Left Termination of the query pattern transpose_in_2(a, 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 QDPOrderProof (⇔)
        ↳22 QDP
        ↳23 DependencyGraphProof (⇔)
        ↳24 TRUE

(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(a,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

row2col14(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151)) :- row2col14(T86, T85, X150, X151).
row2col14([], [], [], []).
p7(T28, T26, T27, X35, X36, T29) :- row2col9(T28, T26, T27, X35, X36).
p7(T28, T26, T27, .(T119, T120), T118, .(T121, T122)) :- ','(row2col9(T28, T26, T27, .(T119, T120), T118), p7(T121, T119, T120, X202, X203, T122)).
p7(T28, T26, T27, T129, T129, []) :- row2col9(T28, T26, T27, T129, T129).
row2col9(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97)) :- row2col14(T61, T60, X96, X97).
transpose1(.(T28, T29), .(T26, T27)) :- p7(T28, T26, T27, X35, X36, T29).
transpose1([], []).

Queries:

transpose1(a,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: (f,b)
p7_in: (f,b,b,f,f,f)
row2col9_in: (f,b,b,f,f)
row2col14_in: (f,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_ag(.(T28, T29), .(T26, T27)) → U7_ag(T28, T29, T26, T27, p7_in_aggaaa(T28, T26, T27, X35, X36, T29))
p7_in_aggaaa(T28, T26, T27, X35, X36, T29) → U2_aggaaa(T28, T26, T27, X35, X36, T29, row2col9_in_aggaa(T28, T26, T27, X35, X36))
row2col9_in_aggaa(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97)) → U6_aggaa(T57, T61, T59, T60, X96, X97, row2col14_in_agaa(T61, T60, X96, X97))
row2col14_in_agaa(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151)) → U1_agaa(T82, T86, T84, T85, X150, X151, row2col14_in_agaa(T86, T85, X150, X151))
row2col14_in_agaa([], [], [], []) → row2col14_out_agaa([], [], [], [])
U1_agaa(T82, T86, T84, T85, X150, X151, row2col14_out_agaa(T86, T85, X150, X151)) → row2col14_out_agaa(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151))
U6_aggaa(T57, T61, T59, T60, X96, X97, row2col14_out_agaa(T61, T60, X96, X97)) → row2col9_out_aggaa(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97))
U2_aggaaa(T28, T26, T27, X35, X36, T29, row2col9_out_aggaa(T28, T26, T27, X35, X36)) → p7_out_aggaaa(T28, T26, T27, X35, X36, T29)
p7_in_aggaaa(T28, T26, T27, .(T119, T120), T118, .(T121, T122)) → U3_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_in_aggaa(T28, T26, T27, .(T119, T120), T118))
U3_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_out_aggaa(T28, T26, T27, .(T119, T120), T118)) → U4_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, p7_in_aggaaa(T121, T119, T120, X202, X203, T122))
p7_in_aggaaa(T28, T26, T27, T129, T129, []) → U5_aggaaa(T28, T26, T27, T129, row2col9_in_aggaa(T28, T26, T27, T129, T129))
U5_aggaaa(T28, T26, T27, T129, row2col9_out_aggaa(T28, T26, T27, T129, T129)) → p7_out_aggaaa(T28, T26, T27, T129, T129, [])
U4_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, p7_out_aggaaa(T121, T119, T120, X202, X203, T122)) → p7_out_aggaaa(T28, T26, T27, .(T119, T120), T118, .(T121, T122))
U7_ag(T28, T29, T26, T27, p7_out_aggaaa(T28, T26, T27, X35, X36, T29)) → transpose1_out_ag(.(T28, T29), .(T26, T27))
transpose1_in_ag([], []) → transpose1_out_ag([], [])

