Left Termination of the query pattern transpose_in_2(g, g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
↳3 PrologToPiTRSProof (⇐)
↳4 PiTRS
↳5 DependencyPairsProof (⇔)
↳6 PiDP
↳7 DependencyGraphProof (⇔)
↳8 AND
    ↳9 PiDP
        ↳10 UsableRulesProof (⇔)
        ↳11 PiDP
        ↳12 PiDPToQDPProof (⇐)
        ↳13 QDP
        ↳14 QDPSizeChangeProof (⇔)
        ↳15 YES
    ↳16 PiDP
        ↳17 UsableRulesProof (⇔)
        ↳18 PiDP
        ↳19 PiDPToQDPProof (⇐)
        ↳20 QDP
        ↳21 QDPSizeChangeProof (⇔)
        ↳22 YES

(0) Obligation:

Clauses:

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

Queries:

transpose(g,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

row2col14(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151)) :- row2col14(T78, T80, X150, X151).
row2col14([], [], [], []).
p7(T23, T25, T26, X35, X36, T24) :- row2col9(T23, T25, T26, X35, X36).
p7(T23, T25, T26, .(T113, T114), T112, .(T110, T111)) :- ','(row2col9(T23, T25, T26, .(T113, T114), T112), p7(T110, T113, T114, X202, X203, T111)).
p7(T23, T25, T26, T121, T121, []) :- row2col9(T23, T25, T26, T121, T121).
row2col9(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97)) :- row2col14(T54, T56, X96, X97).
transpose1(.(T23, T24), .(T25, T26)) :- p7(T23, T25, T26, X35, X36, T24).
transpose1([], []).

Queries:

transpose1(g,g).

(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:
transpose1_in: (b,b)
p7_in: (b,b,b,f,f,b)
row2col9_in: (b,b,b,f,f)
row2col14_in: (b,b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

transpose1_in_gg(.(T23, T24), .(T25, T26)) → U7_gg(T23, T24, T25, T26, p7_in_gggaag(T23, T25, T26, X35, X36, T24))
p7_in_gggaag(T23, T25, T26, X35, X36, T24) → U2_gggaag(T23, T25, T26, X35, X36, T24, row2col9_in_gggaa(T23, T25, T26, X35, X36))
row2col9_in_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97)) → U6_gggaa(T53, T54, T55, T56, X96, X97, row2col14_in_ggaa(T54, T56, X96, X97))
row2col14_in_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151)) → U1_ggaa(T77, T78, T79, T80, X150, X151, row2col14_in_ggaa(T78, T80, X150, X151))
row2col14_in_ggaa([], [], [], []) → row2col14_out_ggaa([], [], [], [])
U1_ggaa(T77, T78, T79, T80, X150, X151, row2col14_out_ggaa(T78, T80, X150, X151)) → row2col14_out_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151))
U6_gggaa(T53, T54, T55, T56, X96, X97, row2col14_out_ggaa(T54, T56, X96, X97)) → row2col9_out_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97))
U2_gggaag(T23, T25, T26, X35, X36, T24, row2col9_out_gggaa(T23, T25, T26, X35, X36)) → p7_out_gggaag(T23, T25, T26, X35, X36, T24)
p7_in_gggaag(T23, T25, T26, .(T113, T114), T112, .(T110, T111)) → U3_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_in_gggaa(T23, T25, T26, .(T113, T114), T112))
U3_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_out_gggaa(T23, T25, T26, .(T113, T114), T112)) → U4_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, p7_in_gggaag(T110, T113, T114, X202, X203, T111))
p7_in_gggaag(T23, T25, T26, T121, T121, []) → U5_gggaag(T23, T25, T26, T121, row2col9_in_gggaa(T23, T25, T26, T121, T121))
U5_gggaag(T23, T25, T26, T121, row2col9_out_gggaa(T23, T25, T26, T121, T121)) → p7_out_gggaag(T23, T25, T26, T121, T121, [])
U4_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, p7_out_gggaag(T110, T113, T114, X202, X203, T111)) → p7_out_gggaag(T23, T25, T26, .(T113, T114), T112, .(T110, T111))
U7_gg(T23, T24, T25, T26, p7_out_gggaag(T23, T25, T26, X35, X36, T24)) → transpose1_out_gg(.(T23, T24), .(T25, T26))
transpose1_in_gg([], []) → transpose1_out_gg([], [])

