(0) Obligation:

Clauses:

goal(X) :- ','(s2t(X, T), tappend(T, X1, X2)).
tappend(nil, Y, Z) :- ','(!, eq(Y, Z)).
tappend(T, T1, node(T1, X, T2)) :- ','(left(T, nil), ','(right(T, T2), value(T, X))).
tappend(T, T2, node(T1, X, T2)) :- ','(left(T, T1), ','(right(T, nil), value(T, X))).
tappend(T, T3, node(U, X, T2)) :- ','(left(T, T1), ','(right(T, T2), ','(value(T, X), tappend(T1, T3, U)))).
tappend(T, T1, node(T1, X, U)) :- ','(left(T, T1), ','(right(T, T2), ','(value(T, X), tappend(T2, T3, U)))).
s2t(0, L) :- ','(!, eq(L, nil)).
s2t(X, node(T, X3, T)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(nil, X4, T)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(T, X5, nil)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(nil, X6, nil)).
left(nil, nil).
left(node(L, X7, X8), L).
right(nil, nil).
right(node(X9, X10, R), R).
value(nil, nil).
value(node(X11, X, X12), X).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

goal(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

s2t4(0, nil).
s2t4(s(T8), node(X34, X35, X34)) :- s2t4(T8, X34).
s2t4(s(T10), node(nil, X48, X49)) :- s2t4(T10, X49).
s2t4(s(T12), node(X62, X63, nil)) :- s2t4(T12, X62).
s2t4(T13, node(nil, X70, nil)).
tappend5(nil, X79, X79).
tappend5(nil, X92, node(X92, nil, nil)).
tappend5(node(nil, T25, T26), X92, node(X92, T25, T26)).
tappend5(node(nil, T36, T37), X92, node(X92, T36, T37)).
tappend5(nil, X149, node(nil, nil, X149)).
tappend5(node(T51, T52, nil), X149, node(T51, T52, X149)).
tappend5(node(T64, T65, nil), X149, node(T64, T65, X149)).
tappend5(nil, X210, node(X211, nil, nil)) :- tappend5(nil, X210, X211).
tappend5(node(T83, T81, T82), X210, node(X211, T81, T82)) :- tappend5(T83, X210, X211).
tappend5(node(T99, T97, T98), X210, node(X211, T97, T98)) :- tappend5(T99, X210, X211).
tappend5(nil, nil, node(nil, nil, X268)) :- tappend5(nil, X265, X268).
tappend5(node(T114, T115, T117), T114, node(T114, T115, X268)) :- tappend5(T117, X265, X268).
tappend5(node(T130, T131, T133), T130, node(T130, T131, X268)) :- tappend5(T133, X265, X268).
goal1(T5) :- s2t4(T5, X15).
goal1(T5) :- s2t4(T5, nil).
goal1(T5) :- s2t4(T5, nil).
goal1(T5) :- s2t4(T5, node(nil, T25, T26)).
goal1(T5) :- s2t4(T5, node(nil, T36, T37)).
goal1(T5) :- s2t4(T5, nil).
goal1(T5) :- s2t4(T5, node(T51, T52, nil)).
goal1(T5) :- s2t4(T5, node(T64, T65, nil)).
goal1(T5) :- ','(s2t4(T5, nil), tappend5(nil, X210, X211)).
goal1(T5) :- ','(s2t4(T5, node(T83, T81, T82)), tappend5(T83, X210, X211)).
goal1(T5) :- ','(s2t4(T5, node(T99, T97, T98)), tappend5(T99, X210, X211)).
goal1(T5) :- ','(s2t4(T5, nil), tappend5(nil, X265, X268)).
goal1(T5) :- ','(s2t4(T5, node(T114, T115, T117)), tappend5(T117, X265, X268)).
goal1(T5) :- ','(s2t4(T5, node(T130, T131, T133)), tappend5(T133, X265, X268)).

Queries:

goal1(g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal1_in: (b)
s2t4_in: (b,f) (b,b)
tappend5_in: (b,f,f) (f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal1_in_g(T5) → U10_g(T5, s2t4_in_ga(T5, X15))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T8), node(X34, X35, X34)) → U1_ga(T8, X34, X35, s2t4_in_ga(T8, X34))
s2t4_in_ga(s(T10), node(nil, X48, X49)) → U2_ga(T10, X48, X49, s2t4_in_ga(T10, X49))
s2t4_in_ga(s(T12), node(X62, X63, nil)) → U3_ga(T12, X62, X63, s2t4_in_ga(T12, X62))
s2t4_in_ga(T13, node(nil, X70, nil)) → s2t4_out_ga(T13, node(nil, X70, nil))
U3_ga(T12, X62, X63, s2t4_out_ga(T12, X62)) → s2t4_out_ga(s(T12), node(X62, X63, nil))
U2_ga(T10, X48, X49, s2t4_out_ga(T10, X49)) → s2t4_out_ga(s(T10), node(nil, X48, X49))
U1_ga(T8, X34, X35, s2t4_out_ga(T8, X34)) → s2t4_out_ga(s(T8), node(X34, X35, X34))
U10_g(T5, s2t4_out_ga(T5, X15)) → goal1_out_g(T5)
goal1_in_g(T5) → U11_g(T5, s2t4_in_gg(T5, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T8), node(X34, X35, X34)) → U1_gg(T8, X34, X35, s2t4_in_gg(T8, X34))
s2t4_in_gg(s(T10), node(nil, X48, X49)) → U2_gg(T10, X48, X49, s2t4_in_gg(T10, X49))
s2t4_in_gg(s(T12), node(X62, X63, nil)) → U3_gg(T12, X62, X63, s2t4_in_gg(T12, X62))
s2t4_in_gg(T13, node(nil, X70, nil)) → s2t4_out_gg(T13, node(nil, X70, nil))
U3_gg(T12, X62, X63, s2t4_out_gg(T12, X62)) → s2t4_out_gg(s(T12), node(X62, X63, nil))
U2_gg(T10, X48, X49, s2t4_out_gg(T10, X49)) → s2t4_out_gg(s(T10), node(nil, X48, X49))
U1_gg(T8, X34, X35, s2t4_out_gg(T8, X34)) → s2t4_out_gg(s(T8), node(X34, X35, X34))
U11_g(T5, s2t4_out_gg(T5, nil)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U12_g(T5, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(nil, X79, X79) → tappend5_out_gaa(nil, X79, X79)
tappend5_in_gaa(nil, X92, node(X92, nil, nil)) → tappend5_out_gaa(nil, X92, node(X92, nil, nil))
tappend5_in_gaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_gaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_gaa(nil, X149, node(nil, nil, X149)) → tappend5_out_gaa(nil, X149, node(nil, nil, X149))
tappend5_in_gaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_gaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_gaa(nil, X210, node(X211, nil, nil)) → U4_gaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_gaa(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
tappend5_in_gaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_gaa(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
tappend5_in_gaa(nil, nil, node(nil, nil, X268)) → U7_gaa(X268, tappend5_in_gaa(nil, X265, X268))
tappend5_in_gaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_gaa(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
tappend5_in_gaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_gaa(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U9_gaa(T130, T131, T133, X268, tappend5_out_gaa(T133, X265, X268)) → tappend5_out_gaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_gaa(T114, T115, T117, X268, tappend5_out_gaa(T117, X265, X268)) → tappend5_out_gaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U7_gaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_gaa(nil, nil, node(nil, nil, X268))
U6_gaa(T99, T97, T98, X210, X211, tappend5_out_gaa(T99, X210, X211)) → tappend5_out_gaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_gaa(T83, T81, T82, X210, X211, tappend5_out_gaa(T83, X210, X211)) → tappend5_out_gaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U4_gaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_gaa(nil, X210, node(X211, nil, nil))
U12_g(T5, tappend5_out_gaa(nil, X210, X211)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U13_g(T5, tappend5_in_gaa(nil, X265, X268))
U13_g(T5, tappend5_out_gaa(nil, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U14_g(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
U14_g(T5, s2t4_out_ga(T5, node(nil, T25, T26))) → goal1_out_g(T5)
goal1_in_g(T5) → U15_g(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
U15_g(T5, s2t4_out_ga(T5, node(nil, T36, T37))) → goal1_out_g(T5)
goal1_in_g(T5) → U16_g(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
U16_g(T5, s2t4_out_ga(T5, node(T51, T52, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U17_g(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
U17_g(T5, s2t4_out_ga(T5, node(T64, T65, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U18_g(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
U18_g(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_g(T5, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(nil, X79, X79) → tappend5_out_aaa(nil, X79, X79)
tappend5_in_aaa(nil, X92, node(X92, nil, nil)) → tappend5_out_aaa(nil, X92, node(X92, nil, nil))
tappend5_in_aaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_aaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_aaa(nil, X149, node(nil, nil, X149)) → tappend5_out_aaa(nil, X149, node(nil, nil, X149))
tappend5_in_aaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_aaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_aaa(nil, X210, node(X211, nil, nil)) → U4_aaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
U4_aaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_aaa(nil, X210, node(X211, nil, nil))
tappend5_in_aaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_aaa(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_aaa(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
tappend5_in_aaa(nil, nil, node(nil, nil, X268)) → U7_aaa(X268, tappend5_in_gaa(nil, X265, X268))
U7_aaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_aaa(nil, nil, node(nil, nil, X268))
tappend5_in_aaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_aaa(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
tappend5_in_aaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_aaa(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
U9_aaa(T130, T131, T133, X268, tappend5_out_aaa(T133, X265, X268)) → tappend5_out_aaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_aaa(T114, T115, T117, X268, tappend5_out_aaa(T117, X265, X268)) → tappend5_out_aaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U6_aaa(T99, T97, T98, X210, X211, tappend5_out_aaa(T99, X210, X211)) → tappend5_out_aaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_aaa(T83, T81, T82, X210, X211, tappend5_out_aaa(T83, X210, X211)) → tappend5_out_aaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U19_g(T5, tappend5_out_aaa(T83, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U20_g(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_g(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_g(T5, tappend5_in_aaa(T99, X210, X211))
U21_g(T5, tappend5_out_aaa(T99, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U22_g(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_g(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_g(T5, tappend5_in_aaa(T117, X265, X268))
U23_g(T5, tappend5_out_aaa(T117, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U24_g(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_g(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_g(T5, tappend5_in_aaa(T133, X265, X268))
U25_g(T5, tappend5_out_aaa(T133, X265, X268)) → goal1_out_g(T5)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U10_g(x1, x2)  =  U10_g(x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
goal1_out_g(x1)  =  goal1_out_g
U11_g(x1, x2)  =  U11_g(x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg
node(x1, x2, x3)  =  node(x1, x2, x3)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U12_g(x1, x2)  =  U12_g(x2)
tappend5_in_gaa(x1, x2, x3)  =  tappend5_in_gaa(x1)
tappend5_out_gaa(x1, x2, x3)  =  tappend5_out_gaa
U4_gaa(x1, x2, x3)  =  U4_gaa(x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x6)
U7_gaa(x1, x2)  =  U7_gaa(x2)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
U13_g(x1, x2)  =  U13_g(x2)
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
tappend5_in_aaa(x1, x2, x3)  =  tappend5_in_aaa
tappend5_out_aaa(x1, x2, x3)  =  tappend5_out_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
U5_aaa(x1, x2, x3, x4, x5, x6)  =  U5_aaa(x6)
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
U7_aaa(x1, x2)  =  U7_aaa(x2)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
U23_g(x1, x2)  =  U23_g(x2)
U24_g(x1, x2)  =  U24_g(x2)
U25_g(x1, x2)  =  U25_g(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

goal1_in_g(T5) → U10_g(T5, s2t4_in_ga(T5, X15))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T8), node(X34, X35, X34)) → U1_ga(T8, X34, X35, s2t4_in_ga(T8, X34))
s2t4_in_ga(s(T10), node(nil, X48, X49)) → U2_ga(T10, X48, X49, s2t4_in_ga(T10, X49))
s2t4_in_ga(s(T12), node(X62, X63, nil)) → U3_ga(T12, X62, X63, s2t4_in_ga(T12, X62))
s2t4_in_ga(T13, node(nil, X70, nil)) → s2t4_out_ga(T13, node(nil, X70, nil))
U3_ga(T12, X62, X63, s2t4_out_ga(T12, X62)) → s2t4_out_ga(s(T12), node(X62, X63, nil))
U2_ga(T10, X48, X49, s2t4_out_ga(T10, X49)) → s2t4_out_ga(s(T10), node(nil, X48, X49))
U1_ga(T8, X34, X35, s2t4_out_ga(T8, X34)) → s2t4_out_ga(s(T8), node(X34, X35, X34))
U10_g(T5, s2t4_out_ga(T5, X15)) → goal1_out_g(T5)
goal1_in_g(T5) → U11_g(T5, s2t4_in_gg(T5, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T8), node(X34, X35, X34)) → U1_gg(T8, X34, X35, s2t4_in_gg(T8, X34))
s2t4_in_gg(s(T10), node(nil, X48, X49)) → U2_gg(T10, X48, X49, s2t4_in_gg(T10, X49))
s2t4_in_gg(s(T12), node(X62, X63, nil)) → U3_gg(T12, X62, X63, s2t4_in_gg(T12, X62))
s2t4_in_gg(T13, node(nil, X70, nil)) → s2t4_out_gg(T13, node(nil, X70, nil))
U3_gg(T12, X62, X63, s2t4_out_gg(T12, X62)) → s2t4_out_gg(s(T12), node(X62, X63, nil))
U2_gg(T10, X48, X49, s2t4_out_gg(T10, X49)) → s2t4_out_gg(s(T10), node(nil, X48, X49))
U1_gg(T8, X34, X35, s2t4_out_gg(T8, X34)) → s2t4_out_gg(s(T8), node(X34, X35, X34))
U11_g(T5, s2t4_out_gg(T5, nil)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U12_g(T5, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(nil, X79, X79) → tappend5_out_gaa(nil, X79, X79)
tappend5_in_gaa(nil, X92, node(X92, nil, nil)) → tappend5_out_gaa(nil, X92, node(X92, nil, nil))
tappend5_in_gaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_gaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_gaa(nil, X149, node(nil, nil, X149)) → tappend5_out_gaa(nil, X149, node(nil, nil, X149))
tappend5_in_gaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_gaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_gaa(nil, X210, node(X211, nil, nil)) → U4_gaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_gaa(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
tappend5_in_gaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_gaa(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
tappend5_in_gaa(nil, nil, node(nil, nil, X268)) → U7_gaa(X268, tappend5_in_gaa(nil, X265, X268))
tappend5_in_gaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_gaa(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
tappend5_in_gaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_gaa(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U9_gaa(T130, T131, T133, X268, tappend5_out_gaa(T133, X265, X268)) → tappend5_out_gaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_gaa(T114, T115, T117, X268, tappend5_out_gaa(T117, X265, X268)) → tappend5_out_gaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U7_gaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_gaa(nil, nil, node(nil, nil, X268))
U6_gaa(T99, T97, T98, X210, X211, tappend5_out_gaa(T99, X210, X211)) → tappend5_out_gaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_gaa(T83, T81, T82, X210, X211, tappend5_out_gaa(T83, X210, X211)) → tappend5_out_gaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U4_gaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_gaa(nil, X210, node(X211, nil, nil))
U12_g(T5, tappend5_out_gaa(nil, X210, X211)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U13_g(T5, tappend5_in_gaa(nil, X265, X268))
U13_g(T5, tappend5_out_gaa(nil, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U14_g(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
U14_g(T5, s2t4_out_ga(T5, node(nil, T25, T26))) → goal1_out_g(T5)
goal1_in_g(T5) → U15_g(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
U15_g(T5, s2t4_out_ga(T5, node(nil, T36, T37))) → goal1_out_g(T5)
goal1_in_g(T5) → U16_g(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
U16_g(T5, s2t4_out_ga(T5, node(T51, T52, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U17_g(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
U17_g(T5, s2t4_out_ga(T5, node(T64, T65, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U18_g(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
U18_g(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_g(T5, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(nil, X79, X79) → tappend5_out_aaa(nil, X79, X79)
tappend5_in_aaa(nil, X92, node(X92, nil, nil)) → tappend5_out_aaa(nil, X92, node(X92, nil, nil))
tappend5_in_aaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_aaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_aaa(nil, X149, node(nil, nil, X149)) → tappend5_out_aaa(nil, X149, node(nil, nil, X149))
tappend5_in_aaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_aaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_aaa(nil, X210, node(X211, nil, nil)) → U4_aaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
U4_aaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_aaa(nil, X210, node(X211, nil, nil))
tappend5_in_aaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_aaa(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_aaa(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
tappend5_in_aaa(nil, nil, node(nil, nil, X268)) → U7_aaa(X268, tappend5_in_gaa(nil, X265, X268))
U7_aaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_aaa(nil, nil, node(nil, nil, X268))
tappend5_in_aaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_aaa(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
tappend5_in_aaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_aaa(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
U9_aaa(T130, T131, T133, X268, tappend5_out_aaa(T133, X265, X268)) → tappend5_out_aaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_aaa(T114, T115, T117, X268, tappend5_out_aaa(T117, X265, X268)) → tappend5_out_aaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U6_aaa(T99, T97, T98, X210, X211, tappend5_out_aaa(T99, X210, X211)) → tappend5_out_aaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_aaa(T83, T81, T82, X210, X211, tappend5_out_aaa(T83, X210, X211)) → tappend5_out_aaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U19_g(T5, tappend5_out_aaa(T83, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U20_g(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_g(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_g(T5, tappend5_in_aaa(T99, X210, X211))
U21_g(T5, tappend5_out_aaa(T99, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U22_g(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_g(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_g(T5, tappend5_in_aaa(T117, X265, X268))
U23_g(T5, tappend5_out_aaa(T117, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U24_g(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_g(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_g(T5, tappend5_in_aaa(T133, X265, X268))
U25_g(T5, tappend5_out_aaa(T133, X265, X268)) → goal1_out_g(T5)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U10_g(x1, x2)  =  U10_g(x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
goal1_out_g(x1)  =  goal1_out_g
U11_g(x1, x2)  =  U11_g(x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg
node(x1, x2, x3)  =  node(x1, x2, x3)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U12_g(x1, x2)  =  U12_g(x2)
tappend5_in_gaa(x1, x2, x3)  =  tappend5_in_gaa(x1)
tappend5_out_gaa(x1, x2, x3)  =  tappend5_out_gaa
U4_gaa(x1, x2, x3)  =  U4_gaa(x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x6)
U7_gaa(x1, x2)  =  U7_gaa(x2)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
U13_g(x1, x2)  =  U13_g(x2)
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
tappend5_in_aaa(x1, x2, x3)  =  tappend5_in_aaa
tappend5_out_aaa(x1, x2, x3)  =  tappend5_out_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
U5_aaa(x1, x2, x3, x4, x5, x6)  =  U5_aaa(x6)
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
U7_aaa(x1, x2)  =  U7_aaa(x2)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
U23_g(x1, x2)  =  U23_g(x2)
U24_g(x1, x2)  =  U24_g(x2)
U25_g(x1, x2)  =  U25_g(x2)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

