(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) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph.
(2) Obligation:
Clauses:row2col29(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) :- row2col29(T182, T183, X397, .([], T181), X398).
transpose_aux1(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) :- row2col29(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346).
Queries:
transpose_aux1(a,g,a).
(3) 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_aux1_in: (f,b,f)
row2col29_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_aux1_in_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) → U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_in_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346))
row2col29_in_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_in_aaaga(T182, T183, X397, .([], T181), X398))
U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_out_aaaga(T182, T183, X397, .([], T181), X398)) → row2col29_out_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398)
U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_out_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346)) → transpose_aux1_out_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156)))))))))
The argument filtering Pi contains the following mapping:
transpose_aux1_in_aga(x1, x2, x3) = transpose_aux1_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U2_aga(x21)
row2col29_in_aaaga(x1, x2, x3, x4, x5) = row2col29_in_aaaga(x4)
U1_aaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U1_aaaga(x8)
row2col29_out_aaaga(x1, x2, x3, x4, x5) = row2col29_out_aaaga(x5)
.(x1, x2) = .(x1, x2)
[] = []
transpose_aux1_out_aga(x1, x2, x3) = transpose_aux1_out_aga
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(4) Obligation:
Pi-finite rewrite system:The TRS R consists of the following rules:
transpose_aux1_in_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) → U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_in_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346))
row2col29_in_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_in_aaaga(T182, T183, X397, .([], T181), X398))
U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_out_aaaga(T182, T183, X397, .([], T181), X398)) → row2col29_out_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398)
U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_out_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346)) → transpose_aux1_out_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156)))))))))
The argument filtering Pi contains the following mapping:
transpose_aux1_in_aga(x1, x2, x3) = transpose_aux1_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U2_aga(x21)
row2col29_in_aaaga(x1, x2, x3, x4, x5) = row2col29_in_aaaga(x4)
U1_aaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U1_aaaga(x8)
row2col29_out_aaaga(x1, x2, x3, x4, x5) = row2col29_out_aaaga(x5)
.(x1, x2) = .(x1, x2)
[] = []
transpose_aux1_out_aga(x1, x2, x3) = transpose_aux1_out_aga
(5) 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_AUX1_IN_AGA(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) → U2_AGA(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_in_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346))
TRANSPOSE_AUX1_IN_AGA(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) → ROW2COL29_IN_AAAGA(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346)
ROW2COL29_IN_AAAGA(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → U1_AAAGA(T177, T182, T179, T183, X397, T181, X398, row2col29_in_aaaga(T182, T183, X397, .([], T181), X398))
ROW2COL29_IN_AAAGA(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → ROW2COL29_IN_AAAGA(T182, T183, X397, .([], T181), X398)
The TRS R consists of the following rules:
transpose_aux1_in_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) → U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_in_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346))
row2col29_in_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_in_aaaga(T182, T183, X397, .([], T181), X398))
U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_out_aaaga(T182, T183, X397, .([], T181), X398)) → row2col29_out_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398)
U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_out_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346)) → transpose_aux1_out_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156)))))))))
The argument filtering Pi contains the following mapping:
transpose_aux1_in_aga(x1, x2, x3) = transpose_aux1_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U2_aga(x21)
row2col29_in_aaaga(x1, x2, x3, x4, x5) = row2col29_in_aaaga(x4)
U1_aaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U1_aaaga(x8)
row2col29_out_aaaga(x1, x2, x3, x4, x5) = row2col29_out_aaaga(x5)
.(x1, x2) = .(x1, x2)
[] = []
transpose_aux1_out_aga(x1, x2, x3) = transpose_aux1_out_aga
TRANSPOSE_AUX1_IN_AGA(x1, x2, x3) = TRANSPOSE_AUX1_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U2_AGA(x21)
ROW2COL29_IN_AAAGA(x1, x2, x3, x4, x5) = ROW2COL29_IN_AAAGA(x4)
U1_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8) = U1_AAAGA(x8)
We have to consider all (P,R,Pi)-chains
(6) Obligation:
Pi DP problem:The TRS P consists of the following rules:
TRANSPOSE_AUX1_IN_AGA(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) → U2_AGA(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_in_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346))
TRANSPOSE_AUX1_IN_AGA(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) → ROW2COL29_IN_AAAGA(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346)
ROW2COL29_IN_AAAGA(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → U1_AAAGA(T177, T182, T179, T183, X397, T181, X398, row2col29_in_aaaga(T182, T183, X397, .([], T181), X398))
ROW2COL29_IN_AAAGA(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → ROW2COL29_IN_AAAGA(T182, T183, X397, .([], T181), X398)
The TRS R consists of the following rules:
transpose_aux1_in_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) → U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_in_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346))
row2col29_in_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_in_aaaga(T182, T183, X397, .([], T181), X398))
U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_out_aaaga(T182, T183, X397, .([], T181), X398)) → row2col29_out_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398)
U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_out_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346)) → transpose_aux1_out_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156)))))))))
The argument filtering Pi contains the following mapping:
transpose_aux1_in_aga(x1, x2, x3) = transpose_aux1_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U2_aga(x21)
row2col29_in_aaaga(x1, x2, x3, x4, x5) = row2col29_in_aaaga(x4)
U1_aaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U1_aaaga(x8)
row2col29_out_aaaga(x1, x2, x3, x4, x5) = row2col29_out_aaaga(x5)
.(x1, x2) = .(x1, x2)
[] = []
transpose_aux1_out_aga(x1, x2, x3) = transpose_aux1_out_aga
TRANSPOSE_AUX1_IN_AGA(x1, x2, x3) = TRANSPOSE_AUX1_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U2_AGA(x21)
ROW2COL29_IN_AAAGA(x1, x2, x3, x4, x5) = ROW2COL29_IN_AAAGA(x4)
U1_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8) = U1_AAAGA(x8)
We have to consider all (P,R,Pi)-chains
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.(8) Obligation:
Pi DP problem:The TRS P consists of the following rules:
ROW2COL29_IN_AAAGA(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → ROW2COL29_IN_AAAGA(T182, T183, X397, .([], T181), X398)
The TRS R consists of the following rules:
transpose_aux1_in_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) → U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_in_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346))
row2col29_in_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_in_aaaga(T182, T183, X397, .([], T181), X398))
U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_out_aaaga(T182, T183, X397, .([], T181), X398)) → row2col29_out_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398)
U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_out_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346)) → transpose_aux1_out_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156)))))))))
The argument filtering Pi contains the following mapping:
transpose_aux1_in_aga(x1, x2, x3) = transpose_aux1_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U2_aga(x21)
row2col29_in_aaaga(x1, x2, x3, x4, x5) = row2col29_in_aaaga(x4)
U1_aaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U1_aaaga(x8)
row2col29_out_aaaga(x1, x2, x3, x4, x5) = row2col29_out_aaaga(x5)
.(x1, x2) = .(x1, x2)
[] = []
transpose_aux1_out_aga(x1, x2, x3) = transpose_aux1_out_aga
ROW2COL29_IN_AAAGA(x1, x2, x3, x4, x5) = ROW2COL29_IN_AAAGA(x4)
We have to consider all (P,R,Pi)-chains
(9) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(10) Obligation:
Pi DP problem:The TRS P consists of the following rules:
ROW2COL29_IN_AAAGA(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → ROW2COL29_IN_AAAGA(T182, T183, X397, .([], T181), X398)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
ROW2COL29_IN_AAAGA(x1, x2, x3, x4, x5) = ROW2COL29_IN_AAAGA(x4)
We have to consider all (P,R,Pi)-chains
(11) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(12) Obligation:
Q DP problem:The TRS P consists of the following rules:
ROW2COL29_IN_AAAGA(T181) → ROW2COL29_IN_AAAGA(.([], T181))
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(13) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule ROW2COL29_IN_AAAGA(T181) → ROW2COL29_IN_AAAGA(.([], T181)) we obtained the following new rules [LPAR04]:
ROW2COL29_IN_AAAGA(.([], z0)) → ROW2COL29_IN_AAAGA(.([], .([], z0)))
(14) Obligation:
Q DP problem:The TRS P consists of the following rules:
ROW2COL29_IN_AAAGA(.([], z0)) → ROW2COL29_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 ROW2COL29_IN_AAAGA(.([], z0)) → ROW2COL29_IN_AAAGA(.([], .([], z0))) we obtained the following new rules [LPAR04]:
ROW2COL29_IN_AAAGA(.([], .([], z0))) → ROW2COL29_IN_AAAGA(.([], .([], .([], z0))))
(16) Obligation:
Q DP problem:The TRS P consists of the following rules:
ROW2COL29_IN_AAAGA(.([], .([], z0))) → ROW2COL29_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 = ROW2COL29_IN_AAAGA(.([], .([], z0))) evaluates to t =ROW2COL29_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 ROW2COL29_IN_AAAGA(.([], .([], z0))) to ROW2COL29_IN_AAAGA(.([], .([], .([], z0)))).
