1
                                         "M!1"
                                         "Q!1*transpose(M!1)"
                                         "n!1")
                                        (("1"
                                          (assert)
                                          (("1"
                                            (case
                                             "square?(Q!1 * transpose(M!1)) AND
                squareMat?(n!1)(Q!1 * transpose(M!1)) AND
                 bijective?(n!1)(T(n!1, n!1)(Q!1 * transpose(M!1)))")
                                            (("1"
                                              (flatten)
                                              (("1"
                                                (assert)
                                                (("1"
                                                  (hide -1 -2 -3)
                                                  (("1"
                                                    (replace -1)
                                                    (("1"
                                                      (lemma
                                                       "mult_assoc_vect")
                                                      (("1"
                                                        (inst
                                                         -1
                                                         "inv(n!1)(Q!1*transpose(M!1))"
                                                         "inv(n!1)(M!1)"
                                                         "M!1*(x!1-c!1)")
                                                        (("1"
                                                          (reveal -8)
                                                          (("1"
                                                            (expand
                                                             "square?")
                                                            (("1"
                                                              (reveal
                                                               -9)
                                                              (("1"
                                                                (replace
                                                                 -2
                                                                 :dir
                                                                 rl
                                                                 :actuals?
                                                                 t
                                                                 :hide?
                                                                 t)
                                                                (("1"
                                                                  (replace
                                                                   -1
                                                                   :actuals?
                                                                   t
                                                                   :hide?
                                                                   t)
                                                                  (("1"
                                                                    (replace
                                                                     -1)
                                                                    (("1"
                                                                      (assert)
                                                                      (("1"
                                                                        (case
                                                                         "inv(n!1)(M!1) * (M!1 * (x!1 - c!1))=x!1-c!1")
                                                                        (("1"
                                                                          (replace
                                                                           -1)
                                                                          (("1"
                                                                            (lemma
                                                                             "prod_inv_oper")
                                                                            (("1"
                                                                              (inst
                                                                               -1
                                                                               "Q!1"
                                                                               "transpose(M!1)"
                                                                               "n!1")
                                                                              (("1"
                                                                                (assert)
                                                                                (("1"
                                                                                  (case
                                                                                   "square?(transpose(M!1)) AND
                      squareMat?(n!1)(transpose(M!1)) AND
                       bijective?(n!1)(T(n!1, n!1)(transpose(M!1)))")
                                                                                  (("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"
                                                                                                            (case
                                                                                                             "(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"
                                                                                                                    (case
                                                                                                                     "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)
                                                                                                          nil
                                                                                                          nil))
                                                                                                        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"
                                                                                          (typepred
                                                                                           "M!1")
                                                                                          (("1"
                                                                                            (grind)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil)
                                                                                       ("2"
                                                                                        (hide-all-but
                                                                                         1)
                                                                                        (("2"
                                                                                          (typepred
                                                                                           "M!1")
                                                                                          (("2"
                                                                                            (grind)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil)
                                                                                       ("3"
                                                                                        (lemma
                                                                                         "bijec_transpose")
                                                                                        (("3"
                                                                                          (inst
                                                                                           -1
                                                                                           "M!1"
                                                                                           "n!1")
                                                                                          (("3"
                                                                                            (assert)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil)
                                                                                   ("3"
                                                                                    (flatten)
                                                                                    (("3"
                                                                                      (hide
                                                                                       2)
                                                                                      (("3"
                                                                                        (grind)
                                                                                        nil
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil)
                                                                         ("2"
                                                                          (hide
                                                                           2)
                                                                          (("2"
                                                                            (assert)
                                                                            (("2"