GOAL1_IN_G(T5) → U10_G(T5, s2t4_in_ga(T5, X15))
GOAL1_IN_G(T5) → S2T4_IN_GA(T5, X15)
S2T4_IN_GA(s(T8), node(X34, X35, X34)) → U1_GA(T8, X34, X35, s2t4_in_ga(T8, X34))
S2T4_IN_GA(s(T8), node(X34, X35, X34)) → S2T4_IN_GA(T8, X34)
S2T4_IN_GA(s(T10), node(nil, X48, X49)) → U2_GA(T10, X48, X49, s2t4_in_ga(T10, X49))
S2T4_IN_GA(s(T10), node(nil, X48, X49)) → S2T4_IN_GA(T10, X49)
S2T4_IN_GA(s(T12), node(X62, X63, nil)) → U3_GA(T12, X62, X63, s2t4_in_ga(T12, X62))
S2T4_IN_GA(s(T12), node(X62, X63, nil)) → S2T4_IN_GA(T12, X62)
GOAL1_IN_G(T5) → U11_G(T5, s2t4_in_gg(T5, nil))
GOAL1_IN_G(T5) → S2T4_IN_GG(T5, nil)
S2T4_IN_GG(s(T8), node(X34, X35, X34)) → U1_GG(T8, X34, X35, s2t4_in_gg(T8, X34))
S2T4_IN_GG(s(T8), node(X34, X35, X34)) → S2T4_IN_GG(T8, X34)
S2T4_IN_GG(s(T10), node(nil, X48, X49)) → U2_GG(T10, X48, X49, s2t4_in_gg(T10, X49))
S2T4_IN_GG(s(T10), node(nil, X48, X49)) → S2T4_IN_GG(T10, X49)
S2T4_IN_GG(s(T12), node(X62, X63, nil)) → U3_GG(T12, X62, X63, s2t4_in_gg(T12, X62))
S2T4_IN_GG(s(T12), node(X62, X63, nil)) → S2T4_IN_GG(T12, X62)
U11_G(T5, s2t4_out_gg(T5, nil)) → U12_G(T5, tappend5_in_gaa(nil, X210, X211))
U11_G(T5, s2t4_out_gg(T5, nil)) → TAPPEND5_IN_GAA(nil, X210, X211)
TAPPEND5_IN_GAA(nil, X210, node(X211, nil, nil)) → U4_GAA(X210, X211, tappend5_in_gaa(nil, X210, X211))
TAPPEND5_IN_GAA(nil, X210, node(X211, nil, nil)) → TAPPEND5_IN_GAA(nil, X210, X211)
TAPPEND5_IN_GAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_GAA(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
TAPPEND5_IN_GAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → TAPPEND5_IN_GAA(T83, X210, X211)
TAPPEND5_IN_GAA(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_GAA(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
TAPPEND5_IN_GAA(nil, nil, node(nil, nil, X268)) → U7_GAA(X268, tappend5_in_gaa(nil, X265, X268))
TAPPEND5_IN_GAA(nil, nil, node(nil, nil, X268)) → TAPPEND5_IN_GAA(nil, X265, X268)
TAPPEND5_IN_GAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_GAA(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
TAPPEND5_IN_GAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → TAPPEND5_IN_GAA(T117, X265, X268)
TAPPEND5_IN_GAA(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_GAA(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U11_G(T5, s2t4_out_gg(T5, nil)) → U13_G(T5, tappend5_in_gaa(nil, X265, X268))
GOAL1_IN_G(T5) → U14_G(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
GOAL1_IN_G(T5) → S2T4_IN_GA(T5, node(nil, T25, T26))
GOAL1_IN_G(T5) → U15_G(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
GOAL1_IN_G(T5) → U16_G(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
GOAL1_IN_G(T5) → S2T4_IN_GA(T5, node(T51, T52, nil))
GOAL1_IN_G(T5) → U17_G(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
GOAL1_IN_G(T5) → U18_G(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
GOAL1_IN_G(T5) → S2T4_IN_GA(T5, node(T83, T81, T82))
U18_G(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_G(T5, tappend5_in_aaa(T83, X210, X211))
U18_G(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → TAPPEND5_IN_AAA(T83, X210, X211)
TAPPEND5_IN_AAA(nil, X210, node(X211, nil, nil)) → U4_AAA(X210, X211, tappend5_in_gaa(nil, X210, X211))
TAPPEND5_IN_AAA(nil, X210, node(X211, nil, nil)) → TAPPEND5_IN_GAA(nil, X210, X211)
TAPPEND5_IN_AAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_AAA(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
TAPPEND5_IN_AAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → TAPPEND5_IN_AAA(T83, X210, X211)
TAPPEND5_IN_AAA(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_AAA(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
TAPPEND5_IN_AAA(nil, nil, node(nil, nil, X268)) → U7_AAA(X268, tappend5_in_gaa(nil, X265, X268))
TAPPEND5_IN_AAA(nil, nil, node(nil, nil, X268)) → TAPPEND5_IN_GAA(nil, X265, X268)
TAPPEND5_IN_AAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_AAA(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
TAPPEND5_IN_AAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → TAPPEND5_IN_AAA(T117, X265, X268)
TAPPEND5_IN_AAA(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_AAA(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
GOAL1_IN_G(T5) → U20_G(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_G(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_G(T5, tappend5_in_aaa(T99, X210, X211))
U20_G(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → TAPPEND5_IN_AAA(T99, X210, X211)
GOAL1_IN_G(T5) → U22_G(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_G(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_G(T5, tappend5_in_aaa(T117, X265, X268))
U22_G(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → TAPPEND5_IN_AAA(T117, X265, X268)
GOAL1_IN_G(T5) → U24_G(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_G(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_G(T5, tappend5_in_aaa(T133, X265, X268))
U24_G(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → TAPPEND5_IN_AAA(T133, X265, X268)

The TRS R consists of the following rules:

goal1_in_g(T5) → U10_g(T5, s2t4_in_ga(T5, X15))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T8), node(X34, X35, X34)) → U1_ga(T8, X34, X35, s2t4_in_ga(T8, X34))
s2t4_in_ga(s(T10), node(nil, X48, X49)) → U2_ga(T10, X48, X49, s2t4_in_ga(T10, X49))
s2t4_in_ga(s(T12), node(X62, X63, nil)) → U3_ga(T12, X62, X63, s2t4_in_ga(T12, X62))
s2t4_in_ga(T13, node(nil, X70, nil)) → s2t4_out_ga(T13, node(nil, X70, nil))
U3_ga(T12, X62, X63, s2t4_out_ga(T12, X62)) → s2t4_out_ga(s(T12), node(X62, X63, nil))
U2_ga(T10, X48, X49, s2t4_out_ga(T10, X49)) → s2t4_out_ga(s(T10), node(nil, X48, X49))
U1_ga(T8, X34, X35, s2t4_out_ga(T8, X34)) → s2t4_out_ga(s(T8), node(X34, X35, X34))
U10_g(T5, s2t4_out_ga(T5, X15)) → goal1_out_g(T5)
goal1_in_g(T5) → U11_g(T5, s2t4_in_gg(T5, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T8), node(X34, X35, X34)) → U1_gg(T8, X34, X35, s2t4_in_gg(T8, X34))
s2t4_in_gg(s(T10), node(nil, X48, X49)) → U2_gg(T10, X48, X49, s2t4_in_gg(T10, X49))
s2t4_in_gg(s(T12), node(X62, X63, nil)) → U3_gg(T12, X62, X63, s2t4_in_gg(T12, X62))
s2t4_in_gg(T13, node(nil, X70, nil)) → s2t4_out_gg(T13, node(nil, X70, nil))
U3_gg(T12, X62, X63, s2t4_out_gg(T12, X62)) → s2t4_out_gg(s(T12), node(X62, X63, nil))
U2_gg(T10, X48, X49, s2t4_out_gg(T10, X49)) → s2t4_out_gg(s(T10), node(nil, X48, X49))
U1_gg(T8, X34, X35, s2t4_out_gg(T8, X34)) → s2t4_out_gg(s(T8), node(X34, X35, X34))
U11_g(T5, s2t4_out_gg(T5, nil)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U12_g(T5, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(nil, X79, X79) → tappend5_out_gaa(nil, X79, X79)
tappend5_in_gaa(nil, X92, node(X92, nil, nil)) → tappend5_out_gaa(nil, X92, node(X92, nil, nil))
tappend5_in_gaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_gaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_gaa(nil, X149, node(nil, nil, X149)) → tappend5_out_gaa(nil, X149, node(nil, nil, X149))
tappend5_in_gaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_gaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_gaa(nil, X210, node(X211, nil, nil)) → U4_gaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_gaa(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
tappend5_in_gaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_gaa(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
tappend5_in_gaa(nil, nil, node(nil, nil, X268)) → U7_gaa(X268, tappend5_in_gaa(nil, X265, X268))
tappend5_in_gaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_gaa(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
tappend5_in_gaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_gaa(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U9_gaa(T130, T131, T133, X268, tappend5_out_gaa(T133, X265, X268)) → tappend5_out_gaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_gaa(T114, T115, T117, X268, tappend5_out_gaa(T117, X265, X268)) → tappend5_out_gaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U7_gaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_gaa(nil, nil, node(nil, nil, X268))
U6_gaa(T99, T97, T98, X210, X211, tappend5_out_gaa(T99, X210, X211)) → tappend5_out_gaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_gaa(T83, T81, T82, X210, X211, tappend5_out_gaa(T83, X210, X211)) → tappend5_out_gaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U4_gaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_gaa(nil, X210, node(X211, nil, nil))
U12_g(T5, tappend5_out_gaa(nil, X210, X211)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U13_g(T5, tappend5_in_gaa(nil, X265, X268))
U13_g(T5, tappend5_out_gaa(nil, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U14_g(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
U14_g(T5, s2t4_out_ga(T5, node(nil, T25, T26))) → goal1_out_g(T5)
goal1_in_g(T5) → U15_g(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
U15_g(T5, s2t4_out_ga(T5, node(nil, T36, T37))) → goal1_out_g(T5)
goal1_in_g(T5) → U16_g(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
U16_g(T5, s2t4_out_ga(T5, node(T51, T52, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U17_g(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
U17_g(T5, s2t4_out_ga(T5, node(T64, T65, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U18_g(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
U18_g(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_g(T5, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(nil, X79, X79) → tappend5_out_aaa(nil, X79, X79)
tappend5_in_aaa(nil, X92, node(X92, nil, nil)) → tappend5_out_aaa(nil, X92, node(X92, nil, nil))
tappend5_in_aaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_aaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_aaa(nil, X149, node(nil, nil, X149)) → tappend5_out_aaa(nil, X149, node(nil, nil, X149))
tappend5_in_aaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_aaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_aaa(nil, X210, node(X211, nil, nil)) → U4_aaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
U4_aaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_aaa(nil, X210, node(X211, nil, nil))
tappend5_in_aaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_aaa(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_aaa(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
tappend5_in_aaa(nil, nil, node(nil, nil, X268)) → U7_aaa(X268, tappend5_in_gaa(nil, X265, X268))
U7_aaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_aaa(nil, nil, node(nil, nil, X268))
tappend5_in_aaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_aaa(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
tappend5_in_aaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_aaa(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
U9_aaa(T130, T131, T133, X268, tappend5_out_aaa(T133, X265, X268)) → tappend5_out_aaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_aaa(T114, T115, T117, X268, tappend5_out_aaa(T117, X265, X268)) → tappend5_out_aaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U6_aaa(T99, T97, T98, X210, X211, tappend5_out_aaa(T99, X210, X211)) → tappend5_out_aaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_aaa(T83, T81, T82, X210, X211, tappend5_out_aaa(T83, X210, X211)) → tappend5_out_aaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U19_g(T5, tappend5_out_aaa(T83, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U20_g(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_g(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_g(T5, tappend5_in_aaa(T99, X210, X211))
U21_g(T5, tappend5_out_aaa(T99, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U22_g(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_g(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_g(T5, tappend5_in_aaa(T117, X265, X268))
U23_g(T5, tappend5_out_aaa(T117, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U24_g(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_g(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_g(T5, tappend5_in_aaa(T133, X265, X268))
U25_g(T5, tappend5_out_aaa(T133, X265, X268)) → goal1_out_g(T5)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U10_g(x1, x2)  =  U10_g(x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
goal1_out_g(x1)  =  goal1_out_g
U11_g(x1, x2)  =  U11_g(x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg
node(x1, x2, x3)  =  node(x1, x2, x3)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U12_g(x1, x2)  =  U12_g(x2)
tappend5_in_gaa(x1, x2, x3)  =  tappend5_in_gaa(x1)
tappend5_out_gaa(x1, x2, x3)  =  tappend5_out_gaa
U4_gaa(x1, x2, x3)  =  U4_gaa(x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x6)
U7_gaa(x1, x2)  =  U7_gaa(x2)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
U13_g(x1, x2)  =  U13_g(x2)
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
tappend5_in_aaa(x1, x2, x3)  =  tappend5_in_aaa
tappend5_out_aaa(x1, x2, x3)  =  tappend5_out_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
U5_aaa(x1, x2, x3, x4, x5, x6)  =  U5_aaa(x6)
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
U7_aaa(x1, x2)  =  U7_aaa(x2)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
U23_g(x1, x2)  =  U23_g(x2)
U24_g(x1, x2)  =  U24_g(x2)
U25_g(x1, x2)  =  U25_g(x2)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U10_G(x1, x2)  =  U10_G(x2)
S2T4_IN_GA(x1, x2)  =  S2T4_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U11_G(x1, x2)  =  U11_G(x2)
S2T4_IN_GG(x1, x2)  =  S2T4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x4)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x4)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x4)
U12_G(x1, x2)  =  U12_G(x2)
TAPPEND5_IN_GAA(x1, x2, x3)  =  TAPPEND5_IN_GAA(x1)
U4_GAA(x1, x2, x3)  =  U4_GAA(x3)
U5_GAA(x1, x2, x3, x4, x5, x6)  =  U5_GAA(x6)
U6_GAA(x1, x2, x3, x4, x5, x6)  =  U6_GAA(x6)
U7_GAA(x1, x2)  =  U7_GAA(x2)
U8_GAA(x1, x2, x3, x4, x5)  =  U8_GAA(x5)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x5)
U13_G(x1, x2)  =  U13_G(x2)
U14_G(x1, x2)  =  U14_G(x2)
U15_G(x1, x2)  =  U15_G(x2)
U16_G(x1, x2)  =  U16_G(x2)
U17_G(x1, x2)  =  U17_G(x2)
U18_G(x1, x2)  =  U18_G(x2)
U19_G(x1, x2)  =  U19_G(x2)
TAPPEND5_IN_AAA(x1, x2, x3)  =  TAPPEND5_IN_AAA
U4_AAA(x1, x2, x3)  =  U4_AAA(x3)
U5_AAA(x1, x2, x3, x4, x5, x6)  =  U5_AAA(x6)
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(x6)
U7_AAA(x1, x2)  =  U7_AAA(x2)
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
U20_G(x1, x2)  =  U20_G(x2)
U21_G(x1, x2)  =  U21_G(x2)
U22_G(x1, x2)  =  U22_G(x2)
U23_G(x1, x2)  =  U23_G(x2)
U24_G(x1, x2)  =  U24_G(x2)
U25_G(x1, x2)  =  U25_G(x2)

