Left Termination of the query pattern shanoi_in_5(g, g, g, g, a) w.r.t. the given Prolog program could not be shown:

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 NonTerminationProof (⇔)
                ↳22 NO
            ↳23 PiDP
                ↳24 UsableRulesProof (⇔)
                ↳25 PiDP
                ↳26 PiDPToQDPProof (⇐)
                ↳27 QDP
                ↳28 QDPSizeChangeProof (⇔)
                ↳29 YES
            ↳30 PiDP
                ↳31 UsableRulesProof (⇔)
                ↳32 PiDP
                ↳33 PiDPToQDPProof (⇐)
                ↳34 QDP
                ↳35 NonTerminationProof (⇔)
                ↳36 NO
            ↳37 PiDP
                ↳38 UsableRulesProof (⇔)
                ↳39 PiDP
                ↳40 PiDPToQDPProof (⇐)
                ↳41 QDP
                ↳42 QDPSizeChangeProof (⇔)
                ↳43 YES
    ↳44 PrologToPiTRSProof (⇐)
        ↳45 PiTRS
        ↳46 DependencyPairsProof (⇔)
        ↳47 PiDP
        ↳48 DependencyGraphProof (⇔)
        ↳49 AND
            ↳50 PiDP
                ↳51 UsableRulesProof (⇔)
                ↳52 PiDP
                ↳53 PiDPToQDPProof (⇐)
                ↳54 QDP
                ↳55 QDPSizeChangeProof (⇔)
                ↳56 YES
            ↳57 PiDP
                ↳58 UsableRulesProof (⇔)
                ↳59 PiDP
                ↳60 PiDPToQDPProof (⇐)
                ↳61 QDP
                ↳62 NonTerminationProof (⇔)
                ↳63 NO
            ↳64 PiDP
                ↳65 UsableRulesProof (⇔)
                ↳66 PiDP
                ↳67 PiDPToQDPProof (⇐)
                ↳68 QDP
                ↳69 QDPSizeChangeProof (⇔)
                ↳70 YES
            ↳71 PiDP
                ↳72 UsableRulesProof (⇔)
                ↳73 PiDP
                ↳74 PiDPToQDPProof (⇐)
                ↳75 QDP
                ↳76 NonTerminationProof (⇔)
                ↳77 NO
            ↳78 PiDP
                ↳79 UsableRulesProof (⇔)
                ↳80 PiDP
                ↳81 PiDPToQDPProof (⇐)
                ↳82 QDP
                ↳83 QDPSizeChangeProof (⇔)
                ↳84 YES

(0) Obligation:

Clauses:

shanoi(s(0), A, B, C, .(mv(A, C), [])).
shanoi(s(s(X)), A, B, C, M) :- ','(eq(N1, s(X)), ','(shanoi(N1, A, C, B, M1), ','(shanoi(N1, B, A, C, M2), ','(append(M1, .(mv(A, C), []), T), append(T, M2, M))))).
append([], L, L).
append(.(H, L), L1, .(H, R)) :- append(L, L1, R).
eq(X, X).

Queries:

shanoi(g,g,g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

shanoi11(0, T57, T58, T59, .(mv(T57, T59), [])).
shanoi11(s(T88), T69, T70, T71, X101) :- shanoi11(T88, T69, T71, T70, X98).
shanoi11(s(T88), T69, T70, T71, X101) :- ','(shanoi11(T88, T69, T71, T70, T95), shanoi11(T88, T70, T69, T71, X99)).
shanoi11(s(T88), T69, T70, T71, X101) :- ','(shanoi11(T88, T69, T71, T70, T95), ','(shanoi11(T88, T70, T69, T71, T108), append28(T95, .(mv(T69, T71), []), X100))).
shanoi11(s(T88), T69, T70, T71, X101) :- ','(shanoi11(T88, T69, T71, T70, T95), ','(shanoi11(T88, T70, T69, T71, T108), ','(append28(T95, .(mv(T69, T71), []), T119), append28(T119, T108, X101)))).
append28([], T130, T130).
append28(.(T137, T138), T139, .(T137, X178)) :- append28(T138, T139, X178).
append40([], T170, T170).
append40(.(T179, T180), T181, .(T179, T183)) :- append40(T180, T181, T183).
shanoi1(s(0), T9, T10, T11, .(mv(T9, T11), [])).
shanoi1(s(s(T31)), T18, T19, T20, T22) :- shanoi11(T31, T18, T20, T19, X26).
shanoi1(s(s(T31)), T18, T19, T20, T22) :- ','(shanoi11(T31, T18, T20, T19, T38), shanoi11(T31, T20, T18, T20, X27)).
shanoi1(s(s(T31)), T18, T19, T20, T22) :- ','(shanoi11(T31, T18, T20, T19, T38), ','(shanoi11(T31, T20, T18, T20, T148), append40(T38, .(mv(T18, T20), []), X28))).
shanoi1(s(s(T31)), T18, T19, T20, T22) :- ','(shanoi11(T31, T18, T20, T19, T38), ','(shanoi11(T31, T20, T18, T20, T148), ','(append40(T38, .(mv(T18, T20), []), T159), append40(T159, T148, T22)))).

Queries:

shanoi1(g,g,g,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:
shanoi1_in: (b,b,b,b,f)
shanoi11_in: (b,b,b,b,f)
append28_in: (f,b,f) (b,f,f)
append40_in: (f,b,f) (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

shanoi1_in_gggga(s(0), T9, T10, T11, .(mv(T9, T11), [])) → shanoi1_out_gggga(s(0), T9, T10, T11, .(mv(T9, T11), []))
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U10_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U10_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, X26)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U11_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U12_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T20, T18, T20, X27)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_gggga(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
append40_in_aga([], T170, T170) → append40_out_aga([], T170, T170)
append40_in_aga(.(T179, T180), T181, .(T179, T183)) → U9_aga(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
U9_aga(T179, T180, T181, T183, append40_out_aga(T180, T181, T183)) → append40_out_aga(.(T179, T180), T181, .(T179, T183))
U14_gggga(T31, T18, T19, T20, T22, append40_out_aga(T38, .(mv(T18, T20), []), X28)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_gggga(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_gggga(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_gggga(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
append40_in_gaa([], T170, T170) → append40_out_gaa([], T170, T170)
append40_in_gaa(.(T179, T180), T181, .(T179, T183)) → U9_gaa(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
U9_gaa(T179, T180, T181, T183, append40_out_gaa(T180, T181, T183)) → append40_out_gaa(.(T179, T180), T181, .(T179, T183))
U16_gggga(T31, T18, T19, T20, T22, append40_out_gaa(T159, T148, T22)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)

The argument filtering Pi contains the following mapping:
shanoi1_in_gggga(x1, x2, x3, x4, x5)  =  shanoi1_in_gggga(x1, x2, x3, x4)
s(x1)  =  s(x1)
0  =  0
shanoi1_out_gggga(x1, x2, x3, x4, x5)  =  shanoi1_out_gggga
U10_gggga(x1, x2, x3, x4, x5, x6)  =  U10_gggga(x6)
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U11_gggga(x1, x2, x3, x4, x5, x6)  =  U11_gggga(x1, x2, x4, x6)
U12_gggga(x1, x2, x3, x4, x5, x6)  =  U12_gggga(x6)
U13_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U13_gggga(x7)
U14_gggga(x1, x2, x3, x4, x5, x6)  =  U14_gggga(x6)
append40_in_aga(x1, x2, x3)  =  append40_in_aga(x2)
append40_out_aga(x1, x2, x3)  =  append40_out_aga(x1, x3)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x5)
U15_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U15_gggga(x7)
U16_gggga(x1, x2, x3, x4, x5, x6)  =  U16_gggga(x6)
append40_in_gaa(x1, x2, x3)  =  append40_in_gaa(x1)
append40_out_gaa(x1, x2, x3)  =  append40_out_gaa
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

shanoi1_in_gggga(s(0), T9, T10, T11, .(mv(T9, T11), [])) → shanoi1_out_gggga(s(0), T9, T10, T11, .(mv(T9, T11), []))
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U10_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U10_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, X26)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U11_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U12_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T20, T18, T20, X27)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_gggga(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
append40_in_aga([], T170, T170) → append40_out_aga([], T170, T170)
append40_in_aga(.(T179, T180), T181, .(T179, T183)) → U9_aga(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
U9_aga(T179, T180, T181, T183, append40_out_aga(T180, T181, T183)) → append40_out_aga(.(T179, T180), T181, .(T179, T183))
U14_gggga(T31, T18, T19, T20, T22, append40_out_aga(T38, .(mv(T18, T20), []), X28)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_gggga(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_gggga(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_gggga(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
append40_in_gaa([], T170, T170) → append40_out_gaa([], T170, T170)
append40_in_gaa(.(T179, T180), T181, .(T179, T183)) → U9_gaa(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
U9_gaa(T179, T180, T181, T183, append40_out_gaa(T180, T181, T183)) → append40_out_gaa(.(T179, T180), T181, .(T179, T183))
U16_gggga(T31, T18, T19, T20, T22, append40_out_gaa(T159, T148, T22)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)

The argument filtering Pi contains the following mapping:
shanoi1_in_gggga(x1, x2, x3, x4, x5)  =  shanoi1_in_gggga(x1, x2, x3, x4)
s(x1)  =  s(x1)
0  =  0
shanoi1_out_gggga(x1, x2, x3, x4, x5)  =  shanoi1_out_gggga
U10_gggga(x1, x2, x3, x4, x5, x6)  =  U10_gggga(x6)
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U11_gggga(x1, x2, x3, x4, x5, x6)  =  U11_gggga(x1, x2, x4, x6)
U12_gggga(x1, x2, x3, x4, x5, x6)  =  U12_gggga(x6)
U13_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U13_gggga(x7)
U14_gggga(x1, x2, x3, x4, x5, x6)  =  U14_gggga(x6)
append40_in_aga(x1, x2, x3)  =  append40_in_aga(x2)
append40_out_aga(x1, x2, x3)  =  append40_out_aga(x1, x3)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x5)
U15_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U15_gggga(x7)
U16_gggga(x1, x2, x3, x4, x5, x6)  =  U16_gggga(x6)
append40_in_gaa(x1, x2, x3)  =  append40_in_gaa(x1)
append40_out_gaa(x1, x2, x3)  =  append40_out_gaa
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

(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:

SHANOI1_IN_GGGGA(s(s(T31)), T18, T19, T20, T22) → U10_GGGGA(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
SHANOI1_IN_GGGGA(s(s(T31)), T18, T19, T20, T22) → SHANOI11_IN_GGGGA(T31, T18, T20, T19, X26)
SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → U1_GGGGA(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → SHANOI11_IN_GGGGA(T88, T69, T71, T70, X98)
SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_GGGGA(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → SHANOI11_IN_GGGGA(T88, T70, T69, T71, X99)
U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_GGGGA(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_GGGGA(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_GGGGA(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
U4_GGGGA(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → APPEND28_IN_AGA(T95, .(mv(T69, T71), []), X100)
APPEND28_IN_AGA(.(T137, T138), T139, .(T137, X178)) → U8_AGA(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
APPEND28_IN_AGA(.(T137, T138), T139, .(T137, X178)) → APPEND28_IN_AGA(T138, T139, X178)
U4_GGGGA(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_GGGGA(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_GGGGA(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_GGGGA(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
U6_GGGGA(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → APPEND28_IN_GAA(T119, T108, X101)
APPEND28_IN_GAA(.(T137, T138), T139, .(T137, X178)) → U8_GAA(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
APPEND28_IN_GAA(.(T137, T138), T139, .(T137, X178)) → APPEND28_IN_GAA(T138, T139, X178)
SHANOI1_IN_GGGGA(s(s(T31)), T18, T19, T20, T22) → U11_GGGGA(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_GGGGA(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_GGGGA(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U11_GGGGA(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → SHANOI11_IN_GGGGA(T31, T20, T18, T20, X27)
U11_GGGGA(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_GGGGA(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_GGGGA(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_GGGGA(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
U13_GGGGA(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → APPEND40_IN_AGA(T38, .(mv(T18, T20), []), X28)
APPEND40_IN_AGA(.(T179, T180), T181, .(T179, T183)) → U9_AGA(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
APPEND40_IN_AGA(.(T179, T180), T181, .(T179, T183)) → APPEND40_IN_AGA(T180, T181, T183)
U13_GGGGA(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_GGGGA(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_GGGGA(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_GGGGA(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
U15_GGGGA(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → APPEND40_IN_GAA(T159, T148, T22)
APPEND40_IN_GAA(.(T179, T180), T181, .(T179, T183)) → U9_GAA(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
APPEND40_IN_GAA(.(T179, T180), T181, .(T179, T183)) → APPEND40_IN_GAA(T180, T181, T183)

