Quelle tensor_product.pvs Sprache: PVS

tensor_product  % [ parameters ]
  : THEORY

  BEGIN



IMPORTING matrices, matrix_props, matrix_inv, linear_dependence,  reals@base_repr, ints@mod_lems, reals@product

m1, m2, n1, n2: VAR nat

a,b: VAR {x:nat | x>1}

Q: VAR Matrix

M, N:  VAR FullMatrix

A, B, A2, B2: VAR PosFullMatrix

AA, BB: VAR Square

v1, v2: VAR Vector

f,g, F: VAR [nat->real]

not_null: LEMMA nonempty?(Q) IMPLIES NOT null?(Q)
mod_int: LEMMA FORALL (k:nat, n:posnat):
               integer_pred((k-mod(k,n))/n) AND (k-mod(k,n))/n>=0

tensor_fun(A,B): [[nat,nat]->real] =
   LAMBDA(i,j:nat): IF i<rows(A)*rows(B) THEN
         IF j<columns(A)*columns(B) THEN
           entry(A)((i-mod(i, rows(B)))/rows(B), (j-mod(j, columns(B)))/columns(B))*
           entry(B)(mod(i, rows(B)), mod(j, columns(B)))
        ELSE 0 ENDIF
      ELSE 0 ENDIF

tensor_prod(A,B): PosFullMatrix = form_matrix(tensor_fun(A,B), rows(A)*rows(B), columns(A)*columns(B))

entry_tensor_prod: LEMMA entry(tensor_prod(A,B)) = tensor_fun(A,B)

tensor_rows: LEMMA rows(tensor_prod(A,B)) = rows(A)*rows(B)

tensor_cols: LEMMA columns(tensor_prod(A,B)) = columns(A)*columns(B)

tensor_mult_entry: LEMMA columns(A) = rows(A2) AND columns(B) = rows(B2)
     IMPLIES entry(tensor_prod(A,B)*tensor_prod(A2, B2))(m1, n1) =
                   entry(A*A2)((m1-mod(m1, rows(B)))/rows(B), (n1-mod(n1, columns(B2)))/columns(B2))*
     entry(B*B2)(mod(m1, rows(B)), mod(n1, columns(B2)))


invertible_tensor: LEMMA invertible?(AA) AND invertible?(BB) IMPLIES
                   invertible?(tensor_prod(AA,BB)) AND
                   inverse(tensor_prod(AA,BB)) = tensor_prod(inverse(AA), inverse(BB))

TQMat: Square = (:(:1,1,1:), (:0,1,-1:), (:0,1,1:):)

TQMatInv: Square = (:(:1,0,-1:), (:0, 1/2, 1/2:), (:0, -1/2, 1/2:):)

invTQ: LEMMA invertible?(TQMat)

is_invTQ: LEMMA inverse(TQMat) = TQMatInv

tensor_power(A:PosFullMatrix,  n:posnat): RECURSIVE PosFullMatrix = IF n=1 THEN A ELSE tensor_prod(A, tensor_power(A, n-1)) ENDIF
                          MEASURE n

invertible_tensor_power: LEMMA FORALL(A:Square, n: posnat): invertible?(A) IMPLIES invertible?(tensor_power(A,n)) AND
                         inverse(tensor_power(A,n)) = tensor_power(inverse(A), n)

tensor_power_rows: LEMMA FORALL(A:PosFullMatrix, n:posnat):
                   rows(A)^n = rows(tensor_power(A,n))

tensor_power_columns: LEMMA FORALL(A:PosFullMatrix, n:posnat):
       columns(A)^n = columns(tensor_power(A,n))

mod_eq_lem_alt: LEMMA FORALL (n,m:posnat,i:nat):
   (mod(i,n)+n*mod((i-mod(i,n))/n,m)-
  mod(mod(i,n)+n*mod((i-mod(i,n))/n,m),n))/n
   =mod((i-mod(i,n))/n,m)

tensor_prod_assoc: LEMMA FORALL (A,B,C:PosFullMatrix):
  tensor_prod(A,tensor_prod(B,C)) = tensor_prod(tensor_prod(A,B),C)

tensor_power_alt(A:PosFullMatrix, n:posnat):  RECURSIVE PosFullMatrix = IF n=1 THEN A ELSE tensor_prod(tensor_power_alt(A, n-1), A) ENDIF
                          MEASURE n

%% Prove this one...

power_assoc: LEMMA FORALL (A:PosFullMatrix, n:posnat):
      tensor_power(A, n) = tensor_power_alt(A, n)

tensor_power_rows_alt: LEMMA FORALL(A:PosFullMatrix, n:posnat):
                   rows(A)^n = rows(tensor_power_alt(A,n))

tensor_power_columns_alt: LEMMA FORALL(A:PosFullMatrix, n:posnat):
       columns(A)^n = columns(tensor_power_alt(A,n))