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

(6) Obligation:

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

GOAL1_IN_G(T5) → U10_G(T5, s2t4_in_ga(T5, X15))
GOAL1_IN_G(T5) → S2T4_IN_GA(T5, X15)
S2T4_IN_GA(s(T8), node(X34, X35, X34)) → U1_GA(T8, X34, X35, s2t4_in_ga(T8, X34))
S2T4_IN_GA(s(T8), node(X34, X35, X34)) → S2T4_IN_GA(T8, X34)
S2T4_IN_GA(s(T10), node(nil, X48, X49)) → U2_GA(T10, X48, X49, s2t4_in_ga(T10, X49))
S2T4_IN_GA(s(T10), node(nil, X48, X49)) → S2T4_IN_GA(T10, X49)
S2T4_IN_GA(s(T12), node(X62, X63, nil)) → U3_GA(T12, X62, X63, s2t4_in_ga(T12, X62))
S2T4_IN_GA(s(T12), node(X62, X63, nil)) → S2T4_IN_GA(T12, X62)
GOAL1_IN_G(T5) → U11_G(T5, s2t4_in_gg(T5, nil))
GOAL1_IN_G(T5) → S2T4_IN_GG(T5, nil)
S2T4_IN_GG(s(T8), node(X34, X35, X34)) → U1_GG(T8, X34, X35, s2t4_in_gg(T8, X34))
S2T4_IN_GG(s(T8), node(X34, X35, X34)) → S2T4_IN_GG(T8, X34)
S2T4_IN_GG(s(T10), node(nil, X48, X49)) → U2_GG(T10, X48, X49, s2t4_in_gg(T10, X49))
S2T4_IN_GG(s(T10), node(nil, X48, X49)) → S2T4_IN_GG(T10, X49)
S2T4_IN_GG(s(T12), node(X62, X63, nil)) → U3_GG(T12, X62, X63, s2t4_in_gg(T12, X62))
S2T4_IN_GG(s(T12), node(X62, X63, nil)) → S2T4_IN_GG(T12, X62)
U11_G(T5, s2t4_out_gg(T5, nil)) → U12_G(T5, tappend5_in_gaa(nil, X210, X211))
U11_G(T5, s2t4_out_gg(T5, nil)) → TAPPEND5_IN_GAA(nil, X210, X211)
TAPPEND5_IN_GAA(nil, X210, node(X211, nil, nil)) → U4_GAA(X210, X211, tappend5_in_gaa(nil, X210, X211))
TAPPEND5_IN_GAA(nil, X210, node(X211, nil, nil)) → TAPPEND5_IN_GAA(nil, X210, X211)
TAPPEND5_IN_GAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_GAA(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
TAPPEND5_IN_GAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → TAPPEND5_IN_GAA(T83, X210, X211)
TAPPEND5_IN_GAA(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_GAA(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
TAPPEND5_IN_GAA(nil, nil, node(nil, nil, X268)) → U7_GAA(X268, tappend5_in_gaa(nil, X265, X268))
TAPPEND5_IN_GAA(nil, nil, node(nil, nil, X268)) → TAPPEND5_IN_GAA(nil, X265, X268)
TAPPEND5_IN_GAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_GAA(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
TAPPEND5_IN_GAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → TAPPEND5_IN_GAA(T117, X265, X268)
TAPPEND5_IN_GAA(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_GAA(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U11_G(T5, s2t4_out_gg(T5, nil)) → U13_G(T5, tappend5_in_gaa(nil, X265, X268))
GOAL1_IN_G(T5) → U14_G(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
GOAL1_IN_G(T5) → S2T4_IN_GA(T5, node(nil, T25, T26))
GOAL1_IN_G(T5) → U15_G(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
GOAL1_IN_G(T5) → U16_G(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
GOAL1_IN_G(T5) → S2T4_IN_GA(T5, node(T51, T52, nil))
GOAL1_IN_G(T5) → U17_G(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
GOAL1_IN_G(T5) → U18_G(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
GOAL1_IN_G(T5) → S2T4_IN_GA(T5, node(T83, T81, T82))
U18_G(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_G(T5, tappend5_in_aaa(T83, X210, X211))
U18_G(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → TAPPEND5_IN_AAA(T83, X210, X211)
TAPPEND5_IN_AAA(nil, X210, node(X211, nil, nil)) → U4_AAA(X210, X211, tappend5_in_gaa(nil, X210, X211))
TAPPEND5_IN_AAA(nil, X210, node(X211, nil, nil)) → TAPPEND5_IN_GAA(nil, X210, X211)
TAPPEND5_IN_AAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_AAA(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
TAPPEND5_IN_AAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → TAPPEND5_IN_AAA(T83, X210, X211)
TAPPEND5_IN_AAA(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_AAA(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
TAPPEND5_IN_AAA(nil, nil, node(nil, nil, X268)) → U7_AAA(X268, tappend5_in_gaa(nil, X265, X268))
TAPPEND5_IN_AAA(nil, nil, node(nil, nil, X268)) → TAPPEND5_IN_GAA(nil, X265, X268)
TAPPEND5_IN_AAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_AAA(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
TAPPEND5_IN_AAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → TAPPEND5_IN_AAA(T117, X265, X268)
TAPPEND5_IN_AAA(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_AAA(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
GOAL1_IN_G(T5) → U20_G(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_G(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_G(T5, tappend5_in_aaa(T99, X210, X211))
U20_G(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → TAPPEND5_IN_AAA(T99, X210, X211)
GOAL1_IN_G(T5) → U22_G(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_G(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_G(T5, tappend5_in_aaa(T117, X265, X268))
U22_G(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → TAPPEND5_IN_AAA(T117, X265, X268)
GOAL1_IN_G(T5) → U24_G(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_G(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_G(T5, tappend5_in_aaa(T133, X265, X268))
U24_G(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → TAPPEND5_IN_AAA(T133, X265, X268)

The TRS R consists of the following rules:

goal1_in_g(T5) → U10_g(T5, s2t4_in_ga(T5, X15))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T8), node(X34, X35, X34)) → U1_ga(T8, X34, X35, s2t4_in_ga(T8, X34))
s2t4_in_ga(s(T10), node(nil, X48, X49)) → U2_ga(T10, X48, X49, s2t4_in_ga(T10, X49))
s2t4_in_ga(s(T12), node(X62, X63, nil)) → U3_ga(T12, X62, X63, s2t4_in_ga(T12, X62))
s2t4_in_ga(T13, node(nil, X70, nil)) → s2t4_out_ga(T13, node(nil, X70, nil))
U3_ga(T12, X62, X63, s2t4_out_ga(T12, X62)) → s2t4_out_ga(s(T12), node(X62, X63, nil))
U2_ga(T10, X48, X49, s2t4_out_ga(T10, X49)) → s2t4_out_ga(s(T10), node(nil, X48, X49))
U1_ga(T8, X34, X35, s2t4_out_ga(T8, X34)) → s2t4_out_ga(s(T8), node(X34, X35, X34))
U10_g(T5, s2t4_out_ga(T5, X15)) → goal1_out_g(T5)
goal1_in_g(T5) → U11_g(T5, s2t4_in_gg(T5, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T8), node(X34, X35, X34)) → U1_gg(T8, X34, X35, s2t4_in_gg(T8, X34))
s2t4_in_gg(s(T10), node(nil, X48, X49)) → U2_gg(T10, X48, X49, s2t4_in_gg(T10, X49))
s2t4_in_gg(s(T12), node(X62, X63, nil)) → U3_gg(T12, X62, X63, s2t4_in_gg(T12, X62))
s2t4_in_gg(T13, node(nil, X70, nil)) → s2t4_out_gg(T13, node(nil, X70, nil))
U3_gg(T12, X62, X63, s2t4_out_gg(T12, X62)) → s2t4_out_gg(s(T12), node(X62, X63, nil))
U2_gg(T10, X48, X49, s2t4_out_gg(T10, X49)) → s2t4_out_gg(s(T10), node(nil, X48, X49))
U1_gg(T8, X34, X35, s2t4_out_gg(T8, X34)) → s2t4_out_gg(s(T8), node(X34, X35, X34))
U11_g(T5, s2t4_out_gg(T5, nil)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U12_g(T5, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(nil, X79, X79) → tappend5_out_gaa(nil, X79, X79)
tappend5_in_gaa(nil, X92, node(X92, nil, nil)) → tappend5_out_gaa(nil, X92, node(X92, nil, nil))
tappend5_in_gaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_gaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_gaa(nil, X149, node(nil, nil, X149)) → tappend5_out_gaa(nil, X149, node(nil, nil, X149))
tappend5_in_gaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_gaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_gaa(nil, X210, node(X211, nil, nil)) → U4_gaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_gaa(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
tappend5_in_gaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_gaa(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
tappend5_in_gaa(nil, nil, node(nil, nil, X268)) → U7_gaa(X268, tappend5_in_gaa(nil, X265, X268))
tappend5_in_gaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_gaa(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
tappend5_in_gaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_gaa(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U9_gaa(T130, T131, T133, X268, tappend5_out_gaa(T133, X265, X268)) → tappend5_out_gaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_gaa(T114, T115, T117, X268, tappend5_out_gaa(T117, X265, X268)) → tappend5_out_gaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U7_gaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_gaa(nil, nil, node(nil, nil, X268))
U6_gaa(T99, T97, T98, X210, X211, tappend5_out_gaa(T99, X210, X211)) → tappend5_out_gaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_gaa(T83, T81, T82, X210, X211, tappend5_out_gaa(T83, X210, X211)) → tappend5_out_gaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U4_gaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_gaa(nil, X210, node(X211, nil, nil))
U12_g(T5, tappend5_out_gaa(nil, X210, X211)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U13_g(T5, tappend5_in_gaa(nil, X265, X268))
U13_g(T5, tappend5_out_gaa(nil, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U14_g(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
U14_g(T5, s2t4_out_ga(T5, node(nil, T25, T26))) → goal1_out_g(T5)
goal1_in_g(T5) → U15_g(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
U15_g(T5, s2t4_out_ga(T5, node(nil, T36, T37))) → goal1_out_g(T5)
goal1_in_g(T5) → U16_g(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
U16_g(T5, s2t4_out_ga(T5, node(T51, T52, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U17_g(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
U17_g(T5, s2t4_out_ga(T5, node(T64, T65, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U18_g(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
U18_g(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_g(T5, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(nil, X79, X79) → tappend5_out_aaa(nil, X79, X79)
tappend5_in_aaa(nil, X92, node(X92, nil, nil)) → tappend5_out_aaa(nil, X92, node(X92, nil, nil))
tappend5_in_aaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_aaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_aaa(nil, X149, node(nil, nil, X149)) → tappend5_out_aaa(nil, X149, node(nil, nil, X149))
tappend5_in_aaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_aaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_aaa(nil, X210, node(X211, nil, nil)) → U4_aaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
U4_aaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_aaa(nil, X210, node(X211, nil, nil))
tappend5_in_aaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_aaa(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_aaa(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
tappend5_in_aaa(nil, nil, node(nil, nil, X268)) → U7_aaa(X268, tappend5_in_gaa(nil, X265, X268))
U7_aaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_aaa(nil, nil, node(nil, nil, X268))
tappend5_in_aaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_aaa(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
tappend5_in_aaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_aaa(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
U9_aaa(T130, T131, T133, X268, tappend5_out_aaa(T133, X265, X268)) → tappend5_out_aaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_aaa(T114, T115, T117, X268, tappend5_out_aaa(T117, X265, X268)) → tappend5_out_aaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U6_aaa(T99, T97, T98, X210, X211, tappend5_out_aaa(T99, X210, X211)) → tappend5_out_aaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_aaa(T83, T81, T82, X210, X211, tappend5_out_aaa(T83, X210, X211)) → tappend5_out_aaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U19_g(T5, tappend5_out_aaa(T83, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U20_g(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_g(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_g(T5, tappend5_in_aaa(T99, X210, X211))
U21_g(T5, tappend5_out_aaa(T99, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U22_g(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_g(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_g(T5, tappend5_in_aaa(T117, X265, X268))
U23_g(T5, tappend5_out_aaa(T117, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U24_g(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_g(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_g(T5, tappend5_in_aaa(T133, X265, X268))
U25_g(T5, tappend5_out_aaa(T133, X265, X268)) → goal1_out_g(T5)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U10_g(x1, x2)  =  U10_g(x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
goal1_out_g(x1)  =  goal1_out_g
U11_g(x1, x2)  =  U11_g(x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg
node(x1, x2, x3)  =  node(x1, x2, x3)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U12_g(x1, x2)  =  U12_g(x2)
tappend5_in_gaa(x1, x2, x3)  =  tappend5_in_gaa(x1)
tappend5_out_gaa(x1, x2, x3)  =  tappend5_out_gaa
U4_gaa(x1, x2, x3)  =  U4_gaa(x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x6)
U7_gaa(x1, x2)  =  U7_gaa(x2)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
U13_g(x1, x2)  =  U13_g(x2)
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
tappend5_in_aaa(x1, x2, x3)  =  tappend5_in_aaa
tappend5_out_aaa(x1, x2, x3)  =  tappend5_out_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
U5_aaa(x1, x2, x3, x4, x5, x6)  =  U5_aaa(x6)
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
U7_aaa(x1, x2)  =  U7_aaa(x2)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
U23_g(x1, x2)  =  U23_g(x2)
U24_g(x1, x2)  =  U24_g(x2)
U25_g(x1, x2)  =  U25_g(x2)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U10_G(x1, x2)  =  U10_G(x2)
S2T4_IN_GA(x1, x2)  =  S2T4_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U11_G(x1, x2)  =  U11_G(x2)
S2T4_IN_GG(x1, x2)  =  S2T4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x4)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x4)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x4)
U12_G(x1, x2)  =  U12_G(x2)
TAPPEND5_IN_GAA(x1, x2, x3)  =  TAPPEND5_IN_GAA(x1)
U4_GAA(x1, x2, x3)  =  U4_GAA(x3)
U5_GAA(x1, x2, x3, x4, x5, x6)  =  U5_GAA(x6)
U6_GAA(x1, x2, x3, x4, x5, x6)  =  U6_GAA(x6)
U7_GAA(x1, x2)  =  U7_GAA(x2)
U8_GAA(x1, x2, x3, x4, x5)  =  U8_GAA(x5)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x5)
U13_G(x1, x2)  =  U13_G(x2)
U14_G(x1, x2)  =  U14_G(x2)
U15_G(x1, x2)  =  U15_G(x2)
U16_G(x1, x2)  =  U16_G(x2)
U17_G(x1, x2)  =  U17_G(x2)
U18_G(x1, x2)  =  U18_G(x2)
U19_G(x1, x2)  =  U19_G(x2)
TAPPEND5_IN_AAA(x1, x2, x3)  =  TAPPEND5_IN_AAA
U4_AAA(x1, x2, x3)  =  U4_AAA(x3)
U5_AAA(x1, x2, x3, x4, x5, x6)  =  U5_AAA(x6)
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(x6)
U7_AAA(x1, x2)  =  U7_AAA(x2)
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
U20_G(x1, x2)  =  U20_G(x2)
U21_G(x1, x2)  =  U21_G(x2)
U22_G(x1, x2)  =  U22_G(x2)
U23_G(x1, x2)  =  U23_G(x2)
U24_G(x1, x2)  =  U24_G(x2)
U25_G(x1, x2)  =  U25_G(x2)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 46 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