The TRS R consists of the following rules:

shanoi1_in_gggga(s(0), T9, T10, T11, .(mv(T9, T11), [])) → shanoi1_out_gggga(s(0), T9, T10, T11, .(mv(T9, T11), []))
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U10_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U10_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, X26)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U11_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U12_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T20, T18, T20, X27)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_gggga(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
append40_in_aga([], T170, T170) → append40_out_aga([], T170, T170)
append40_in_aga(.(T179, T180), T181, .(T179, T183)) → U9_aga(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
U9_aga(T179, T180, T181, T183, append40_out_aga(T180, T181, T183)) → append40_out_aga(.(T179, T180), T181, .(T179, T183))
U14_gggga(T31, T18, T19, T20, T22, append40_out_aga(T38, .(mv(T18, T20), []), X28)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_gggga(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_gggga(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_gggga(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
append40_in_gaa([], T170, T170) → append40_out_gaa([], T170, T170)
append40_in_gaa(.(T179, T180), T181, .(T179, T183)) → U9_gaa(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
U9_gaa(T179, T180, T181, T183, append40_out_gaa(T180, T181, T183)) → append40_out_gaa(.(T179, T180), T181, .(T179, T183))
U16_gggga(T31, T18, T19, T20, T22, append40_out_gaa(T159, T148, T22)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)

The argument filtering Pi contains the following mapping:
shanoi1_in_gggga(x1, x2, x3, x4, x5)  =  shanoi1_in_gggga(x1, x2, x3, x4)
s(x1)  =  s(x1)
0  =  0
shanoi1_out_gggga(x1, x2, x3, x4, x5)  =  shanoi1_out_gggga
U10_gggga(x1, x2, x3, x4, x5, x6)  =  U10_gggga(x6)
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U11_gggga(x1, x2, x3, x4, x5, x6)  =  U11_gggga(x1, x2, x4, x6)
U12_gggga(x1, x2, x3, x4, x5, x6)  =  U12_gggga(x6)
U13_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U13_gggga(x7)
U14_gggga(x1, x2, x3, x4, x5, x6)  =  U14_gggga(x6)
append40_in_aga(x1, x2, x3)  =  append40_in_aga(x2)
append40_out_aga(x1, x2, x3)  =  append40_out_aga(x1, x3)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x5)
U15_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U15_gggga(x7)
U16_gggga(x1, x2, x3, x4, x5, x6)  =  U16_gggga(x6)
append40_in_gaa(x1, x2, x3)  =  append40_in_gaa(x1)
append40_out_gaa(x1, x2, x3)  =  append40_out_gaa
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
SHANOI1_IN_GGGGA(x1, x2, x3, x4, x5)  =  SHANOI1_IN_GGGGA(x1, x2, x3, x4)
U10_GGGGA(x1, x2, x3, x4, x5, x6)  =  U10_GGGGA(x6)
SHANOI11_IN_GGGGA(x1, x2, x3, x4, x5)  =  SHANOI11_IN_GGGGA(x1, x2, x3, x4)
U1_GGGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGGA(x6)
U2_GGGGA(x1, x2, x3, x4, x5, x6)  =  U2_GGGGA(x1, x2, x3, x4, x6)
U3_GGGGA(x1, x2, x3, x4, x5, x6)  =  U3_GGGGA(x6)
U4_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GGGGA(x7)
U5_GGGGA(x1, x2, x3, x4, x5, x6)  =  U5_GGGGA(x6)
APPEND28_IN_AGA(x1, x2, x3)  =  APPEND28_IN_AGA(x2)
U8_AGA(x1, x2, x3, x4, x5)  =  U8_AGA(x5)
U6_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GGGGA(x7)
U7_GGGGA(x1, x2, x3, x4, x5, x6)  =  U7_GGGGA(x6)
APPEND28_IN_GAA(x1, x2, x3)  =  APPEND28_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4, x5)  =  U8_GAA(x5)
U11_GGGGA(x1, x2, x3, x4, x5, x6)  =  U11_GGGGA(x1, x2, x4, x6)
U12_GGGGA(x1, x2, x3, x4, x5, x6)  =  U12_GGGGA(x6)
U13_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U13_GGGGA(x7)
U14_GGGGA(x1, x2, x3, x4, x5, x6)  =  U14_GGGGA(x6)
APPEND40_IN_AGA(x1, x2, x3)  =  APPEND40_IN_AGA(x2)
U9_AGA(x1, x2, x3, x4, x5)  =  U9_AGA(x5)
U15_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U15_GGGGA(x7)
U16_GGGGA(x1, x2, x3, x4, x5, x6)  =  U16_GGGGA(x6)
APPEND40_IN_GAA(x1, x2, x3)  =  APPEND40_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x5)

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

(6) Obligation:

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

SHANOI1_IN_GGGGA(s(s(T31)), T18, T19, T20, T22) → U10_GGGGA(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
SHANOI1_IN_GGGGA(s(s(T31)), T18, T19, T20, T22) → SHANOI11_IN_GGGGA(T31, T18, T20, T19, X26)
SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → U1_GGGGA(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → SHANOI11_IN_GGGGA(T88, T69, T71, T70, X98)
SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_GGGGA(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → SHANOI11_IN_GGGGA(T88, T70, T69, T71, X99)
U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_GGGGA(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_GGGGA(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_GGGGA(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
U4_GGGGA(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → APPEND28_IN_AGA(T95, .(mv(T69, T71), []), X100)
APPEND28_IN_AGA(.(T137, T138), T139, .(T137, X178)) → U8_AGA(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
APPEND28_IN_AGA(.(T137, T138), T139, .(T137, X178)) → APPEND28_IN_AGA(T138, T139, X178)
U4_GGGGA(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_GGGGA(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_GGGGA(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_GGGGA(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
U6_GGGGA(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → APPEND28_IN_GAA(T119, T108, X101)
APPEND28_IN_GAA(.(T137, T138), T139, .(T137, X178)) → U8_GAA(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
APPEND28_IN_GAA(.(T137, T138), T139, .(T137, X178)) → APPEND28_IN_GAA(T138, T139, X178)
SHANOI1_IN_GGGGA(s(s(T31)), T18, T19, T20, T22) → U11_GGGGA(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_GGGGA(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_GGGGA(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U11_GGGGA(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → SHANOI11_IN_GGGGA(T31, T20, T18, T20, X27)
U11_GGGGA(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_GGGGA(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_GGGGA(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_GGGGA(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
U13_GGGGA(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → APPEND40_IN_AGA(T38, .(mv(T18, T20), []), X28)
APPEND40_IN_AGA(.(T179, T180), T181, .(T179, T183)) → U9_AGA(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
APPEND40_IN_AGA(.(T179, T180), T181, .(T179, T183)) → APPEND40_IN_AGA(T180, T181, T183)
U13_GGGGA(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_GGGGA(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_GGGGA(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_GGGGA(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
U15_GGGGA(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → APPEND40_IN_GAA(T159, T148, T22)
APPEND40_IN_GAA(.(T179, T180), T181, .(T179, T183)) → U9_GAA(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
APPEND40_IN_GAA(.(T179, T180), T181, .(T179, T183)) → APPEND40_IN_GAA(T180, T181, T183)

The TRS R consists of the following rules:

shanoi1_in_gggga(s(0), T9, T10, T11, .(mv(T9, T11), [])) → shanoi1_out_gggga(s(0), T9, T10, T11, .(mv(T9, T11), []))
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U10_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U10_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, X26)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U11_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U12_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T20, T18, T20, X27)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_gggga(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
append40_in_aga([], T170, T170) → append40_out_aga([], T170, T170)
append40_in_aga(.(T179, T180), T181, .(T179, T183)) → U9_aga(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
U9_aga(T179, T180, T181, T183, append40_out_aga(T180, T181, T183)) → append40_out_aga(.(T179, T180), T181, .(T179, T183))
U14_gggga(T31, T18, T19, T20, T22, append40_out_aga(T38, .(mv(T18, T20), []), X28)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_gggga(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_gggga(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_gggga(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
append40_in_gaa([], T170, T170) → append40_out_gaa([], T170, T170)
append40_in_gaa(.(T179, T180), T181, .(T179, T183)) → U9_gaa(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
U9_gaa(T179, T180, T181, T183, append40_out_gaa(T180, T181, T183)) → append40_out_gaa(.(T179, T180), T181, .(T179, T183))
U16_gggga(T31, T18, T19, T20, T22, append40_out_gaa(T159, T148, T22)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)

The argument filtering Pi contains the following mapping:
shanoi1_in_gggga(x1, x2, x3, x4, x5)  =  shanoi1_in_gggga(x1, x2, x3, x4)
s(x1)  =  s(x1)
0  =  0
shanoi1_out_gggga(x1, x2, x3, x4, x5)  =  shanoi1_out_gggga
U10_gggga(x1, x2, x3, x4, x5, x6)  =  U10_gggga(x6)
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U11_gggga(x1, x2, x3, x4, x5, x6)  =  U11_gggga(x1, x2, x4, x6)
U12_gggga(x1, x2, x3, x4, x5, x6)  =  U12_gggga(x6)
U13_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U13_gggga(x7)
U14_gggga(x1, x2, x3, x4, x5, x6)  =  U14_gggga(x6)
append40_in_aga(x1, x2, x3)  =  append40_in_aga(x2)
append40_out_aga(x1, x2, x3)  =  append40_out_aga(x1, x3)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x5)
U15_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U15_gggga(x7)
U16_gggga(x1, x2, x3, x4, x5, x6)  =  U16_gggga(x6)
append40_in_gaa(x1, x2, x3)  =  append40_in_gaa(x1)
append40_out_gaa(x1, x2, x3)  =  append40_out_gaa
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
SHANOI1_IN_GGGGA(x1, x2, x3, x4, x5)  =  SHANOI1_IN_GGGGA(x1, x2, x3, x4)
U10_GGGGA(x1, x2, x3, x4, x5, x6)  =  U10_GGGGA(x6)
SHANOI11_IN_GGGGA(x1, x2, x3, x4, x5)  =  SHANOI11_IN_GGGGA(x1, x2, x3, x4)
U1_GGGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGGA(x6)
U2_GGGGA(x1, x2, x3, x4, x5, x6)  =  U2_GGGGA(x1, x2, x3, x4, x6)
U3_GGGGA(x1, x2, x3, x4, x5, x6)  =  U3_GGGGA(x6)
U4_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GGGGA(x7)
U5_GGGGA(x1, x2, x3, x4, x5, x6)  =  U5_GGGGA(x6)
APPEND28_IN_AGA(x1, x2, x3)  =  APPEND28_IN_AGA(x2)
U8_AGA(x1, x2, x3, x4, x5)  =  U8_AGA(x5)
U6_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GGGGA(x7)
U7_GGGGA(x1, x2, x3, x4, x5, x6)  =  U7_GGGGA(x6)
APPEND28_IN_GAA(x1, x2, x3)  =  APPEND28_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4, x5)  =  U8_GAA(x5)
U11_GGGGA(x1, x2, x3, x4, x5, x6)  =  U11_GGGGA(x1, x2, x4, x6)
U12_GGGGA(x1, x2, x3, x4, x5, x6)  =  U12_GGGGA(x6)
U13_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U13_GGGGA(x7)
U14_GGGGA(x1, x2, x3, x4, x5, x6)  =  U14_GGGGA(x6)
APPEND40_IN_AGA(x1, x2, x3)  =  APPEND40_IN_AGA(x2)
U9_AGA(x1, x2, x3, x4, x5)  =  U9_AGA(x5)
U15_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U15_GGGGA(x7)
U16_GGGGA(x1, x2, x3, x4, x5, x6)  =  U16_GGGGA(x6)
APPEND40_IN_GAA(x1, x2, x3)  =  APPEND40_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 23 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APPEND40_IN_GAA(.(T179, T180), T181, .(T179, T183)) → APPEND40_IN_GAA(T180, T181, T183)