                                                                              (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))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil)
                                                                                 ("2"
                                                                                  (hide
                                                                                   2)
                                                                                  (("2"
                                                                                    (hide-all-but
                                                                                     1)
                                                                                    (("2"
                                                                                      (typepred
                                                                                       "M!1")
                                                                                      (("2"
                                                                                        (grind)
                                                                                        nil
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil)
                                                                         ("3"
                                                                          (hide-all-but
                                                                           1)
                                                                          (("3"
                                                                            (typepred
                                                                             "M!1")
                                                                            (("3"
                                                                              (grind)
                                                                              nil
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (hide-all-but
                                                           1)
                                                          (("2"
                                                            (typepred
                                                             "M!1")
                                                            (("2"
                                                              (grind)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("3"
                                                          (hide-all-but
                                                           1)
                                                          (("3"
                                                            (typepred
                                                             "M!1")
                                                            (("3"
                                                              (grind)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("2"
                                              (split)
                                              (("1"
                                                (hide-all-but 1)
                                                (("1"
                                                  (grind)
                                                  (("1"
                                                    (typepred "M!1")
                                                    (("1"
                                                      (typepred "Q!1")
                                                      (("1"
                                                        (grind)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (hide-all-but 1)
                                                (("2"
                                                  (typepred "M!1")
                                                  (("2"
                                                    (typepred "Q!1")
                                                    (("2"
                                                      (grind)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("3"
                                                (lemma "bijec_prod")
                                                (("3"
                                                  (inst
                                                   -1
                                                   "Q!1"
                                                   "transpose(M!1)"
                                                   "n!1")
                                                  (("1"
                                                    (assert)
                                                    (("1"
                                                      (split)
                                                      (("1"
                                                        (hide-all-but
                                                         1)
                                                        (("1"
                                                          (typepred
                                                           "M!1")
                                                          (("1"
                                                            (grind)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (hide-all-but
                                                         1)
                                                        (("2"
                                                          (typepred
                                                           "M!1")
                                                          (("2"
                                                            (grind)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("3"
                                                        (lemma
                                                         "bijec_transpose")
                                                        (("3"
                                                          (inst
                                                           -1
                                                           "M!1"
                                                           "n!1")
                                                          (("3"
                                                            (assert)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (hide-all-but 1)
                                                    (("2"
                                                      (typepred "M!1")
                                                      (("2"
                                                        (grind)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("3"
                                              (flatten)
                                              (("3"
                                                (hide 2)
                                                (("3"
                                                  (hide-all-but 1)
                                                  (("3"
                                                    (typepred "M!1")
                                                    (("3"
                                                      (typepred "Q!1")
                                                      (("3"
                                                        (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" (hide-all-but 1)
              (("2" (typepred "M!1") (("2" (grind) nil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((add_cancel_neg2 formula-decl nil vectors "vectors/")
    (prod_inv_oper formula-decl nil matrix_operator nil)
    (bijec_prod formula-decl nil matrix_operator nil)
    (mult_assoc_vect formula-decl nil matrices nil)
    (sigma def-decl "real" sigma "reals/")
    (iso_inv formula-decl nil matrix_operator nil)
    (ident_vect formula-decl nil matrices nil)
    (identity? const-decl "bool" matrices nil)
    (I const-decl "(identity?)" matrices nil)
    (trans_mat_scal formula-decl nil matrices nil)
    (tran_inv_oper formula-decl nil matrix_operator nil)
    (transpose2 formula-decl nil matrices nil)
    (inverse const-decl "Maping" linear_map nil)
    (L const-decl "[Map_linear(n, m) -> Mat(m, n)]" matrix_operator
     nil)
    (bijec_transpose formula-decl nil matrix_operator nil)
    (- const-decl "real" vectors "vectors/")
    (inv const-decl "[Matrix_inv(n) -> Matrix_inv(n)]" matrix_operator
     nil)
    (Matrix_inv type-eq-decl nil matrix_operator nil)
    (T const-decl "[Mat(m, n) -> Map_linear(n, m)]" matrix_operator
     nil)
    (Map_linear type-eq-decl nil linear_map_def nil)
    (Mat type-eq-decl nil matrices nil)
    (bijective? const-decl "bool" linear_map nil)
    (Map type-eq-decl nil linear_map nil)
    (Maping type-eq-decl nil linear_map nil)
    (transpose const-decl "Matrix" matrices nil)
    (left_distributive_vect formula-decl nil matrices nil)
    (* const-decl "Vector[M`rows]" matrices nil)
    (Vector type-eq-decl nil vectors "vectors/")
    (Index type-eq-decl nil vectors "vectors/")
    (Matrix type-eq-decl nil matrices nil)
    (square? const-decl "bool" matrices nil)
    (Square type-eq-decl nil matrices nil)
    (squareMat? const-decl "bool" matrices nil)
    (SquareMat type-eq-decl nil matrices nil))
   shostak)
  (ellipsoid-2 "thm" 3521286610
   ("" (skosimp)
    (("" (replace -4)
      (("" (assert)
        (("" (lemma "add_cancel_neg2[n!1]")
          (("" (inst -1 "b!1" "M!1 * x!1")
            (("1" (typepred "M!1")
              (("1" (expand "squareMat?")
                (("1" (replace -2 :actuals? t :hide? t)
                  (("1" (hide -1)
                    (("1" (replace -1 :hide? t)
                      (("1" (lemma "left_distributive_vect")
                        (("1" (inst -1 "M!1" "x!1" "c!1")
                          (("1" (typepred "M!1")
                            (("1" (expand "squareMat?")