The argument filtering Pi contains the following mapping:
transpose1_in_gg(x1, x2)  =  transpose1_in_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x1, x2, x3, x4, x5)
p7_in_gggaag(x1, x2, x3, x4, x5, x6)  =  p7_in_gggaag(x1, x2, x3, x6)
U2_gggaag(x1, x2, x3, x4, x5, x6, x7)  =  U2_gggaag(x1, x2, x3, x6, x7)
row2col9_in_gggaa(x1, x2, x3, x4, x5)  =  row2col9_in_gggaa(x1, x2, x3)
U6_gggaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggaa(x1, x2, x3, x4, x7)
row2col14_in_ggaa(x1, x2, x3, x4)  =  row2col14_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U1_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2col14_out_ggaa(x1, x2, x3, x4)  =  row2col14_out_ggaa(x1, x2, x3, x4)
row2col9_out_gggaa(x1, x2, x3, x4, x5)  =  row2col9_out_gggaa(x1, x2, x3, x4, x5)
p7_out_gggaag(x1, x2, x3, x4, x5, x6)  =  p7_out_gggaag(x1, x2, x3, x4, x5, x6)
U3_gggaag(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_gggaag(x1, x2, x3, x7, x8, x9)
U4_gggaag(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_gggaag(x1, x2, x3, x4, x5, x6, x7, x8, x9)
U5_gggaag(x1, x2, x3, x4, x5)  =  U5_gggaag(x1, x2, x3, x5)
transpose1_out_gg(x1, x2)  =  transpose1_out_gg(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

transpose1_in_gg(.(T23, T24), .(T25, T26)) → U7_gg(T23, T24, T25, T26, p7_in_gggaag(T23, T25, T26, X35, X36, T24))
p7_in_gggaag(T23, T25, T26, X35, X36, T24) → U2_gggaag(T23, T25, T26, X35, X36, T24, row2col9_in_gggaa(T23, T25, T26, X35, X36))
row2col9_in_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97)) → U6_gggaa(T53, T54, T55, T56, X96, X97, row2col14_in_ggaa(T54, T56, X96, X97))
row2col14_in_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151)) → U1_ggaa(T77, T78, T79, T80, X150, X151, row2col14_in_ggaa(T78, T80, X150, X151))
row2col14_in_ggaa([], [], [], []) → row2col14_out_ggaa([], [], [], [])
U1_ggaa(T77, T78, T79, T80, X150, X151, row2col14_out_ggaa(T78, T80, X150, X151)) → row2col14_out_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151))
U6_gggaa(T53, T54, T55, T56, X96, X97, row2col14_out_ggaa(T54, T56, X96, X97)) → row2col9_out_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97))
U2_gggaag(T23, T25, T26, X35, X36, T24, row2col9_out_gggaa(T23, T25, T26, X35, X36)) → p7_out_gggaag(T23, T25, T26, X35, X36, T24)
p7_in_gggaag(T23, T25, T26, .(T113, T114), T112, .(T110, T111)) → U3_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_in_gggaa(T23, T25, T26, .(T113, T114), T112))
U3_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_out_gggaa(T23, T25, T26, .(T113, T114), T112)) → U4_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, p7_in_gggaag(T110, T113, T114, X202, X203, T111))
p7_in_gggaag(T23, T25, T26, T121, T121, []) → U5_gggaag(T23, T25, T26, T121, row2col9_in_gggaa(T23, T25, T26, T121, T121))
U5_gggaag(T23, T25, T26, T121, row2col9_out_gggaa(T23, T25, T26, T121, T121)) → p7_out_gggaag(T23, T25, T26, T121, T121, [])
U4_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, p7_out_gggaag(T110, T113, T114, X202, X203, T111)) → p7_out_gggaag(T23, T25, T26, .(T113, T114), T112, .(T110, T111))
U7_gg(T23, T24, T25, T26, p7_out_gggaag(T23, T25, T26, X35, X36, T24)) → transpose1_out_gg(.(T23, T24), .(T25, T26))
transpose1_in_gg([], []) → transpose1_out_gg([], [])

The argument filtering Pi contains the following mapping:
transpose1_in_gg(x1, x2)  =  transpose1_in_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x1, x2, x3, x4, x5)
p7_in_gggaag(x1, x2, x3, x4, x5, x6)  =  p7_in_gggaag(x1, x2, x3, x6)
U2_gggaag(x1, x2, x3, x4, x5, x6, x7)  =  U2_gggaag(x1, x2, x3, x6, x7)
row2col9_in_gggaa(x1, x2, x3, x4, x5)  =  row2col9_in_gggaa(x1, x2, x3)
U6_gggaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggaa(x1, x2, x3, x4, x7)
row2col14_in_ggaa(x1, x2, x3, x4)  =  row2col14_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U1_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2col14_out_ggaa(x1, x2, x3, x4)  =  row2col14_out_ggaa(x1, x2, x3, x4)
row2col9_out_gggaa(x1, x2, x3, x4, x5)  =  row2col9_out_gggaa(x1, x2, x3, x4, x5)
p7_out_gggaag(x1, x2, x3, x4, x5, x6)  =  p7_out_gggaag(x1, x2, x3, x4, x5, x6)
U3_gggaag(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_gggaag(x1, x2, x3, x7, x8, x9)
U4_gggaag(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_gggaag(x1, x2, x3, x4, x5, x6, x7, x8, x9)
U5_gggaag(x1, x2, x3, x4, x5)  =  U5_gggaag(x1, x2, x3, x5)
transpose1_out_gg(x1, x2)  =  transpose1_out_gg(x1, x2)

(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:

TRANSPOSE1_IN_GG(.(T23, T24), .(T25, T26)) → U7_GG(T23, T24, T25, T26, p7_in_gggaag(T23, T25, T26, X35, X36, T24))
TRANSPOSE1_IN_GG(.(T23, T24), .(T25, T26)) → P7_IN_GGGAAG(T23, T25, T26, X35, X36, T24)
P7_IN_GGGAAG(T23, T25, T26, X35, X36, T24) → U2_GGGAAG(T23, T25, T26, X35, X36, T24, row2col9_in_gggaa(T23, T25, T26, X35, X36))
P7_IN_GGGAAG(T23, T25, T26, X35, X36, T24) → ROW2COL9_IN_GGGAA(T23, T25, T26, X35, X36)
ROW2COL9_IN_GGGAA(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97)) → U6_GGGAA(T53, T54, T55, T56, X96, X97, row2col14_in_ggaa(T54, T56, X96, X97))
ROW2COL9_IN_GGGAA(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97)) → ROW2COL14_IN_GGAA(T54, T56, X96, X97)
ROW2COL14_IN_GGAA(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151)) → U1_GGAA(T77, T78, T79, T80, X150, X151, row2col14_in_ggaa(T78, T80, X150, X151))
ROW2COL14_IN_GGAA(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151)) → ROW2COL14_IN_GGAA(T78, T80, X150, X151)
P7_IN_GGGAAG(T23, T25, T26, .(T113, T114), T112, .(T110, T111)) → U3_GGGAAG(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_in_gggaa(T23, T25, T26, .(T113, T114), T112))
P7_IN_GGGAAG(T23, T25, T26, .(T113, T114), T112, .(T110, T111)) → ROW2COL9_IN_GGGAA(T23, T25, T26, .(T113, T114), T112)
U3_GGGAAG(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_out_gggaa(T23, T25, T26, .(T113, T114), T112)) → U4_GGGAAG(T23, T25, T26, T113, T114, T112, T110, T111, p7_in_gggaag(T110, T113, T114, X202, X203, T111))
U3_GGGAAG(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_out_gggaa(T23, T25, T26, .(T113, T114), T112)) → P7_IN_GGGAAG(T110, T113, T114, X202, X203, T111)
P7_IN_GGGAAG(T23, T25, T26, T121, T121, []) → U5_GGGAAG(T23, T25, T26, T121, row2col9_in_gggaa(T23, T25, T26, T121, T121))
P7_IN_GGGAAG(T23, T25, T26, T121, T121, []) → ROW2COL9_IN_GGGAA(T23, T25, T26, T121, T121)

The TRS R consists of the following rules:

transpose1_in_gg(.(T23, T24), .(T25, T26)) → U7_gg(T23, T24, T25, T26, p7_in_gggaag(T23, T25, T26, X35, X36, T24))
p7_in_gggaag(T23, T25, T26, X35, X36, T24) → U2_gggaag(T23, T25, T26, X35, X36, T24, row2col9_in_gggaa(T23, T25, T26, X35, X36))
row2col9_in_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97)) → U6_gggaa(T53, T54, T55, T56, X96, X97, row2col14_in_ggaa(T54, T56, X96, X97))
row2col14_in_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151)) → U1_ggaa(T77, T78, T79, T80, X150, X151, row2col14_in_ggaa(T78, T80, X150, X151))
row2col14_in_ggaa([], [], [], []) → row2col14_out_ggaa([], [], [], [])
U1_ggaa(T77, T78, T79, T80, X150, X151, row2col14_out_ggaa(T78, T80, X150, X151)) → row2col14_out_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151))
U6_gggaa(T53, T54, T55, T56, X96, X97, row2col14_out_ggaa(T54, T56, X96, X97)) → row2col9_out_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97))
U2_gggaag(T23, T25, T26, X35, X36, T24, row2col9_out_gggaa(T23, T25, T26, X35, X36)) → p7_out_gggaag(T23, T25, T26, X35, X36, T24)
p7_in_gggaag(T23, T25, T26, .(T113, T114), T112, .(T110, T111)) → U3_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_in_gggaa(T23, T25, T26, .(T113, T114), T112))
U3_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_out_gggaa(T23, T25, T26, .(T113, T114), T112)) → U4_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, p7_in_gggaag(T110, T113, T114, X202, X203, T111))
p7_in_gggaag(T23, T25, T26, T121, T121, []) → U5_gggaag(T23, T25, T26, T121, row2col9_in_gggaa(T23, T25, T26, T121, T121))
U5_gggaag(T23, T25, T26, T121, row2col9_out_gggaa(T23, T25, T26, T121, T121)) → p7_out_gggaag(T23, T25, T26, T121, T121, [])
U4_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, p7_out_gggaag(T110, T113, T114, X202, X203, T111)) → p7_out_gggaag(T23, T25, T26, .(T113, T114), T112, .(T110, T111))
U7_gg(T23, T24, T25, T26, p7_out_gggaag(T23, T25, T26, X35, X36, T24)) → transpose1_out_gg(.(T23, T24), .(T25, T26))
transpose1_in_gg([], []) → transpose1_out_gg([], [])