The TRS R consists of the following rules:

shanoi1_in_gggga(s(0), T9, T10, T11, .(mv(T9, T11), [])) → shanoi1_out_gggga(s(0), T9, T10, T11, .(mv(T9, T11), []))
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U10_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U10_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, X26)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U11_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U12_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T20, T18, T20, X27)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_gggga(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
append40_in_aga([], T170, T170) → append40_out_aga([], T170, T170)
append40_in_aga(.(T179, T180), T181, .(T179, T183)) → U9_aga(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
U9_aga(T179, T180, T181, T183, append40_out_aga(T180, T181, T183)) → append40_out_aga(.(T179, T180), T181, .(T179, T183))
U14_gggga(T31, T18, T19, T20, T22, append40_out_aga(T38, .(mv(T18, T20), []), X28)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_gggga(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_gggga(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_gggga(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
append40_in_gaa([], T170, T170) → append40_out_gaa([], T170, T170)
append40_in_gaa(.(T179, T180), T181, .(T179, T183)) → U9_gaa(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
U9_gaa(T179, T180, T181, T183, append40_out_gaa(T180, T181, T183)) → append40_out_gaa(.(T179, T180), T181, .(T179, T183))
U16_gggga(T31, T18, T19, T20, T22, append40_out_gaa(T159, T148, T22)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)

The argument filtering Pi contains the following mapping:
shanoi1_in_gggga(x1, x2, x3, x4, x5)  =  shanoi1_in_gggga(x1, x2, x3, x4)
s(x1)  =  s(x1)
0  =  0
shanoi1_out_gggga(x1, x2, x3, x4, x5)  =  shanoi1_out_gggga
U10_gggga(x1, x2, x3, x4, x5, x6)  =  U10_gggga(x6)
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U11_gggga(x1, x2, x3, x4, x5, x6)  =  U11_gggga(x1, x2, x4, x6)
U12_gggga(x1, x2, x3, x4, x5, x6)  =  U12_gggga(x6)
U13_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U13_gggga(x7)
U14_gggga(x1, x2, x3, x4, x5, x6)  =  U14_gggga(x6)
append40_in_aga(x1, x2, x3)  =  append40_in_aga(x2)
append40_out_aga(x1, x2, x3)  =  append40_out_aga(x1, x3)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x5)
U15_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U15_gggga(x7)
U16_gggga(x1, x2, x3, x4, x5, x6)  =  U16_gggga(x6)
append40_in_gaa(x1, x2, x3)  =  append40_in_gaa(x1)
append40_out_gaa(x1, x2, x3)  =  append40_out_gaa
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
APPEND40_IN_GAA(x1, x2, x3)  =  APPEND40_IN_GAA(x1)

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:

APPEND40_IN_GAA(.(T179, T180), T181, .(T179, T183)) → APPEND40_IN_GAA(T180, T181, T183)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND40_IN_GAA(x1, x2, x3)  =  APPEND40_IN_GAA(x1)

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:

APPEND40_IN_GAA(.(T180)) → APPEND40_IN_GAA(T180)

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:

  • APPEND40_IN_GAA(.(T180)) → APPEND40_IN_GAA(T180)
    The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

APPEND40_IN_AGA(.(T179, T180), T181, .(T179, T183)) → APPEND40_IN_AGA(T180, T181, T183)

The TRS R consists of the following rules:

shanoi1_in_gggga(s(0), T9, T10, T11, .(mv(T9, T11), [])) → shanoi1_out_gggga(s(0), T9, T10, T11, .(mv(T9, T11), []))
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U10_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U10_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, X26)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U11_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U12_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T20, T18, T20, X27)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_gggga(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
append40_in_aga([], T170, T170) → append40_out_aga([], T170, T170)
append40_in_aga(.(T179, T180), T181, .(T179, T183)) → U9_aga(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
U9_aga(T179, T180, T181, T183, append40_out_aga(T180, T181, T183)) → append40_out_aga(.(T179, T180), T181, .(T179, T183))
U14_gggga(T31, T18, T19, T20, T22, append40_out_aga(T38, .(mv(T18, T20), []), X28)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_gggga(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_gggga(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_gggga(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
append40_in_gaa([], T170, T170) → append40_out_gaa([], T170, T170)
append40_in_gaa(.(T179, T180), T181, .(T179, T183)) → U9_gaa(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
U9_gaa(T179, T180, T181, T183, append40_out_gaa(T180, T181, T183)) → append40_out_gaa(.(T179, T180), T181, .(T179, T183))
U16_gggga(T31, T18, T19, T20, T22, append40_out_gaa(T159, T148, T22)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)

The argument filtering Pi contains the following mapping:
shanoi1_in_gggga(x1, x2, x3, x4, x5)  =  shanoi1_in_gggga(x1, x2, x3, x4)
s(x1)  =  s(x1)
0  =  0
shanoi1_out_gggga(x1, x2, x3, x4, x5)  =  shanoi1_out_gggga
U10_gggga(x1, x2, x3, x4, x5, x6)  =  U10_gggga(x6)
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U11_gggga(x1, x2, x3, x4, x5, x6)  =  U11_gggga(x1, x2, x4, x6)
U12_gggga(x1, x2, x3, x4, x5, x6)  =  U12_gggga(x6)
U13_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U13_gggga(x7)
U14_gggga(x1, x2, x3, x4, x5, x6)  =  U14_gggga(x6)
append40_in_aga(x1, x2, x3)  =  append40_in_aga(x2)
append40_out_aga(x1, x2, x3)  =  append40_out_aga(x1, x3)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x5)
U15_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U15_gggga(x7)
U16_gggga(x1, x2, x3, x4, x5, x6)  =  U16_gggga(x6)
append40_in_gaa(x1, x2, x3)  =  append40_in_gaa(x1)
append40_out_gaa(x1, x2, x3)  =  append40_out_gaa
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
APPEND40_IN_AGA(x1, x2, x3)  =  APPEND40_IN_AGA(x2)

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:

APPEND40_IN_AGA(.(T179, T180), T181, .(T179, T183)) → APPEND40_IN_AGA(T180, T181, T183)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND40_IN_AGA(x1, x2, x3)  =  APPEND40_IN_AGA(x2)

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:

APPEND40_IN_AGA(T181) → APPEND40_IN_AGA(T181)

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

(21) 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 = APPEND40_IN_AGA(T181) evaluates to t =APPEND40_IN_AGA(T181)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND40_IN_AGA(T181) to APPEND40_IN_AGA(T181).



(22) NO

(23) Obligation:

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

APPEND28_IN_GAA(.(T137, T138), T139, .(T137, X178)) → APPEND28_IN_GAA(T138, T139, X178)

The TRS R consists of the following rules:

shanoi1_in_gggga(s(0), T9, T10, T11, .(mv(T9, T11), [])) → shanoi1_out_gggga(s(0), T9, T10, T11, .(mv(T9, T11), []))
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U10_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U10_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, X26)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U11_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U12_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T20, T18, T20, X27)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_gggga(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
append40_in_aga([], T170, T170) → append40_out_aga([], T170, T170)
append40_in_aga(.(T179, T180), T181, .(T179, T183)) → U9_aga(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
U9_aga(T179, T180, T181, T183, append40_out_aga(T180, T181, T183)) → append40_out_aga(.(T179, T180), T181, .(T179, T183))
U14_gggga(T31, T18, T19, T20, T22, append40_out_aga(T38, .(mv(T18, T20), []), X28)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_gggga(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_gggga(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_gggga(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
append40_in_gaa([], T170, T170) → append40_out_gaa([], T170, T170)
append40_in_gaa(.(T179, T180), T181, .(T179, T183)) → U9_gaa(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
U9_gaa(T179, T180, T181, T183, append40_out_gaa(T180, T181, T183)) → append40_out_gaa(.(T179, T180), T181, .(T179, T183))
U16_gggga(T31, T18, T19, T20, T22, append40_out_gaa(T159, T148, T22)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)

The argument filtering Pi contains the following mapping:
shanoi1_in_gggga(x1, x2, x3, x4, x5)  =  shanoi1_in_gggga(x1, x2, x3, x4)
s(x1)  =  s(x1)
0  =  0
shanoi1_out_gggga(x1, x2, x3, x4, x5)  =  shanoi1_out_gggga
U10_gggga(x1, x2, x3, x4, x5, x6)  =  U10_gggga(x6)
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U11_gggga(x1, x2, x3, x4, x5, x6)  =  U11_gggga(x1, x2, x4, x6)
U12_gggga(x1, x2, x3, x4, x5, x6)  =  U12_gggga(x6)
U13_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U13_gggga(x7)
U14_gggga(x1, x2, x3, x4, x5, x6)  =  U14_gggga(x6)
append40_in_aga(x1, x2, x3)  =  append40_in_aga(x2)
append40_out_aga(x1, x2, x3)  =  append40_out_aga(x1, x3)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x5)
U15_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U15_gggga(x7)
U16_gggga(x1, x2, x3, x4, x5, x6)  =  U16_gggga(x6)
append40_in_gaa(x1, x2, x3)  =  append40_in_gaa(x1)
append40_out_gaa(x1, x2, x3)  =  append40_out_gaa
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
APPEND28_IN_GAA(x1, x2, x3)  =  APPEND28_IN_GAA(x1)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

APPEND28_IN_GAA(.(T137, T138), T139, .(T137, X178)) → APPEND28_IN_GAA(T138, T139, X178)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND28_IN_GAA(x1, x2, x3)  =  APPEND28_IN_GAA(x1)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

APPEND28_IN_GAA(.(T138)) → APPEND28_IN_GAA(T138)

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

(28) 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:

  • APPEND28_IN_GAA(.(T138)) → APPEND28_IN_GAA(T138)
    The graph contains the following edges 1 > 1

(29) YES

(30) Obligation:

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

APPEND28_IN_AGA(.(T137, T138), T139, .(T137, X178)) → APPEND28_IN_AGA(T138, T139, X178)

The TRS R consists of the following rules:

shanoi1_in_gggga(s(0), T9, T10, T11, .(mv(T9, T11), [])) → shanoi1_out_gggga(s(0), T9, T10, T11, .(mv(T9, T11), []))
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U10_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U10_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, X26)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U11_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U12_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T20, T18, T20, X27)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_gggga(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
append40_in_aga([], T170, T170) → append40_out_aga([], T170, T170)
append40_in_aga(.(T179, T180), T181, .(T179, T183)) → U9_aga(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
U9_aga(T179, T180, T181, T183, append40_out_aga(T180, T181, T183)) → append40_out_aga(.(T179, T180), T181, .(T179, T183))
U14_gggga(T31, T18, T19, T20, T22, append40_out_aga(T38, .(mv(T18, T20), []), X28)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_gggga(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_gggga(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_gggga(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
append40_in_gaa([], T170, T170) → append40_out_gaa([], T170, T170)
append40_in_gaa(.(T179, T180), T181, .(T179, T183)) → U9_gaa(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
U9_gaa(T179, T180, T181, T183, append40_out_gaa(T180, T181, T183)) → append40_out_gaa(.(T179, T180), T181, .(T179, T183))
U16_gggga(T31, T18, T19, T20, T22, append40_out_gaa(T159, T148, T22)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)

The argument filtering Pi contains the following mapping:
shanoi1_in_gggga(x1, x2, x3, x4, x5)  =  shanoi1_in_gggga(x1, x2, x3, x4)
s(x1)  =  s(x1)
0  =  0
shanoi1_out_gggga(x1, x2, x3, x4, x5)  =  shanoi1_out_gggga
U10_gggga(x1, x2, x3, x4, x5, x6)  =  U10_gggga(x6)
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U11_gggga(x1, x2, x3, x4, x5, x6)  =  U11_gggga(x1, x2, x4, x6)
U12_gggga(x1, x2, x3, x4, x5, x6)  =  U12_gggga(x6)
U13_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U13_gggga(x7)
U14_gggga(x1, x2, x3, x4, x5, x6)  =  U14_gggga(x6)
append40_in_aga(x1, x2, x3)  =  append40_in_aga(x2)
append40_out_aga(x1, x2, x3)  =  append40_out_aga(x1, x3)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x5)
U15_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U15_gggga(x7)
U16_gggga(x1, x2, x3, x4, x5, x6)  =  U16_gggga(x6)
append40_in_gaa(x1, x2, x3)  =  append40_in_gaa(x1)
append40_out_gaa(x1, x2, x3)  =  append40_out_gaa
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
APPEND28_IN_AGA(x1, x2, x3)  =  APPEND28_IN_AGA(x2)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

APPEND28_IN_AGA(.(T137, T138), T139, .(T137, X178)) → APPEND28_IN_AGA(T138, T139, X178)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND28_IN_AGA(x1, x2, x3)  =  APPEND28_IN_AGA(x2)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

APPEND28_IN_AGA(T139) → APPEND28_IN_AGA(T139)

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

(35) NonTerminationProof (EQUIVALENT transformation)

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

s = APPEND28_IN_AGA(T139) evaluates to t =APPEND28_IN_AGA(T139)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND28_IN_AGA(T139) to APPEND28_IN_AGA(T139).