                              (("1"
                                (replace -2 :actuals? t :hide? t)
                                (("1"
                                  (hide -1)
                                  (("1"
                                    (replace -1)
                                    (("1"
                                      (lemma "prod_inv_oper")
                                      (("1"
                                        (inst
                                         -1
                                         "M!1"
                                         "Q!1*transpose(M!1)"
                                         "n!1")
                                        (("1"
                                          (assert)
                                          (("1"
                                            (case
                                             "square?(Q!1 * transpose(M!1)) AND
                squareMat?(n!1)(Q!1 * transpose(M!1)) AND
                 bijective?(n!1)(T(n!1, n!1)(Q!1 * transpose(M!1)))")
                                            (("1"
                                              (flatten)
                                              (("1"
                                                (assert)
                                                (("1"
                                                  (hide -1 -2 -3)
                                                  (("1"
                                                    (replace -1)
                                                    (("1"
                                                      (lemma
                                                       "mult_assoc_vect")
                                                      (("1"
                                                        (inst
                                                         -1
                                                         "inv(n!1)(Q!1*transpose(M!1))"
                                                         "inv(n!1)(M!1)"
                                                         "M!1*(x!1-c!1)")
                                                        (("1"
                                                          (reveal -8)
                                                          (("1"
                                                            (expand
                                                             "square?")
                                                            (("1"
                                                              (reveal
                                                               -9)
                                                              (("1"
                                                                (replace
                                                                 -2
                                                                 :dir
                                                                 rl
                                                                 :actuals?
                                                                 t
                                                                 :hide?
                                                                 t)
                                                                (("1"
                                                                  (replace
                                                                   -1
                                                                   :actuals?
                                                                   t
                                                                   :hide?
                                                                   t)
                                                                  (("1"
                                                                    (replace
                                                                     -1)
                                                                    (("1"
                                                                      (assert)
                                                                      (("1"
                                                                        (case
                                                                         "inv(n!1)(M!1) * (M!1 * (x!1 - c!1))=x!1-c!1")
                                                                        (("1"
                                                                          (replace
                                                                           -1)
                                                                          (("1"
                                                                            (lemma
                                                                             "prod_inv_oper")
                                                                            (("1"
                                                                              (inst
                                                                               -1
                                                                               "Q!1"
                                                                               "transpose(M!1)"
                                                                               "n!1")
                                                                              (("1"
                                                                                (assert)
                                                                                (("1"
                                                                                  (case
                                                                                   "square?(transpose(M!1)) AND
                      squareMat?(n!1)(transpose(M!1)) AND
                       bijective?(n!1)(T(n!1, n!1)(transpose(M!1)))")
                                                                                  (("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"
                                                                                                            (case
                                                                                                             "(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"
                                                                                                                    (case
                                                                                                                     "transpose(inv(n!1)(transpose(M!1)))`rows = n!1")
                                                                                                                    (("1"
                                                                                                                      (replace
                                                                                                                       -1
                                                                                                                       :actuals?
                                                                                                                       t
                                                                                                                       :hide?
                                                                                                                       t)
                                                                                                                      (("1"
                                                                                                                        (propax)
                                                                                                                        nil)))
                                                                                                                     ("2"
                                                                                                                      (assert)
                                                                                                                      (("2"
                                                                                                                        (hide-all-but
                                                                                                                         1)
                                                                                                                        (("2"
                                                                                                                          (grind)
                                                                                                                          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)))))))))))))))))))
                                                                                                             ("3"
                                                                                                              (hide-all-but
                                                                                                               1)
                                                                                                              (("3"
                                                                                                                (grind)
                                                                                                                nil)))))))))
                                                                                                       ("2"
                                                                                                        (hide-all-but
                                                                                                         1)
                                                                                                        (("2"
                                                                                                          (grind)
                                                                                                          nil)))))
                                                                                                     ("2"
                                                                                                      (hide-all-but
                                                                                                       1)
                                                                                                      (("2"
                                                                                                        (grind)
                                                                                                        (("1"
                                                                                                          (reveal
                                                                                                           -24)
                                                                                                          (("1"
                                                                                                            (assert)
                                                                                                            nil)))
                                                                                                         ("2"
                                                                                                          (reveal
                                                                                                           -24)
                                                                                                          (("2"
                                                                                                            (assert)
                                                                                                            nil)))))))))))))))))))))))))
                                                                                   ("2"
                                                                                    (hide
                                                                                     2)
                                                                                    (("2"
                                                                                      (split)
                                                                                      (("1"
                                                                                        (hide-all-but
                                                                                         1)
                                                                                        (("1"
                                                                                          (typepred
                                                                                           "M!1")
                                                                                          (("1"
                                                                                            (grind)
                                                                                            nil)))))