The argument filtering Pi contains the following mapping:
transpose1_in_ag(x1, x2)  =  transpose1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
U7_ag(x1, x2, x3, x4, x5)  =  U7_ag(x3, x4, x5)
p7_in_aggaaa(x1, x2, x3, x4, x5, x6)  =  p7_in_aggaaa(x2, x3)
U2_aggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U2_aggaaa(x2, x3, x7)
row2col9_in_aggaa(x1, x2, x3, x4, x5)  =  row2col9_in_aggaa(x2, x3)
U6_aggaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_aggaa(x1, x3, x4, x7)
row2col14_in_agaa(x1, x2, x3, x4)  =  row2col14_in_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U1_agaa(x1, x3, x4, x7)
[]  =  []
row2col14_out_agaa(x1, x2, x3, x4)  =  row2col14_out_agaa(x1, x2, x3, x4)
row2col9_out_aggaa(x1, x2, x3, x4, x5)  =  row2col9_out_aggaa(x1, x2, x3, x4, x5)
p7_out_aggaaa(x1, x2, x3, x4, x5, x6)  =  p7_out_aggaaa(x1, x2, x3, x4, x5)
U3_aggaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_aggaaa(x2, x3, x9)
U4_aggaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_aggaaa(x1, x2, x3, x4, x5, x6, x9)
U5_aggaaa(x1, x2, x3, x4, x5)  =  U5_aggaaa(x2, x3, x5)
transpose1_out_ag(x1, x2)  =  transpose1_out_ag(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_ag(.(T28, T29), .(T26, T27)) → U7_ag(T28, T29, T26, T27, p7_in_aggaaa(T28, T26, T27, X35, X36, T29))
p7_in_aggaaa(T28, T26, T27, X35, X36, T29) → U2_aggaaa(T28, T26, T27, X35, X36, T29, row2col9_in_aggaa(T28, T26, T27, X35, X36))
row2col9_in_aggaa(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97)) → U6_aggaa(T57, T61, T59, T60, X96, X97, row2col14_in_agaa(T61, T60, X96, X97))
row2col14_in_agaa(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151)) → U1_agaa(T82, T86, T84, T85, X150, X151, row2col14_in_agaa(T86, T85, X150, X151))
row2col14_in_agaa([], [], [], []) → row2col14_out_agaa([], [], [], [])
U1_agaa(T82, T86, T84, T85, X150, X151, row2col14_out_agaa(T86, T85, X150, X151)) → row2col14_out_agaa(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151))
U6_aggaa(T57, T61, T59, T60, X96, X97, row2col14_out_agaa(T61, T60, X96, X97)) → row2col9_out_aggaa(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97))
U2_aggaaa(T28, T26, T27, X35, X36, T29, row2col9_out_aggaa(T28, T26, T27, X35, X36)) → p7_out_aggaaa(T28, T26, T27, X35, X36, T29)
p7_in_aggaaa(T28, T26, T27, .(T119, T120), T118, .(T121, T122)) → U3_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_in_aggaa(T28, T26, T27, .(T119, T120), T118))
U3_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_out_aggaa(T28, T26, T27, .(T119, T120), T118)) → U4_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, p7_in_aggaaa(T121, T119, T120, X202, X203, T122))
p7_in_aggaaa(T28, T26, T27, T129, T129, []) → U5_aggaaa(T28, T26, T27, T129, row2col9_in_aggaa(T28, T26, T27, T129, T129))
U5_aggaaa(T28, T26, T27, T129, row2col9_out_aggaa(T28, T26, T27, T129, T129)) → p7_out_aggaaa(T28, T26, T27, T129, T129, [])
U4_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, p7_out_aggaaa(T121, T119, T120, X202, X203, T122)) → p7_out_aggaaa(T28, T26, T27, .(T119, T120), T118, .(T121, T122))
U7_ag(T28, T29, T26, T27, p7_out_aggaaa(T28, T26, T27, X35, X36, T29)) → transpose1_out_ag(.(T28, T29), .(T26, T27))
transpose1_in_ag([], []) → transpose1_out_ag([], [])

The argument filtering Pi contains the following mapping:
transpose1_in_ag(x1, x2)  =  transpose1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
U7_ag(x1, x2, x3, x4, x5)  =  U7_ag(x3, x4, x5)
p7_in_aggaaa(x1, x2, x3, x4, x5, x6)  =  p7_in_aggaaa(x2, x3)
U2_aggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U2_aggaaa(x2, x3, x7)
row2col9_in_aggaa(x1, x2, x3, x4, x5)  =  row2col9_in_aggaa(x2, x3)
U6_aggaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_aggaa(x1, x3, x4, x7)
row2col14_in_agaa(x1, x2, x3, x4)  =  row2col14_in_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U1_agaa(x1, x3, x4, x7)
[]  =  []
row2col14_out_agaa(x1, x2, x3, x4)  =  row2col14_out_agaa(x1, x2, x3, x4)
row2col9_out_aggaa(x1, x2, x3, x4, x5)  =  row2col9_out_aggaa(x1, x2, x3, x4, x5)
p7_out_aggaaa(x1, x2, x3, x4, x5, x6)  =  p7_out_aggaaa(x1, x2, x3, x4, x5)
U3_aggaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_aggaaa(x2, x3, x9)
U4_aggaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_aggaaa(x1, x2, x3, x4, x5, x6, x9)
U5_aggaaa(x1, x2, x3, x4, x5)  =  U5_aggaaa(x2, x3, x5)
transpose1_out_ag(x1, x2)  =  transpose1_out_ag(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_AG(.(T28, T29), .(T26, T27)) → U7_AG(T28, T29, T26, T27, p7_in_aggaaa(T28, T26, T27, X35, X36, T29))
TRANSPOSE1_IN_AG(.(T28, T29), .(T26, T27)) → P7_IN_AGGAAA(T28, T26, T27, X35, X36, T29)
P7_IN_AGGAAA(T28, T26, T27, X35, X36, T29) → U2_AGGAAA(T28, T26, T27, X35, X36, T29, row2col9_in_aggaa(T28, T26, T27, X35, X36))
P7_IN_AGGAAA(T28, T26, T27, X35, X36, T29) → ROW2COL9_IN_AGGAA(T28, T26, T27, X35, X36)
ROW2COL9_IN_AGGAA(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97)) → U6_AGGAA(T57, T61, T59, T60, X96, X97, row2col14_in_agaa(T61, T60, X96, X97))
ROW2COL9_IN_AGGAA(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97)) → ROW2COL14_IN_AGAA(T61, T60, X96, X97)
ROW2COL14_IN_AGAA(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151)) → U1_AGAA(T82, T86, T84, T85, X150, X151, row2col14_in_agaa(T86, T85, X150, X151))
ROW2COL14_IN_AGAA(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151)) → ROW2COL14_IN_AGAA(T86, T85, X150, X151)
P7_IN_AGGAAA(T28, T26, T27, .(T119, T120), T118, .(T121, T122)) → U3_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_in_aggaa(T28, T26, T27, .(T119, T120), T118))
P7_IN_AGGAAA(T28, T26, T27, .(T119, T120), T118, .(T121, T122)) → ROW2COL9_IN_AGGAA(T28, T26, T27, .(T119, T120), T118)
U3_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_out_aggaa(T28, T26, T27, .(T119, T120), T118)) → U4_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, p7_in_aggaaa(T121, T119, T120, X202, X203, T122))
U3_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_out_aggaa(T28, T26, T27, .(T119, T120), T118)) → P7_IN_AGGAAA(T121, T119, T120, X202, X203, T122)
P7_IN_AGGAAA(T28, T26, T27, T129, T129, []) → U5_AGGAAA(T28, T26, T27, T129, row2col9_in_aggaa(T28, T26, T27, T129, T129))
P7_IN_AGGAAA(T28, T26, T27, T129, T129, []) → ROW2COL9_IN_AGGAA(T28, T26, T27, T129, T129)