The argument filtering Pi contains the following mapping:
transpose1_in_gg(x1, x2)  =  transpose1_in_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x1, x2, x3, x4, x5)
p7_in_gggaag(x1, x2, x3, x4, x5, x6)  =  p7_in_gggaag(x1, x2, x3, x6)
U2_gggaag(x1, x2, x3, x4, x5, x6, x7)  =  U2_gggaag(x1, x2, x3, x6, x7)
row2col9_in_gggaa(x1, x2, x3, x4, x5)  =  row2col9_in_gggaa(x1, x2, x3)
U6_gggaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggaa(x1, x2, x3, x4, x7)
row2col14_in_ggaa(x1, x2, x3, x4)  =  row2col14_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U1_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2col14_out_ggaa(x1, x2, x3, x4)  =  row2col14_out_ggaa(x1, x2, x3, x4)
row2col9_out_gggaa(x1, x2, x3, x4, x5)  =  row2col9_out_gggaa(x1, x2, x3, x4, x5)
p7_out_gggaag(x1, x2, x3, x4, x5, x6)  =  p7_out_gggaag(x1, x2, x3, x4, x5, x6)
U3_gggaag(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_gggaag(x1, x2, x3, x7, x8, x9)
U4_gggaag(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_gggaag(x1, x2, x3, x4, x5, x6, x7, x8, x9)
U5_gggaag(x1, x2, x3, x4, x5)  =  U5_gggaag(x1, x2, x3, x5)
transpose1_out_gg(x1, x2)  =  transpose1_out_gg(x1, x2)
TRANSPOSE1_IN_GG(x1, x2)  =  TRANSPOSE1_IN_GG(x1, x2)
U7_GG(x1, x2, x3, x4, x5)  =  U7_GG(x1, x2, x3, x4, x5)
P7_IN_GGGAAG(x1, x2, x3, x4, x5, x6)  =  P7_IN_GGGAAG(x1, x2, x3, x6)
U2_GGGAAG(x1, x2, x3, x4, x5, x6, x7)  =  U2_GGGAAG(x1, x2, x3, x6, x7)
ROW2COL9_IN_GGGAA(x1, x2, x3, x4, x5)  =  ROW2COL9_IN_GGGAA(x1, x2, x3)
U6_GGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GGGAA(x1, x2, x3, x4, x7)
ROW2COL14_IN_GGAA(x1, x2, x3, x4)  =  ROW2COL14_IN_GGAA(x1, x2)
U1_GGAA(x1, x2, x3, x4, x5, x6, x7)  =  U1_GGAA(x1, x2, x3, x4, x7)
U3_GGGAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_GGGAAG(x1, x2, x3, x7, x8, x9)
U4_GGGAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_GGGAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9)
U5_GGGAAG(x1, x2, x3, x4, x5)  =  U5_GGGAAG(x1, x2, x3, x5)

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

(6) Obligation:

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

TRANSPOSE1_IN_GG(.(T23, T24), .(T25, T26)) → U7_GG(T23, T24, T25, T26, p7_in_gggaag(T23, T25, T26, X35, X36, T24))
TRANSPOSE1_IN_GG(.(T23, T24), .(T25, T26)) → P7_IN_GGGAAG(T23, T25, T26, X35, X36, T24)
P7_IN_GGGAAG(T23, T25, T26, X35, X36, T24) → U2_GGGAAG(T23, T25, T26, X35, X36, T24, row2col9_in_gggaa(T23, T25, T26, X35, X36))
P7_IN_GGGAAG(T23, T25, T26, X35, X36, T24) → ROW2COL9_IN_GGGAA(T23, T25, T26, X35, X36)
ROW2COL9_IN_GGGAA(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97)) → U6_GGGAA(T53, T54, T55, T56, X96, X97, row2col14_in_ggaa(T54, T56, X96, X97))
ROW2COL9_IN_GGGAA(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97)) → ROW2COL14_IN_GGAA(T54, T56, X96, X97)
ROW2COL14_IN_GGAA(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151)) → U1_GGAA(T77, T78, T79, T80, X150, X151, row2col14_in_ggaa(T78, T80, X150, X151))
ROW2COL14_IN_GGAA(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151)) → ROW2COL14_IN_GGAA(T78, T80, X150, X151)
P7_IN_GGGAAG(T23, T25, T26, .(T113, T114), T112, .(T110, T111)) → U3_GGGAAG(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_in_gggaa(T23, T25, T26, .(T113, T114), T112))
P7_IN_GGGAAG(T23, T25, T26, .(T113, T114), T112, .(T110, T111)) → ROW2COL9_IN_GGGAA(T23, T25, T26, .(T113, T114), T112)
U3_GGGAAG(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_out_gggaa(T23, T25, T26, .(T113, T114), T112)) → U4_GGGAAG(T23, T25, T26, T113, T114, T112, T110, T111, p7_in_gggaag(T110, T113, T114, X202, X203, T111))
U3_GGGAAG(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_out_gggaa(T23, T25, T26, .(T113, T114), T112)) → P7_IN_GGGAAG(T110, T113, T114, X202, X203, T111)
P7_IN_GGGAAG(T23, T25, T26, T121, T121, []) → U5_GGGAAG(T23, T25, T26, T121, row2col9_in_gggaa(T23, T25, T26, T121, T121))
P7_IN_GGGAAG(T23, T25, T26, T121, T121, []) → ROW2COL9_IN_GGGAA(T23, T25, T26, T121, T121)