                                                                                       ("2"
                                                                                        (hide-all-but
                                                                                         1)
                                                                                        (("2"
                                                                                          (typepred
                                                                                           "M!1")
                                                                                          (("2"
                                                                                            (grind)
                                                                                            nil)))))
                                                                                       ("3"
                                                                                        (lemma
                                                                                         "bijec_transpose")
                                                                                        (("3"
                                                                                          (inst
                                                                                           -1
                                                                                           "M!1"
                                                                                           "n!1")
                                                                                          (("3"
                                                                                            (assert)
                                                                                            nil)))))))))
                                                                                   ("3"
                                                                                    (flatten)
                                                                                    (("3"
                                                                                      (hide
                                                                                       2)
                                                                                      (("3"
                                                                                        (grind)
                                                                                        nil)))))))))))))))
                                                                         ("2"
                                                                          (hide
                                                                           2)
                                                                          (("2"
                                                                            (assert)
                                                                            (("2"
                                                                              (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)))))))))))))))))))
                                                                                 ("2"
                                                                                  (hide
                                                                                   2)
                                                                                  (("2"
                                                                                    (hide-all-but
                                                                                     1)
                                                                                    (("2"
                                                                                      (typepred
                                                                                       "M!1")
                                                                                      (("2"
                                                                                        (grind)
                                                                                        nil)))))))))))))))
                                                                         ("3"
                                                                          (hide-all-but
                                                                           1)
                                                                          (("3"
                                                                            (typepred
                                                                             "M!1")
                                                                            (("3"
                                                                              (grind)
                                                                              nil)))))))))))))))))))))
                                                         ("2"
                                                          (hide-all-but
                                                           1)
                                                          (("2"
                                                            (typepred
                                                             "M!1")
                                                            (("2"
                                                              (grind)
                                                              nil)))))
                                                         ("3"
                                                          (hide-all-but
                                                           1)
                                                          (("3"
                                                            (typepred
                                                             "M!1")
                                                            (("3"
                                                              (grind)
                                                              nil)))))))))))))))))
                                             ("2"
                                              (split)
                                              (("1"
                                                (hide-all-but 1)
                                                (("1"
                                                  (grind)
                                                  (("1"
                                                    (typepred "M!1")
                                                    (("1"
                                                      (typepred "Q!1")
                                                      (("1"
                                                        (grind)
                                                        nil)))))))))
                                               ("2"
                                                (hide-all-but 1)
                                                (("2"
                                                  (typepred "M!1")
                                                  (("2"
                                                    (typepred "Q!1")
                                                    (("2"
                                                      (grind)
                                                      nil)))))))
                                               ("3"
                                                (lemma "bijec_prod")
                                                (("3"
                                                  (inst
                                                   -1
                                                   "Q!1"
                                                   "transpose(M!1)"
                                                   "n!1")
                                                  (("1"
                                                    (assert)
                                                    (("1"
                                                      (split)
                                                      (("1"
                                                        (hide-all-but
                                                         1)
                                                        (("1"
                                                          (typepred
                                                           "M!1")
                                                          (("1"
                                                            (grind)
                                                            nil)))))
                                                       ("2"
                                                        (hide-all-but
                                                         1)
                                                        (("2"
                                                          (typepred
                                                           "M!1")
                                                          (("2"
                                                            (grind)
                                                            nil)))))
                                                       ("3"
                                                        (lemma
                                                         "bijec_transpose")
                                                        (("3"
                                                          (inst
                                                           -1
                                                           "M!1"
                                                           "n!1")
                                                          (("3"
                                                            (assert)
                                                            nil)))))))))
                                                   ("2"
                                                    (hide-all-but 1)
                                                    (("2"
                                                      (typepred "M!1")
                                                      (("2"
                                                        (grind)
                                                        nil)))))))))))
                                             ("3"
                                              (flatten)
                                              (("3"
                                                (hide 2)
                                                (("3"
                                                  (hide-all-but 1)
                                                  (("3"
                                                    (typepred "M!1")
                                                    (("3"
                                                      (typepred "Q!1")
                                                      (("3"
                                                        (grind)
                                                        nil)))))))))))))))))))))))))))))))))))))))))))
             ("2" (hide-all-but 1)
              (("2" (typepred "M!1") (("2" (grind) nil))))))))))))))
    nil)
   ((SquareMat type-eq-decl nil matrices nil)
    (squareMat? const-decl "bool" matrices nil)
    (Square type-eq-decl nil matrices nil)
    (square? const-decl "bool" matrices nil)
    (Matrix type-eq-decl nil matrices nil)
    (Index type-eq-decl nil vectors "vectors/")
    (Vector type-eq-decl nil vectors "vectors/")
    (* const-decl "Vector[M`rows]" matrices nil)
    (left_distributive_vect formula-decl nil matrices nil)
    (transpose const-decl "Matrix" matrices nil)
    (Maping type-eq-decl nil linear_map nil)
    (Map type-eq-decl nil linear_map nil)
    (bijective? const-decl "bool" linear_map nil)
    (Mat type-eq-decl nil matrices nil)
    (Map_linear type-eq-decl nil linear_map_def nil)
    (T const-decl "[Mat(m, n) -> Map_linear(n, m)]" matrix_operator
     nil)
    (Matrix_inv type-eq-decl nil matrix_operator nil)
    (inv const-decl "[Matrix_inv(n) -> Matrix_inv(n)]" matrix_operator
     nil)
    (- const-decl "real" vectors "vectors/")
    (bijec_transpose formula-decl nil matrix_operator nil)
    (L const-decl "[Map_linear(n, m) -> Mat(m, n)]" matrix_operator
     nil)
    (inverse const-decl "Maping" linear_map nil)
    (transpose2 formula-decl nil matrices nil)
    (tran_inv_oper formula-decl nil matrix_operator nil)
    (trans_mat_scal formula-decl nil matrices nil)
    (I const-decl "(identity?)" matrices nil)
    (identity? const-decl "bool" matrices nil)
    (ident_vect formula-decl nil matrices nil)
    (iso_inv formula-decl nil matrix_operator nil)
    (sigma def-decl "real" sigma "reals/")
    (mult_assoc_vect formula-decl nil matrices nil)
    (bijec_prod formula-decl nil matrix_operator nil)
    (prod_inv_oper formula-decl nil matrix_operator nil)
    (add_cancel_neg2 formula-decl nil vectors "vectors/"))
   nil)
  (thm "thm" 3520609934
   ("" (skosimp)
    (("" (replace -4)
      (("" (assert)
        (("" (lemma "add_cancel_neg2[n!1]")
          (("" (inst -1 "b!1" "M!1 * x!1")
            (("1" (typepred "M!1")
              (("1" (expand "squareMat?")