The TRS R consists of the following rules:

transpose1_in_ag(.(T28, T29), .(T26, T27)) → U7_ag(T28, T29, T26, T27, p7_in_aggaaa(T28, T26, T27, X35, X36, T29))
p7_in_aggaaa(T28, T26, T27, X35, X36, T29) → U2_aggaaa(T28, T26, T27, X35, X36, T29, row2col9_in_aggaa(T28, T26, T27, X35, X36))
row2col9_in_aggaa(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97)) → U6_aggaa(T57, T61, T59, T60, X96, X97, row2col14_in_agaa(T61, T60, X96, X97))
row2col14_in_agaa(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151)) → U1_agaa(T82, T86, T84, T85, X150, X151, row2col14_in_agaa(T86, T85, X150, X151))
row2col14_in_agaa([], [], [], []) → row2col14_out_agaa([], [], [], [])
U1_agaa(T82, T86, T84, T85, X150, X151, row2col14_out_agaa(T86, T85, X150, X151)) → row2col14_out_agaa(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151))
U6_aggaa(T57, T61, T59, T60, X96, X97, row2col14_out_agaa(T61, T60, X96, X97)) → row2col9_out_aggaa(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97))
U2_aggaaa(T28, T26, T27, X35, X36, T29, row2col9_out_aggaa(T28, T26, T27, X35, X36)) → p7_out_aggaaa(T28, T26, T27, X35, X36, T29)
p7_in_aggaaa(T28, T26, T27, .(T119, T120), T118, .(T121, T122)) → U3_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_in_aggaa(T28, T26, T27, .(T119, T120), T118))
U3_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_out_aggaa(T28, T26, T27, .(T119, T120), T118)) → U4_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, p7_in_aggaaa(T121, T119, T120, X202, X203, T122))
p7_in_aggaaa(T28, T26, T27, T129, T129, []) → U5_aggaaa(T28, T26, T27, T129, row2col9_in_aggaa(T28, T26, T27, T129, T129))
U5_aggaaa(T28, T26, T27, T129, row2col9_out_aggaa(T28, T26, T27, T129, T129)) → p7_out_aggaaa(T28, T26, T27, T129, T129, [])
U4_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, p7_out_aggaaa(T121, T119, T120, X202, X203, T122)) → p7_out_aggaaa(T28, T26, T27, .(T119, T120), T118, .(T121, T122))
U7_ag(T28, T29, T26, T27, p7_out_aggaaa(T28, T26, T27, X35, X36, T29)) → transpose1_out_ag(.(T28, T29), .(T26, T27))
transpose1_in_ag([], []) → transpose1_out_ag([], [])

