Left Termination of the query pattern transpose_aux_in_3(a, g, a) w.r.t. the given Prolog program could not be shown:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 PiDP
    ↳7 UsableRulesProof (⇔)
    ↳8 PiDP
    ↳9 PiDPToQDPProof (⇐)
    ↳10 QDP
      ↳11 Instantiation (⇔)
        ↳12 QDP
      ↳13 Instantiation (⇔)
        ↳14 QDP
        ↳15 Instantiation (⇔)
        ↳16 QDP
        ↳17 NonTerminationProof (⇔)
        ↳18 FALSE
  ↳19 PrologToPiTRSProof (⇐)
    ↳20 PiTRS
    ↳21 DependencyPairsProof (⇔)
    ↳22 PiDP
    ↳23 DependencyGraphProof (⇔)
    ↳24 PiDP
    ↳25 UsableRulesProof (⇔)
    ↳26 PiDP
    ↳27 PiDPToQDPProof (⇐)
    ↳28 QDP
      ↳29 Instantiation (⇔)
        ↳30 QDP
      ↳31 Instantiation (⇔)
        ↳32 QDP
        ↳33 Instantiation (⇔)
        ↳34 QDP
        ↳35 NonTerminationProof (⇔)
        ↳36 FALSE

(0) Obligation:

Clauses:

transpose_aux(.(R, Rs), X1, .(C, Cs)) :- row2col(R, .(C, Cs), Cols1, [], Accm).
row2col(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) :- row2col(Xs, Cols, Cols1, .([], A), B).

Queries:

transpose_aux(a,g,a).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
transpose_aux_in: (f,b,f)
row2col_in: (f,f,f,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) → U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x3, x6)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x6, x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x4, x5)
.(x1, x2)  =  .(x1, x2)
[]  =  []
transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) → U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x3, x6)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x6, x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x4, x5)
.(x1, x2)  =  .(x1, x2)
[]  =  []
transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga(x2)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

TRANSPOSE_AUX_IN_AGA(.(R, Rs), X1, .(C, Cs)) → U1_AGA(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
TRANSPOSE_AUX_IN_AGA(.(R, Rs), X1, .(C, Cs)) → ROW2COL_IN_AAAGA(R, .(C, Cs), Cols1, [], Accm)
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_AAAGA(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)

The TRS R consists of the following rules:

transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) → U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x3, x6)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x6, x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x4, x5)
.(x1, x2)  =  .(x1, x2)
[]  =  []
transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga(x2)
TRANSPOSE_AUX_IN_AGA(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5, x6)  =  U1_AGA(x3, x6)
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)
U2_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_AAAGA(x6, x8)

We have to consider all (P,R,Pi)-chains

(4) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

TRANSPOSE_AUX_IN_AGA(.(R, Rs), X1, .(C, Cs)) → U1_AGA(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
TRANSPOSE_AUX_IN_AGA(.(R, Rs), X1, .(C, Cs)) → ROW2COL_IN_AAAGA(R, .(C, Cs), Cols1, [], Accm)
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_AAAGA(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)

The TRS R consists of the following rules:

transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) → U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x3, x6)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x6, x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x4, x5)
.(x1, x2)  =  .(x1, x2)
[]  =  []
transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga(x2)
TRANSPOSE_AUX_IN_AGA(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5, x6)  =  U1_AGA(x3, x6)
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)
U2_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_AAAGA(x6, x8)

We have to consider all (P,R,Pi)-chains

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)

The TRS R consists of the following rules:

transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) → U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x3, x6)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x6, x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x4, x5)
.(x1, x2)  =  .(x1, x2)
[]  =  []
transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga(x2)
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)

We have to consider all (P,R,Pi)-chains

(7) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(8) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)

We have to consider all (P,R,Pi)-chains

(9) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ROW2COL_IN_AAAGA(A) → ROW2COL_IN_AAAGA(.([], A))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(11) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule ROW2COL_IN_AAAGA(A) → ROW2COL_IN_AAAGA(.([], A)) we obtained the following new rules [LPAR04]:

ROW2COL_IN_AAAGA(.([], z0)) → ROW2COL_IN_AAAGA(.([], .([], z0)))

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ROW2COL_IN_AAAGA(.([], z0)) → ROW2COL_IN_AAAGA(.([], .([], z0)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(13) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule ROW2COL_IN_AAAGA(A) → ROW2COL_IN_AAAGA(.([], A)) we obtained the following new rules [LPAR04]:

ROW2COL_IN_AAAGA(.([], z0)) → ROW2COL_IN_AAAGA(.([], .([], z0)))