                (("1" (replace -2 :actuals? t :hide? t)
                  (("1" (hide -1)
                    (("1" (replace -1 :hide? t)
                      (("1" (lemma "left_distributive_vect")
                        (("1" (inst -1 "M!1" "x!1" "c!1")
                          (("1" (typepred "M!1")
                            (("1" (expand "squareMat?")
                              (("1"
                                (replace -2 :actuals? t :hide? t)
                                (("1"
                                  (hide -1)
                                  (("1"
                                    (replace -1)
                                    (("1"
                                      (lemma "prod_inv_oper")
                                      (("1"
                                        (inst
                                         -1
                                         "M!1"
                                         "Q!1*transpose(M!1)"
                                         "n!1")
                                        (("1"
                                          (assert)
                                          (("1"
                                            (case
                                             "square?(Q!1 * transpose(M!1)) AND
       squareMat?(n!1)(Q!1 * transpose(M!1)) AND
        bijective?(n!1)(T(n!1, n!1)(Q!1 * transpose(M!1)))")
                                            (("1"
                                              (flatten)
                                              (("1"
                                                (assert)
                                                (("1"
                                                  (hide -1 -2 -3)
                                                  (("1"
                                                    (replace -1)
                                                    (("1"
                                                      (lemma
                                                       "mult_assoc_vect")
                                                      (("1"
                                                        (inst
                                                         -1
                                                         "inv(n!1)(Q!1*transpose(M!1))"
                                                         "inv(n!1)(M!1)"
                                                         "M!1*(x!1-c!1)")
                                                        (("1"
                                                          (reveal -8)
                                                          (("1"
                                                            (expand
                                                             "square?")
                                                            (("1"
                                                              (reveal
                                                               -9)
                                                              (("1"
                                                                (replace
                                                                 -2
                                                                 :dir
                                                                 rl
                                                                 :actuals?
                                                                 t
                                                                 :hide?
                                                                 t)
                                                                (("1"
                                                                  (replace
                                                                   -1
                                                                   :actuals?
                                                                   t
                                                                   :hide?
                                                                   t)
                                                                  (("1"
                                                                    (replace
                                                                     -1)
                                                                    (("1"
                                                                      (assert)
                                                                      (("1"
                                                                        (case
                                                                         "inv(n!1)(M!1) * (M!1 * (x!1 - c!1))=x!1-c!1")
                                                                        (("1"
                                                                          (replace
                                                                           -1)
                                                                          (("1"
                                                                            (lemma
                                                                             "prod_inv_oper")
                                                                            (("1"
                                                                              (inst
                                                                               -1
                                                                               "Q!1"
                                                                               "transpose(M!1)"
                                                                               "n!1")
                                                                              (("1"
                                                                                (assert)
                                                                                (("1"
                                                                                  (case
                                                                                   "square?(transpose(M!1)) AND
       squareMat?(n!1)(transpose(M!1)) AND
        bijective?(n!1)(T(n!1, n!1)(transpose(M!1)))")
                                                                                  (("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"
                                                                                                            (case
                                                                                                             "(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"
                                                                                                                    (case
                                                                                                                     "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)
                                                                                                          nil
                                                                                                          nil))
                                                                                                        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"
                                                                                          (typepred
                                                                                           "M!1")
                                                                                          (("1"
                                                                                            (grind)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil)
                                                                                       ("2"
                                                                                        (hide-all-but
                                                                                         1)
                                                                                        (("2"
                                                                                          (typepred
                                                                                           "M!1")
                                                                                          (("2"
                                                                                            (grind)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil)
                                                                                       ("3"
                                                                                        (lemma
                                                                                         "bijec_transpose")
                                                                                        (("3"
                                                                                          (inst
                                                                                           -1
                                                                                           "M!1"
                                                                                           "n!1")
                                                                                          (("3"