(18) NO
(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_aux1_in: (f,b,f)
row2col29_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_aux1_in_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) → U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_in_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346))
row2col29_in_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_in_aaaga(T182, T183, X397, .([], T181), X398))
U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_out_aaaga(T182, T183, X397, .([], T181), X398)) → row2col29_out_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398)
U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_out_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346)) → transpose_aux1_out_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156)))))))))
The argument filtering Pi contains the following mapping:
transpose_aux1_in_aga(x1, x2, x3) = transpose_aux1_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U2_aga(x11, x21)
row2col29_in_aaaga(x1, x2, x3, x4, x5) = row2col29_in_aaaga(x4)
U1_aaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U1_aaaga(x6, x8)
row2col29_out_aaaga(x1, x2, x3, x4, x5) = row2col29_out_aaaga(x4, x5)
.(x1, x2) = .(x1, x2)
[] = []
transpose_aux1_out_aga(x1, x2, x3) = transpose_aux1_out_aga(x2)
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_aux1_in_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) → U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_in_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346))
row2col29_in_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_in_aaaga(T182, T183, X397, .([], T181), X398))
U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_out_aaaga(T182, T183, X397, .([], T181), X398)) → row2col29_out_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398)
U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_out_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346)) → transpose_aux1_out_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156)))))))))
The argument filtering Pi contains the following mapping:
transpose_aux1_in_aga(x1, x2, x3) = transpose_aux1_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U2_aga(x11, x21)
row2col29_in_aaaga(x1, x2, x3, x4, x5) = row2col29_in_aaaga(x4)
U1_aaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U1_aaaga(x6, x8)
row2col29_out_aaaga(x1, x2, x3, x4, x5) = row2col29_out_aaaga(x4, x5)
.(x1, x2) = .(x1, x2)
[] = []
transpose_aux1_out_aga(x1, x2, x3) = transpose_aux1_out_aga(x2)
(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_AUX1_IN_AGA(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) → U2_AGA(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_in_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346))
TRANSPOSE_AUX1_IN_AGA(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) → ROW2COL29_IN_AAAGA(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346)
ROW2COL29_IN_AAAGA(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → U1_AAAGA(T177, T182, T179, T183, X397, T181, X398, row2col29_in_aaaga(T182, T183, X397, .([], T181), X398))
ROW2COL29_IN_AAAGA(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → ROW2COL29_IN_AAAGA(T182, T183, X397, .([], T181), X398)
The TRS R consists of the following rules:
transpose_aux1_in_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) → U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_in_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346))
row2col29_in_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_in_aaaga(T182, T183, X397, .([], T181), X398))
U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_out_aaaga(T182, T183, X397, .([], T181), X398)) → row2col29_out_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398)
U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_out_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346)) → transpose_aux1_out_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156)))))))))
The argument filtering Pi contains the following mapping:
transpose_aux1_in_aga(x1, x2, x3) = transpose_aux1_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U2_aga(x11, x21)
row2col29_in_aaaga(x1, x2, x3, x4, x5) = row2col29_in_aaaga(x4)
U1_aaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U1_aaaga(x6, x8)
row2col29_out_aaaga(x1, x2, x3, x4, x5) = row2col29_out_aaaga(x4, x5)
.(x1, x2) = .(x1, x2)
[] = []
transpose_aux1_out_aga(x1, x2, x3) = transpose_aux1_out_aga(x2)
TRANSPOSE_AUX1_IN_AGA(x1, x2, x3) = TRANSPOSE_AUX1_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U2_AGA(x11, x21)
ROW2COL29_IN_AAAGA(x1, x2, x3, x4, x5) = ROW2COL29_IN_AAAGA(x4)
U1_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8) = U1_AAAGA(x6, x8)
We have to consider all (P,R,Pi)-chains
(22) Obligation:
Pi DP problem:The TRS P consists of the following rules:
TRANSPOSE_AUX1_IN_AGA(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) → U2_AGA(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_in_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346))
TRANSPOSE_AUX1_IN_AGA(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) → ROW2COL29_IN_AAAGA(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346)