The argument filtering Pi contains the following mapping:
transpose1_in_ag(x1, x2)  =  transpose1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
U7_ag(x1, x2, x3, x4, x5)  =  U7_ag(x3, x4, x5)
p7_in_aggaaa(x1, x2, x3, x4, x5, x6)  =  p7_in_aggaaa(x2, x3)
U2_aggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U2_aggaaa(x2, x3, x7)
row2col9_in_aggaa(x1, x2, x3, x4, x5)  =  row2col9_in_aggaa(x2, x3)
U6_aggaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_aggaa(x1, x3, x4, x7)
row2col14_in_agaa(x1, x2, x3, x4)  =  row2col14_in_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U1_agaa(x1, x3, x4, x7)
[]  =  []
row2col14_out_agaa(x1, x2, x3, x4)  =  row2col14_out_agaa(x1, x2, x3, x4)
row2col9_out_aggaa(x1, x2, x3, x4, x5)  =  row2col9_out_aggaa(x1, x2, x3, x4, x5)
p7_out_aggaaa(x1, x2, x3, x4, x5, x6)  =  p7_out_aggaaa(x1, x2, x3, x4, x5)
U3_aggaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_aggaaa(x2, x3, x9)
U4_aggaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_aggaaa(x1, x2, x3, x4, x5, x6, x9)
U5_aggaaa(x1, x2, x3, x4, x5)  =  U5_aggaaa(x2, x3, x5)
transpose1_out_ag(x1, x2)  =  transpose1_out_ag(x2)
TRANSPOSE1_IN_AG(x1, x2)  =  TRANSPOSE1_IN_AG(x2)
U7_AG(x1, x2, x3, x4, x5)  =  U7_AG(x3, x4, x5)
P7_IN_AGGAAA(x1, x2, x3, x4, x5, x6)  =  P7_IN_AGGAAA(x2, x3)
U2_AGGAAA(x1, x2, x3, x4, x5, x6, x7)  =  U2_AGGAAA(x2, x3, x7)
ROW2COL9_IN_AGGAA(x1, x2, x3, x4, x5)  =  ROW2COL9_IN_AGGAA(x2, x3)
U6_AGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_AGGAA(x1, x3, x4, x7)
ROW2COL14_IN_AGAA(x1, x2, x3, x4)  =  ROW2COL14_IN_AGAA(x2)
U1_AGAA(x1, x2, x3, x4, x5, x6, x7)  =  U1_AGAA(x1, x3, x4, x7)
U3_AGGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_AGGAAA(x2, x3, x9)
U4_AGGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_AGGAAA(x1, x2, x3, x4, x5, x6, x9)
U5_AGGAAA(x1, x2, x3, x4, x5)  =  U5_AGGAAA(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_AG(.(T28, T29), .(T26, T27)) → U7_AG(T28, T29, T26, T27, p7_in_aggaaa(T28, T26, T27, X35, X36, T29))
TRANSPOSE1_IN_AG(.(T28, T29), .(T26, T27)) → P7_IN_AGGAAA(T28, T26, T27, X35, X36, T29)
P7_IN_AGGAAA(T28, T26, T27, X35, X36, T29) → U2_AGGAAA(T28, T26, T27, X35, X36, T29, row2col9_in_aggaa(T28, T26, T27, X35, X36))
P7_IN_AGGAAA(T28, T26, T27, X35, X36, T29) → ROW2COL9_IN_AGGAA(T28, T26, T27, X35, X36)
ROW2COL9_IN_AGGAA(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97)) → U6_AGGAA(T57, T61, T59, T60, X96, X97, row2col14_in_agaa(T61, T60, X96, X97))
ROW2COL9_IN_AGGAA(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97)) → ROW2COL14_IN_AGAA(T61, T60, X96, X97)
ROW2COL14_IN_AGAA(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151)) → U1_AGAA(T82, T86, T84, T85, X150, X151, row2col14_in_agaa(T86, T85, X150, X151))
ROW2COL14_IN_AGAA(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151)) → ROW2COL14_IN_AGAA(T86, T85, X150, X151)
P7_IN_AGGAAA(T28, T26, T27, .(T119, T120), T118, .(T121, T122)) → U3_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_in_aggaa(T28, T26, T27, .(T119, T120), T118))
P7_IN_AGGAAA(T28, T26, T27, .(T119, T120), T118, .(T121, T122)) → ROW2COL9_IN_AGGAA(T28, T26, T27, .(T119, T120), T118)
U3_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_out_aggaa(T28, T26, T27, .(T119, T120), T118)) → U4_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, p7_in_aggaaa(T121, T119, T120, X202, X203, T122))
U3_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_out_aggaa(T28, T26, T27, .(T119, T120), T118)) → P7_IN_AGGAAA(T121, T119, T120, X202, X203, T122)
P7_IN_AGGAAA(T28, T26, T27, T129, T129, []) → U5_AGGAAA(T28, T26, T27, T129, row2col9_in_aggaa(T28, T26, T27, T129, T129))
P7_IN_AGGAAA(T28, T26, T27, T129, T129, []) → ROW2COL9_IN_AGGAA(T28, T26, T27, T129, T129)