TQXL:    Square = (: (: 1,   1,   1, 0, 0, 0 :),
                    (: 0,   1,  -1, 0, 0, 0 :),
       (: 0,   1,   1, 0, 0, 0 :),
       (: 0,  -1,  -1, 1, 0, 0 :),
       (:-1,  -1,   0, 0, 1, 0 :),
       (:-1,   0,  -1, 0, 0, 1 :) :)

TQXLinv: Square = (: (: 1,   0,  -1, 0, 0, 0 :),
                    (: 0, 1/2, 1/2, 0, 0, 0 :),
       (: 0,-1/2, 1/2, 0, 0, 0 :),
       (: 0,   0,   1, 1, 0, 0 :),
       (: 1, 1/2,-1/2, 0, 1, 0 :),
       (: 1,-1/2,-1/2, 0, 0, 1 :) :)

invTQXL: LEMMA invertible?(TQXL) AND invertible?(TQXLinv)

is_invTQXL: LEMMA inverse(TQXL) = TQXLinv AND inverse(TQXLinv) = TQXL

RowToMat(M:PosFullMatrix, k:below(rows(M))): PosFullMatrix = (:row(M)(k):)

RtM: LEMMA FORALL (M:PosFullMatrix, k:below(rows(M))):
     RowToMat(M, k)  = row2mat(M, k)

RowToMat_rows: LEMMA FORALL (M:PosFullMatrix, k:below(rows(M))):
         rows(RowToMat(M,k))=1

RowToMat_columns: LEMMA FORALL (M:PosFullMatrix, k:below(rows(M))):
         columns(RowToMat(M,k))=columns(M)

RowToMat_entry: LEMMA FORALL (M:PosFullMatrix, k:below(rows(M)), n:below(columns(M))):
             entry(RowToMat(M, k))(0,n) = entry(M)(k,n)

RowToMat_tensor_prod: LEMMA FORALL(M, N:PosFullMatrix, k:below(rows(M)*rows(N))):
         RowToMat(tensor_prod(M, N), k) =
  tensor_prod(RowToMat(M, (k-mod(k,rows(N)))/rows(N)), RowToMat(N, mod(k,rows(N))))

RowTensor(M:PosFullMatrix, n: posnat, L:{ LL: list[below(rows(M))] | length(LL)=n}): RECURSIVE PosFullMatrix =
       IF n=1 THEN RowToMat(M, car(L))
       ELSE tensor_prod( RowTensor(M, n-1, cdr(L)), RowToMat(M,car(L))) ENDIF
       MEASURE n

RowTensorAlt(M:PosFullMatrix, n:posnat, f:[nat->below(rows(M))]): RECURSIVE PosFullMatrix =
              IF n=1 THEN row2mat(M, f(0))
       ELSE tensor_prod(RowTensorAlt(M, n-1, LAMBDA(j:nat):f(j+1)), row2mat(M, f(0)))
       ENDIF
       MEASURE n

RowTensors_same: LEMMA FORALL (M:{AA:PosFullMatrix|rows(AA)>1}, n:posnat, k:below(rows(M)^n)):
   RowTensor(M, n, base_list(rows(M), k, n)) = RowTensorAlt(M, n, base_n(rows(M), k))

RowTensor_is_TensorRow: LEMMA FORALL (M:PosFullMatrix, n:posnat, k:below(rows(M)^n)):
             rows(M)>1 IMPLIES
      RowToMat(tensor_power_alt(M, n), k) = RowTensor(M, n, base_list(rows(M), k , n))

RowTensor_is_TensorRow2: LEMMA FORALL (M:{AA:PosFullMatrix|rows(AA)>1}, n:posnat, k:below(rows(M)^n)):
    row2mat(tensor_power(M, n), k) = RowTensorAlt(M, n, base_n(rows(M), k))

entry_pick(n, rdim, cdim:posnat, A:{AA:PosFullMatrix |rows(AA)=rdim AND columns(AA)=cdim},
       R, C: [nat->nat])(i:nat):
          real = entry(A)(R(i), C(i))

tensor_entry: LEMMA FORALL (A:{M:PosFullMatrix | rows(M)>1 AND columns(M)>1}, n:posnat, k:below(rows(A)^n), m:below(columns(A)^n)):
       LET R = base_n(rows(A), k),
           C = base_n(columns(A), m) IN
           entry(tensor_power_alt(A,n))(k,m) = product[nat](0, n-1, entry_pick(n, rows(A), columns(A), A, R, C))

tensor_entry_alt: LEMMA FORALL (A:{M:PosFullMatrix | rows(M)>1 AND columns(M)>1}, n:posnat, k:below(rows(A)^n), m:below(columns(A)^n)):
       LET R = base_n(rows(A), k),
           C = base_n(columns(A), m) IN
           entry(tensor_power_alt(A,n))(k,m) = product[nat](0, n-1, entry_pick(n, rows(A), columns(A), A, R, C))





  END tensor_product

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Beweissystem der NASA

Beweissystem Isabelle

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Cephes Mathematical Library

Wiener Entwicklungsmethode

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