(36) NO

(37) Obligation:

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

SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → SHANOI11_IN_GGGGA(T88, T70, T69, T71, X99)
SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → SHANOI11_IN_GGGGA(T88, T69, T71, T70, X98)

The TRS R consists of the following rules:

shanoi1_in_gggga(s(0), T9, T10, T11, .(mv(T9, T11), [])) → shanoi1_out_gggga(s(0), T9, T10, T11, .(mv(T9, T11), []))
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U10_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U10_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, X26)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U11_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U12_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T20, T18, T20, X27)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_gggga(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
append40_in_aga([], T170, T170) → append40_out_aga([], T170, T170)
append40_in_aga(.(T179, T180), T181, .(T179, T183)) → U9_aga(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
U9_aga(T179, T180, T181, T183, append40_out_aga(T180, T181, T183)) → append40_out_aga(.(T179, T180), T181, .(T179, T183))
U14_gggga(T31, T18, T19, T20, T22, append40_out_aga(T38, .(mv(T18, T20), []), X28)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_gggga(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_gggga(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_gggga(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
append40_in_gaa([], T170, T170) → append40_out_gaa([], T170, T170)
append40_in_gaa(.(T179, T180), T181, .(T179, T183)) → U9_gaa(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
U9_gaa(T179, T180, T181, T183, append40_out_gaa(T180, T181, T183)) → append40_out_gaa(.(T179, T180), T181, .(T179, T183))
U16_gggga(T31, T18, T19, T20, T22, append40_out_gaa(T159, T148, T22)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)

The argument filtering Pi contains the following mapping:
shanoi1_in_gggga(x1, x2, x3, x4, x5)  =  shanoi1_in_gggga(x1, x2, x3, x4)
s(x1)  =  s(x1)
0  =  0
shanoi1_out_gggga(x1, x2, x3, x4, x5)  =  shanoi1_out_gggga
U10_gggga(x1, x2, x3, x4, x5, x6)  =  U10_gggga(x6)
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U11_gggga(x1, x2, x3, x4, x5, x6)  =  U11_gggga(x1, x2, x4, x6)
U12_gggga(x1, x2, x3, x4, x5, x6)  =  U12_gggga(x6)
U13_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U13_gggga(x7)
U14_gggga(x1, x2, x3, x4, x5, x6)  =  U14_gggga(x6)
append40_in_aga(x1, x2, x3)  =  append40_in_aga(x2)
append40_out_aga(x1, x2, x3)  =  append40_out_aga(x1, x3)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x5)
U15_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U15_gggga(x7)
U16_gggga(x1, x2, x3, x4, x5, x6)  =  U16_gggga(x6)
append40_in_gaa(x1, x2, x3)  =  append40_in_gaa(x1)
append40_out_gaa(x1, x2, x3)  =  append40_out_gaa
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
SHANOI11_IN_GGGGA(x1, x2, x3, x4, x5)  =  SHANOI11_IN_GGGGA(x1, x2, x3, x4)
U2_GGGGA(x1, x2, x3, x4, x5, x6)  =  U2_GGGGA(x1, x2, x3, x4, x6)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → SHANOI11_IN_GGGGA(T88, T70, T69, T71, X99)
SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → SHANOI11_IN_GGGGA(T88, T69, T71, T70, X98)

The TRS R consists of the following rules:

shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
SHANOI11_IN_GGGGA(x1, x2, x3, x4, x5)  =  SHANOI11_IN_GGGGA(x1, x2, x3, x4)
U2_GGGGA(x1, x2, x3, x4, x5, x6)  =  U2_GGGGA(x1, x2, x3, x4, x6)

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

SHANOI11_IN_GGGGA(s(T88), T69, T70, T71) → U2_GGGGA(T88, T69, T70, T71, shanoi11_in_gggga(T88, T69, T71, T70))
U2_GGGGA(T88, T69, T70, T71, shanoi11_out_gggga) → SHANOI11_IN_GGGGA(T88, T70, T69, T71)
SHANOI11_IN_GGGGA(s(T88), T69, T70, T71) → SHANOI11_IN_GGGGA(T88, T69, T71, T70)

The TRS R consists of the following rules:

shanoi11_in_gggga(0, T57, T58, T59) → shanoi11_out_gggga
shanoi11_in_gggga(s(T88), T69, T70, T71) → U1_gggga(shanoi11_in_gggga(T88, T69, T71, T70))
shanoi11_in_gggga(s(T88), T69, T70, T71) → U2_gggga(T88, T69, T70, T71, shanoi11_in_gggga(T88, T69, T71, T70))
U1_gggga(shanoi11_out_gggga) → shanoi11_out_gggga
U2_gggga(T88, T69, T70, T71, shanoi11_out_gggga) → U3_gggga(shanoi11_in_gggga(T88, T70, T69, T71))
U2_gggga(T88, T69, T70, T71, shanoi11_out_gggga) → U4_gggga(shanoi11_in_gggga(T88, T70, T69, T71))
U3_gggga(shanoi11_out_gggga) → shanoi11_out_gggga
U4_gggga(shanoi11_out_gggga) → U5_gggga(append28_in_aga(.([])))
U4_gggga(shanoi11_out_gggga) → U6_gggga(append28_in_aga(.([])))
U5_gggga(append28_out_aga(T95, X100)) → shanoi11_out_gggga
U6_gggga(append28_out_aga(T95, T119)) → U7_gggga(append28_in_gaa(T119))
append28_in_aga(T130) → append28_out_aga([], T130)
append28_in_aga(T139) → U8_aga(append28_in_aga(T139))
U7_gggga(append28_out_gaa) → shanoi11_out_gggga
U8_aga(append28_out_aga(T138, X178)) → append28_out_aga(.(T138), .(X178))
append28_in_gaa([]) → append28_out_gaa
append28_in_gaa(.(T138)) → U8_gaa(append28_in_gaa(T138))
U8_gaa(append28_out_gaa) → append28_out_gaa

The set Q consists of the following terms:

shanoi11_in_gggga(x0, x1, x2, x3)
U1_gggga(x0)
U2_gggga(x0, x1, x2, x3, x4)
U3_gggga(x0)
U4_gggga(x0)
U5_gggga(x0)
U6_gggga(x0)
append28_in_aga(x0)
U7_gggga(x0)
U8_aga(x0)
append28_in_gaa(x0)
U8_gaa(x0)

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

(42) 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:

  • U2_GGGGA(T88, T69, T70, T71, shanoi11_out_gggga) → SHANOI11_IN_GGGGA(T88, T70, T69, T71)
    The graph contains the following edges 1 >= 1, 3 >= 2, 2 >= 3, 4 >= 4

  • SHANOI11_IN_GGGGA(s(T88), T69, T70, T71) → SHANOI11_IN_GGGGA(T88, T69, T71, T70)
    The graph contains the following edges 1 > 1, 2 >= 2, 4 >= 3, 3 >= 4

  • SHANOI11_IN_GGGGA(s(T88), T69, T70, T71) → U2_GGGGA(T88, T69, T70, T71, shanoi11_in_gggga(T88, T69, T71, T70))
    The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4

(43) YES

(44) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
shanoi1_in: (b,b,b,b,f)
shanoi11_in: (b,b,b,b,f)
append28_in: (f,b,f) (b,f,f)
append40_in: (f,b,f) (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

shanoi1_in_gggga(s(0), T9, T10, T11, .(mv(T9, T11), [])) → shanoi1_out_gggga(s(0), T9, T10, T11, .(mv(T9, T11), []))
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U10_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U10_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, X26)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U11_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U12_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T20, T18, T20, X27)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_gggga(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
append40_in_aga([], T170, T170) → append40_out_aga([], T170, T170)
append40_in_aga(.(T179, T180), T181, .(T179, T183)) → U9_aga(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
U9_aga(T179, T180, T181, T183, append40_out_aga(T180, T181, T183)) → append40_out_aga(.(T179, T180), T181, .(T179, T183))
U14_gggga(T31, T18, T19, T20, T22, append40_out_aga(T38, .(mv(T18, T20), []), X28)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_gggga(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_gggga(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_gggga(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
append40_in_gaa([], T170, T170) → append40_out_gaa([], T170, T170)
append40_in_gaa(.(T179, T180), T181, .(T179, T183)) → U9_gaa(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
U9_gaa(T179, T180, T181, T183, append40_out_gaa(T180, T181, T183)) → append40_out_gaa(.(T179, T180), T181, .(T179, T183))
U16_gggga(T31, T18, T19, T20, T22, append40_out_gaa(T159, T148, T22)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)

The argument filtering Pi contains the following mapping:
shanoi1_in_gggga(x1, x2, x3, x4, x5)  =  shanoi1_in_gggga(x1, x2, x3, x4)
s(x1)  =  s(x1)
0  =  0
shanoi1_out_gggga(x1, x2, x3, x4, x5)  =  shanoi1_out_gggga(x1, x2, x3, x4)
U10_gggga(x1, x2, x3, x4, x5, x6)  =  U10_gggga(x1, x2, x3, x4, x6)
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga(x1, x2, x3, x4)
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x1, x2, x3, x4, x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x1, x2, x3, x4, x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x1, x2, x3, x4, x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x1, x2, x3, x4, x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x2, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x3, x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x1, x2, x3, x4, x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x1, x2, x3, x4, x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa(x1)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x2, x5)
U11_gggga(x1, x2, x3, x4, x5, x6)  =  U11_gggga(x1, x2, x3, x4, x6)
U12_gggga(x1, x2, x3, x4, x5, x6)  =  U12_gggga(x1, x2, x3, x4, x6)
U13_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U13_gggga(x1, x2, x3, x4, x7)
U14_gggga(x1, x2, x3, x4, x5, x6)  =  U14_gggga(x1, x2, x3, x4, x6)
append40_in_aga(x1, x2, x3)  =  append40_in_aga(x2)
append40_out_aga(x1, x2, x3)  =  append40_out_aga(x1, x2, x3)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x3, x5)
U15_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U15_gggga(x1, x2, x3, x4, x7)
U16_gggga(x1, x2, x3, x4, x5, x6)  =  U16_gggga(x1, x2, x3, x4, x6)
append40_in_gaa(x1, x2, x3)  =  append40_in_gaa(x1)
append40_out_gaa(x1, x2, x3)  =  append40_out_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x2, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(45) Obligation:

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

shanoi1_in_gggga(s(0), T9, T10, T11, .(mv(T9, T11), [])) → shanoi1_out_gggga(s(0), T9, T10, T11, .(mv(T9, T11), []))
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U10_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U10_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, X26)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U11_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U12_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T20, T18, T20, X27)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_gggga(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
append40_in_aga([], T170, T170) → append40_out_aga([], T170, T170)
append40_in_aga(.(T179, T180), T181, .(T179, T183)) → U9_aga(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
U9_aga(T179, T180, T181, T183, append40_out_aga(T180, T181, T183)) → append40_out_aga(.(T179, T180), T181, .(T179, T183))
U14_gggga(T31, T18, T19, T20, T22, append40_out_aga(T38, .(mv(T18, T20), []), X28)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_gggga(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_gggga(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_gggga(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
append40_in_gaa([], T170, T170) → append40_out_gaa([], T170, T170)
append40_in_gaa(.(T179, T180), T181, .(T179, T183)) → U9_gaa(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
U9_gaa(T179, T180, T181, T183, append40_out_gaa(T180, T181, T183)) → append40_out_gaa(.(T179, T180), T181, .(T179, T183))
U16_gggga(T31, T18, T19, T20, T22, append40_out_gaa(T159, T148, T22)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)