The TRS R consists of the following rules:

transpose1_in_ag(.(T28, T29), .(T26, T27)) → U7_ag(T28, T29, T26, T27, p7_in_aggaaa(T28, T26, T27, X35, X36, T29))
p7_in_aggaaa(T28, T26, T27, X35, X36, T29) → U2_aggaaa(T28, T26, T27, X35, X36, T29, row2col9_in_aggaa(T28, T26, T27, X35, X36))
row2col9_in_aggaa(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97)) → U6_aggaa(T57, T61, T59, T60, X96, X97, row2col14_in_agaa(T61, T60, X96, X97))
row2col14_in_agaa(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151)) → U1_agaa(T82, T86, T84, T85, X150, X151, row2col14_in_agaa(T86, T85, X150, X151))
row2col14_in_agaa([], [], [], []) → row2col14_out_agaa([], [], [], [])
U1_agaa(T82, T86, T84, T85, X150, X151, row2col14_out_agaa(T86, T85, X150, X151)) → row2col14_out_agaa(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151))
U6_aggaa(T57, T61, T59, T60, X96, X97, row2col14_out_agaa(T61, T60, X96, X97)) → row2col9_out_aggaa(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97))
U2_aggaaa(T28, T26, T27, X35, X36, T29, row2col9_out_aggaa(T28, T26, T27, X35, X36)) → p7_out_aggaaa(T28, T26, T27, X35, X36, T29)
p7_in_aggaaa(T28, T26, T27, .(T119, T120), T118, .(T121, T122)) → U3_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_in_aggaa(T28, T26, T27, .(T119, T120), T118))
U3_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_out_aggaa(T28, T26, T27, .(T119, T120), T118)) → U4_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, p7_in_aggaaa(T121, T119, T120, X202, X203, T122))
p7_in_aggaaa(T28, T26, T27, T129, T129, []) → U5_aggaaa(T28, T26, T27, T129, row2col9_in_aggaa(T28, T26, T27, T129, T129))
U5_aggaaa(T28, T26, T27, T129, row2col9_out_aggaa(T28, T26, T27, T129, T129)) → p7_out_aggaaa(T28, T26, T27, T129, T129, [])
U4_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, p7_out_aggaaa(T121, T119, T120, X202, X203, T122)) → p7_out_aggaaa(T28, T26, T27, .(T119, T120), T118, .(T121, T122))
U7_ag(T28, T29, T26, T27, p7_out_aggaaa(T28, T26, T27, X35, X36, T29)) → transpose1_out_ag(.(T28, T29), .(T26, T27))
transpose1_in_ag([], []) → transpose1_out_ag([], [])

The argument filtering Pi contains the following mapping:
transpose1_in_ag(x1, x2)  =  transpose1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
U7_ag(x1, x2, x3, x4, x5)  =  U7_ag(x3, x4, x5)
p7_in_aggaaa(x1, x2, x3, x4, x5, x6)  =  p7_in_aggaaa(x2, x3)
U2_aggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U2_aggaaa(x2, x3, x7)
row2col9_in_aggaa(x1, x2, x3, x4, x5)  =  row2col9_in_aggaa(x2, x3)
U6_aggaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_aggaa(x1, x3, x4, x7)
row2col14_in_agaa(x1, x2, x3, x4)  =  row2col14_in_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U1_agaa(x1, x3, x4, x7)
[]  =  []
row2col14_out_agaa(x1, x2, x3, x4)  =  row2col14_out_agaa(x1, x2, x3, x4)
row2col9_out_aggaa(x1, x2, x3, x4, x5)  =  row2col9_out_aggaa(x1, x2, x3, x4, x5)
p7_out_aggaaa(x1, x2, x3, x4, x5, x6)  =  p7_out_aggaaa(x1, x2, x3, x4, x5)
U3_aggaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_aggaaa(x2, x3, x9)
U4_aggaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_aggaaa(x1, x2, x3, x4, x5, x6, x9)
U5_aggaaa(x1, x2, x3, x4, x5)  =  U5_aggaaa(x2, x3, x5)
transpose1_out_ag(x1, x2)  =  transpose1_out_ag(x2)
TRANSPOSE1_IN_AG(x1, x2)  =  TRANSPOSE1_IN_AG(x2)
U7_AG(x1, x2, x3, x4, x5)  =  U7_AG(x3, x4, x5)
P7_IN_AGGAAA(x1, x2, x3, x4, x5, x6)  =  P7_IN_AGGAAA(x2, x3)
U2_AGGAAA(x1, x2, x3, x4, x5, x6, x7)  =  U2_AGGAAA(x2, x3, x7)
ROW2COL9_IN_AGGAA(x1, x2, x3, x4, x5)  =  ROW2COL9_IN_AGGAA(x2, x3)
U6_AGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_AGGAA(x1, x3, x4, x7)
ROW2COL14_IN_AGAA(x1, x2, x3, x4)  =  ROW2COL14_IN_AGAA(x2)
U1_AGAA(x1, x2, x3, x4, x5, x6, x7)  =  U1_AGAA(x1, x3, x4, x7)
U3_AGGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_AGGAAA(x2, x3, x9)
U4_AGGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_AGGAAA(x1, x2, x3, x4, x5, x6, x9)
U5_AGGAAA(x1, x2, x3, x4, x5)  =  U5_AGGAAA(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_AGAA(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151)) → ROW2COL14_IN_AGAA(T86, T85, X150, X151)