                                                                                            (assert)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil)
                                                                                   ("3"
                                                                                    (flatten)
                                                                                    (("3"
                                                                                      (hide
                                                                                       2)
                                                                                      (("3"
                                                                                        (grind)
                                                                                        nil
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil)
                                                                         ("2"
                                                                          (hide
                                                                           2)
                                                                          (("2"
                                                                            (assert)
                                                                            (("2"
                                                                              (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))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil)
                                                                                 ("2"
                                                                                  (hide
                                                                                   2)
                                                                                  (("2"
                                                                                    (hide-all-but
                                                                                     1)
                                                                                    (("2"
                                                                                      (typepred
                                                                                       "M!1")
                                                                                      (("2"
                                                                                        (grind)
                                                                                        nil
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil)
                                                                         ("3"
                                                                          (hide-all-but
                                                                           1)
                                                                          (("3"
                                                                            (typepred
                                                                             "M!1")
                                                                            (("3"
                                                                              (grind)
                                                                              nil
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (hide-all-but
                                                           1)
                                                          (("2"
                                                            (typepred
                                                             "M!1")
                                                            (("2"
                                                              (grind)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("3"
                                                          (hide-all-but
                                                           1)
                                                          (("3"
                                                            (typepred
                                                             "M!1")
                                                            (("3"
                                                              (grind)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("2"
                                              (split)
                                              (("1"
                                                (hide-all-but 1)
                                                (("1"
                                                  (grind)
                                                  (("1"
                                                    (typepred "M!1")
                                                    (("1"
                                                      (typepred "Q!1")
                                                      (("1"
                                                        (grind)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (hide-all-but 1)
                                                (("2"
                                                  (typepred "M!1")
                                                  (("2"
                                                    (typepred "Q!1")
                                                    (("2"
                                                      (grind)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("3"
                                                (lemma "bijec_prod")
                                                (("3"
                                                  (inst
                                                   -1
                                                   "Q!1"
                                                   "transpose(M!1)"
                                                   "n!1")
                                                  (("1"
                                                    (assert)
                                                    (("1"
                                                      (split)
                                                      (("1"
                                                        (hide-all-but
                                                         1)
                                                        (("1"
                                                          (typepred
                                                           "M!1")
                                                          (("1"
                                                            (grind)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (hide-all-but
                                                         1)
                                                        (("2"
                                                          (typepred
                                                           "M!1")
                                                          (("2"
                                                            (grind)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("3"
                                                        (lemma
                                                         "bijec_transpose")
                                                        (("3"
                                                          (inst
                                                           -1
                                                           "M!1"
                                                           "n!1")
                                                          (("3"
                                                            (assert)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (hide-all-but 1)
                                                    (("2"
                                                      (typepred "M!1")
                                                      (("2"
                                                        (grind)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("3"
                                              (flatten)
                                              (("3"
                                                (hide 2)
                                                (("3"
                                                  (hide-all-but 1)
                                                  (("3"
                                                    (typepred "M!1")
                                                    (("3"
                                                      (typepred "Q!1")
                                                      (("3"
                                                        (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" (hide-all-but 1)
              (("2" (typepred "M!1") (("2" (grind) nil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((SquareMat type-eq-decl nil matrices nil)
    (squareMat? const-decl "bool" matrices nil)
    (Square type-eq-decl nil matrices nil)
    (square? const-decl "bool" matrices nil)
    (Matrix type-eq-decl nil matrices nil)
    (Index type-eq-decl nil vectors "vectors/")
    (Vector type-eq-decl nil vectors "vectors/")
    (* const-decl "Vector[M`rows]" matrices nil)
    (left_distributive_vect formula-decl nil matrices nil)
    (transpose const-decl "Matrix" matrices nil)
    (Maping type-eq-decl nil linear_map nil)