The argument filtering Pi contains the following mapping:
shanoi1_in_gggga(x1, x2, x3, x4, x5)  =  shanoi1_in_gggga(x1, x2, x3, x4)
s(x1)  =  s(x1)
0  =  0
shanoi1_out_gggga(x1, x2, x3, x4, x5)  =  shanoi1_out_gggga(x1, x2, x3, x4)
U10_gggga(x1, x2, x3, x4, x5, x6)  =  U10_gggga(x1, x2, x3, x4, x6)
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga(x1, x2, x3, x4)
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x1, x2, x3, x4, x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x1, x2, x3, x4, x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x1, x2, x3, x4, x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x1, x2, x3, x4, x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x2, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x3, x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x1, x2, x3, x4, x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x1, x2, x3, x4, x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa(x1)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x2, x5)
U11_gggga(x1, x2, x3, x4, x5, x6)  =  U11_gggga(x1, x2, x3, x4, x6)
U12_gggga(x1, x2, x3, x4, x5, x6)  =  U12_gggga(x1, x2, x3, x4, x6)
U13_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U13_gggga(x1, x2, x3, x4, x7)
U14_gggga(x1, x2, x3, x4, x5, x6)  =  U14_gggga(x1, x2, x3, x4, x6)
append40_in_aga(x1, x2, x3)  =  append40_in_aga(x2)
append40_out_aga(x1, x2, x3)  =  append40_out_aga(x1, x2, x3)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x3, x5)
U15_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U15_gggga(x1, x2, x3, x4, x7)
U16_gggga(x1, x2, x3, x4, x5, x6)  =  U16_gggga(x1, x2, x3, x4, x6)
append40_in_gaa(x1, x2, x3)  =  append40_in_gaa(x1)
append40_out_gaa(x1, x2, x3)  =  append40_out_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x2, x5)

(46) 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:

SHANOI1_IN_GGGGA(s(s(T31)), T18, T19, T20, T22) → U10_GGGGA(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
SHANOI1_IN_GGGGA(s(s(T31)), T18, T19, T20, T22) → SHANOI11_IN_GGGGA(T31, T18, T20, T19, X26)
SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → U1_GGGGA(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → SHANOI11_IN_GGGGA(T88, T69, T71, T70, X98)
SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_GGGGA(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → SHANOI11_IN_GGGGA(T88, T70, T69, T71, X99)
U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_GGGGA(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_GGGGA(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_GGGGA(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
U4_GGGGA(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → APPEND28_IN_AGA(T95, .(mv(T69, T71), []), X100)
APPEND28_IN_AGA(.(T137, T138), T139, .(T137, X178)) → U8_AGA(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
APPEND28_IN_AGA(.(T137, T138), T139, .(T137, X178)) → APPEND28_IN_AGA(T138, T139, X178)
U4_GGGGA(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_GGGGA(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_GGGGA(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_GGGGA(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
U6_GGGGA(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → APPEND28_IN_GAA(T119, T108, X101)
APPEND28_IN_GAA(.(T137, T138), T139, .(T137, X178)) → U8_GAA(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
APPEND28_IN_GAA(.(T137, T138), T139, .(T137, X178)) → APPEND28_IN_GAA(T138, T139, X178)
SHANOI1_IN_GGGGA(s(s(T31)), T18, T19, T20, T22) → U11_GGGGA(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_GGGGA(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_GGGGA(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U11_GGGGA(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → SHANOI11_IN_GGGGA(T31, T20, T18, T20, X27)
U11_GGGGA(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_GGGGA(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_GGGGA(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_GGGGA(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
U13_GGGGA(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → APPEND40_IN_AGA(T38, .(mv(T18, T20), []), X28)
APPEND40_IN_AGA(.(T179, T180), T181, .(T179, T183)) → U9_AGA(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
APPEND40_IN_AGA(.(T179, T180), T181, .(T179, T183)) → APPEND40_IN_AGA(T180, T181, T183)
U13_GGGGA(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_GGGGA(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_GGGGA(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_GGGGA(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
U15_GGGGA(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → APPEND40_IN_GAA(T159, T148, T22)
APPEND40_IN_GAA(.(T179, T180), T181, .(T179, T183)) → U9_GAA(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
APPEND40_IN_GAA(.(T179, T180), T181, .(T179, T183)) → APPEND40_IN_GAA(T180, T181, T183)

The TRS R consists of the following rules:

shanoi1_in_gggga(s(0), T9, T10, T11, .(mv(T9, T11), [])) → shanoi1_out_gggga(s(0), T9, T10, T11, .(mv(T9, T11), []))
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U10_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U10_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, X26)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U11_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U12_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T20, T18, T20, X27)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_gggga(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
append40_in_aga([], T170, T170) → append40_out_aga([], T170, T170)
append40_in_aga(.(T179, T180), T181, .(T179, T183)) → U9_aga(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
U9_aga(T179, T180, T181, T183, append40_out_aga(T180, T181, T183)) → append40_out_aga(.(T179, T180), T181, .(T179, T183))
U14_gggga(T31, T18, T19, T20, T22, append40_out_aga(T38, .(mv(T18, T20), []), X28)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_gggga(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_gggga(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_gggga(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
append40_in_gaa([], T170, T170) → append40_out_gaa([], T170, T170)
append40_in_gaa(.(T179, T180), T181, .(T179, T183)) → U9_gaa(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
U9_gaa(T179, T180, T181, T183, append40_out_gaa(T180, T181, T183)) → append40_out_gaa(.(T179, T180), T181, .(T179, T183))
U16_gggga(T31, T18, T19, T20, T22, append40_out_gaa(T159, T148, T22)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)

The argument filtering Pi contains the following mapping:
shanoi1_in_gggga(x1, x2, x3, x4, x5)  =  shanoi1_in_gggga(x1, x2, x3, x4)
s(x1)  =  s(x1)
0  =  0
shanoi1_out_gggga(x1, x2, x3, x4, x5)  =  shanoi1_out_gggga(x1, x2, x3, x4)
U10_gggga(x1, x2, x3, x4, x5, x6)  =  U10_gggga(x1, x2, x3, x4, x6)
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga(x1, x2, x3, x4)
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x1, x2, x3, x4, x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x1, x2, x3, x4, x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x1, x2, x3, x4, x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x1, x2, x3, x4, x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x2, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x3, x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x1, x2, x3, x4, x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x1, x2, x3, x4, x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa(x1)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x2, x5)
U11_gggga(x1, x2, x3, x4, x5, x6)  =  U11_gggga(x1, x2, x3, x4, x6)
U12_gggga(x1, x2, x3, x4, x5, x6)  =  U12_gggga(x1, x2, x3, x4, x6)
U13_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U13_gggga(x1, x2, x3, x4, x7)
U14_gggga(x1, x2, x3, x4, x5, x6)  =  U14_gggga(x1, x2, x3, x4, x6)
append40_in_aga(x1, x2, x3)  =  append40_in_aga(x2)
append40_out_aga(x1, x2, x3)  =  append40_out_aga(x1, x2, x3)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x3, x5)
U15_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U15_gggga(x1, x2, x3, x4, x7)
U16_gggga(x1, x2, x3, x4, x5, x6)  =  U16_gggga(x1, x2, x3, x4, x6)
append40_in_gaa(x1, x2, x3)  =  append40_in_gaa(x1)
append40_out_gaa(x1, x2, x3)  =  append40_out_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x2, x5)
SHANOI1_IN_GGGGA(x1, x2, x3, x4, x5)  =  SHANOI1_IN_GGGGA(x1, x2, x3, x4)
U10_GGGGA(x1, x2, x3, x4, x5, x6)  =  U10_GGGGA(x1, x2, x3, x4, x6)
SHANOI11_IN_GGGGA(x1, x2, x3, x4, x5)  =  SHANOI11_IN_GGGGA(x1, x2, x3, x4)
U1_GGGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGGA(x1, x2, x3, x4, x6)
U2_GGGGA(x1, x2, x3, x4, x5, x6)  =  U2_GGGGA(x1, x2, x3, x4, x6)
U3_GGGGA(x1, x2, x3, x4, x5, x6)  =  U3_GGGGA(x1, x2, x3, x4, x6)
U4_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GGGGA(x1, x2, x3, x4, x7)
U5_GGGGA(x1, x2, x3, x4, x5, x6)  =  U5_GGGGA(x1, x2, x3, x4, x6)
APPEND28_IN_AGA(x1, x2, x3)  =  APPEND28_IN_AGA(x2)
U8_AGA(x1, x2, x3, x4, x5)  =  U8_AGA(x3, x5)
U6_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GGGGA(x1, x2, x3, x4, x7)
U7_GGGGA(x1, x2, x3, x4, x5, x6)  =  U7_GGGGA(x1, x2, x3, x4, x6)
APPEND28_IN_GAA(x1, x2, x3)  =  APPEND28_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4, x5)  =  U8_GAA(x2, x5)
U11_GGGGA(x1, x2, x3, x4, x5, x6)  =  U11_GGGGA(x1, x2, x3, x4, x6)
U12_GGGGA(x1, x2, x3, x4, x5, x6)  =  U12_GGGGA(x1, x2, x3, x4, x6)
U13_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U13_GGGGA(x1, x2, x3, x4, x7)
U14_GGGGA(x1, x2, x3, x4, x5, x6)  =  U14_GGGGA(x1, x2, x3, x4, x6)
APPEND40_IN_AGA(x1, x2, x3)  =  APPEND40_IN_AGA(x2)
U9_AGA(x1, x2, x3, x4, x5)  =  U9_AGA(x3, x5)
U15_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U15_GGGGA(x1, x2, x3, x4, x7)
U16_GGGGA(x1, x2, x3, x4, x5, x6)  =  U16_GGGGA(x1, x2, x3, x4, x6)
APPEND40_IN_GAA(x1, x2, x3)  =  APPEND40_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x2, x5)

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

(47) Obligation:

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

SHANOI1_IN_GGGGA(s(s(T31)), T18, T19, T20, T22) → U10_GGGGA(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
SHANOI1_IN_GGGGA(s(s(T31)), T18, T19, T20, T22) → SHANOI11_IN_GGGGA(T31, T18, T20, T19, X26)
SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → U1_GGGGA(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → SHANOI11_IN_GGGGA(T88, T69, T71, T70, X98)
SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_GGGGA(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → SHANOI11_IN_GGGGA(T88, T70, T69, T71, X99)
U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_GGGGA(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_GGGGA(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_GGGGA(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
U4_GGGGA(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → APPEND28_IN_AGA(T95, .(mv(T69, T71), []), X100)
APPEND28_IN_AGA(.(T137, T138), T139, .(T137, X178)) → U8_AGA(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
APPEND28_IN_AGA(.(T137, T138), T139, .(T137, X178)) → APPEND28_IN_AGA(T138, T139, X178)
U4_GGGGA(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_GGGGA(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_GGGGA(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_GGGGA(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
U6_GGGGA(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → APPEND28_IN_GAA(T119, T108, X101)
APPEND28_IN_GAA(.(T137, T138), T139, .(T137, X178)) → U8_GAA(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
APPEND28_IN_GAA(.(T137, T138), T139, .(T137, X178)) → APPEND28_IN_GAA(T138, T139, X178)
SHANOI1_IN_GGGGA(s(s(T31)), T18, T19, T20, T22) → U11_GGGGA(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_GGGGA(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_GGGGA(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U11_GGGGA(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → SHANOI11_IN_GGGGA(T31, T20, T18, T20, X27)
U11_GGGGA(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_GGGGA(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_GGGGA(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_GGGGA(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
U13_GGGGA(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → APPEND40_IN_AGA(T38, .(mv(T18, T20), []), X28)
APPEND40_IN_AGA(.(T179, T180), T181, .(T179, T183)) → U9_AGA(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
APPEND40_IN_AGA(.(T179, T180), T181, .(T179, T183)) → APPEND40_IN_AGA(T180, T181, T183)
U13_GGGGA(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_GGGGA(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_GGGGA(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_GGGGA(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
U15_GGGGA(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → APPEND40_IN_GAA(T159, T148, T22)
APPEND40_IN_GAA(.(T179, T180), T181, .(T179, T183)) → U9_GAA(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
APPEND40_IN_GAA(.(T179, T180), T181, .(T179, T183)) → APPEND40_IN_GAA(T180, T181, T183)