The TRS R consists of the following rules:

transpose1_in_ag(.(T28, T29), .(T26, T27)) → U7_ag(T28, T29, T26, T27, p7_in_aggaaa(T28, T26, T27, X35, X36, T29))
p7_in_aggaaa(T28, T26, T27, X35, X36, T29) → U2_aggaaa(T28, T26, T27, X35, X36, T29, row2col9_in_aggaa(T28, T26, T27, X35, X36))
row2col9_in_aggaa(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97)) → U6_aggaa(T57, T61, T59, T60, X96, X97, row2col14_in_agaa(T61, T60, X96, X97))
row2col14_in_agaa(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151)) → U1_agaa(T82, T86, T84, T85, X150, X151, row2col14_in_agaa(T86, T85, X150, X151))
row2col14_in_agaa([], [], [], []) → row2col14_out_agaa([], [], [], [])
U1_agaa(T82, T86, T84, T85, X150, X151, row2col14_out_agaa(T86, T85, X150, X151)) → row2col14_out_agaa(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151))
U6_aggaa(T57, T61, T59, T60, X96, X97, row2col14_out_agaa(T61, T60, X96, X97)) → row2col9_out_aggaa(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97))
U2_aggaaa(T28, T26, T27, X35, X36, T29, row2col9_out_aggaa(T28, T26, T27, X35, X36)) → p7_out_aggaaa(T28, T26, T27, X35, X36, T29)
p7_in_aggaaa(T28, T26, T27, .(T119, T120), T118, .(T121, T122)) → U3_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_in_aggaa(T28, T26, T27, .(T119, T120), T118))
U3_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_out_aggaa(T28, T26, T27, .(T119, T120), T118)) → U4_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, p7_in_aggaaa(T121, T119, T120, X202, X203, T122))
p7_in_aggaaa(T28, T26, T27, T129, T129, []) → U5_aggaaa(T28, T26, T27, T129, row2col9_in_aggaa(T28, T26, T27, T129, T129))
U5_aggaaa(T28, T26, T27, T129, row2col9_out_aggaa(T28, T26, T27, T129, T129)) → p7_out_aggaaa(T28, T26, T27, T129, T129, [])
U4_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, p7_out_aggaaa(T121, T119, T120, X202, X203, T122)) → p7_out_aggaaa(T28, T26, T27, .(T119, T120), T118, .(T121, T122))
U7_ag(T28, T29, T26, T27, p7_out_aggaaa(T28, T26, T27, X35, X36, T29)) → transpose1_out_ag(.(T28, T29), .(T26, T27))
transpose1_in_ag([], []) → transpose1_out_ag([], [])

The argument filtering Pi contains the following mapping:
transpose1_in_ag(x1, x2)  =  transpose1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
U7_ag(x1, x2, x3, x4, x5)  =  U7_ag(x3, x4, x5)
p7_in_aggaaa(x1, x2, x3, x4, x5, x6)  =  p7_in_aggaaa(x2, x3)
U2_aggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U2_aggaaa(x2, x3, x7)
row2col9_in_aggaa(x1, x2, x3, x4, x5)  =  row2col9_in_aggaa(x2, x3)
U6_aggaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_aggaa(x1, x3, x4, x7)
row2col14_in_agaa(x1, x2, x3, x4)  =  row2col14_in_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U1_agaa(x1, x3, x4, x7)
[]  =  []
row2col14_out_agaa(x1, x2, x3, x4)  =  row2col14_out_agaa(x1, x2, x3, x4)
row2col9_out_aggaa(x1, x2, x3, x4, x5)  =  row2col9_out_aggaa(x1, x2, x3, x4, x5)
p7_out_aggaaa(x1, x2, x3, x4, x5, x6)  =  p7_out_aggaaa(x1, x2, x3, x4, x5)
U3_aggaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_aggaaa(x2, x3, x9)
U4_aggaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_aggaaa(x1, x2, x3, x4, x5, x6, x9)
U5_aggaaa(x1, x2, x3, x4, x5)  =  U5_aggaaa(x2, x3, x5)
transpose1_out_ag(x1, x2)  =  transpose1_out_ag(x2)
ROW2COL14_IN_AGAA(x1, x2, x3, x4)  =  ROW2COL14_IN_AGAA(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_AGAA(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151)) → ROW2COL14_IN_AGAA(T86, T85, X150, X151)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
ROW2COL14_IN_AGAA(x1, x2, x3, x4)  =  ROW2COL14_IN_AGAA(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_AGAA(.(.(T82, T84), T85)) → ROW2COL14_IN_AGAA(T85)

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_AGAA(.(.(T82, T84), T85)) → ROW2COL14_IN_AGAA(T85)
    The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

P7_IN_AGGAAA(T28, T26, T27, .(T119, T120), T118, .(T121, T122)) → U3_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_in_aggaa(T28, T26, T27, .(T119, T120), T118))
U3_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_out_aggaa(T28, T26, T27, .(T119, T120), T118)) → P7_IN_AGGAAA(T121, T119, T120, X202, X203, T122)