The TRS R consists of the following rules:

transpose1_in_gg(.(T23, T24), .(T25, T26)) → U7_gg(T23, T24, T25, T26, p7_in_gggaag(T23, T25, T26, X35, X36, T24))
p7_in_gggaag(T23, T25, T26, X35, X36, T24) → U2_gggaag(T23, T25, T26, X35, X36, T24, row2col9_in_gggaa(T23, T25, T26, X35, X36))
row2col9_in_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97)) → U6_gggaa(T53, T54, T55, T56, X96, X97, row2col14_in_ggaa(T54, T56, X96, X97))
row2col14_in_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151)) → U1_ggaa(T77, T78, T79, T80, X150, X151, row2col14_in_ggaa(T78, T80, X150, X151))
row2col14_in_ggaa([], [], [], []) → row2col14_out_ggaa([], [], [], [])
U1_ggaa(T77, T78, T79, T80, X150, X151, row2col14_out_ggaa(T78, T80, X150, X151)) → row2col14_out_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151))
U6_gggaa(T53, T54, T55, T56, X96, X97, row2col14_out_ggaa(T54, T56, X96, X97)) → row2col9_out_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97))
U2_gggaag(T23, T25, T26, X35, X36, T24, row2col9_out_gggaa(T23, T25, T26, X35, X36)) → p7_out_gggaag(T23, T25, T26, X35, X36, T24)
p7_in_gggaag(T23, T25, T26, .(T113, T114), T112, .(T110, T111)) → U3_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_in_gggaa(T23, T25, T26, .(T113, T114), T112))
U3_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_out_gggaa(T23, T25, T26, .(T113, T114), T112)) → U4_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, p7_in_gggaag(T110, T113, T114, X202, X203, T111))
p7_in_gggaag(T23, T25, T26, T121, T121, []) → U5_gggaag(T23, T25, T26, T121, row2col9_in_gggaa(T23, T25, T26, T121, T121))
U5_gggaag(T23, T25, T26, T121, row2col9_out_gggaa(T23, T25, T26, T121, T121)) → p7_out_gggaag(T23, T25, T26, T121, T121, [])
U4_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, p7_out_gggaag(T110, T113, T114, X202, X203, T111)) → p7_out_gggaag(T23, T25, T26, .(T113, T114), T112, .(T110, T111))
U7_gg(T23, T24, T25, T26, p7_out_gggaag(T23, T25, T26, X35, X36, T24)) → transpose1_out_gg(.(T23, T24), .(T25, T26))
transpose1_in_gg([], []) → transpose1_out_gg([], [])

The argument filtering Pi contains the following mapping:
transpose1_in_gg(x1, x2)  =  transpose1_in_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x1, x2, x3, x4, x5)
p7_in_gggaag(x1, x2, x3, x4, x5, x6)  =  p7_in_gggaag(x1, x2, x3, x6)
U2_gggaag(x1, x2, x3, x4, x5, x6, x7)  =  U2_gggaag(x1, x2, x3, x6, x7)
row2col9_in_gggaa(x1, x2, x3, x4, x5)  =  row2col9_in_gggaa(x1, x2, x3)
U6_gggaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggaa(x1, x2, x3, x4, x7)
row2col14_in_ggaa(x1, x2, x3, x4)  =  row2col14_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U1_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2col14_out_ggaa(x1, x2, x3, x4)  =  row2col14_out_ggaa(x1, x2, x3, x4)
row2col9_out_gggaa(x1, x2, x3, x4, x5)  =  row2col9_out_gggaa(x1, x2, x3, x4, x5)
p7_out_gggaag(x1, x2, x3, x4, x5, x6)  =  p7_out_gggaag(x1, x2, x3, x4, x5, x6)
U3_gggaag(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_gggaag(x1, x2, x3, x7, x8, x9)
U4_gggaag(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_gggaag(x1, x2, x3, x4, x5, x6, x7, x8, x9)
U5_gggaag(x1, x2, x3, x4, x5)  =  U5_gggaag(x1, x2, x3, x5)
transpose1_out_gg(x1, x2)  =  transpose1_out_gg(x1, x2)
TRANSPOSE1_IN_GG(x1, x2)  =  TRANSPOSE1_IN_GG(x1, x2)
U7_GG(x1, x2, x3, x4, x5)  =  U7_GG(x1, x2, x3, x4, x5)
P7_IN_GGGAAG(x1, x2, x3, x4, x5, x6)  =  P7_IN_GGGAAG(x1, x2, x3, x6)
U2_GGGAAG(x1, x2, x3, x4, x5, x6, x7)  =  U2_GGGAAG(x1, x2, x3, x6, x7)
ROW2COL9_IN_GGGAA(x1, x2, x3, x4, x5)  =  ROW2COL9_IN_GGGAA(x1, x2, x3)
U6_GGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GGGAA(x1, x2, x3, x4, x7)
ROW2COL14_IN_GGAA(x1, x2, x3, x4)  =  ROW2COL14_IN_GGAA(x1, x2)
U1_GGAA(x1, x2, x3, x4, x5, x6, x7)  =  U1_GGAA(x1, x2, x3, x4, x7)
U3_GGGAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_GGGAAG(x1, x2, x3, x7, x8, x9)
U4_GGGAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_GGGAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9)
U5_GGGAAG(x1, x2, x3, x4, x5)  =  U5_GGGAAG(x1, x2, x3, x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 11 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