The TRS R consists of the following rules:

shanoi1_in_gggga(s(0), T9, T10, T11, .(mv(T9, T11), [])) → shanoi1_out_gggga(s(0), T9, T10, T11, .(mv(T9, T11), []))
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U10_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U10_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, X26)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U11_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U12_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T20, T18, T20, X27)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_gggga(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
append40_in_aga([], T170, T170) → append40_out_aga([], T170, T170)
append40_in_aga(.(T179, T180), T181, .(T179, T183)) → U9_aga(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
U9_aga(T179, T180, T181, T183, append40_out_aga(T180, T181, T183)) → append40_out_aga(.(T179, T180), T181, .(T179, T183))
U14_gggga(T31, T18, T19, T20, T22, append40_out_aga(T38, .(mv(T18, T20), []), X28)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_gggga(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_gggga(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_gggga(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
append40_in_gaa([], T170, T170) → append40_out_gaa([], T170, T170)
append40_in_gaa(.(T179, T180), T181, .(T179, T183)) → U9_gaa(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
U9_gaa(T179, T180, T181, T183, append40_out_gaa(T180, T181, T183)) → append40_out_gaa(.(T179, T180), T181, .(T179, T183))
U16_gggga(T31, T18, T19, T20, T22, append40_out_gaa(T159, T148, T22)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)

The argument filtering Pi contains the following mapping:
shanoi1_in_gggga(x1, x2, x3, x4, x5)  =  shanoi1_in_gggga(x1, x2, x3, x4)
s(x1)  =  s(x1)
0  =  0
shanoi1_out_gggga(x1, x2, x3, x4, x5)  =  shanoi1_out_gggga(x1, x2, x3, x4)
U10_gggga(x1, x2, x3, x4, x5, x6)  =  U10_gggga(x1, x2, x3, x4, x6)
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga(x1, x2, x3, x4)
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x1, x2, x3, x4, x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x1, x2, x3, x4, x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x1, x2, x3, x4, x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x1, x2, x3, x4, x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x2, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x3, x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x1, x2, x3, x4, x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x1, x2, x3, x4, x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa(x1)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x2, x5)
U11_gggga(x1, x2, x3, x4, x5, x6)  =  U11_gggga(x1, x2, x3, x4, x6)
U12_gggga(x1, x2, x3, x4, x5, x6)  =  U12_gggga(x1, x2, x3, x4, x6)
U13_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U13_gggga(x1, x2, x3, x4, x7)
U14_gggga(x1, x2, x3, x4, x5, x6)  =  U14_gggga(x1, x2, x3, x4, x6)
append40_in_aga(x1, x2, x3)  =  append40_in_aga(x2)
append40_out_aga(x1, x2, x3)  =  append40_out_aga(x1, x2, x3)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x3, x5)
U15_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U15_gggga(x1, x2, x3, x4, x7)
U16_gggga(x1, x2, x3, x4, x5, x6)  =  U16_gggga(x1, x2, x3, x4, x6)
append40_in_gaa(x1, x2, x3)  =  append40_in_gaa(x1)
append40_out_gaa(x1, x2, x3)  =  append40_out_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x2, x5)
SHANOI1_IN_GGGGA(x1, x2, x3, x4, x5)  =  SHANOI1_IN_GGGGA(x1, x2, x3, x4)
U10_GGGGA(x1, x2, x3, x4, x5, x6)  =  U10_GGGGA(x1, x2, x3, x4, x6)
SHANOI11_IN_GGGGA(x1, x2, x3, x4, x5)  =  SHANOI11_IN_GGGGA(x1, x2, x3, x4)
U1_GGGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGGA(x1, x2, x3, x4, x6)
U2_GGGGA(x1, x2, x3, x4, x5, x6)  =  U2_GGGGA(x1, x2, x3, x4, x6)
U3_GGGGA(x1, x2, x3, x4, x5, x6)  =  U3_GGGGA(x1, x2, x3, x4, x6)
U4_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GGGGA(x1, x2, x3, x4, x7)
U5_GGGGA(x1, x2, x3, x4, x5, x6)  =  U5_GGGGA(x1, x2, x3, x4, x6)
APPEND28_IN_AGA(x1, x2, x3)  =  APPEND28_IN_AGA(x2)
U8_AGA(x1, x2, x3, x4, x5)  =  U8_AGA(x3, x5)
U6_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GGGGA(x1, x2, x3, x4, x7)
U7_GGGGA(x1, x2, x3, x4, x5, x6)  =  U7_GGGGA(x1, x2, x3, x4, x6)
APPEND28_IN_GAA(x1, x2, x3)  =  APPEND28_IN_GAA(x1)
U8_GAA(x1, x2, x3, x4, x5)  =  U8_GAA(x2, x5)
U11_GGGGA(x1, x2, x3, x4, x5, x6)  =  U11_GGGGA(x1, x2, x3, x4, x6)
U12_GGGGA(x1, x2, x3, x4, x5, x6)  =  U12_GGGGA(x1, x2, x3, x4, x6)
U13_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U13_GGGGA(x1, x2, x3, x4, x7)
U14_GGGGA(x1, x2, x3, x4, x5, x6)  =  U14_GGGGA(x1, x2, x3, x4, x6)
APPEND40_IN_AGA(x1, x2, x3)  =  APPEND40_IN_AGA(x2)
U9_AGA(x1, x2, x3, x4, x5)  =  U9_AGA(x3, x5)
U15_GGGGA(x1, x2, x3, x4, x5, x6, x7)  =  U15_GGGGA(x1, x2, x3, x4, x7)
U16_GGGGA(x1, x2, x3, x4, x5, x6)  =  U16_GGGGA(x1, x2, x3, x4, x6)
APPEND40_IN_GAA(x1, x2, x3)  =  APPEND40_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x2, x5)

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

(48) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 23 less nodes.

(49) Complex Obligation (AND)

(50) Obligation:

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

APPEND40_IN_GAA(.(T179, T180), T181, .(T179, T183)) → APPEND40_IN_GAA(T180, T181, T183)

The TRS R consists of the following rules:

shanoi1_in_gggga(s(0), T9, T10, T11, .(mv(T9, T11), [])) → shanoi1_out_gggga(s(0), T9, T10, T11, .(mv(T9, T11), []))
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U10_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U10_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, X26)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U11_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U12_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T20, T18, T20, X27)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_gggga(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
append40_in_aga([], T170, T170) → append40_out_aga([], T170, T170)
append40_in_aga(.(T179, T180), T181, .(T179, T183)) → U9_aga(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
U9_aga(T179, T180, T181, T183, append40_out_aga(T180, T181, T183)) → append40_out_aga(.(T179, T180), T181, .(T179, T183))
U14_gggga(T31, T18, T19, T20, T22, append40_out_aga(T38, .(mv(T18, T20), []), X28)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_gggga(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_gggga(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_gggga(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
append40_in_gaa([], T170, T170) → append40_out_gaa([], T170, T170)
append40_in_gaa(.(T179, T180), T181, .(T179, T183)) → U9_gaa(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
U9_gaa(T179, T180, T181, T183, append40_out_gaa(T180, T181, T183)) → append40_out_gaa(.(T179, T180), T181, .(T179, T183))
U16_gggga(T31, T18, T19, T20, T22, append40_out_gaa(T159, T148, T22)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)

The argument filtering Pi contains the following mapping:
shanoi1_in_gggga(x1, x2, x3, x4, x5)  =  shanoi1_in_gggga(x1, x2, x3, x4)
s(x1)  =  s(x1)
0  =  0
shanoi1_out_gggga(x1, x2, x3, x4, x5)  =  shanoi1_out_gggga(x1, x2, x3, x4)
U10_gggga(x1, x2, x3, x4, x5, x6)  =  U10_gggga(x1, x2, x3, x4, x6)
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga(x1, x2, x3, x4)
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x1, x2, x3, x4, x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x1, x2, x3, x4, x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x1, x2, x3, x4, x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x1, x2, x3, x4, x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x2, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x3, x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x1, x2, x3, x4, x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x1, x2, x3, x4, x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa(x1)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x2, x5)
U11_gggga(x1, x2, x3, x4, x5, x6)  =  U11_gggga(x1, x2, x3, x4, x6)
U12_gggga(x1, x2, x3, x4, x5, x6)  =  U12_gggga(x1, x2, x3, x4, x6)
U13_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U13_gggga(x1, x2, x3, x4, x7)
U14_gggga(x1, x2, x3, x4, x5, x6)  =  U14_gggga(x1, x2, x3, x4, x6)
append40_in_aga(x1, x2, x3)  =  append40_in_aga(x2)
append40_out_aga(x1, x2, x3)  =  append40_out_aga(x1, x2, x3)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x3, x5)
U15_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U15_gggga(x1, x2, x3, x4, x7)
U16_gggga(x1, x2, x3, x4, x5, x6)  =  U16_gggga(x1, x2, x3, x4, x6)
append40_in_gaa(x1, x2, x3)  =  append40_in_gaa(x1)
append40_out_gaa(x1, x2, x3)  =  append40_out_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x2, x5)
APPEND40_IN_GAA(x1, x2, x3)  =  APPEND40_IN_GAA(x1)

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

(51) UsableRulesProof (EQUIVALENT transformation)

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

(52) Obligation:

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

APPEND40_IN_GAA(.(T179, T180), T181, .(T179, T183)) → APPEND40_IN_GAA(T180, T181, T183)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND40_IN_GAA(x1, x2, x3)  =  APPEND40_IN_GAA(x1)

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

(53) PiDPToQDPProof (SOUND transformation)

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

(54) Obligation:

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

APPEND40_IN_GAA(.(T180)) → APPEND40_IN_GAA(T180)

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

(55) 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:

  • APPEND40_IN_GAA(.(T180)) → APPEND40_IN_GAA(T180)
    The graph contains the following edges 1 > 1

(56) YES

(57) Obligation:

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

APPEND40_IN_AGA(.(T179, T180), T181, .(T179, T183)) → APPEND40_IN_AGA(T180, T181, T183)

The TRS R consists of the following rules:

shanoi1_in_gggga(s(0), T9, T10, T11, .(mv(T9, T11), [])) → shanoi1_out_gggga(s(0), T9, T10, T11, .(mv(T9, T11), []))
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U10_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U10_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, X26)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U11_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U12_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T20, T18, T20, X27)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_gggga(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
append40_in_aga([], T170, T170) → append40_out_aga([], T170, T170)
append40_in_aga(.(T179, T180), T181, .(T179, T183)) → U9_aga(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
U9_aga(T179, T180, T181, T183, append40_out_aga(T180, T181, T183)) → append40_out_aga(.(T179, T180), T181, .(T179, T183))
U14_gggga(T31, T18, T19, T20, T22, append40_out_aga(T38, .(mv(T18, T20), []), X28)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_gggga(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_gggga(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_gggga(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
append40_in_gaa([], T170, T170) → append40_out_gaa([], T170, T170)
append40_in_gaa(.(T179, T180), T181, .(T179, T183)) → U9_gaa(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
U9_gaa(T179, T180, T181, T183, append40_out_gaa(T180, T181, T183)) → append40_out_gaa(.(T179, T180), T181, .(T179, T183))
U16_gggga(T31, T18, T19, T20, T22, append40_out_gaa(T159, T148, T22)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)