The TRS R consists of the following rules:

transpose1_in_ag(.(T28, T29), .(T26, T27)) → U7_ag(T28, T29, T26, T27, p7_in_aggaaa(T28, T26, T27, X35, X36, T29))
p7_in_aggaaa(T28, T26, T27, X35, X36, T29) → U2_aggaaa(T28, T26, T27, X35, X36, T29, row2col9_in_aggaa(T28, T26, T27, X35, X36))
row2col9_in_aggaa(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97)) → U6_aggaa(T57, T61, T59, T60, X96, X97, row2col14_in_agaa(T61, T60, X96, X97))
row2col14_in_agaa(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151)) → U1_agaa(T82, T86, T84, T85, X150, X151, row2col14_in_agaa(T86, T85, X150, X151))
row2col14_in_agaa([], [], [], []) → row2col14_out_agaa([], [], [], [])
U1_agaa(T82, T86, T84, T85, X150, X151, row2col14_out_agaa(T86, T85, X150, X151)) → row2col14_out_agaa(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151))
U6_aggaa(T57, T61, T59, T60, X96, X97, row2col14_out_agaa(T61, T60, X96, X97)) → row2col9_out_aggaa(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97))
U2_aggaaa(T28, T26, T27, X35, X36, T29, row2col9_out_aggaa(T28, T26, T27, X35, X36)) → p7_out_aggaaa(T28, T26, T27, X35, X36, T29)
p7_in_aggaaa(T28, T26, T27, .(T119, T120), T118, .(T121, T122)) → U3_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_in_aggaa(T28, T26, T27, .(T119, T120), T118))
U3_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_out_aggaa(T28, T26, T27, .(T119, T120), T118)) → U4_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, p7_in_aggaaa(T121, T119, T120, X202, X203, T122))
p7_in_aggaaa(T28, T26, T27, T129, T129, []) → U5_aggaaa(T28, T26, T27, T129, row2col9_in_aggaa(T28, T26, T27, T129, T129))
U5_aggaaa(T28, T26, T27, T129, row2col9_out_aggaa(T28, T26, T27, T129, T129)) → p7_out_aggaaa(T28, T26, T27, T129, T129, [])
U4_aggaaa(T28, T26, T27, T119, T120, T118, T121, T122, p7_out_aggaaa(T121, T119, T120, X202, X203, T122)) → p7_out_aggaaa(T28, T26, T27, .(T119, T120), T118, .(T121, T122))
U7_ag(T28, T29, T26, T27, p7_out_aggaaa(T28, T26, T27, X35, X36, T29)) → transpose1_out_ag(.(T28, T29), .(T26, T27))
transpose1_in_ag([], []) → transpose1_out_ag([], [])

The argument filtering Pi contains the following mapping:
transpose1_in_ag(x1, x2)  =  transpose1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
U7_ag(x1, x2, x3, x4, x5)  =  U7_ag(x3, x4, x5)
p7_in_aggaaa(x1, x2, x3, x4, x5, x6)  =  p7_in_aggaaa(x2, x3)
U2_aggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U2_aggaaa(x2, x3, x7)
row2col9_in_aggaa(x1, x2, x3, x4, x5)  =  row2col9_in_aggaa(x2, x3)
U6_aggaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_aggaa(x1, x3, x4, x7)
row2col14_in_agaa(x1, x2, x3, x4)  =  row2col14_in_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U1_agaa(x1, x3, x4, x7)
[]  =  []
row2col14_out_agaa(x1, x2, x3, x4)  =  row2col14_out_agaa(x1, x2, x3, x4)
row2col9_out_aggaa(x1, x2, x3, x4, x5)  =  row2col9_out_aggaa(x1, x2, x3, x4, x5)
p7_out_aggaaa(x1, x2, x3, x4, x5, x6)  =  p7_out_aggaaa(x1, x2, x3, x4, x5)
U3_aggaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_aggaaa(x2, x3, x9)
U4_aggaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_aggaaa(x1, x2, x3, x4, x5, x6, x9)
U5_aggaaa(x1, x2, x3, x4, x5)  =  U5_aggaaa(x2, x3, x5)
transpose1_out_ag(x1, x2)  =  transpose1_out_ag(x2)
P7_IN_AGGAAA(x1, x2, x3, x4, x5, x6)  =  P7_IN_AGGAAA(x2, x3)
U3_AGGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_AGGAAA(x2, x3, 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_AGGAAA(T28, T26, T27, .(T119, T120), T118, .(T121, T122)) → U3_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_in_aggaa(T28, T26, T27, .(T119, T120), T118))
U3_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, row2col9_out_aggaa(T28, T26, T27, .(T119, T120), T118)) → P7_IN_AGGAAA(T121, T119, T120, X202, X203, T122)