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ROW2COL_IN_AAAGA(.([], z0)) → ROW2COL_IN_AAAGA(.([], .([], z0)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(15) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule ROW2COL_IN_AAAGA(.([], z0)) → ROW2COL_IN_AAAGA(.([], .([], z0))) we obtained the following new rules [LPAR04]:

ROW2COL_IN_AAAGA(.([], .([], z0))) → ROW2COL_IN_AAAGA(.([], .([], .([], z0))))

(16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ROW2COL_IN_AAAGA(.([], .([], z0))) → ROW2COL_IN_AAAGA(.([], .([], .([], z0))))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(17) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = ROW2COL_IN_AAAGA(.([], .([], z0))) evaluates to t =ROW2COL_IN_AAAGA(.([], .([], .([], z0))))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [z0 / .([], z0)]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ROW2COL_IN_AAAGA(.([], .([], z0))) to ROW2COL_IN_AAAGA(.([], .([], .([], z0)))).



(18) FALSE

(19) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
transpose_aux_in: (f,b,f)
row2col_in: (f,f,f,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) → U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x6)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x5)
.(x1, x2)  =  .(x1, x2)
[]  =  []
transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(20) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) → U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x6)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x5)
.(x1, x2)  =  .(x1, x2)
[]  =  []
transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga

(21) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

TRANSPOSE_AUX_IN_AGA(.(R, Rs), X1, .(C, Cs)) → U1_AGA(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
TRANSPOSE_AUX_IN_AGA(.(R, Rs), X1, .(C, Cs)) → ROW2COL_IN_AAAGA(R, .(C, Cs), Cols1, [], Accm)
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_AAAGA(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)

The TRS R consists of the following rules:

transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) → U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x6)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x5)
.(x1, x2)  =  .(x1, x2)
[]  =  []
transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga
TRANSPOSE_AUX_IN_AGA(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5, x6)  =  U1_AGA(x6)
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)
U2_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_AAAGA(x8)

We have to consider all (P,R,Pi)-chains

(22) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

TRANSPOSE_AUX_IN_AGA(.(R, Rs), X1, .(C, Cs)) → U1_AGA(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
TRANSPOSE_AUX_IN_AGA(.(R, Rs), X1, .(C, Cs)) → ROW2COL_IN_AAAGA(R, .(C, Cs), Cols1, [], Accm)
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_AAAGA(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)

The TRS R consists of the following rules:

transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) → U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x6)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x5)
.(x1, x2)  =  .(x1, x2)
[]  =  []
transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga
TRANSPOSE_AUX_IN_AGA(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5, x6)  =  U1_AGA(x6)
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)
U2_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_AAAGA(x8)

We have to consider all (P,R,Pi)-chains

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(24) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)

The TRS R consists of the following rules:

transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) → U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x6)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x5)
.(x1, x2)  =  .(x1, x2)
[]  =  []
transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)

We have to consider all (P,R,Pi)-chains

(25) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(26) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)

We have to consider all (P,R,Pi)-chains

(27) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(28) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ROW2COL_IN_AAAGA(A) → ROW2COL_IN_AAAGA(.([], A))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(29) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule ROW2COL_IN_AAAGA(A) → ROW2COL_IN_AAAGA(.([], A)) we obtained the following new rules [LPAR04]:

ROW2COL_IN_AAAGA(.([], z0)) → ROW2COL_IN_AAAGA(.([], .([], z0)))

(30) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ROW2COL_IN_AAAGA(.([], z0)) → ROW2COL_IN_AAAGA(.([], .([], z0)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(31) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule ROW2COL_IN_AAAGA(A) → ROW2COL_IN_AAAGA(.([], A)) we obtained the following new rules [LPAR04]:

ROW2COL_IN_AAAGA(.([], z0)) → ROW2COL_IN_AAAGA(.([], .([], z0)))

(32) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ROW2COL_IN_AAAGA(.([], z0)) → ROW2COL_IN_AAAGA(.([], .([], z0)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(33) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule ROW2COL_IN_AAAGA(.([], z0)) → ROW2COL_IN_AAAGA(.([], .([], z0))) we obtained the following new rules [LPAR04]:

ROW2COL_IN_AAAGA(.([], .([], z0))) → ROW2COL_IN_AAAGA(.([], .([], .([], z0))))

(34) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ROW2COL_IN_AAAGA(.([], .([], z0))) → ROW2COL_IN_AAAGA(.([], .([], .([], z0))))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(35) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = ROW2COL_IN_AAAGA(.([], .([], z0))) evaluates to t =ROW2COL_IN_AAAGA(.([], .([], .([], z0))))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [z0 / .([], z0)]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ROW2COL_IN_AAAGA(.([], .([], z0))) to ROW2COL_IN_AAAGA(.([], .([], .([], z0)))).



(36) FALSE