The argument filtering Pi contains the following mapping:
shanoi1_in_gggga(x1, x2, x3, x4, x5)  =  shanoi1_in_gggga(x1, x2, x3, x4)
s(x1)  =  s(x1)
0  =  0
shanoi1_out_gggga(x1, x2, x3, x4, x5)  =  shanoi1_out_gggga(x1, x2, x3, x4)
U10_gggga(x1, x2, x3, x4, x5, x6)  =  U10_gggga(x1, x2, x3, x4, x6)
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga(x1, x2, x3, x4)
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x1, x2, x3, x4, x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x1, x2, x3, x4, x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x1, x2, x3, x4, x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x1, x2, x3, x4, x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x2, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x3, x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x1, x2, x3, x4, x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x1, x2, x3, x4, x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa(x1)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x2, x5)
U11_gggga(x1, x2, x3, x4, x5, x6)  =  U11_gggga(x1, x2, x3, x4, x6)
U12_gggga(x1, x2, x3, x4, x5, x6)  =  U12_gggga(x1, x2, x3, x4, x6)
U13_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U13_gggga(x1, x2, x3, x4, x7)
U14_gggga(x1, x2, x3, x4, x5, x6)  =  U14_gggga(x1, x2, x3, x4, x6)
append40_in_aga(x1, x2, x3)  =  append40_in_aga(x2)
append40_out_aga(x1, x2, x3)  =  append40_out_aga(x1, x2, x3)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x3, x5)
U15_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U15_gggga(x1, x2, x3, x4, x7)
U16_gggga(x1, x2, x3, x4, x5, x6)  =  U16_gggga(x1, x2, x3, x4, x6)
append40_in_gaa(x1, x2, x3)  =  append40_in_gaa(x1)
append40_out_gaa(x1, x2, x3)  =  append40_out_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x2, x5)
APPEND40_IN_AGA(x1, x2, x3)  =  APPEND40_IN_AGA(x2)

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

(58) UsableRulesProof (EQUIVALENT transformation)

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

(59) Obligation:

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

APPEND40_IN_AGA(.(T179, T180), T181, .(T179, T183)) → APPEND40_IN_AGA(T180, T181, T183)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND40_IN_AGA(x1, x2, x3)  =  APPEND40_IN_AGA(x2)

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

(60) PiDPToQDPProof (SOUND transformation)

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

(61) Obligation:

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

APPEND40_IN_AGA(T181) → APPEND40_IN_AGA(T181)

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

(62) 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 = APPEND40_IN_AGA(T181) evaluates to t =APPEND40_IN_AGA(T181)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND40_IN_AGA(T181) to APPEND40_IN_AGA(T181).



(63) NO

(64) Obligation:

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

APPEND28_IN_GAA(.(T137, T138), T139, .(T137, X178)) → APPEND28_IN_GAA(T138, T139, X178)

The TRS R consists of the following rules:

shanoi1_in_gggga(s(0), T9, T10, T11, .(mv(T9, T11), [])) → shanoi1_out_gggga(s(0), T9, T10, T11, .(mv(T9, T11), []))
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U10_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U10_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, X26)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U11_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U12_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T20, T18, T20, X27)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_gggga(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
append40_in_aga([], T170, T170) → append40_out_aga([], T170, T170)
append40_in_aga(.(T179, T180), T181, .(T179, T183)) → U9_aga(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
U9_aga(T179, T180, T181, T183, append40_out_aga(T180, T181, T183)) → append40_out_aga(.(T179, T180), T181, .(T179, T183))
U14_gggga(T31, T18, T19, T20, T22, append40_out_aga(T38, .(mv(T18, T20), []), X28)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_gggga(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_gggga(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_gggga(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
append40_in_gaa([], T170, T170) → append40_out_gaa([], T170, T170)
append40_in_gaa(.(T179, T180), T181, .(T179, T183)) → U9_gaa(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
U9_gaa(T179, T180, T181, T183, append40_out_gaa(T180, T181, T183)) → append40_out_gaa(.(T179, T180), T181, .(T179, T183))
U16_gggga(T31, T18, T19, T20, T22, append40_out_gaa(T159, T148, T22)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)

The argument filtering Pi contains the following mapping:
shanoi1_in_gggga(x1, x2, x3, x4, x5)  =  shanoi1_in_gggga(x1, x2, x3, x4)
s(x1)  =  s(x1)
0  =  0
shanoi1_out_gggga(x1, x2, x3, x4, x5)  =  shanoi1_out_gggga(x1, x2, x3, x4)
U10_gggga(x1, x2, x3, x4, x5, x6)  =  U10_gggga(x1, x2, x3, x4, x6)
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga(x1, x2, x3, x4)
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x1, x2, x3, x4, x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x1, x2, x3, x4, x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x1, x2, x3, x4, x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x1, x2, x3, x4, x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x2, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x3, x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x1, x2, x3, x4, x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x1, x2, x3, x4, x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa(x1)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x2, x5)
U11_gggga(x1, x2, x3, x4, x5, x6)  =  U11_gggga(x1, x2, x3, x4, x6)
U12_gggga(x1, x2, x3, x4, x5, x6)  =  U12_gggga(x1, x2, x3, x4, x6)
U13_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U13_gggga(x1, x2, x3, x4, x7)
U14_gggga(x1, x2, x3, x4, x5, x6)  =  U14_gggga(x1, x2, x3, x4, x6)
append40_in_aga(x1, x2, x3)  =  append40_in_aga(x2)
append40_out_aga(x1, x2, x3)  =  append40_out_aga(x1, x2, x3)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x3, x5)
U15_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U15_gggga(x1, x2, x3, x4, x7)
U16_gggga(x1, x2, x3, x4, x5, x6)  =  U16_gggga(x1, x2, x3, x4, x6)
append40_in_gaa(x1, x2, x3)  =  append40_in_gaa(x1)
append40_out_gaa(x1, x2, x3)  =  append40_out_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x2, x5)
APPEND28_IN_GAA(x1, x2, x3)  =  APPEND28_IN_GAA(x1)

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

(65) UsableRulesProof (EQUIVALENT transformation)

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

(66) Obligation:

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

APPEND28_IN_GAA(.(T137, T138), T139, .(T137, X178)) → APPEND28_IN_GAA(T138, T139, X178)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND28_IN_GAA(x1, x2, x3)  =  APPEND28_IN_GAA(x1)

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

(67) PiDPToQDPProof (SOUND transformation)

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

(68) Obligation:

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

APPEND28_IN_GAA(.(T138)) → APPEND28_IN_GAA(T138)

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

(69) 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:

  • APPEND28_IN_GAA(.(T138)) → APPEND28_IN_GAA(T138)
    The graph contains the following edges 1 > 1

(70) YES

(71) Obligation:

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

APPEND28_IN_AGA(.(T137, T138), T139, .(T137, X178)) → APPEND28_IN_AGA(T138, T139, X178)

The TRS R consists of the following rules:

shanoi1_in_gggga(s(0), T9, T10, T11, .(mv(T9, T11), [])) → shanoi1_out_gggga(s(0), T9, T10, T11, .(mv(T9, T11), []))
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U10_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U10_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, X26)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U11_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U12_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T20, T18, T20, X27)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_gggga(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
append40_in_aga([], T170, T170) → append40_out_aga([], T170, T170)
append40_in_aga(.(T179, T180), T181, .(T179, T183)) → U9_aga(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
U9_aga(T179, T180, T181, T183, append40_out_aga(T180, T181, T183)) → append40_out_aga(.(T179, T180), T181, .(T179, T183))
U14_gggga(T31, T18, T19, T20, T22, append40_out_aga(T38, .(mv(T18, T20), []), X28)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_gggga(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_gggga(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_gggga(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
append40_in_gaa([], T170, T170) → append40_out_gaa([], T170, T170)
append40_in_gaa(.(T179, T180), T181, .(T179, T183)) → U9_gaa(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
U9_gaa(T179, T180, T181, T183, append40_out_gaa(T180, T181, T183)) → append40_out_gaa(.(T179, T180), T181, .(T179, T183))
U16_gggga(T31, T18, T19, T20, T22, append40_out_gaa(T159, T148, T22)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)

The argument filtering Pi contains the following mapping:
shanoi1_in_gggga(x1, x2, x3, x4, x5)  =  shanoi1_in_gggga(x1, x2, x3, x4)
s(x1)  =  s(x1)
0  =  0
shanoi1_out_gggga(x1, x2, x3, x4, x5)  =  shanoi1_out_gggga(x1, x2, x3, x4)
U10_gggga(x1, x2, x3, x4, x5, x6)  =  U10_gggga(x1, x2, x3, x4, x6)
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga(x1, x2, x3, x4)
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x1, x2, x3, x4, x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x1, x2, x3, x4, x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x1, x2, x3, x4, x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x1, x2, x3, x4, x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x2, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x3, x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x1, x2, x3, x4, x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x1, x2, x3, x4, x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa(x1)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x2, x5)
U11_gggga(x1, x2, x3, x4, x5, x6)  =  U11_gggga(x1, x2, x3, x4, x6)
U12_gggga(x1, x2, x3, x4, x5, x6)  =  U12_gggga(x1, x2, x3, x4, x6)
U13_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U13_gggga(x1, x2, x3, x4, x7)
U14_gggga(x1, x2, x3, x4, x5, x6)  =  U14_gggga(x1, x2, x3, x4, x6)
append40_in_aga(x1, x2, x3)  =  append40_in_aga(x2)
append40_out_aga(x1, x2, x3)  =  append40_out_aga(x1, x2, x3)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x3, x5)
U15_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U15_gggga(x1, x2, x3, x4, x7)
U16_gggga(x1, x2, x3, x4, x5, x6)  =  U16_gggga(x1, x2, x3, x4, x6)
append40_in_gaa(x1, x2, x3)  =  append40_in_gaa(x1)
append40_out_gaa(x1, x2, x3)  =  append40_out_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x2, x5)
APPEND28_IN_AGA(x1, x2, x3)  =  APPEND28_IN_AGA(x2)

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

(72) UsableRulesProof (EQUIVALENT transformation)

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

(73) Obligation:

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

APPEND28_IN_AGA(.(T137, T138), T139, .(T137, X178)) → APPEND28_IN_AGA(T138, T139, X178)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND28_IN_AGA(x1, x2, x3)  =  APPEND28_IN_AGA(x2)

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

(74) PiDPToQDPProof (SOUND transformation)

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

(75) Obligation:

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

APPEND28_IN_AGA(T139) → APPEND28_IN_AGA(T139)

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

(76) 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 = APPEND28_IN_AGA(T139) evaluates to t =APPEND28_IN_AGA(T139)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND28_IN_AGA(T139) to APPEND28_IN_AGA(T139).



(77) NO

(78) Obligation:

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

SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → SHANOI11_IN_GGGGA(T88, T70, T69, T71, X99)
SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → SHANOI11_IN_GGGGA(T88, T69, T71, T70, X98)

The TRS R consists of the following rules:

shanoi1_in_gggga(s(0), T9, T10, T11, .(mv(T9, T11), [])) → shanoi1_out_gggga(s(0), T9, T10, T11, .(mv(T9, T11), []))
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U10_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, X26))
shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U10_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, X26)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
shanoi1_in_gggga(s(s(T31)), T18, T19, T20, T22) → U11_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T18, T20, T19, T38))
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U12_gggga(T31, T18, T19, T20, T22, shanoi11_in_gggga(T31, T20, T18, T20, X27))
U12_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T20, T18, T20, X27)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U11_gggga(T31, T18, T19, T20, T22, shanoi11_out_gggga(T31, T18, T20, T19, T38)) → U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_in_gggga(T31, T20, T18, T20, T148))
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U14_gggga(T31, T18, T19, T20, T22, append40_in_aga(T38, .(mv(T18, T20), []), X28))
append40_in_aga([], T170, T170) → append40_out_aga([], T170, T170)
append40_in_aga(.(T179, T180), T181, .(T179, T183)) → U9_aga(T179, T180, T181, T183, append40_in_aga(T180, T181, T183))
U9_aga(T179, T180, T181, T183, append40_out_aga(T180, T181, T183)) → append40_out_aga(.(T179, T180), T181, .(T179, T183))
U14_gggga(T31, T18, T19, T20, T22, append40_out_aga(T38, .(mv(T18, T20), []), X28)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)
U13_gggga(T31, T18, T19, T20, T22, T38, shanoi11_out_gggga(T31, T20, T18, T20, T148)) → U15_gggga(T31, T18, T19, T20, T22, T148, append40_in_aga(T38, .(mv(T18, T20), []), T159))
U15_gggga(T31, T18, T19, T20, T22, T148, append40_out_aga(T38, .(mv(T18, T20), []), T159)) → U16_gggga(T31, T18, T19, T20, T22, append40_in_gaa(T159, T148, T22))
append40_in_gaa([], T170, T170) → append40_out_gaa([], T170, T170)
append40_in_gaa(.(T179, T180), T181, .(T179, T183)) → U9_gaa(T179, T180, T181, T183, append40_in_gaa(T180, T181, T183))
U9_gaa(T179, T180, T181, T183, append40_out_gaa(T180, T181, T183)) → append40_out_gaa(.(T179, T180), T181, .(T179, T183))
U16_gggga(T31, T18, T19, T20, T22, append40_out_gaa(T159, T148, T22)) → shanoi1_out_gggga(s(s(T31)), T18, T19, T20, T22)