ROW2COL14_IN_GGAA(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151)) → ROW2COL14_IN_GGAA(T78, T80, X150, X151)

The TRS R consists of the following rules:

transpose1_in_gg(.(T23, T24), .(T25, T26)) → U7_gg(T23, T24, T25, T26, p7_in_gggaag(T23, T25, T26, X35, X36, T24))
p7_in_gggaag(T23, T25, T26, X35, X36, T24) → U2_gggaag(T23, T25, T26, X35, X36, T24, row2col9_in_gggaa(T23, T25, T26, X35, X36))
row2col9_in_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97)) → U6_gggaa(T53, T54, T55, T56, X96, X97, row2col14_in_ggaa(T54, T56, X96, X97))
row2col14_in_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151)) → U1_ggaa(T77, T78, T79, T80, X150, X151, row2col14_in_ggaa(T78, T80, X150, X151))
row2col14_in_ggaa([], [], [], []) → row2col14_out_ggaa([], [], [], [])
U1_ggaa(T77, T78, T79, T80, X150, X151, row2col14_out_ggaa(T78, T80, X150, X151)) → row2col14_out_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151))
U6_gggaa(T53, T54, T55, T56, X96, X97, row2col14_out_ggaa(T54, T56, X96, X97)) → row2col9_out_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97))
U2_gggaag(T23, T25, T26, X35, X36, T24, row2col9_out_gggaa(T23, T25, T26, X35, X36)) → p7_out_gggaag(T23, T25, T26, X35, X36, T24)
p7_in_gggaag(T23, T25, T26, .(T113, T114), T112, .(T110, T111)) → U3_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_in_gggaa(T23, T25, T26, .(T113, T114), T112))
U3_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_out_gggaa(T23, T25, T26, .(T113, T114), T112)) → U4_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, p7_in_gggaag(T110, T113, T114, X202, X203, T111))
p7_in_gggaag(T23, T25, T26, T121, T121, []) → U5_gggaag(T23, T25, T26, T121, row2col9_in_gggaa(T23, T25, T26, T121, T121))
U5_gggaag(T23, T25, T26, T121, row2col9_out_gggaa(T23, T25, T26, T121, T121)) → p7_out_gggaag(T23, T25, T26, T121, T121, [])
U4_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, p7_out_gggaag(T110, T113, T114, X202, X203, T111)) → p7_out_gggaag(T23, T25, T26, .(T113, T114), T112, .(T110, T111))
U7_gg(T23, T24, T25, T26, p7_out_gggaag(T23, T25, T26, X35, X36, T24)) → transpose1_out_gg(.(T23, T24), .(T25, T26))
transpose1_in_gg([], []) → transpose1_out_gg([], [])

The argument filtering Pi contains the following mapping:
transpose1_in_gg(x1, x2)  =  transpose1_in_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x1, x2, x3, x4, x5)
p7_in_gggaag(x1, x2, x3, x4, x5, x6)  =  p7_in_gggaag(x1, x2, x3, x6)
U2_gggaag(x1, x2, x3, x4, x5, x6, x7)  =  U2_gggaag(x1, x2, x3, x6, x7)
row2col9_in_gggaa(x1, x2, x3, x4, x5)  =  row2col9_in_gggaa(x1, x2, x3)
U6_gggaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggaa(x1, x2, x3, x4, x7)
row2col14_in_ggaa(x1, x2, x3, x4)  =  row2col14_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U1_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2col14_out_ggaa(x1, x2, x3, x4)  =  row2col14_out_ggaa(x1, x2, x3, x4)
row2col9_out_gggaa(x1, x2, x3, x4, x5)  =  row2col9_out_gggaa(x1, x2, x3, x4, x5)
p7_out_gggaag(x1, x2, x3, x4, x5, x6)  =  p7_out_gggaag(x1, x2, x3, x4, x5, x6)
U3_gggaag(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_gggaag(x1, x2, x3, x7, x8, x9)
U4_gggaag(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_gggaag(x1, x2, x3, x4, x5, x6, x7, x8, x9)
U5_gggaag(x1, x2, x3, x4, x5)  =  U5_gggaag(x1, x2, x3, x5)
transpose1_out_gg(x1, x2)  =  transpose1_out_gg(x1, x2)
ROW2COL14_IN_GGAA(x1, x2, x3, x4)  =  ROW2COL14_IN_GGAA(x1, x2)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