    (Map type-eq-decl nil linear_map nil)
    (bijective? const-decl "bool" linear_map nil)
    (Mat type-eq-decl nil matrices nil)
    (Map_linear type-eq-decl nil linear_map_def nil)
    (T const-decl "[Mat(m, n) -> Map_linear(n, m)]" matrix_operator
     nil)
    (Matrix_inv type-eq-decl nil matrix_operator nil)
    (inv const-decl "[Matrix_inv(n) -> Matrix_inv(n)]" matrix_operator
     nil)
    (- const-decl "real" vectors "vectors/")
    (bijec_transpose formula-decl nil matrix_operator nil)
    (L const-decl "[Map_linear(n, m) -> Mat(m, n)]" matrix_operator
     nil)
    (inverse const-decl "Maping" linear_map nil)
    (transpose2 formula-decl nil matrices nil)
    (tran_inv_oper formula-decl nil matrix_operator nil)
    (trans_mat_scal formula-decl nil matrices nil)
    (I const-decl "(identity?)" matrices nil)
    (identity? const-decl "bool" matrices nil)
    (ident_vect formula-decl nil matrices nil)
    (iso_inv formula-decl nil matrix_operator nil)
    (sigma def-decl "real" sigma "reals/")
    (mult_assoc_vect formula-decl nil matrices nil)
    (bijec_prod formula-decl nil matrix_operator nil)
    (prod_inv_oper formula-decl nil matrix_operator nil)
    (add_cancel_neg2 formula-decl nil vectors "vectors/"))
   shostak)
  (ellipsoid-1 nil 3504886332
   ("" (skosimp)
    (("" (replace -4)
      (("" (assert)
        (("" (lemma "left_distributive_vect_n")
          (("" (inst -1 "M!1" "x!1" "c!1")
            (("" (swap-rel -1)
              (("" (replace -1)
                (("" (lemma "prod_inv_oper")
                  (("" (inst -1 "M!1" "Q!1*transpose(M!1)" "n")
                    (("" (assert)
                      ((""
                        (case "    square?(M!1) AND squareMat?(n)(M!1)
       AND square?(Q!1 * transpose(M!1))
       AND squareMat?(n)(Q!1 * transpose(M!1))
       AND bijective?(n)(T(n, n)(Q!1 * transpose(M!1)))")
                        (("1" (flatten)
                          (("1" (assert)
                            (("1" (hide -1 -2 -3 -4 -5)
                              (("1"
                                (replace -1)
                                (("1"
                                  (lemma "mult_assoc_vect_n")
                                  (("1"
                                    (inst
                                     -1
                                     "inv(n)(Q!1*transpose(M!1))"
                                     "inv(n)(M!1)"
                                     "M!1*(x!1-c!1)")
                                    (("1"
                                      (replace -1)
                                      (("1"
                                        (assert)
                                        (("1"
                                          (case
                                           "inv(n)(M!1) * (M!1 * (x!1 - c!1))=x!1-c!1")
                                          (("1"
                                            (replace -1)
                                            (("1"
                                              (lemma "prod_inv_oper")
                                              (("1"
                                                (inst
                                                 -1
                                                 "Q!1"
                                                 "transpose(M!1)"
                                                 "n")
                                                (("1"
                                                  (assert)
                                                  (("1"
                                                    (case
                                                     "square?(Q!1) AND squareMat?(n)(Q!1) AND square?(transpose(M!1))
       AND squareMat?(n)(transpose(M!1))
       AND bijective?(n)(T(n, n)(transpose(M!1)))")
                                                    (("1"
                                                      (flatten)
                                                      (("1"
                                                        (assert)
                                                        (("1"
                                                          (hide
                                                           -1
                                                           -2
                                                           -3
                                                           -4
                                                           -5)
                                                          (("1"
                                                            (replace
                                                             -1)
                                                            (("1"
                                                              (lemma
                                                               "mult_assoc_vect_n")
                                                              (("1"
                                                                (inst
                                                                 -1
                                                                 "inv(n)(transpose(M!1))"
                                                                 "inv(n)(Q!1)"
                                                                 "x!1-c!1")
                                                                (("1"
                                                                  (replace
                                                                   -1)
                                                                  (("1"
                                                                    (lemma
                                                                     "trans_mat_scal_n")
                                                                    (("1"
                                                                      (inst
                                                                       -1
                                                                       "inv(n)(transpose(M!1))"
                                                                       "M!1*(x!1-c!1)"
                                                                       "inv(n)(Q!1)*(x!1-c!1)")
                                                                      (("1"
                                                                        (replace
                                                                         -1)
                                                                        (("1"
                                                                          (case
                                                                           "(transpose(inv(n)(transpose(M!1))) * (M!1 * (x!1 - c!1)))=x!1-c!1")
                                                                          (("1"
                                                                            (replace
                                                                             -1)
                                                                            (("1"
                                                                              (propax)
                                                                              nil
                                                                              nil))
                                                                            nil)
                                                                           ("2"
                                                                            (hide
                                                                             -1
                                                                             -2
                                                                             -3)
                                                                            (("2"
                                                                              (hide
                                                                               2)
                                                                              (("2"
                                                                                (lemma
                                                                                 "tran_inv_oper")
                                                                                (("2"
                                                                                  (inst
                                                                                   -1
                                                                                   "M!1"
                                                                                   "n")
                                                                                  (("2"
                                                                                    (assert)
                                                                                    (("2"
                                                                                      (replace
                                                                                       -1)
                                                                                      (("2"
                                                                                        (lemma
                                                                                         "transpose2")