The argument filtering Pi contains the following mapping:
shanoi1_in_gggga(x1, x2, x3, x4, x5)  =  shanoi1_in_gggga(x1, x2, x3, x4)
s(x1)  =  s(x1)
0  =  0
shanoi1_out_gggga(x1, x2, x3, x4, x5)  =  shanoi1_out_gggga(x1, x2, x3, x4)
U10_gggga(x1, x2, x3, x4, x5, x6)  =  U10_gggga(x1, x2, x3, x4, x6)
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga(x1, x2, x3, x4)
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x1, x2, x3, x4, x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x1, x2, x3, x4, x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x1, x2, x3, x4, x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x1, x2, x3, x4, x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x2, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x3, x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x1, x2, x3, x4, x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x1, x2, x3, x4, x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa(x1)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x2, x5)
U11_gggga(x1, x2, x3, x4, x5, x6)  =  U11_gggga(x1, x2, x3, x4, x6)
U12_gggga(x1, x2, x3, x4, x5, x6)  =  U12_gggga(x1, x2, x3, x4, x6)
U13_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U13_gggga(x1, x2, x3, x4, x7)
U14_gggga(x1, x2, x3, x4, x5, x6)  =  U14_gggga(x1, x2, x3, x4, x6)
append40_in_aga(x1, x2, x3)  =  append40_in_aga(x2)
append40_out_aga(x1, x2, x3)  =  append40_out_aga(x1, x2, x3)
U9_aga(x1, x2, x3, x4, x5)  =  U9_aga(x3, x5)
U15_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U15_gggga(x1, x2, x3, x4, x7)
U16_gggga(x1, x2, x3, x4, x5, x6)  =  U16_gggga(x1, x2, x3, x4, x6)
append40_in_gaa(x1, x2, x3)  =  append40_in_gaa(x1)
append40_out_gaa(x1, x2, x3)  =  append40_out_gaa(x1)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x2, x5)
SHANOI11_IN_GGGGA(x1, x2, x3, x4, x5)  =  SHANOI11_IN_GGGGA(x1, x2, x3, x4)
U2_GGGGA(x1, x2, x3, x4, x5, x6)  =  U2_GGGGA(x1, x2, x3, x4, x6)

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

(79) UsableRulesProof (EQUIVALENT transformation)

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

(80) Obligation:

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

SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U2_GGGGA(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → SHANOI11_IN_GGGGA(T88, T70, T69, T71, X99)
SHANOI11_IN_GGGGA(s(T88), T69, T70, T71, X101) → SHANOI11_IN_GGGGA(T88, T69, T71, T70, X98)

The TRS R consists of the following rules:

shanoi11_in_gggga(0, T57, T58, T59, .(mv(T57, T59), [])) → shanoi11_out_gggga(0, T57, T58, T59, .(mv(T57, T59), []))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U1_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, X98))
shanoi11_in_gggga(s(T88), T69, T70, T71, X101) → U2_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T69, T71, T70, T95))
U1_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, X98)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U3_gggga(T88, T69, T70, T71, X101, shanoi11_in_gggga(T88, T70, T69, T71, X99))
U2_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T69, T71, T70, T95)) → U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_in_gggga(T88, T70, T69, T71, T108))
U3_gggga(T88, T69, T70, T71, X101, shanoi11_out_gggga(T88, T70, T69, T71, X99)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U5_gggga(T88, T69, T70, T71, X101, append28_in_aga(T95, .(mv(T69, T71), []), X100))
U4_gggga(T88, T69, T70, T71, X101, T95, shanoi11_out_gggga(T88, T70, T69, T71, T108)) → U6_gggga(T88, T69, T70, T71, X101, T108, append28_in_aga(T95, .(mv(T69, T71), []), T119))
U5_gggga(T88, T69, T70, T71, X101, append28_out_aga(T95, .(mv(T69, T71), []), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U6_gggga(T88, T69, T70, T71, X101, T108, append28_out_aga(T95, .(mv(T69, T71), []), T119)) → U7_gggga(T88, T69, T70, T71, X101, append28_in_gaa(T119, T108, X101))
append28_in_aga([], T130, T130) → append28_out_aga([], T130, T130)
append28_in_aga(.(T137, T138), T139, .(T137, X178)) → U8_aga(T137, T138, T139, X178, append28_in_aga(T138, T139, X178))
U7_gggga(T88, T69, T70, T71, X101, append28_out_gaa(T119, T108, X101)) → shanoi11_out_gggga(s(T88), T69, T70, T71, X101)
U8_aga(T137, T138, T139, X178, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T137, T138), T139, .(T137, X178))
append28_in_gaa([], T130, T130) → append28_out_gaa([], T130, T130)
append28_in_gaa(.(T137, T138), T139, .(T137, X178)) → U8_gaa(T137, T138, T139, X178, append28_in_gaa(T138, T139, X178))
U8_gaa(T137, T138, T139, X178, append28_out_gaa(T138, T139, X178)) → append28_out_gaa(.(T137, T138), T139, .(T137, X178))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
shanoi11_in_gggga(x1, x2, x3, x4, x5)  =  shanoi11_in_gggga(x1, x2, x3, x4)
shanoi11_out_gggga(x1, x2, x3, x4, x5)  =  shanoi11_out_gggga(x1, x2, x3, x4)
U1_gggga(x1, x2, x3, x4, x5, x6)  =  U1_gggga(x1, x2, x3, x4, x6)
U2_gggga(x1, x2, x3, x4, x5, x6)  =  U2_gggga(x1, x2, x3, x4, x6)
U3_gggga(x1, x2, x3, x4, x5, x6)  =  U3_gggga(x1, x2, x3, x4, x6)
U4_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U4_gggga(x1, x2, x3, x4, x7)
U5_gggga(x1, x2, x3, x4, x5, x6)  =  U5_gggga(x1, x2, x3, x4, x6)
append28_in_aga(x1, x2, x3)  =  append28_in_aga(x2)
append28_out_aga(x1, x2, x3)  =  append28_out_aga(x1, x2, x3)
U8_aga(x1, x2, x3, x4, x5)  =  U8_aga(x3, x5)
.(x1, x2)  =  .(x2)
[]  =  []
U6_gggga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggga(x1, x2, x3, x4, x7)
U7_gggga(x1, x2, x3, x4, x5, x6)  =  U7_gggga(x1, x2, x3, x4, x6)
append28_in_gaa(x1, x2, x3)  =  append28_in_gaa(x1)
append28_out_gaa(x1, x2, x3)  =  append28_out_gaa(x1)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x2, x5)
SHANOI11_IN_GGGGA(x1, x2, x3, x4, x5)  =  SHANOI11_IN_GGGGA(x1, x2, x3, x4)
U2_GGGGA(x1, x2, x3, x4, x5, x6)  =  U2_GGGGA(x1, x2, x3, x4, x6)

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

(81) PiDPToQDPProof (SOUND transformation)

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

(82) Obligation:

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

SHANOI11_IN_GGGGA(s(T88), T69, T70, T71) → U2_GGGGA(T88, T69, T70, T71, shanoi11_in_gggga(T88, T69, T71, T70))
U2_GGGGA(T88, T69, T70, T71, shanoi11_out_gggga(T88, T69, T71, T70)) → SHANOI11_IN_GGGGA(T88, T70, T69, T71)
SHANOI11_IN_GGGGA(s(T88), T69, T70, T71) → SHANOI11_IN_GGGGA(T88, T69, T71, T70)

The TRS R consists of the following rules:

shanoi11_in_gggga(0, T57, T58, T59) → shanoi11_out_gggga(0, T57, T58, T59)
shanoi11_in_gggga(s(T88), T69, T70, T71) → U1_gggga(T88, T69, T70, T71, shanoi11_in_gggga(T88, T69, T71, T70))
shanoi11_in_gggga(s(T88), T69, T70, T71) → U2_gggga(T88, T69, T70, T71, shanoi11_in_gggga(T88, T69, T71, T70))
U1_gggga(T88, T69, T70, T71, shanoi11_out_gggga(T88, T69, T71, T70)) → shanoi11_out_gggga(s(T88), T69, T70, T71)
U2_gggga(T88, T69, T70, T71, shanoi11_out_gggga(T88, T69, T71, T70)) → U3_gggga(T88, T69, T70, T71, shanoi11_in_gggga(T88, T70, T69, T71))
U2_gggga(T88, T69, T70, T71, shanoi11_out_gggga(T88, T69, T71, T70)) → U4_gggga(T88, T69, T70, T71, shanoi11_in_gggga(T88, T70, T69, T71))
U3_gggga(T88, T69, T70, T71, shanoi11_out_gggga(T88, T70, T69, T71)) → shanoi11_out_gggga(s(T88), T69, T70, T71)
U4_gggga(T88, T69, T70, T71, shanoi11_out_gggga(T88, T70, T69, T71)) → U5_gggga(T88, T69, T70, T71, append28_in_aga(.([])))
U4_gggga(T88, T69, T70, T71, shanoi11_out_gggga(T88, T70, T69, T71)) → U6_gggga(T88, T69, T70, T71, append28_in_aga(.([])))
U5_gggga(T88, T69, T70, T71, append28_out_aga(T95, .([]), X100)) → shanoi11_out_gggga(s(T88), T69, T70, T71)
U6_gggga(T88, T69, T70, T71, append28_out_aga(T95, .([]), T119)) → U7_gggga(T88, T69, T70, T71, append28_in_gaa(T119))
append28_in_aga(T130) → append28_out_aga([], T130, T130)
append28_in_aga(T139) → U8_aga(T139, append28_in_aga(T139))
U7_gggga(T88, T69, T70, T71, append28_out_gaa(T119)) → shanoi11_out_gggga(s(T88), T69, T70, T71)
U8_aga(T139, append28_out_aga(T138, T139, X178)) → append28_out_aga(.(T138), T139, .(X178))
append28_in_gaa([]) → append28_out_gaa([])
append28_in_gaa(.(T138)) → U8_gaa(T138, append28_in_gaa(T138))
U8_gaa(T138, append28_out_gaa(T138)) → append28_out_gaa(.(T138))

The set Q consists of the following terms:

shanoi11_in_gggga(x0, x1, x2, x3)
U1_gggga(x0, x1, x2, x3, x4)
U2_gggga(x0, x1, x2, x3, x4)
U3_gggga(x0, x1, x2, x3, x4)
U4_gggga(x0, x1, x2, x3, x4)
U5_gggga(x0, x1, x2, x3, x4)
U6_gggga(x0, x1, x2, x3, x4)
append28_in_aga(x0)
U7_gggga(x0, x1, x2, x3, x4)
U8_aga(x0, x1)
append28_in_gaa(x0)
U8_gaa(x0, x1)

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

(83) 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:

  • U2_GGGGA(T88, T69, T70, T71, shanoi11_out_gggga(T88, T69, T71, T70)) → SHANOI11_IN_GGGGA(T88, T70, T69, T71)
    The graph contains the following edges 1 >= 1, 5 > 1, 3 >= 2, 5 > 2, 2 >= 3, 5 > 3, 4 >= 4, 5 > 4

  • SHANOI11_IN_GGGGA(s(T88), T69, T70, T71) → SHANOI11_IN_GGGGA(T88, T69, T71, T70)
    The graph contains the following edges 1 > 1, 2 >= 2, 4 >= 3, 3 >= 4

  • SHANOI11_IN_GGGGA(s(T88), T69, T70, T71) → U2_GGGGA(T88, T69, T70, T71, shanoi11_in_gggga(T88, T69, T71, T70))
    The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4

(84) YES