ROW2COL14_IN_GGAA(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151)) → ROW2COL14_IN_GGAA(T78, T80, X150, X151)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
ROW2COL14_IN_GGAA(x1, x2, x3, x4)  =  ROW2COL14_IN_GGAA(x1, x2)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

ROW2COL14_IN_GGAA(.(T77, T78), .(.(T77, T79), T80)) → ROW2COL14_IN_GGAA(T78, T80)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • ROW2COL14_IN_GGAA(.(T77, T78), .(.(T77, T79), T80)) → ROW2COL14_IN_GGAA(T78, T80)
    The graph contains the following edges 1 > 1, 2 > 2

(15) YES

(16) Obligation:

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

P7_IN_GGGAAG(T23, T25, T26, .(T113, T114), T112, .(T110, T111)) → U3_GGGAAG(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_in_gggaa(T23, T25, T26, .(T113, T114), T112))
U3_GGGAAG(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_out_gggaa(T23, T25, T26, .(T113, T114), T112)) → P7_IN_GGGAAG(T110, T113, T114, X202, X203, T111)

The TRS R consists of the following rules:

transpose1_in_gg(.(T23, T24), .(T25, T26)) → U7_gg(T23, T24, T25, T26, p7_in_gggaag(T23, T25, T26, X35, X36, T24))
p7_in_gggaag(T23, T25, T26, X35, X36, T24) → U2_gggaag(T23, T25, T26, X35, X36, T24, row2col9_in_gggaa(T23, T25, T26, X35, X36))
row2col9_in_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97)) → U6_gggaa(T53, T54, T55, T56, X96, X97, row2col14_in_ggaa(T54, T56, X96, X97))
row2col14_in_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151)) → U1_ggaa(T77, T78, T79, T80, X150, X151, row2col14_in_ggaa(T78, T80, X150, X151))
row2col14_in_ggaa([], [], [], []) → row2col14_out_ggaa([], [], [], [])
U1_ggaa(T77, T78, T79, T80, X150, X151, row2col14_out_ggaa(T78, T80, X150, X151)) → row2col14_out_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151))
U6_gggaa(T53, T54, T55, T56, X96, X97, row2col14_out_ggaa(T54, T56, X96, X97)) → row2col9_out_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97))
U2_gggaag(T23, T25, T26, X35, X36, T24, row2col9_out_gggaa(T23, T25, T26, X35, X36)) → p7_out_gggaag(T23, T25, T26, X35, X36, T24)
p7_in_gggaag(T23, T25, T26, .(T113, T114), T112, .(T110, T111)) → U3_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_in_gggaa(T23, T25, T26, .(T113, T114), T112))
U3_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_out_gggaa(T23, T25, T26, .(T113, T114), T112)) → U4_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, p7_in_gggaag(T110, T113, T114, X202, X203, T111))
p7_in_gggaag(T23, T25, T26, T121, T121, []) → U5_gggaag(T23, T25, T26, T121, row2col9_in_gggaa(T23, T25, T26, T121, T121))
U5_gggaag(T23, T25, T26, T121, row2col9_out_gggaa(T23, T25, T26, T121, T121)) → p7_out_gggaag(T23, T25, T26, T121, T121, [])
U4_gggaag(T23, T25, T26, T113, T114, T112, T110, T111, p7_out_gggaag(T110, T113, T114, X202, X203, T111)) → p7_out_gggaag(T23, T25, T26, .(T113, T114), T112, .(T110, T111))
U7_gg(T23, T24, T25, T26, p7_out_gggaag(T23, T25, T26, X35, X36, T24)) → transpose1_out_gg(.(T23, T24), .(T25, T26))
transpose1_in_gg([], []) → transpose1_out_gg([], [])

The argument filtering Pi contains the following mapping:
transpose1_in_gg(x1, x2)  =  transpose1_in_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U7_gg(x1, x2, x3, x4, x5)  =  U7_gg(x1, x2, x3, x4, x5)
p7_in_gggaag(x1, x2, x3, x4, x5, x6)  =  p7_in_gggaag(x1, x2, x3, x6)
U2_gggaag(x1, x2, x3, x4, x5, x6, x7)  =  U2_gggaag(x1, x2, x3, x6, x7)
row2col9_in_gggaa(x1, x2, x3, x4, x5)  =  row2col9_in_gggaa(x1, x2, x3)
U6_gggaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggaa(x1, x2, x3, x4, x7)
row2col14_in_ggaa(x1, x2, x3, x4)  =  row2col14_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U1_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2col14_out_ggaa(x1, x2, x3, x4)  =  row2col14_out_ggaa(x1, x2, x3, x4)
row2col9_out_gggaa(x1, x2, x3, x4, x5)  =  row2col9_out_gggaa(x1, x2, x3, x4, x5)
p7_out_gggaag(x1, x2, x3, x4, x5, x6)  =  p7_out_gggaag(x1, x2, x3, x4, x5, x6)
U3_gggaag(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_gggaag(x1, x2, x3, x7, x8, x9)
U4_gggaag(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_gggaag(x1, x2, x3, x4, x5, x6, x7, x8, x9)
U5_gggaag(x1, x2, x3, x4, x5)  =  U5_gggaag(x1, x2, x3, x5)
transpose1_out_gg(x1, x2)  =  transpose1_out_gg(x1, x2)
P7_IN_GGGAAG(x1, x2, x3, x4, x5, x6)  =  P7_IN_GGGAAG(x1, x2, x3, x6)
U3_GGGAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_GGGAAG(x1, x2, x3, x7, x8, x9)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