                                                                                        (("2"
                                                                                          (inst
                                                                                           -1
                                                                                           "inv(n)(M!1)")
                                                                                          (("2"
                                                                                            (replace
                                                                                             -1)
                                                                                            (("2"
                                                                                              (propax)
                                                                                              nil
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil)
                                                                 ("2"
                                                                  (hide
                                                                   2)
                                                                  (("2"
                                                                    (grind)
                                                                    nil
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (hide 2)
                                                      (("2"
                                                        (split)
                                                        (("1"
                                                          (assert)
                                                          (("1"
                                                            (grind)
                                                            nil
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (grind)
                                                          nil
                                                          nil)
                                                         ("3"
                                                          (grind)
                                                          nil
                                                          nil)
                                                         ("4"
                                                          (grind)
                                                          nil
                                                          nil)
                                                         ("5"
                                                          (lemma
                                                           "bijec_transpose")
                                                          (("5"
                                                            (inst
                                                             -1
                                                             "M!1"
                                                             "n")
                                                            (("5"
                                                              (assert)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("3"
                                                      (flatten)
                                                      (("3"
                                                        (hide 2)
                                                        (("3"
                                                          (grind)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (hide 2)
                                            (("2"
                                              (assert)
                                              (("2"
                                                (lemma
                                                 "mult_assoc_vect_n")
                                                (("2"
                                                  (inst
                                                   -1
                                                   "inv(n)(M!1)"
                                                   "M!1"
                                                   "x!1-c!1")
                                                  (("2"
                                                    (swap-rel -1)
                                                    (("2"
                                                      (replace -1)
                                                      (("2"
                                                        (assert)
                                                        (("2"
                                                          (lemma
                                                           "iso_inv")
                                                          (("2"
                                                            (inst
                                                             -1
                                                             "M!1"
                                                             "n")
                                                            (("2"
                                                              (assert)
                                                              (("2"
                                                                (flatten)
                                                                (("2"
                                                                  (replace
                                                                   -1)
                                                                  (("2"
                                                                    (assert)
                                                                    (("2"
                                                                      (postpone)
                                                                      nil
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil)
                                     ("2"
                                      (hide 2)
                                      (("2" (grind) nil nil))
                                      nil)
                                     ("3"
                                      (hide 2)
                                      (("3" (grind) nil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (split)
                          (("1" (grind) nil nil) ("2" (grind) nil nil)
                           ("3" (grind) nil nil) ("4" (grind) nil nil)
                           ("5" (lemma "bijec_prod")
                            (("5" (inst -1 "Q!1" "transpose(M!1)" "n")
                              (("1"
                                (assert)
                                (("1"
                                  (split)
                                  (("1" (grind) nil nil)
                                   ("2" (grind) nil nil)
                                   ("3" (grind) nil nil)
                                   ("4" (grind) nil nil)
                                   ("5"
                                    (lemma "bijec_transpose")
                                    (("5"
                                      (inst -1 "M!1" "n")
                                      (("5"
                                        (assert)
                                        (("5"
                                          (hide 2 3 4)
                                          (("5" (grind) nil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("2" (grind) nil nil))
                              nil))
                            nil))
                          nil)
                         ("3" (flatten)
                          (("3" (hide 2) (("3" (grind) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((add_cancel_neg2 formula-decl nil vectors "vectors/")
    (Matrix type-eq-decl nil matrices nil)
    (Mat type-eq-decl nil matrices nil)
    (Index type-eq-decl nil vectors "vectors/")
    (Vector type-eq-decl nil vectors "vectors/")
    (transpose const-decl "Matrix" matrices nil)
    (square? const-decl "bool" matrices nil)
    (Square type-eq-decl nil matrices nil)
    (squareMat? const-decl "bool" matrices nil)
    (Maping type-eq-decl nil linear_map nil)
    (Map type-eq-decl nil linear_map nil)
    (bijective? const-decl "bool" linear_map nil)
    (T const-decl "[Mat(m, n) -> Map_linear(n, m)]" matrix_operator
     nil)
    (Matrix_inv type-eq-decl nil matrix_operator nil)
    (inv const-decl "[Matrix_inv(n) -> Matrix_inv(n)]" matrix_operator
     nil)
    (I const-decl "(identity?)" matrices nil)
    (identity? const-decl "bool" matrices nil)
    (iso_inv formula-decl nil matrix_operator nil)
    (tran_inv_oper formula-decl nil matrix_operator nil)
    (transpose2 formula-decl nil matrices nil)
    (L const-decl "[Map_linear(n, m) -> Mat(m, n)]" matrix_operator
     nil)
    (inverse const-decl "Maping" linear_map nil)
    (sigma def-decl "real" sigma "reals/")
    (bijec_transpose formula-decl nil matrix_operator nil)
    (bijec_prod formula-decl nil matrix_operator nil)
    (prod_inv_oper formula-decl nil matrix_operator nil)
    (- const-decl "real" vectors "vectors/"))
   shostak)))


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

¤ Dauer der Verarbeitung: 0.278 Sekunden  (vorverarbeitet am  2026-04-25) ¤

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