ROW2COL29_IN_AAAGA(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → U1_AAAGA(T177, T182, T179, T183, X397, T181, X398, row2col29_in_aaaga(T182, T183, X397, .([], T181), X398))
ROW2COL29_IN_AAAGA(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → ROW2COL29_IN_AAAGA(T182, T183, X397, .([], T181), X398)
The TRS R consists of the following rules:
transpose_aux1_in_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) → U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_in_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346))
row2col29_in_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_in_aaaga(T182, T183, X397, .([], T181), X398))
U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_out_aaaga(T182, T183, X397, .([], T181), X398)) → row2col29_out_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398)
U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_out_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346)) → transpose_aux1_out_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156)))))))))
The argument filtering Pi contains the following mapping:
transpose_aux1_in_aga(x1, x2, x3) = transpose_aux1_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U2_aga(x11, x21)
row2col29_in_aaaga(x1, x2, x3, x4, x5) = row2col29_in_aaaga(x4)
U1_aaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U1_aaaga(x6, x8)
row2col29_out_aaaga(x1, x2, x3, x4, x5) = row2col29_out_aaaga(x4, x5)
.(x1, x2) = .(x1, x2)
[] = []
transpose_aux1_out_aga(x1, x2, x3) = transpose_aux1_out_aga(x2)
TRANSPOSE_AUX1_IN_AGA(x1, x2, x3) = TRANSPOSE_AUX1_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U2_AGA(x11, x21)
ROW2COL29_IN_AAAGA(x1, x2, x3, x4, x5) = ROW2COL29_IN_AAAGA(x4)
U1_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8) = U1_AAAGA(x6, 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:
ROW2COL29_IN_AAAGA(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → ROW2COL29_IN_AAAGA(T182, T183, X397, .([], T181), X398)
The TRS R consists of the following rules:
transpose_aux1_in_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156))))))))) → U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_in_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346))
row2col29_in_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_in_aaaga(T182, T183, X397, .([], T181), X398))
U1_aaaga(T177, T182, T179, T183, X397, T181, X398, row2col29_out_aaaga(T182, T183, X397, .([], T181), X398)) → row2col29_out_aaaga(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398)
U2_aga(T25, T43, T61, T79, T97, T115, T133, T151, T155, T10, T11, T27, T45, T63, T81, T99, T117, T135, T153, T156, row2col29_out_aaaga(T155, T156, X345, .([], .([], .([], .([], .([], .([], .([], []))))))), X346)) → transpose_aux1_out_aga(.(.(T25, .(T43, .(T61, .(T79, .(T97, .(T115, .(T133, .(T151, T155)))))))), T10), T11, .(.(T25, T27), .(.(T43, T45), .(.(T61, T63), .(.(T79, T81), .(.(T97, T99), .(.(T115, T117), .(.(T133, T135), .(.(T151, T153), T156)))))))))
The argument filtering Pi contains the following mapping:
transpose_aux1_in_aga(x1, x2, x3) = transpose_aux1_in_aga(x2)
U2_aga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) = U2_aga(x11, x21)
row2col29_in_aaaga(x1, x2, x3, x4, x5) = row2col29_in_aaaga(x4)
U1_aaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U1_aaaga(x6, x8)
row2col29_out_aaaga(x1, x2, x3, x4, x5) = row2col29_out_aaaga(x4, x5)
.(x1, x2) = .(x1, x2)
[] = []
transpose_aux1_out_aga(x1, x2, x3) = transpose_aux1_out_aga(x2)
ROW2COL29_IN_AAAGA(x1, x2, x3, x4, x5) = ROW2COL29_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:
ROW2COL29_IN_AAAGA(.(T177, T182), .(.(T177, T179), T183), .(T179, X397), T181, X398) → ROW2COL29_IN_AAAGA(T182, T183, X397, .([], T181), X398)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
ROW2COL29_IN_AAAGA(x1, x2, x3, x4, x5) = ROW2COL29_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:
ROW2COL29_IN_AAAGA(T181) → ROW2COL29_IN_AAAGA(.([], T181))
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(29) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule ROW2COL29_IN_AAAGA(T181) → ROW2COL29_IN_AAAGA(.([], T181)) we obtained the following new rules [LPAR04]:
ROW2COL29_IN_AAAGA(.([], z0)) → ROW2COL29_IN_AAAGA(.([], .([], z0)))
(30) Obligation:
Q DP problem:The TRS P consists of the following rules:
ROW2COL29_IN_AAAGA(.([], z0)) → ROW2COL29_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 ROW2COL29_IN_AAAGA(.([], z0)) → ROW2COL29_IN_AAAGA(.([], .([], z0))) we obtained the following new rules [LPAR04]:
ROW2COL29_IN_AAAGA(.([], .([], z0))) → ROW2COL29_IN_AAAGA(.([], .([], .([], z0))))
(32) Obligation:
Q DP problem:The TRS P consists of the following rules:
ROW2COL29_IN_AAAGA(.([], .([], z0))) → ROW2COL29_IN_AAAGA(.([], .([], .([], z0))))
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(33) 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 = ROW2COL29_IN_AAAGA(.([], .([], z0))) evaluates to t =ROW2COL29_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 ROW2COL29_IN_AAAGA(.([], .([], z0))) to ROW2COL29_IN_AAAGA(.([], .([], .([], z0)))).