TAPPEND5_IN_GAA(nil, nil, node(nil, nil, X268)) → TAPPEND5_IN_GAA(nil, X265, X268)
TAPPEND5_IN_GAA(nil, X210, node(X211, nil, nil)) → TAPPEND5_IN_GAA(nil, X210, X211)

The TRS R consists of the following rules:

goal1_in_g(T5) → U10_g(T5, s2t4_in_ga(T5, X15))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T8), node(X34, X35, X34)) → U1_ga(T8, X34, X35, s2t4_in_ga(T8, X34))
s2t4_in_ga(s(T10), node(nil, X48, X49)) → U2_ga(T10, X48, X49, s2t4_in_ga(T10, X49))
s2t4_in_ga(s(T12), node(X62, X63, nil)) → U3_ga(T12, X62, X63, s2t4_in_ga(T12, X62))
s2t4_in_ga(T13, node(nil, X70, nil)) → s2t4_out_ga(T13, node(nil, X70, nil))
U3_ga(T12, X62, X63, s2t4_out_ga(T12, X62)) → s2t4_out_ga(s(T12), node(X62, X63, nil))
U2_ga(T10, X48, X49, s2t4_out_ga(T10, X49)) → s2t4_out_ga(s(T10), node(nil, X48, X49))
U1_ga(T8, X34, X35, s2t4_out_ga(T8, X34)) → s2t4_out_ga(s(T8), node(X34, X35, X34))
U10_g(T5, s2t4_out_ga(T5, X15)) → goal1_out_g(T5)
goal1_in_g(T5) → U11_g(T5, s2t4_in_gg(T5, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T8), node(X34, X35, X34)) → U1_gg(T8, X34, X35, s2t4_in_gg(T8, X34))
s2t4_in_gg(s(T10), node(nil, X48, X49)) → U2_gg(T10, X48, X49, s2t4_in_gg(T10, X49))
s2t4_in_gg(s(T12), node(X62, X63, nil)) → U3_gg(T12, X62, X63, s2t4_in_gg(T12, X62))
s2t4_in_gg(T13, node(nil, X70, nil)) → s2t4_out_gg(T13, node(nil, X70, nil))
U3_gg(T12, X62, X63, s2t4_out_gg(T12, X62)) → s2t4_out_gg(s(T12), node(X62, X63, nil))
U2_gg(T10, X48, X49, s2t4_out_gg(T10, X49)) → s2t4_out_gg(s(T10), node(nil, X48, X49))
U1_gg(T8, X34, X35, s2t4_out_gg(T8, X34)) → s2t4_out_gg(s(T8), node(X34, X35, X34))
U11_g(T5, s2t4_out_gg(T5, nil)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U12_g(T5, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(nil, X79, X79) → tappend5_out_gaa(nil, X79, X79)
tappend5_in_gaa(nil, X92, node(X92, nil, nil)) → tappend5_out_gaa(nil, X92, node(X92, nil, nil))
tappend5_in_gaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_gaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_gaa(nil, X149, node(nil, nil, X149)) → tappend5_out_gaa(nil, X149, node(nil, nil, X149))
tappend5_in_gaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_gaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_gaa(nil, X210, node(X211, nil, nil)) → U4_gaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_gaa(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
tappend5_in_gaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_gaa(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
tappend5_in_gaa(nil, nil, node(nil, nil, X268)) → U7_gaa(X268, tappend5_in_gaa(nil, X265, X268))
tappend5_in_gaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_gaa(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
tappend5_in_gaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_gaa(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U9_gaa(T130, T131, T133, X268, tappend5_out_gaa(T133, X265, X268)) → tappend5_out_gaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_gaa(T114, T115, T117, X268, tappend5_out_gaa(T117, X265, X268)) → tappend5_out_gaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U7_gaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_gaa(nil, nil, node(nil, nil, X268))
U6_gaa(T99, T97, T98, X210, X211, tappend5_out_gaa(T99, X210, X211)) → tappend5_out_gaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_gaa(T83, T81, T82, X210, X211, tappend5_out_gaa(T83, X210, X211)) → tappend5_out_gaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U4_gaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_gaa(nil, X210, node(X211, nil, nil))
U12_g(T5, tappend5_out_gaa(nil, X210, X211)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U13_g(T5, tappend5_in_gaa(nil, X265, X268))
U13_g(T5, tappend5_out_gaa(nil, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U14_g(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
U14_g(T5, s2t4_out_ga(T5, node(nil, T25, T26))) → goal1_out_g(T5)
goal1_in_g(T5) → U15_g(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
U15_g(T5, s2t4_out_ga(T5, node(nil, T36, T37))) → goal1_out_g(T5)
goal1_in_g(T5) → U16_g(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
U16_g(T5, s2t4_out_ga(T5, node(T51, T52, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U17_g(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
U17_g(T5, s2t4_out_ga(T5, node(T64, T65, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U18_g(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
U18_g(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_g(T5, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(nil, X79, X79) → tappend5_out_aaa(nil, X79, X79)
tappend5_in_aaa(nil, X92, node(X92, nil, nil)) → tappend5_out_aaa(nil, X92, node(X92, nil, nil))
tappend5_in_aaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_aaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_aaa(nil, X149, node(nil, nil, X149)) → tappend5_out_aaa(nil, X149, node(nil, nil, X149))
tappend5_in_aaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_aaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_aaa(nil, X210, node(X211, nil, nil)) → U4_aaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
U4_aaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_aaa(nil, X210, node(X211, nil, nil))
tappend5_in_aaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_aaa(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_aaa(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
tappend5_in_aaa(nil, nil, node(nil, nil, X268)) → U7_aaa(X268, tappend5_in_gaa(nil, X265, X268))
U7_aaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_aaa(nil, nil, node(nil, nil, X268))
tappend5_in_aaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_aaa(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
tappend5_in_aaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_aaa(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
U9_aaa(T130, T131, T133, X268, tappend5_out_aaa(T133, X265, X268)) → tappend5_out_aaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_aaa(T114, T115, T117, X268, tappend5_out_aaa(T117, X265, X268)) → tappend5_out_aaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U6_aaa(T99, T97, T98, X210, X211, tappend5_out_aaa(T99, X210, X211)) → tappend5_out_aaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_aaa(T83, T81, T82, X210, X211, tappend5_out_aaa(T83, X210, X211)) → tappend5_out_aaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U19_g(T5, tappend5_out_aaa(T83, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U20_g(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_g(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_g(T5, tappend5_in_aaa(T99, X210, X211))
U21_g(T5, tappend5_out_aaa(T99, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U22_g(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_g(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_g(T5, tappend5_in_aaa(T117, X265, X268))
U23_g(T5, tappend5_out_aaa(T117, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U24_g(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_g(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_g(T5, tappend5_in_aaa(T133, X265, X268))
U25_g(T5, tappend5_out_aaa(T133, X265, X268)) → goal1_out_g(T5)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U10_g(x1, x2)  =  U10_g(x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
goal1_out_g(x1)  =  goal1_out_g
U11_g(x1, x2)  =  U11_g(x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg
node(x1, x2, x3)  =  node(x1, x2, x3)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U12_g(x1, x2)  =  U12_g(x2)
tappend5_in_gaa(x1, x2, x3)  =  tappend5_in_gaa(x1)
tappend5_out_gaa(x1, x2, x3)  =  tappend5_out_gaa
U4_gaa(x1, x2, x3)  =  U4_gaa(x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x6)
U7_gaa(x1, x2)  =  U7_gaa(x2)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
U13_g(x1, x2)  =  U13_g(x2)
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
tappend5_in_aaa(x1, x2, x3)  =  tappend5_in_aaa
tappend5_out_aaa(x1, x2, x3)  =  tappend5_out_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
U5_aaa(x1, x2, x3, x4, x5, x6)  =  U5_aaa(x6)
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
U7_aaa(x1, x2)  =  U7_aaa(x2)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
U23_g(x1, x2)  =  U23_g(x2)
U24_g(x1, x2)  =  U24_g(x2)
U25_g(x1, x2)  =  U25_g(x2)
TAPPEND5_IN_GAA(x1, x2, x3)  =  TAPPEND5_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:

TAPPEND5_IN_GAA(nil, nil, node(nil, nil, X268)) → TAPPEND5_IN_GAA(nil, X265, X268)
TAPPEND5_IN_GAA(nil, X210, node(X211, nil, nil)) → TAPPEND5_IN_GAA(nil, X210, X211)

R is empty.
The argument filtering Pi contains the following mapping:
nil  =  nil
node(x1, x2, x3)  =  node(x1, x2, x3)
TAPPEND5_IN_GAA(x1, x2, x3)  =  TAPPEND5_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:

TAPPEND5_IN_GAA(nil) → TAPPEND5_IN_GAA(nil)

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

(14) 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 = TAPPEND5_IN_GAA(nil) evaluates to t =TAPPEND5_IN_GAA(nil)

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 TAPPEND5_IN_GAA(nil) to TAPPEND5_IN_GAA(nil).



(15) FALSE

(16) Obligation:

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

TAPPEND5_IN_AAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → TAPPEND5_IN_AAA(T117, X265, X268)
TAPPEND5_IN_AAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → TAPPEND5_IN_AAA(T83, X210, X211)

The TRS R consists of the following rules:

goal1_in_g(T5) → U10_g(T5, s2t4_in_ga(T5, X15))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T8), node(X34, X35, X34)) → U1_ga(T8, X34, X35, s2t4_in_ga(T8, X34))
s2t4_in_ga(s(T10), node(nil, X48, X49)) → U2_ga(T10, X48, X49, s2t4_in_ga(T10, X49))
s2t4_in_ga(s(T12), node(X62, X63, nil)) → U3_ga(T12, X62, X63, s2t4_in_ga(T12, X62))
s2t4_in_ga(T13, node(nil, X70, nil)) → s2t4_out_ga(T13, node(nil, X70, nil))
U3_ga(T12, X62, X63, s2t4_out_ga(T12, X62)) → s2t4_out_ga(s(T12), node(X62, X63, nil))
U2_ga(T10, X48, X49, s2t4_out_ga(T10, X49)) → s2t4_out_ga(s(T10), node(nil, X48, X49))
U1_ga(T8, X34, X35, s2t4_out_ga(T8, X34)) → s2t4_out_ga(s(T8), node(X34, X35, X34))
U10_g(T5, s2t4_out_ga(T5, X15)) → goal1_out_g(T5)
goal1_in_g(T5) → U11_g(T5, s2t4_in_gg(T5, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T8), node(X34, X35, X34)) → U1_gg(T8, X34, X35, s2t4_in_gg(T8, X34))
s2t4_in_gg(s(T10), node(nil, X48, X49)) → U2_gg(T10, X48, X49, s2t4_in_gg(T10, X49))
s2t4_in_gg(s(T12), node(X62, X63, nil)) → U3_gg(T12, X62, X63, s2t4_in_gg(T12, X62))
s2t4_in_gg(T13, node(nil, X70, nil)) → s2t4_out_gg(T13, node(nil, X70, nil))
U3_gg(T12, X62, X63, s2t4_out_gg(T12, X62)) → s2t4_out_gg(s(T12), node(X62, X63, nil))
U2_gg(T10, X48, X49, s2t4_out_gg(T10, X49)) → s2t4_out_gg(s(T10), node(nil, X48, X49))
U1_gg(T8, X34, X35, s2t4_out_gg(T8, X34)) → s2t4_out_gg(s(T8), node(X34, X35, X34))
U11_g(T5, s2t4_out_gg(T5, nil)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U12_g(T5, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(nil, X79, X79) → tappend5_out_gaa(nil, X79, X79)
tappend5_in_gaa(nil, X92, node(X92, nil, nil)) → tappend5_out_gaa(nil, X92, node(X92, nil, nil))
tappend5_in_gaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_gaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_gaa(nil, X149, node(nil, nil, X149)) → tappend5_out_gaa(nil, X149, node(nil, nil, X149))
tappend5_in_gaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_gaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_gaa(nil, X210, node(X211, nil, nil)) → U4_gaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_gaa(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
tappend5_in_gaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_gaa(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
tappend5_in_gaa(nil, nil, node(nil, nil, X268)) → U7_gaa(X268, tappend5_in_gaa(nil, X265, X268))
tappend5_in_gaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_gaa(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
tappend5_in_gaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_gaa(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U9_gaa(T130, T131, T133, X268, tappend5_out_gaa(T133, X265, X268)) → tappend5_out_gaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_gaa(T114, T115, T117, X268, tappend5_out_gaa(T117, X265, X268)) → tappend5_out_gaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U7_gaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_gaa(nil, nil, node(nil, nil, X268))
U6_gaa(T99, T97, T98, X210, X211, tappend5_out_gaa(T99, X210, X211)) → tappend5_out_gaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_gaa(T83, T81, T82, X210, X211, tappend5_out_gaa(T83, X210, X211)) → tappend5_out_gaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U4_gaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_gaa(nil, X210, node(X211, nil, nil))
U12_g(T5, tappend5_out_gaa(nil, X210, X211)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U13_g(T5, tappend5_in_gaa(nil, X265, X268))
U13_g(T5, tappend5_out_gaa(nil, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U14_g(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
U14_g(T5, s2t4_out_ga(T5, node(nil, T25, T26))) → goal1_out_g(T5)
goal1_in_g(T5) → U15_g(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
U15_g(T5, s2t4_out_ga(T5, node(nil, T36, T37))) → goal1_out_g(T5)
goal1_in_g(T5) → U16_g(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
U16_g(T5, s2t4_out_ga(T5, node(T51, T52, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U17_g(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
U17_g(T5, s2t4_out_ga(T5, node(T64, T65, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U18_g(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
U18_g(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_g(T5, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(nil, X79, X79) → tappend5_out_aaa(nil, X79, X79)
tappend5_in_aaa(nil, X92, node(X92, nil, nil)) → tappend5_out_aaa(nil, X92, node(X92, nil, nil))
tappend5_in_aaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_aaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_aaa(nil, X149, node(nil, nil, X149)) → tappend5_out_aaa(nil, X149, node(nil, nil, X149))
tappend5_in_aaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_aaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_aaa(nil, X210, node(X211, nil, nil)) → U4_aaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
U4_aaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_aaa(nil, X210, node(X211, nil, nil))
tappend5_in_aaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_aaa(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_aaa(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
tappend5_in_aaa(nil, nil, node(nil, nil, X268)) → U7_aaa(X268, tappend5_in_gaa(nil, X265, X268))
U7_aaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_aaa(nil, nil, node(nil, nil, X268))
tappend5_in_aaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_aaa(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
tappend5_in_aaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_aaa(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
U9_aaa(T130, T131, T133, X268, tappend5_out_aaa(T133, X265, X268)) → tappend5_out_aaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_aaa(T114, T115, T117, X268, tappend5_out_aaa(T117, X265, X268)) → tappend5_out_aaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U6_aaa(T99, T97, T98, X210, X211, tappend5_out_aaa(T99, X210, X211)) → tappend5_out_aaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_aaa(T83, T81, T82, X210, X211, tappend5_out_aaa(T83, X210, X211)) → tappend5_out_aaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U19_g(T5, tappend5_out_aaa(T83, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U20_g(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_g(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_g(T5, tappend5_in_aaa(T99, X210, X211))
U21_g(T5, tappend5_out_aaa(T99, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U22_g(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_g(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_g(T5, tappend5_in_aaa(T117, X265, X268))
U23_g(T5, tappend5_out_aaa(T117, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U24_g(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_g(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_g(T5, tappend5_in_aaa(T133, X265, X268))
U25_g(T5, tappend5_out_aaa(T133, X265, X268)) → goal1_out_g(T5)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U10_g(x1, x2)  =  U10_g(x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
goal1_out_g(x1)  =  goal1_out_g
U11_g(x1, x2)  =  U11_g(x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg
node(x1, x2, x3)  =  node(x1, x2, x3)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U12_g(x1, x2)  =  U12_g(x2)
tappend5_in_gaa(x1, x2, x3)  =  tappend5_in_gaa(x1)
tappend5_out_gaa(x1, x2, x3)  =  tappend5_out_gaa
U4_gaa(x1, x2, x3)  =  U4_gaa(x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x6)
U7_gaa(x1, x2)  =  U7_gaa(x2)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
U13_g(x1, x2)  =  U13_g(x2)
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
tappend5_in_aaa(x1, x2, x3)  =  tappend5_in_aaa
tappend5_out_aaa(x1, x2, x3)  =  tappend5_out_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
U5_aaa(x1, x2, x3, x4, x5, x6)  =  U5_aaa(x6)
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
U7_aaa(x1, x2)  =  U7_aaa(x2)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
U23_g(x1, x2)  =  U23_g(x2)
U24_g(x1, x2)  =  U24_g(x2)
U25_g(x1, x2)  =  U25_g(x2)
TAPPEND5_IN_AAA(x1, x2, x3)  =  TAPPEND5_IN_AAA

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:

TAPPEND5_IN_AAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → TAPPEND5_IN_AAA(T117, X265, X268)
TAPPEND5_IN_AAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → TAPPEND5_IN_AAA(T83, X210, X211)

R is empty.
The argument filtering Pi contains the following mapping:
node(x1, x2, x3)  =  node(x1, x2, x3)
TAPPEND5_IN_AAA(x1, x2, x3)  =  TAPPEND5_IN_AAA

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:

TAPPEND5_IN_AAATAPPEND5_IN_AAA

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 = TAPPEND5_IN_AAA evaluates to t =TAPPEND5_IN_AAA

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 TAPPEND5_IN_AAA to TAPPEND5_IN_AAA.



(22) FALSE

(23) Obligation:

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

TAPPEND5_IN_GAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → TAPPEND5_IN_GAA(T117, X265, X268)
TAPPEND5_IN_GAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → TAPPEND5_IN_GAA(T83, X210, X211)

The TRS R consists of the following rules:

goal1_in_g(T5) → U10_g(T5, s2t4_in_ga(T5, X15))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T8), node(X34, X35, X34)) → U1_ga(T8, X34, X35, s2t4_in_ga(T8, X34))
s2t4_in_ga(s(T10), node(nil, X48, X49)) → U2_ga(T10, X48, X49, s2t4_in_ga(T10, X49))
s2t4_in_ga(s(T12), node(X62, X63, nil)) → U3_ga(T12, X62, X63, s2t4_in_ga(T12, X62))
s2t4_in_ga(T13, node(nil, X70, nil)) → s2t4_out_ga(T13, node(nil, X70, nil))
U3_ga(T12, X62, X63, s2t4_out_ga(T12, X62)) → s2t4_out_ga(s(T12), node(X62, X63, nil))
U2_ga(T10, X48, X49, s2t4_out_ga(T10, X49)) → s2t4_out_ga(s(T10), node(nil, X48, X49))
U1_ga(T8, X34, X35, s2t4_out_ga(T8, X34)) → s2t4_out_ga(s(T8), node(X34, X35, X34))
U10_g(T5, s2t4_out_ga(T5, X15)) → goal1_out_g(T5)
goal1_in_g(T5) → U11_g(T5, s2t4_in_gg(T5, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T8), node(X34, X35, X34)) → U1_gg(T8, X34, X35, s2t4_in_gg(T8, X34))
s2t4_in_gg(s(T10), node(nil, X48, X49)) → U2_gg(T10, X48, X49, s2t4_in_gg(T10, X49))
s2t4_in_gg(s(T12), node(X62, X63, nil)) → U3_gg(T12, X62, X63, s2t4_in_gg(T12, X62))
s2t4_in_gg(T13, node(nil, X70, nil)) → s2t4_out_gg(T13, node(nil, X70, nil))
U3_gg(T12, X62, X63, s2t4_out_gg(T12, X62)) → s2t4_out_gg(s(T12), node(X62, X63, nil))
U2_gg(T10, X48, X49, s2t4_out_gg(T10, X49)) → s2t4_out_gg(s(T10), node(nil, X48, X49))
U1_gg(T8, X34, X35, s2t4_out_gg(T8, X34)) → s2t4_out_gg(s(T8), node(X34, X35, X34))
U11_g(T5, s2t4_out_gg(T5, nil)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U12_g(T5, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(nil, X79, X79) → tappend5_out_gaa(nil, X79, X79)
tappend5_in_gaa(nil, X92, node(X92, nil, nil)) → tappend5_out_gaa(nil, X92, node(X92, nil, nil))
tappend5_in_gaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_gaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_gaa(nil, X149, node(nil, nil, X149)) → tappend5_out_gaa(nil, X149, node(nil, nil, X149))
tappend5_in_gaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_gaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_gaa(nil, X210, node(X211, nil, nil)) → U4_gaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_gaa(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
tappend5_in_gaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_gaa(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
tappend5_in_gaa(nil, nil, node(nil, nil, X268)) → U7_gaa(X268, tappend5_in_gaa(nil, X265, X268))
tappend5_in_gaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_gaa(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
tappend5_in_gaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_gaa(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U9_gaa(T130, T131, T133, X268, tappend5_out_gaa(T133, X265, X268)) → tappend5_out_gaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_gaa(T114, T115, T117, X268, tappend5_out_gaa(T117, X265, X268)) → tappend5_out_gaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U7_gaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_gaa(nil, nil, node(nil, nil, X268))
U6_gaa(T99, T97, T98, X210, X211, tappend5_out_gaa(T99, X210, X211)) → tappend5_out_gaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_gaa(T83, T81, T82, X210, X211, tappend5_out_gaa(T83, X210, X211)) → tappend5_out_gaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U4_gaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_gaa(nil, X210, node(X211, nil, nil))
U12_g(T5, tappend5_out_gaa(nil, X210, X211)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U13_g(T5, tappend5_in_gaa(nil, X265, X268))
U13_g(T5, tappend5_out_gaa(nil, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U14_g(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
U14_g(T5, s2t4_out_ga(T5, node(nil, T25, T26))) → goal1_out_g(T5)
goal1_in_g(T5) → U15_g(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
U15_g(T5, s2t4_out_ga(T5, node(nil, T36, T37))) → goal1_out_g(T5)
goal1_in_g(T5) → U16_g(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
U16_g(T5, s2t4_out_ga(T5, node(T51, T52, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U17_g(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
U17_g(T5, s2t4_out_ga(T5, node(T64, T65, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U18_g(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
U18_g(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_g(T5, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(nil, X79, X79) → tappend5_out_aaa(nil, X79, X79)
tappend5_in_aaa(nil, X92, node(X92, nil, nil)) → tappend5_out_aaa(nil, X92, node(X92, nil, nil))
tappend5_in_aaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_aaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_aaa(nil, X149, node(nil, nil, X149)) → tappend5_out_aaa(nil, X149, node(nil, nil, X149))
tappend5_in_aaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_aaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_aaa(nil, X210, node(X211, nil, nil)) → U4_aaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
U4_aaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_aaa(nil, X210, node(X211, nil, nil))
tappend5_in_aaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_aaa(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_aaa(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
tappend5_in_aaa(nil, nil, node(nil, nil, X268)) → U7_aaa(X268, tappend5_in_gaa(nil, X265, X268))
U7_aaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_aaa(nil, nil, node(nil, nil, X268))
tappend5_in_aaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_aaa(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
tappend5_in_aaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_aaa(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
U9_aaa(T130, T131, T133, X268, tappend5_out_aaa(T133, X265, X268)) → tappend5_out_aaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_aaa(T114, T115, T117, X268, tappend5_out_aaa(T117, X265, X268)) → tappend5_out_aaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U6_aaa(T99, T97, T98, X210, X211, tappend5_out_aaa(T99, X210, X211)) → tappend5_out_aaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_aaa(T83, T81, T82, X210, X211, tappend5_out_aaa(T83, X210, X211)) → tappend5_out_aaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U19_g(T5, tappend5_out_aaa(T83, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U20_g(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_g(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_g(T5, tappend5_in_aaa(T99, X210, X211))
U21_g(T5, tappend5_out_aaa(T99, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U22_g(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_g(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_g(T5, tappend5_in_aaa(T117, X265, X268))
U23_g(T5, tappend5_out_aaa(T117, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U24_g(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_g(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_g(T5, tappend5_in_aaa(T133, X265, X268))
U25_g(T5, tappend5_out_aaa(T133, X265, X268)) → goal1_out_g(T5)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U10_g(x1, x2)  =  U10_g(x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
goal1_out_g(x1)  =  goal1_out_g
U11_g(x1, x2)  =  U11_g(x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg
node(x1, x2, x3)  =  node(x1, x2, x3)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U12_g(x1, x2)  =  U12_g(x2)
tappend5_in_gaa(x1, x2, x3)  =  tappend5_in_gaa(x1)
tappend5_out_gaa(x1, x2, x3)  =  tappend5_out_gaa
U4_gaa(x1, x2, x3)  =  U4_gaa(x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x6)
U7_gaa(x1, x2)  =  U7_gaa(x2)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
U13_g(x1, x2)  =  U13_g(x2)
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
tappend5_in_aaa(x1, x2, x3)  =  tappend5_in_aaa
tappend5_out_aaa(x1, x2, x3)  =  tappend5_out_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
U5_aaa(x1, x2, x3, x4, x5, x6)  =  U5_aaa(x6)
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
U7_aaa(x1, x2)  =  U7_aaa(x2)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
U23_g(x1, x2)  =  U23_g(x2)
U24_g(x1, x2)  =  U24_g(x2)
U25_g(x1, x2)  =  U25_g(x2)
TAPPEND5_IN_GAA(x1, x2, x3)  =  TAPPEND5_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:

TAPPEND5_IN_GAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → TAPPEND5_IN_GAA(T117, X265, X268)
TAPPEND5_IN_GAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → TAPPEND5_IN_GAA(T83, X210, X211)

R is empty.
The argument filtering Pi contains the following mapping:
node(x1, x2, x3)  =  node(x1, x2, x3)
TAPPEND5_IN_GAA(x1, x2, x3)  =  TAPPEND5_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:

TAPPEND5_IN_GAA(node(T114, T115, T117)) → TAPPEND5_IN_GAA(T117)
TAPPEND5_IN_GAA(node(T83, T81, T82)) → TAPPEND5_IN_GAA(T83)

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:

  • TAPPEND5_IN_GAA(node(T114, T115, T117)) → TAPPEND5_IN_GAA(T117)
    The graph contains the following edges 1 > 1

  • TAPPEND5_IN_GAA(node(T83, T81, T82)) → TAPPEND5_IN_GAA(T83)
    The graph contains the following edges 1 > 1

(29) TRUE

(30) Obligation:

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

S2T4_IN_GG(s(T10), node(nil, X48, X49)) → S2T4_IN_GG(T10, X49)
S2T4_IN_GG(s(T8), node(X34, X35, X34)) → S2T4_IN_GG(T8, X34)
S2T4_IN_GG(s(T12), node(X62, X63, nil)) → S2T4_IN_GG(T12, X62)