P7_IN_GGGAAG(T23, T25, T26, .(T113, T114), T112, .(T110, T111)) → U3_GGGAAG(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_in_gggaa(T23, T25, T26, .(T113, T114), T112))
U3_GGGAAG(T23, T25, T26, T113, T114, T112, T110, T111, row2col9_out_gggaa(T23, T25, T26, .(T113, T114), T112)) → P7_IN_GGGAAG(T110, T113, T114, X202, X203, T111)

The TRS R consists of the following rules:

row2col9_in_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97)) → U6_gggaa(T53, T54, T55, T56, X96, X97, row2col14_in_ggaa(T54, T56, X96, X97))
U6_gggaa(T53, T54, T55, T56, X96, X97, row2col14_out_ggaa(T54, T56, X96, X97)) → row2col9_out_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97))
row2col14_in_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151)) → U1_ggaa(T77, T78, T79, T80, X150, X151, row2col14_in_ggaa(T78, T80, X150, X151))
row2col14_in_ggaa([], [], [], []) → row2col14_out_ggaa([], [], [], [])
U1_ggaa(T77, T78, T79, T80, X150, X151, row2col14_out_ggaa(T78, T80, X150, X151)) → row2col14_out_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
row2col9_in_gggaa(x1, x2, x3, x4, x5)  =  row2col9_in_gggaa(x1, x2, x3)
U6_gggaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggaa(x1, x2, x3, x4, x7)
row2col14_in_ggaa(x1, x2, x3, x4)  =  row2col14_in_ggaa(x1, x2)
U1_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U1_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2col14_out_ggaa(x1, x2, x3, x4)  =  row2col14_out_ggaa(x1, x2, x3, x4)
row2col9_out_gggaa(x1, x2, x3, x4, x5)  =  row2col9_out_gggaa(x1, x2, x3, x4, x5)
P7_IN_GGGAAG(x1, x2, x3, x4, x5, x6)  =  P7_IN_GGGAAG(x1, x2, x3, x6)
U3_GGGAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_GGGAAG(x1, x2, x3, x7, x8, x9)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

P7_IN_GGGAAG(T23, T25, T26, .(T110, T111)) → U3_GGGAAG(T23, T25, T26, T110, T111, row2col9_in_gggaa(T23, T25, T26))
U3_GGGAAG(T23, T25, T26, T110, T111, row2col9_out_gggaa(T23, T25, T26, .(T113, T114), T112)) → P7_IN_GGGAAG(T110, T113, T114, T111)

The TRS R consists of the following rules:

row2col9_in_gggaa(.(T53, T54), .(T53, T55), T56) → U6_gggaa(T53, T54, T55, T56, row2col14_in_ggaa(T54, T56))
U6_gggaa(T53, T54, T55, T56, row2col14_out_ggaa(T54, T56, X96, X97)) → row2col9_out_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X96), .([], X97))
row2col14_in_ggaa(.(T77, T78), .(.(T77, T79), T80)) → U1_ggaa(T77, T78, T79, T80, row2col14_in_ggaa(T78, T80))
row2col14_in_ggaa([], []) → row2col14_out_ggaa([], [], [], [])
U1_ggaa(T77, T78, T79, T80, row2col14_out_ggaa(T78, T80, X150, X151)) → row2col14_out_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X150), .([], X151))

The set Q consists of the following terms:

row2col9_in_gggaa(x0, x1, x2)
U6_gggaa(x0, x1, x2, x3, x4)
row2col14_in_ggaa(x0, x1)
U1_ggaa(x0, x1, x2, x3, x4)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • U3_GGGAAG(T23, T25, T26, T110, T111, row2col9_out_gggaa(T23, T25, T26, .(T113, T114), T112)) → P7_IN_GGGAAG(T110, T113, T114, T111)
    The graph contains the following edges 4 >= 1, 6 > 2, 6 > 3, 5 >= 4

  • P7_IN_GGGAAG(T23, T25, T26, .(T110, T111)) → U3_GGGAAG(T23, T25, T26, T110, T111, row2col9_in_gggaa(T23, T25, T26))
    The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 4 > 5

(22) YES