Quelle tarski_query_matrix.pvs Sprache: PVS

tarski_query_matrix: THEORY
BEGIN

IMPORTING tarski_query

  a,r,g : VAR [nat->int]
  n : VAR posnat
  d,i,j,k : VAR nat
  m: VAR posnat
  %m : VAR posnat
  x,y,c,b : VAR real
  rel: VAR [[real,real]->bool]
  babove,bbelow,bbelow2,babove2: VAR bool
  p : VAR nat

  GF: VAR [nat->[nat->int]]
  KF: VAR [nat->nat] % degrees
  GP: VAR [nat->subrange(0,2)]
  RelF,RelF1,RelF2: VAR [nat->subrange(-1,1)]
  ll: VAR list[int]

  TQ_vect3k(k,a,(n|a(n)/=0),GF,(KF|FORALL (i:upto(k)):GF(i)(KF(i))/=0)): {v: PosFullMatrix | rows(v)=3^(k+1) AND columns(v)=1} =
    LET GPFun = (LAMBDA (i,j): base_n(3,i)),
     eij = (LAMBDA (i,j): IF i>=3^(k+1) OR j>=1 THEN 0 ELSE TQ_fam(k,a,n,GF,KF,GPFun(i,j)) ENDIF)
    IN form_matrix(eij,3^(k+1),1)

  TQ_vect6k(k,a,(n|a(n)/=0),GF,(KF|FORALL (i:upto(k)):GF(i)(KF(i))/=0)): {v: PosFullMatrix | rows(v)=6^(k+1) AND columns(v)=1} =
    LET GPFun = (LAMBDA (i,j): base_n(6,i)),
     eij = (LAMBDA (i,j): IF i>=6^(k+1) OR j>=1 OR (EXISTS (p:upto(k)): GPFun(i,j)(p)>2) THEN 0 ELSE TQ_fam(k,a,n,GF,KF,GPFun(i,j)) ENDIF)
    IN form_matrix(eij,6^(k+1),1)

  NSol_vect3k(k,a,(n|a(n)/=0),GF,(KF|FORALL (i:upto(k)):GF(i)(KF(i))/=0)): {v: PosFullMatrix | rows(v)=3^(k+1) AND columns(v)=1} =
    LET eij = (LAMBDA (i,j): IF i>=3^(k+1) OR j>=1 THEN 0 ELSE NSol(k,a,n,GF,KF,sig_seq(base_n(3,i))) ENDIF)
    IN form_matrix(eij,3^(k+1),1)

  NSol_vect6k(k,a,(n|a(n)/=0),GF,(KF|FORALL (i:upto(k)):GF(i)(KF(i))/=0)): {v: PosFullMatrix | rows(v)=6^(k+1) AND columns(v)=1} =
    LET eij = (LAMBDA (i,j): IF i>=6^(k+1) OR j>=1 THEN 0 ELSE NSol_all(k,a,n,GF,KF,base_n(6,i)) ENDIF)
    IN form_matrix(eij,6^(k+1),1)

  A66_inv: {M: PosFullMatrix | rows(M)=6 AND columns(M)=6} =
    (: (: 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 :) :)

  A66: {M: PosFullMatrix | rows(M)=6 AND columns(M)=6 AND M = inverse(A66_inv)} =
   (: (: 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 :) :)

  multi_sturm_tarski_6by6: LEMMA a(n)/=0 AND
    (FORALL (i:upto(k)):GF(i)(KF(i))/=0)
    IMPLIES
    TQ_vect6k(k,a,n,GF, KF) = tensor_power_alt(A66_inv,k+1)*
    NSol_vect6k(k,a,n,GF,KF)

  multi_sturm_tarski_NSol: LEMMA a(n)/=0 AND
    (FORALL (i:upto(k)):GF(i)(KF(i))/=0)
    IMPLIES
    NSol_vect6k(k,a,n,GF,KF) = tensor_power_alt(A66,k+1)*
    TQ_vect6k(k,a,n,GF, KF)

  A63: {M: PosFullMatrix | rows(M)=6 AND columns(M)=3} =
   (: (: 1, 0, -1  :), (: 0, 1/2, 1/2  :),
   (: 0, -1/2, 1/2  :), (: 0, 0, 1 :),
   (: 1, 1/2, -1/2 :), (: 1, -1/2, -1/2 :) :)

  A63_def: LEMMA FORALL (i:upto(5),j:upto(2)):
    entry(A63)(i,j) = entry(A66)(i,j)

  multi_sturm_tarski_NSol63: LEMMA a(n)/=0 AND
    (FORALL (i:upto(k)):GF(i)(KF(i))/=0)
    IMPLIES
    NSol_vect6k(k,a,n,GF,KF) = tensor_power_alt(A63,k+1)*
    TQ_vect3k(k,a,n,GF, KF)

END tarski_query_matrix

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