The TRS R consists of the following rules:

goal1_in_g(T5) → U10_g(T5, s2t4_in_ga(T5, X15))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T8), node(X34, X35, X34)) → U1_ga(T8, X34, X35, s2t4_in_ga(T8, X34))
s2t4_in_ga(s(T10), node(nil, X48, X49)) → U2_ga(T10, X48, X49, s2t4_in_ga(T10, X49))
s2t4_in_ga(s(T12), node(X62, X63, nil)) → U3_ga(T12, X62, X63, s2t4_in_ga(T12, X62))
s2t4_in_ga(T13, node(nil, X70, nil)) → s2t4_out_ga(T13, node(nil, X70, nil))
U3_ga(T12, X62, X63, s2t4_out_ga(T12, X62)) → s2t4_out_ga(s(T12), node(X62, X63, nil))
U2_ga(T10, X48, X49, s2t4_out_ga(T10, X49)) → s2t4_out_ga(s(T10), node(nil, X48, X49))
U1_ga(T8, X34, X35, s2t4_out_ga(T8, X34)) → s2t4_out_ga(s(T8), node(X34, X35, X34))
U10_g(T5, s2t4_out_ga(T5, X15)) → goal1_out_g(T5)
goal1_in_g(T5) → U11_g(T5, s2t4_in_gg(T5, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T8), node(X34, X35, X34)) → U1_gg(T8, X34, X35, s2t4_in_gg(T8, X34))
s2t4_in_gg(s(T10), node(nil, X48, X49)) → U2_gg(T10, X48, X49, s2t4_in_gg(T10, X49))
s2t4_in_gg(s(T12), node(X62, X63, nil)) → U3_gg(T12, X62, X63, s2t4_in_gg(T12, X62))
s2t4_in_gg(T13, node(nil, X70, nil)) → s2t4_out_gg(T13, node(nil, X70, nil))
U3_gg(T12, X62, X63, s2t4_out_gg(T12, X62)) → s2t4_out_gg(s(T12), node(X62, X63, nil))
U2_gg(T10, X48, X49, s2t4_out_gg(T10, X49)) → s2t4_out_gg(s(T10), node(nil, X48, X49))
U1_gg(T8, X34, X35, s2t4_out_gg(T8, X34)) → s2t4_out_gg(s(T8), node(X34, X35, X34))
U11_g(T5, s2t4_out_gg(T5, nil)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U12_g(T5, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(nil, X79, X79) → tappend5_out_gaa(nil, X79, X79)
tappend5_in_gaa(nil, X92, node(X92, nil, nil)) → tappend5_out_gaa(nil, X92, node(X92, nil, nil))
tappend5_in_gaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_gaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_gaa(nil, X149, node(nil, nil, X149)) → tappend5_out_gaa(nil, X149, node(nil, nil, X149))
tappend5_in_gaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_gaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_gaa(nil, X210, node(X211, nil, nil)) → U4_gaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_gaa(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
tappend5_in_gaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_gaa(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
tappend5_in_gaa(nil, nil, node(nil, nil, X268)) → U7_gaa(X268, tappend5_in_gaa(nil, X265, X268))
tappend5_in_gaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_gaa(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
tappend5_in_gaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_gaa(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U9_gaa(T130, T131, T133, X268, tappend5_out_gaa(T133, X265, X268)) → tappend5_out_gaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_gaa(T114, T115, T117, X268, tappend5_out_gaa(T117, X265, X268)) → tappend5_out_gaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U7_gaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_gaa(nil, nil, node(nil, nil, X268))
U6_gaa(T99, T97, T98, X210, X211, tappend5_out_gaa(T99, X210, X211)) → tappend5_out_gaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_gaa(T83, T81, T82, X210, X211, tappend5_out_gaa(T83, X210, X211)) → tappend5_out_gaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U4_gaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_gaa(nil, X210, node(X211, nil, nil))
U12_g(T5, tappend5_out_gaa(nil, X210, X211)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U13_g(T5, tappend5_in_gaa(nil, X265, X268))
U13_g(T5, tappend5_out_gaa(nil, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U14_g(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
U14_g(T5, s2t4_out_ga(T5, node(nil, T25, T26))) → goal1_out_g(T5)
goal1_in_g(T5) → U15_g(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
U15_g(T5, s2t4_out_ga(T5, node(nil, T36, T37))) → goal1_out_g(T5)
goal1_in_g(T5) → U16_g(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
U16_g(T5, s2t4_out_ga(T5, node(T51, T52, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U17_g(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
U17_g(T5, s2t4_out_ga(T5, node(T64, T65, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U18_g(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
U18_g(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_g(T5, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(nil, X79, X79) → tappend5_out_aaa(nil, X79, X79)
tappend5_in_aaa(nil, X92, node(X92, nil, nil)) → tappend5_out_aaa(nil, X92, node(X92, nil, nil))
tappend5_in_aaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_aaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_aaa(nil, X149, node(nil, nil, X149)) → tappend5_out_aaa(nil, X149, node(nil, nil, X149))
tappend5_in_aaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_aaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_aaa(nil, X210, node(X211, nil, nil)) → U4_aaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
U4_aaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_aaa(nil, X210, node(X211, nil, nil))
tappend5_in_aaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_aaa(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_aaa(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
tappend5_in_aaa(nil, nil, node(nil, nil, X268)) → U7_aaa(X268, tappend5_in_gaa(nil, X265, X268))
U7_aaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_aaa(nil, nil, node(nil, nil, X268))
tappend5_in_aaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_aaa(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
tappend5_in_aaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_aaa(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
U9_aaa(T130, T131, T133, X268, tappend5_out_aaa(T133, X265, X268)) → tappend5_out_aaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_aaa(T114, T115, T117, X268, tappend5_out_aaa(T117, X265, X268)) → tappend5_out_aaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U6_aaa(T99, T97, T98, X210, X211, tappend5_out_aaa(T99, X210, X211)) → tappend5_out_aaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_aaa(T83, T81, T82, X210, X211, tappend5_out_aaa(T83, X210, X211)) → tappend5_out_aaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U19_g(T5, tappend5_out_aaa(T83, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U20_g(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_g(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_g(T5, tappend5_in_aaa(T99, X210, X211))
U21_g(T5, tappend5_out_aaa(T99, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U22_g(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_g(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_g(T5, tappend5_in_aaa(T117, X265, X268))
U23_g(T5, tappend5_out_aaa(T117, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U24_g(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_g(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_g(T5, tappend5_in_aaa(T133, X265, X268))
U25_g(T5, tappend5_out_aaa(T133, X265, X268)) → goal1_out_g(T5)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U10_g(x1, x2)  =  U10_g(x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
goal1_out_g(x1)  =  goal1_out_g
U11_g(x1, x2)  =  U11_g(x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg
node(x1, x2, x3)  =  node(x1, x2, x3)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U12_g(x1, x2)  =  U12_g(x2)
tappend5_in_gaa(x1, x2, x3)  =  tappend5_in_gaa(x1)
tappend5_out_gaa(x1, x2, x3)  =  tappend5_out_gaa
U4_gaa(x1, x2, x3)  =  U4_gaa(x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x6)
U7_gaa(x1, x2)  =  U7_gaa(x2)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
U13_g(x1, x2)  =  U13_g(x2)
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
tappend5_in_aaa(x1, x2, x3)  =  tappend5_in_aaa
tappend5_out_aaa(x1, x2, x3)  =  tappend5_out_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
U5_aaa(x1, x2, x3, x4, x5, x6)  =  U5_aaa(x6)
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
U7_aaa(x1, x2)  =  U7_aaa(x2)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
U23_g(x1, x2)  =  U23_g(x2)
U24_g(x1, x2)  =  U24_g(x2)
U25_g(x1, x2)  =  U25_g(x2)
S2T4_IN_GG(x1, x2)  =  S2T4_IN_GG(x1, 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:

S2T4_IN_GG(s(T10), node(nil, X48, X49)) → S2T4_IN_GG(T10, X49)
S2T4_IN_GG(s(T8), node(X34, X35, X34)) → S2T4_IN_GG(T8, X34)
S2T4_IN_GG(s(T12), node(X62, X63, nil)) → S2T4_IN_GG(T12, X62)

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

(33) PiDPToQDPProof (EQUIVALENT 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:

S2T4_IN_GG(s(T10), node(nil, X48, X49)) → S2T4_IN_GG(T10, X49)
S2T4_IN_GG(s(T8), node(X34, X35, X34)) → S2T4_IN_GG(T8, X34)
S2T4_IN_GG(s(T12), node(X62, X63, nil)) → S2T4_IN_GG(T12, X62)

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

(35) 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:

  • S2T4_IN_GG(s(T10), node(nil, X48, X49)) → S2T4_IN_GG(T10, X49)
    The graph contains the following edges 1 > 1, 2 > 2

  • S2T4_IN_GG(s(T8), node(X34, X35, X34)) → S2T4_IN_GG(T8, X34)
    The graph contains the following edges 1 > 1, 2 > 2

  • S2T4_IN_GG(s(T12), node(X62, X63, nil)) → S2T4_IN_GG(T12, X62)
    The graph contains the following edges 1 > 1, 2 > 2

(36) TRUE

(37) Obligation:

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

S2T4_IN_GA(s(T10), node(nil, X48, X49)) → S2T4_IN_GA(T10, X49)
S2T4_IN_GA(s(T8), node(X34, X35, X34)) → S2T4_IN_GA(T8, X34)
S2T4_IN_GA(s(T12), node(X62, X63, nil)) → S2T4_IN_GA(T12, X62)

The TRS R consists of the following rules:

goal1_in_g(T5) → U10_g(T5, s2t4_in_ga(T5, X15))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T8), node(X34, X35, X34)) → U1_ga(T8, X34, X35, s2t4_in_ga(T8, X34))
s2t4_in_ga(s(T10), node(nil, X48, X49)) → U2_ga(T10, X48, X49, s2t4_in_ga(T10, X49))
s2t4_in_ga(s(T12), node(X62, X63, nil)) → U3_ga(T12, X62, X63, s2t4_in_ga(T12, X62))
s2t4_in_ga(T13, node(nil, X70, nil)) → s2t4_out_ga(T13, node(nil, X70, nil))
U3_ga(T12, X62, X63, s2t4_out_ga(T12, X62)) → s2t4_out_ga(s(T12), node(X62, X63, nil))
U2_ga(T10, X48, X49, s2t4_out_ga(T10, X49)) → s2t4_out_ga(s(T10), node(nil, X48, X49))
U1_ga(T8, X34, X35, s2t4_out_ga(T8, X34)) → s2t4_out_ga(s(T8), node(X34, X35, X34))
U10_g(T5, s2t4_out_ga(T5, X15)) → goal1_out_g(T5)
goal1_in_g(T5) → U11_g(T5, s2t4_in_gg(T5, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T8), node(X34, X35, X34)) → U1_gg(T8, X34, X35, s2t4_in_gg(T8, X34))
s2t4_in_gg(s(T10), node(nil, X48, X49)) → U2_gg(T10, X48, X49, s2t4_in_gg(T10, X49))
s2t4_in_gg(s(T12), node(X62, X63, nil)) → U3_gg(T12, X62, X63, s2t4_in_gg(T12, X62))
s2t4_in_gg(T13, node(nil, X70, nil)) → s2t4_out_gg(T13, node(nil, X70, nil))
U3_gg(T12, X62, X63, s2t4_out_gg(T12, X62)) → s2t4_out_gg(s(T12), node(X62, X63, nil))
U2_gg(T10, X48, X49, s2t4_out_gg(T10, X49)) → s2t4_out_gg(s(T10), node(nil, X48, X49))
U1_gg(T8, X34, X35, s2t4_out_gg(T8, X34)) → s2t4_out_gg(s(T8), node(X34, X35, X34))
U11_g(T5, s2t4_out_gg(T5, nil)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U12_g(T5, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(nil, X79, X79) → tappend5_out_gaa(nil, X79, X79)
tappend5_in_gaa(nil, X92, node(X92, nil, nil)) → tappend5_out_gaa(nil, X92, node(X92, nil, nil))
tappend5_in_gaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_gaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_gaa(nil, X149, node(nil, nil, X149)) → tappend5_out_gaa(nil, X149, node(nil, nil, X149))
tappend5_in_gaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_gaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_gaa(nil, X210, node(X211, nil, nil)) → U4_gaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_gaa(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
tappend5_in_gaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_gaa(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
tappend5_in_gaa(nil, nil, node(nil, nil, X268)) → U7_gaa(X268, tappend5_in_gaa(nil, X265, X268))
tappend5_in_gaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_gaa(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
tappend5_in_gaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_gaa(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U9_gaa(T130, T131, T133, X268, tappend5_out_gaa(T133, X265, X268)) → tappend5_out_gaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_gaa(T114, T115, T117, X268, tappend5_out_gaa(T117, X265, X268)) → tappend5_out_gaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U7_gaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_gaa(nil, nil, node(nil, nil, X268))
U6_gaa(T99, T97, T98, X210, X211, tappend5_out_gaa(T99, X210, X211)) → tappend5_out_gaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_gaa(T83, T81, T82, X210, X211, tappend5_out_gaa(T83, X210, X211)) → tappend5_out_gaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U4_gaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_gaa(nil, X210, node(X211, nil, nil))
U12_g(T5, tappend5_out_gaa(nil, X210, X211)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U13_g(T5, tappend5_in_gaa(nil, X265, X268))
U13_g(T5, tappend5_out_gaa(nil, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U14_g(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
U14_g(T5, s2t4_out_ga(T5, node(nil, T25, T26))) → goal1_out_g(T5)
goal1_in_g(T5) → U15_g(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
U15_g(T5, s2t4_out_ga(T5, node(nil, T36, T37))) → goal1_out_g(T5)
goal1_in_g(T5) → U16_g(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
U16_g(T5, s2t4_out_ga(T5, node(T51, T52, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U17_g(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
U17_g(T5, s2t4_out_ga(T5, node(T64, T65, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U18_g(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
U18_g(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_g(T5, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(nil, X79, X79) → tappend5_out_aaa(nil, X79, X79)
tappend5_in_aaa(nil, X92, node(X92, nil, nil)) → tappend5_out_aaa(nil, X92, node(X92, nil, nil))
tappend5_in_aaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_aaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_aaa(nil, X149, node(nil, nil, X149)) → tappend5_out_aaa(nil, X149, node(nil, nil, X149))
tappend5_in_aaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_aaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_aaa(nil, X210, node(X211, nil, nil)) → U4_aaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
U4_aaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_aaa(nil, X210, node(X211, nil, nil))
tappend5_in_aaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_aaa(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_aaa(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
tappend5_in_aaa(nil, nil, node(nil, nil, X268)) → U7_aaa(X268, tappend5_in_gaa(nil, X265, X268))
U7_aaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_aaa(nil, nil, node(nil, nil, X268))
tappend5_in_aaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_aaa(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
tappend5_in_aaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_aaa(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
U9_aaa(T130, T131, T133, X268, tappend5_out_aaa(T133, X265, X268)) → tappend5_out_aaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_aaa(T114, T115, T117, X268, tappend5_out_aaa(T117, X265, X268)) → tappend5_out_aaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U6_aaa(T99, T97, T98, X210, X211, tappend5_out_aaa(T99, X210, X211)) → tappend5_out_aaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_aaa(T83, T81, T82, X210, X211, tappend5_out_aaa(T83, X210, X211)) → tappend5_out_aaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U19_g(T5, tappend5_out_aaa(T83, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U20_g(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_g(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_g(T5, tappend5_in_aaa(T99, X210, X211))
U21_g(T5, tappend5_out_aaa(T99, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U22_g(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_g(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_g(T5, tappend5_in_aaa(T117, X265, X268))
U23_g(T5, tappend5_out_aaa(T117, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U24_g(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_g(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_g(T5, tappend5_in_aaa(T133, X265, X268))
U25_g(T5, tappend5_out_aaa(T133, X265, X268)) → goal1_out_g(T5)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U10_g(x1, x2)  =  U10_g(x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
goal1_out_g(x1)  =  goal1_out_g
U11_g(x1, x2)  =  U11_g(x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg
node(x1, x2, x3)  =  node(x1, x2, x3)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x4)
U12_g(x1, x2)  =  U12_g(x2)
tappend5_in_gaa(x1, x2, x3)  =  tappend5_in_gaa(x1)
tappend5_out_gaa(x1, x2, x3)  =  tappend5_out_gaa
U4_gaa(x1, x2, x3)  =  U4_gaa(x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x6)
U7_gaa(x1, x2)  =  U7_gaa(x2)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x5)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)
U13_g(x1, x2)  =  U13_g(x2)
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
tappend5_in_aaa(x1, x2, x3)  =  tappend5_in_aaa
tappend5_out_aaa(x1, x2, x3)  =  tappend5_out_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
U5_aaa(x1, x2, x3, x4, x5, x6)  =  U5_aaa(x6)
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
U7_aaa(x1, x2)  =  U7_aaa(x2)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U20_g(x1, x2)  =  U20_g(x2)
U21_g(x1, x2)  =  U21_g(x2)
U22_g(x1, x2)  =  U22_g(x2)
U23_g(x1, x2)  =  U23_g(x2)
U24_g(x1, x2)  =  U24_g(x2)
U25_g(x1, x2)  =  U25_g(x2)
S2T4_IN_GA(x1, x2)  =  S2T4_IN_GA(x1)