The TRS R consists of the following rules:

row2col9_in_aggaa(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97)) → U6_aggaa(T57, T61, T59, T60, X96, X97, row2col14_in_agaa(T61, T60, X96, X97))
U6_aggaa(T57, T61, T59, T60, X96, X97, row2col14_out_agaa(T61, T60, X96, X97)) → row2col9_out_aggaa(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97))
row2col14_in_agaa(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151)) → U1_agaa(T82, T86, T84, T85, X150, X151, row2col14_in_agaa(T86, T85, X150, X151))
row2col14_in_agaa([], [], [], []) → row2col14_out_agaa([], [], [], [])
U1_agaa(T82, T86, T84, T85, X150, X151, row2col14_out_agaa(T86, T85, X150, X151)) → row2col14_out_agaa(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
row2col9_in_aggaa(x1, x2, x3, x4, x5)  =  row2col9_in_aggaa(x2, x3)
U6_aggaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_aggaa(x1, x3, x4, x7)
row2col14_in_agaa(x1, x2, x3, x4)  =  row2col14_in_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U1_agaa(x1, x3, x4, x7)
[]  =  []
row2col14_out_agaa(x1, x2, x3, x4)  =  row2col14_out_agaa(x1, x2, x3, x4)
row2col9_out_aggaa(x1, x2, x3, x4, x5)  =  row2col9_out_aggaa(x1, x2, x3, x4, x5)
P7_IN_AGGAAA(x1, x2, x3, x4, x5, x6)  =  P7_IN_AGGAAA(x2, x3)
U3_AGGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_AGGAAA(x2, x3, 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_AGGAAA(T26, T27) → U3_AGGAAA(T26, T27, row2col9_in_aggaa(T26, T27))
U3_AGGAAA(T26, T27, row2col9_out_aggaa(T28, T26, T27, .(T119, T120), T118)) → P7_IN_AGGAAA(T119, T120)

The TRS R consists of the following rules:

row2col9_in_aggaa(.(T57, T59), T60) → U6_aggaa(T57, T59, T60, row2col14_in_agaa(T60))
U6_aggaa(T57, T59, T60, row2col14_out_agaa(T61, T60, X96, X97)) → row2col9_out_aggaa(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97))
row2col14_in_agaa(.(.(T82, T84), T85)) → U1_agaa(T82, T84, T85, row2col14_in_agaa(T85))
row2col14_in_agaa([]) → row2col14_out_agaa([], [], [], [])
U1_agaa(T82, T84, T85, row2col14_out_agaa(T86, T85, X150, X151)) → row2col14_out_agaa(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151))

The set Q consists of the following terms:

row2col9_in_aggaa(x0, x1)
U6_aggaa(x0, x1, x2, x3)
row2col14_in_agaa(x0)
U1_agaa(x0, x1, x2, x3)

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


P7_IN_AGGAAA(T26, T27) → U3_AGGAAA(T26, T27, row2col9_in_aggaa(T26, T27))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x1 + x2   
POL(P7_IN_AGGAAA(x1, x2)) = 1 + x1 + x2   
POL(U1_agaa(x1, x2, x3, x4)) = 1 + x2 + x4   
POL(U3_AGGAAA(x1, x2, x3)) = x3   
POL(U6_aggaa(x1, x2, x3, x4)) = 1 + x2 + x4   
POL([]) = 0   
POL(row2col14_in_agaa(x1)) = x1   
POL(row2col14_out_agaa(x1, x2, x3, x4)) = x3   
POL(row2col9_in_aggaa(x1, x2)) = x1 + x2   
POL(row2col9_out_aggaa(x1, x2, x3, x4, x5)) = x4   

The following usable rules [FROCOS05] were oriented:

row2col9_in_aggaa(.(T57, T59), T60) → U6_aggaa(T57, T59, T60, row2col14_in_agaa(T60))
row2col14_in_agaa(.(.(T82, T84), T85)) → U1_agaa(T82, T84, T85, row2col14_in_agaa(T85))
row2col14_in_agaa([]) → row2col14_out_agaa([], [], [], [])
U6_aggaa(T57, T59, T60, row2col14_out_agaa(T61, T60, X96, X97)) → row2col9_out_aggaa(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97))
U1_agaa(T82, T84, T85, row2col14_out_agaa(T86, T85, X150, X151)) → row2col14_out_agaa(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151))

(22) Obligation:

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

U3_AGGAAA(T26, T27, row2col9_out_aggaa(T28, T26, T27, .(T119, T120), T118)) → P7_IN_AGGAAA(T119, T120)

The TRS R consists of the following rules:

row2col9_in_aggaa(.(T57, T59), T60) → U6_aggaa(T57, T59, T60, row2col14_in_agaa(T60))
U6_aggaa(T57, T59, T60, row2col14_out_agaa(T61, T60, X96, X97)) → row2col9_out_aggaa(.(T57, T61), .(T57, T59), T60, .(T59, X96), .([], X97))
row2col14_in_agaa(.(.(T82, T84), T85)) → U1_agaa(T82, T84, T85, row2col14_in_agaa(T85))
row2col14_in_agaa([]) → row2col14_out_agaa([], [], [], [])
U1_agaa(T82, T84, T85, row2col14_out_agaa(T86, T85, X150, X151)) → row2col14_out_agaa(.(T82, T86), .(.(T82, T84), T85), .(T84, X150), .([], X151))

The set Q consists of the following terms:

row2col9_in_aggaa(x0, x1)
U6_aggaa(x0, x1, x2, x3)
row2col14_in_agaa(x0)
U1_agaa(x0, x1, x2, x3)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(24) TRUE