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:

S2T4_IN_GA(s(T10), node(nil, X48, X49)) → S2T4_IN_GA(T10, X49)
S2T4_IN_GA(s(T8), node(X34, X35, X34)) → S2T4_IN_GA(T8, X34)
S2T4_IN_GA(s(T12), node(X62, X63, nil)) → S2T4_IN_GA(T12, X62)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
nil  =  nil
node(x1, x2, x3)  =  node(x1, x2, x3)
S2T4_IN_GA(x1, x2)  =  S2T4_IN_GA(x1)

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:

S2T4_IN_GA(s(T10)) → S2T4_IN_GA(T10)

R is empty.
Q is empty.
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:

  • S2T4_IN_GA(s(T10)) → S2T4_IN_GA(T10)
    The graph contains the following edges 1 > 1

(43) TRUE

(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:
goal1_in: (b)
s2t4_in: (b,f) (b,b)
tappend5_in: (b,f,f) (f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal1_in_g(T5) → U10_g(T5, s2t4_in_ga(T5, X15))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T8), node(X34, X35, X34)) → U1_ga(T8, X34, X35, s2t4_in_ga(T8, X34))
s2t4_in_ga(s(T10), node(nil, X48, X49)) → U2_ga(T10, X48, X49, s2t4_in_ga(T10, X49))
s2t4_in_ga(s(T12), node(X62, X63, nil)) → U3_ga(T12, X62, X63, s2t4_in_ga(T12, X62))
s2t4_in_ga(T13, node(nil, X70, nil)) → s2t4_out_ga(T13, node(nil, X70, nil))
U3_ga(T12, X62, X63, s2t4_out_ga(T12, X62)) → s2t4_out_ga(s(T12), node(X62, X63, nil))
U2_ga(T10, X48, X49, s2t4_out_ga(T10, X49)) → s2t4_out_ga(s(T10), node(nil, X48, X49))
U1_ga(T8, X34, X35, s2t4_out_ga(T8, X34)) → s2t4_out_ga(s(T8), node(X34, X35, X34))
U10_g(T5, s2t4_out_ga(T5, X15)) → goal1_out_g(T5)
goal1_in_g(T5) → U11_g(T5, s2t4_in_gg(T5, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T8), node(X34, X35, X34)) → U1_gg(T8, X34, X35, s2t4_in_gg(T8, X34))
s2t4_in_gg(s(T10), node(nil, X48, X49)) → U2_gg(T10, X48, X49, s2t4_in_gg(T10, X49))
s2t4_in_gg(s(T12), node(X62, X63, nil)) → U3_gg(T12, X62, X63, s2t4_in_gg(T12, X62))
s2t4_in_gg(T13, node(nil, X70, nil)) → s2t4_out_gg(T13, node(nil, X70, nil))
U3_gg(T12, X62, X63, s2t4_out_gg(T12, X62)) → s2t4_out_gg(s(T12), node(X62, X63, nil))
U2_gg(T10, X48, X49, s2t4_out_gg(T10, X49)) → s2t4_out_gg(s(T10), node(nil, X48, X49))
U1_gg(T8, X34, X35, s2t4_out_gg(T8, X34)) → s2t4_out_gg(s(T8), node(X34, X35, X34))
U11_g(T5, s2t4_out_gg(T5, nil)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U12_g(T5, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(nil, X79, X79) → tappend5_out_gaa(nil, X79, X79)
tappend5_in_gaa(nil, X92, node(X92, nil, nil)) → tappend5_out_gaa(nil, X92, node(X92, nil, nil))
tappend5_in_gaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_gaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_gaa(nil, X149, node(nil, nil, X149)) → tappend5_out_gaa(nil, X149, node(nil, nil, X149))
tappend5_in_gaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_gaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_gaa(nil, X210, node(X211, nil, nil)) → U4_gaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_gaa(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
tappend5_in_gaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_gaa(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
tappend5_in_gaa(nil, nil, node(nil, nil, X268)) → U7_gaa(X268, tappend5_in_gaa(nil, X265, X268))
tappend5_in_gaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_gaa(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
tappend5_in_gaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_gaa(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U9_gaa(T130, T131, T133, X268, tappend5_out_gaa(T133, X265, X268)) → tappend5_out_gaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_gaa(T114, T115, T117, X268, tappend5_out_gaa(T117, X265, X268)) → tappend5_out_gaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U7_gaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_gaa(nil, nil, node(nil, nil, X268))
U6_gaa(T99, T97, T98, X210, X211, tappend5_out_gaa(T99, X210, X211)) → tappend5_out_gaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_gaa(T83, T81, T82, X210, X211, tappend5_out_gaa(T83, X210, X211)) → tappend5_out_gaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U4_gaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_gaa(nil, X210, node(X211, nil, nil))
U12_g(T5, tappend5_out_gaa(nil, X210, X211)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U13_g(T5, tappend5_in_gaa(nil, X265, X268))
U13_g(T5, tappend5_out_gaa(nil, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U14_g(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
U14_g(T5, s2t4_out_ga(T5, node(nil, T25, T26))) → goal1_out_g(T5)
goal1_in_g(T5) → U15_g(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
U15_g(T5, s2t4_out_ga(T5, node(nil, T36, T37))) → goal1_out_g(T5)
goal1_in_g(T5) → U16_g(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
U16_g(T5, s2t4_out_ga(T5, node(T51, T52, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U17_g(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
U17_g(T5, s2t4_out_ga(T5, node(T64, T65, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U18_g(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
U18_g(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_g(T5, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(nil, X79, X79) → tappend5_out_aaa(nil, X79, X79)
tappend5_in_aaa(nil, X92, node(X92, nil, nil)) → tappend5_out_aaa(nil, X92, node(X92, nil, nil))
tappend5_in_aaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_aaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_aaa(nil, X149, node(nil, nil, X149)) → tappend5_out_aaa(nil, X149, node(nil, nil, X149))
tappend5_in_aaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_aaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_aaa(nil, X210, node(X211, nil, nil)) → U4_aaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
U4_aaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_aaa(nil, X210, node(X211, nil, nil))
tappend5_in_aaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_aaa(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_aaa(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
tappend5_in_aaa(nil, nil, node(nil, nil, X268)) → U7_aaa(X268, tappend5_in_gaa(nil, X265, X268))
U7_aaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_aaa(nil, nil, node(nil, nil, X268))
tappend5_in_aaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_aaa(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
tappend5_in_aaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_aaa(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
U9_aaa(T130, T131, T133, X268, tappend5_out_aaa(T133, X265, X268)) → tappend5_out_aaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_aaa(T114, T115, T117, X268, tappend5_out_aaa(T117, X265, X268)) → tappend5_out_aaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U6_aaa(T99, T97, T98, X210, X211, tappend5_out_aaa(T99, X210, X211)) → tappend5_out_aaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_aaa(T83, T81, T82, X210, X211, tappend5_out_aaa(T83, X210, X211)) → tappend5_out_aaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U19_g(T5, tappend5_out_aaa(T83, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U20_g(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_g(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_g(T5, tappend5_in_aaa(T99, X210, X211))
U21_g(T5, tappend5_out_aaa(T99, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U22_g(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_g(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_g(T5, tappend5_in_aaa(T117, X265, X268))
U23_g(T5, tappend5_out_aaa(T117, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U24_g(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_g(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_g(T5, tappend5_in_aaa(T133, X265, X268))
U25_g(T5, tappend5_out_aaa(T133, X265, X268)) → goal1_out_g(T5)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
goal1_out_g(x1)  =  goal1_out_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg(x1, x2)
node(x1, x2, x3)  =  node(x1, x2, x3)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x3, x4)
U12_g(x1, x2)  =  U12_g(x1, x2)
tappend5_in_gaa(x1, x2, x3)  =  tappend5_in_gaa(x1)
tappend5_out_gaa(x1, x2, x3)  =  tappend5_out_gaa(x1)
U4_gaa(x1, x2, x3)  =  U4_gaa(x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x1, x2, x3, x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x2, x3, x6)
U7_gaa(x1, x2)  =  U7_gaa(x2)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x1, x2, x3, x5)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x3, x5)
U13_g(x1, x2)  =  U13_g(x1, x2)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
tappend5_in_aaa(x1, x2, x3)  =  tappend5_in_aaa
tappend5_out_aaa(x1, x2, x3)  =  tappend5_out_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
U5_aaa(x1, x2, x3, x4, x5, x6)  =  U5_aaa(x6)
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
U7_aaa(x1, x2)  =  U7_aaa(x2)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U20_g(x1, x2)  =  U20_g(x1, x2)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
U23_g(x1, x2)  =  U23_g(x1, x2)
U24_g(x1, x2)  =  U24_g(x1, x2)
U25_g(x1, x2)  =  U25_g(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(45) Obligation:

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

goal1_in_g(T5) → U10_g(T5, s2t4_in_ga(T5, X15))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T8), node(X34, X35, X34)) → U1_ga(T8, X34, X35, s2t4_in_ga(T8, X34))
s2t4_in_ga(s(T10), node(nil, X48, X49)) → U2_ga(T10, X48, X49, s2t4_in_ga(T10, X49))
s2t4_in_ga(s(T12), node(X62, X63, nil)) → U3_ga(T12, X62, X63, s2t4_in_ga(T12, X62))
s2t4_in_ga(T13, node(nil, X70, nil)) → s2t4_out_ga(T13, node(nil, X70, nil))
U3_ga(T12, X62, X63, s2t4_out_ga(T12, X62)) → s2t4_out_ga(s(T12), node(X62, X63, nil))
U2_ga(T10, X48, X49, s2t4_out_ga(T10, X49)) → s2t4_out_ga(s(T10), node(nil, X48, X49))
U1_ga(T8, X34, X35, s2t4_out_ga(T8, X34)) → s2t4_out_ga(s(T8), node(X34, X35, X34))
U10_g(T5, s2t4_out_ga(T5, X15)) → goal1_out_g(T5)
goal1_in_g(T5) → U11_g(T5, s2t4_in_gg(T5, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T8), node(X34, X35, X34)) → U1_gg(T8, X34, X35, s2t4_in_gg(T8, X34))
s2t4_in_gg(s(T10), node(nil, X48, X49)) → U2_gg(T10, X48, X49, s2t4_in_gg(T10, X49))
s2t4_in_gg(s(T12), node(X62, X63, nil)) → U3_gg(T12, X62, X63, s2t4_in_gg(T12, X62))
s2t4_in_gg(T13, node(nil, X70, nil)) → s2t4_out_gg(T13, node(nil, X70, nil))
U3_gg(T12, X62, X63, s2t4_out_gg(T12, X62)) → s2t4_out_gg(s(T12), node(X62, X63, nil))
U2_gg(T10, X48, X49, s2t4_out_gg(T10, X49)) → s2t4_out_gg(s(T10), node(nil, X48, X49))
U1_gg(T8, X34, X35, s2t4_out_gg(T8, X34)) → s2t4_out_gg(s(T8), node(X34, X35, X34))
U11_g(T5, s2t4_out_gg(T5, nil)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U12_g(T5, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(nil, X79, X79) → tappend5_out_gaa(nil, X79, X79)
tappend5_in_gaa(nil, X92, node(X92, nil, nil)) → tappend5_out_gaa(nil, X92, node(X92, nil, nil))
tappend5_in_gaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_gaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_gaa(nil, X149, node(nil, nil, X149)) → tappend5_out_gaa(nil, X149, node(nil, nil, X149))
tappend5_in_gaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_gaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_gaa(nil, X210, node(X211, nil, nil)) → U4_gaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_gaa(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
tappend5_in_gaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_gaa(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
tappend5_in_gaa(nil, nil, node(nil, nil, X268)) → U7_gaa(X268, tappend5_in_gaa(nil, X265, X268))
tappend5_in_gaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_gaa(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
tappend5_in_gaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_gaa(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U9_gaa(T130, T131, T133, X268, tappend5_out_gaa(T133, X265, X268)) → tappend5_out_gaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_gaa(T114, T115, T117, X268, tappend5_out_gaa(T117, X265, X268)) → tappend5_out_gaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U7_gaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_gaa(nil, nil, node(nil, nil, X268))
U6_gaa(T99, T97, T98, X210, X211, tappend5_out_gaa(T99, X210, X211)) → tappend5_out_gaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_gaa(T83, T81, T82, X210, X211, tappend5_out_gaa(T83, X210, X211)) → tappend5_out_gaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U4_gaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_gaa(nil, X210, node(X211, nil, nil))
U12_g(T5, tappend5_out_gaa(nil, X210, X211)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U13_g(T5, tappend5_in_gaa(nil, X265, X268))
U13_g(T5, tappend5_out_gaa(nil, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U14_g(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
U14_g(T5, s2t4_out_ga(T5, node(nil, T25, T26))) → goal1_out_g(T5)
goal1_in_g(T5) → U15_g(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
U15_g(T5, s2t4_out_ga(T5, node(nil, T36, T37))) → goal1_out_g(T5)
goal1_in_g(T5) → U16_g(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
U16_g(T5, s2t4_out_ga(T5, node(T51, T52, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U17_g(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
U17_g(T5, s2t4_out_ga(T5, node(T64, T65, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U18_g(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
U18_g(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_g(T5, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(nil, X79, X79) → tappend5_out_aaa(nil, X79, X79)
tappend5_in_aaa(nil, X92, node(X92, nil, nil)) → tappend5_out_aaa(nil, X92, node(X92, nil, nil))
tappend5_in_aaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_aaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_aaa(nil, X149, node(nil, nil, X149)) → tappend5_out_aaa(nil, X149, node(nil, nil, X149))
tappend5_in_aaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_aaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_aaa(nil, X210, node(X211, nil, nil)) → U4_aaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
U4_aaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_aaa(nil, X210, node(X211, nil, nil))
tappend5_in_aaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_aaa(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_aaa(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
tappend5_in_aaa(nil, nil, node(nil, nil, X268)) → U7_aaa(X268, tappend5_in_gaa(nil, X265, X268))
U7_aaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_aaa(nil, nil, node(nil, nil, X268))
tappend5_in_aaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_aaa(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
tappend5_in_aaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_aaa(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
U9_aaa(T130, T131, T133, X268, tappend5_out_aaa(T133, X265, X268)) → tappend5_out_aaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_aaa(T114, T115, T117, X268, tappend5_out_aaa(T117, X265, X268)) → tappend5_out_aaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U6_aaa(T99, T97, T98, X210, X211, tappend5_out_aaa(T99, X210, X211)) → tappend5_out_aaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_aaa(T83, T81, T82, X210, X211, tappend5_out_aaa(T83, X210, X211)) → tappend5_out_aaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U19_g(T5, tappend5_out_aaa(T83, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U20_g(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_g(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_g(T5, tappend5_in_aaa(T99, X210, X211))
U21_g(T5, tappend5_out_aaa(T99, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U22_g(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_g(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_g(T5, tappend5_in_aaa(T117, X265, X268))
U23_g(T5, tappend5_out_aaa(T117, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U24_g(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_g(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_g(T5, tappend5_in_aaa(T133, X265, X268))
U25_g(T5, tappend5_out_aaa(T133, X265, X268)) → goal1_out_g(T5)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
goal1_out_g(x1)  =  goal1_out_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg(x1, x2)
node(x1, x2, x3)  =  node(x1, x2, x3)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x3, x4)
U12_g(x1, x2)  =  U12_g(x1, x2)
tappend5_in_gaa(x1, x2, x3)  =  tappend5_in_gaa(x1)
tappend5_out_gaa(x1, x2, x3)  =  tappend5_out_gaa(x1)
U4_gaa(x1, x2, x3)  =  U4_gaa(x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x1, x2, x3, x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x2, x3, x6)
U7_gaa(x1, x2)  =  U7_gaa(x2)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x1, x2, x3, x5)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x3, x5)
U13_g(x1, x2)  =  U13_g(x1, x2)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
tappend5_in_aaa(x1, x2, x3)  =  tappend5_in_aaa
tappend5_out_aaa(x1, x2, x3)  =  tappend5_out_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
U5_aaa(x1, x2, x3, x4, x5, x6)  =  U5_aaa(x6)
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
U7_aaa(x1, x2)  =  U7_aaa(x2)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U20_g(x1, x2)  =  U20_g(x1, x2)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
U23_g(x1, x2)  =  U23_g(x1, x2)
U24_g(x1, x2)  =  U24_g(x1, x2)
U25_g(x1, x2)  =  U25_g(x1, x2)

(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:

GOAL1_IN_G(T5) → U10_G(T5, s2t4_in_ga(T5, X15))
GOAL1_IN_G(T5) → S2T4_IN_GA(T5, X15)
S2T4_IN_GA(s(T8), node(X34, X35, X34)) → U1_GA(T8, X34, X35, s2t4_in_ga(T8, X34))
S2T4_IN_GA(s(T8), node(X34, X35, X34)) → S2T4_IN_GA(T8, X34)
S2T4_IN_GA(s(T10), node(nil, X48, X49)) → U2_GA(T10, X48, X49, s2t4_in_ga(T10, X49))
S2T4_IN_GA(s(T10), node(nil, X48, X49)) → S2T4_IN_GA(T10, X49)
S2T4_IN_GA(s(T12), node(X62, X63, nil)) → U3_GA(T12, X62, X63, s2t4_in_ga(T12, X62))
S2T4_IN_GA(s(T12), node(X62, X63, nil)) → S2T4_IN_GA(T12, X62)
GOAL1_IN_G(T5) → U11_G(T5, s2t4_in_gg(T5, nil))
GOAL1_IN_G(T5) → S2T4_IN_GG(T5, nil)
S2T4_IN_GG(s(T8), node(X34, X35, X34)) → U1_GG(T8, X34, X35, s2t4_in_gg(T8, X34))
S2T4_IN_GG(s(T8), node(X34, X35, X34)) → S2T4_IN_GG(T8, X34)
S2T4_IN_GG(s(T10), node(nil, X48, X49)) → U2_GG(T10, X48, X49, s2t4_in_gg(T10, X49))
S2T4_IN_GG(s(T10), node(nil, X48, X49)) → S2T4_IN_GG(T10, X49)
S2T4_IN_GG(s(T12), node(X62, X63, nil)) → U3_GG(T12, X62, X63, s2t4_in_gg(T12, X62))
S2T4_IN_GG(s(T12), node(X62, X63, nil)) → S2T4_IN_GG(T12, X62)
U11_G(T5, s2t4_out_gg(T5, nil)) → U12_G(T5, tappend5_in_gaa(nil, X210, X211))
U11_G(T5, s2t4_out_gg(T5, nil)) → TAPPEND5_IN_GAA(nil, X210, X211)
TAPPEND5_IN_GAA(nil, X210, node(X211, nil, nil)) → U4_GAA(X210, X211, tappend5_in_gaa(nil, X210, X211))
TAPPEND5_IN_GAA(nil, X210, node(X211, nil, nil)) → TAPPEND5_IN_GAA(nil, X210, X211)
TAPPEND5_IN_GAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_GAA(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
TAPPEND5_IN_GAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → TAPPEND5_IN_GAA(T83, X210, X211)
TAPPEND5_IN_GAA(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_GAA(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
TAPPEND5_IN_GAA(nil, nil, node(nil, nil, X268)) → U7_GAA(X268, tappend5_in_gaa(nil, X265, X268))
TAPPEND5_IN_GAA(nil, nil, node(nil, nil, X268)) → TAPPEND5_IN_GAA(nil, X265, X268)
TAPPEND5_IN_GAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_GAA(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
TAPPEND5_IN_GAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → TAPPEND5_IN_GAA(T117, X265, X268)
TAPPEND5_IN_GAA(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_GAA(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U11_G(T5, s2t4_out_gg(T5, nil)) → U13_G(T5, tappend5_in_gaa(nil, X265, X268))
GOAL1_IN_G(T5) → U14_G(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
GOAL1_IN_G(T5) → S2T4_IN_GA(T5, node(nil, T25, T26))
GOAL1_IN_G(T5) → U15_G(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
GOAL1_IN_G(T5) → U16_G(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
GOAL1_IN_G(T5) → S2T4_IN_GA(T5, node(T51, T52, nil))
GOAL1_IN_G(T5) → U17_G(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
GOAL1_IN_G(T5) → U18_G(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
GOAL1_IN_G(T5) → S2T4_IN_GA(T5, node(T83, T81, T82))
U18_G(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_G(T5, tappend5_in_aaa(T83, X210, X211))
U18_G(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → TAPPEND5_IN_AAA(T83, X210, X211)
TAPPEND5_IN_AAA(nil, X210, node(X211, nil, nil)) → U4_AAA(X210, X211, tappend5_in_gaa(nil, X210, X211))
TAPPEND5_IN_AAA(nil, X210, node(X211, nil, nil)) → TAPPEND5_IN_GAA(nil, X210, X211)
TAPPEND5_IN_AAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_AAA(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
TAPPEND5_IN_AAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → TAPPEND5_IN_AAA(T83, X210, X211)
TAPPEND5_IN_AAA(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_AAA(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
TAPPEND5_IN_AAA(nil, nil, node(nil, nil, X268)) → U7_AAA(X268, tappend5_in_gaa(nil, X265, X268))
TAPPEND5_IN_AAA(nil, nil, node(nil, nil, X268)) → TAPPEND5_IN_GAA(nil, X265, X268)
TAPPEND5_IN_AAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_AAA(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
TAPPEND5_IN_AAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → TAPPEND5_IN_AAA(T117, X265, X268)
TAPPEND5_IN_AAA(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_AAA(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
GOAL1_IN_G(T5) → U20_G(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_G(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_G(T5, tappend5_in_aaa(T99, X210, X211))
U20_G(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → TAPPEND5_IN_AAA(T99, X210, X211)
GOAL1_IN_G(T5) → U22_G(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_G(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_G(T5, tappend5_in_aaa(T117, X265, X268))
U22_G(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → TAPPEND5_IN_AAA(T117, X265, X268)
GOAL1_IN_G(T5) → U24_G(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_G(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_G(T5, tappend5_in_aaa(T133, X265, X268))
U24_G(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → TAPPEND5_IN_AAA(T133, X265, X268)

The TRS R consists of the following rules:

goal1_in_g(T5) → U10_g(T5, s2t4_in_ga(T5, X15))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T8), node(X34, X35, X34)) → U1_ga(T8, X34, X35, s2t4_in_ga(T8, X34))
s2t4_in_ga(s(T10), node(nil, X48, X49)) → U2_ga(T10, X48, X49, s2t4_in_ga(T10, X49))
s2t4_in_ga(s(T12), node(X62, X63, nil)) → U3_ga(T12, X62, X63, s2t4_in_ga(T12, X62))
s2t4_in_ga(T13, node(nil, X70, nil)) → s2t4_out_ga(T13, node(nil, X70, nil))
U3_ga(T12, X62, X63, s2t4_out_ga(T12, X62)) → s2t4_out_ga(s(T12), node(X62, X63, nil))
U2_ga(T10, X48, X49, s2t4_out_ga(T10, X49)) → s2t4_out_ga(s(T10), node(nil, X48, X49))
U1_ga(T8, X34, X35, s2t4_out_ga(T8, X34)) → s2t4_out_ga(s(T8), node(X34, X35, X34))
U10_g(T5, s2t4_out_ga(T5, X15)) → goal1_out_g(T5)
goal1_in_g(T5) → U11_g(T5, s2t4_in_gg(T5, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T8), node(X34, X35, X34)) → U1_gg(T8, X34, X35, s2t4_in_gg(T8, X34))
s2t4_in_gg(s(T10), node(nil, X48, X49)) → U2_gg(T10, X48, X49, s2t4_in_gg(T10, X49))
s2t4_in_gg(s(T12), node(X62, X63, nil)) → U3_gg(T12, X62, X63, s2t4_in_gg(T12, X62))
s2t4_in_gg(T13, node(nil, X70, nil)) → s2t4_out_gg(T13, node(nil, X70, nil))
U3_gg(T12, X62, X63, s2t4_out_gg(T12, X62)) → s2t4_out_gg(s(T12), node(X62, X63, nil))
U2_gg(T10, X48, X49, s2t4_out_gg(T10, X49)) → s2t4_out_gg(s(T10), node(nil, X48, X49))
U1_gg(T8, X34, X35, s2t4_out_gg(T8, X34)) → s2t4_out_gg(s(T8), node(X34, X35, X34))
U11_g(T5, s2t4_out_gg(T5, nil)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U12_g(T5, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(nil, X79, X79) → tappend5_out_gaa(nil, X79, X79)
tappend5_in_gaa(nil, X92, node(X92, nil, nil)) → tappend5_out_gaa(nil, X92, node(X92, nil, nil))
tappend5_in_gaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_gaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_gaa(nil, X149, node(nil, nil, X149)) → tappend5_out_gaa(nil, X149, node(nil, nil, X149))
tappend5_in_gaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_gaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_gaa(nil, X210, node(X211, nil, nil)) → U4_gaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_gaa(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
tappend5_in_gaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_gaa(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
tappend5_in_gaa(nil, nil, node(nil, nil, X268)) → U7_gaa(X268, tappend5_in_gaa(nil, X265, X268))
tappend5_in_gaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_gaa(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
tappend5_in_gaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_gaa(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U9_gaa(T130, T131, T133, X268, tappend5_out_gaa(T133, X265, X268)) → tappend5_out_gaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_gaa(T114, T115, T117, X268, tappend5_out_gaa(T117, X265, X268)) → tappend5_out_gaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U7_gaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_gaa(nil, nil, node(nil, nil, X268))
U6_gaa(T99, T97, T98, X210, X211, tappend5_out_gaa(T99, X210, X211)) → tappend5_out_gaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_gaa(T83, T81, T82, X210, X211, tappend5_out_gaa(T83, X210, X211)) → tappend5_out_gaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U4_gaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_gaa(nil, X210, node(X211, nil, nil))
U12_g(T5, tappend5_out_gaa(nil, X210, X211)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U13_g(T5, tappend5_in_gaa(nil, X265, X268))
U13_g(T5, tappend5_out_gaa(nil, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U14_g(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
U14_g(T5, s2t4_out_ga(T5, node(nil, T25, T26))) → goal1_out_g(T5)
goal1_in_g(T5) → U15_g(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
U15_g(T5, s2t4_out_ga(T5, node(nil, T36, T37))) → goal1_out_g(T5)
goal1_in_g(T5) → U16_g(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
U16_g(T5, s2t4_out_ga(T5, node(T51, T52, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U17_g(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
U17_g(T5, s2t4_out_ga(T5, node(T64, T65, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U18_g(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
U18_g(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_g(T5, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(nil, X79, X79) → tappend5_out_aaa(nil, X79, X79)
tappend5_in_aaa(nil, X92, node(X92, nil, nil)) → tappend5_out_aaa(nil, X92, node(X92, nil, nil))
tappend5_in_aaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_aaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_aaa(nil, X149, node(nil, nil, X149)) → tappend5_out_aaa(nil, X149, node(nil, nil, X149))
tappend5_in_aaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_aaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_aaa(nil, X210, node(X211, nil, nil)) → U4_aaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
U4_aaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_aaa(nil, X210, node(X211, nil, nil))
tappend5_in_aaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_aaa(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_aaa(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
tappend5_in_aaa(nil, nil, node(nil, nil, X268)) → U7_aaa(X268, tappend5_in_gaa(nil, X265, X268))
U7_aaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_aaa(nil, nil, node(nil, nil, X268))
tappend5_in_aaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_aaa(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
tappend5_in_aaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_aaa(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
U9_aaa(T130, T131, T133, X268, tappend5_out_aaa(T133, X265, X268)) → tappend5_out_aaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_aaa(T114, T115, T117, X268, tappend5_out_aaa(T117, X265, X268)) → tappend5_out_aaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U6_aaa(T99, T97, T98, X210, X211, tappend5_out_aaa(T99, X210, X211)) → tappend5_out_aaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_aaa(T83, T81, T82, X210, X211, tappend5_out_aaa(T83, X210, X211)) → tappend5_out_aaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U19_g(T5, tappend5_out_aaa(T83, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U20_g(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_g(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_g(T5, tappend5_in_aaa(T99, X210, X211))
U21_g(T5, tappend5_out_aaa(T99, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U22_g(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_g(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_g(T5, tappend5_in_aaa(T117, X265, X268))
U23_g(T5, tappend5_out_aaa(T117, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U24_g(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_g(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_g(T5, tappend5_in_aaa(T133, X265, X268))
U25_g(T5, tappend5_out_aaa(T133, X265, X268)) → goal1_out_g(T5)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
goal1_out_g(x1)  =  goal1_out_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg(x1, x2)
node(x1, x2, x3)  =  node(x1, x2, x3)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x3, x4)
U12_g(x1, x2)  =  U12_g(x1, x2)
tappend5_in_gaa(x1, x2, x3)  =  tappend5_in_gaa(x1)
tappend5_out_gaa(x1, x2, x3)  =  tappend5_out_gaa(x1)
U4_gaa(x1, x2, x3)  =  U4_gaa(x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x1, x2, x3, x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x2, x3, x6)
U7_gaa(x1, x2)  =  U7_gaa(x2)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x1, x2, x3, x5)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x3, x5)
U13_g(x1, x2)  =  U13_g(x1, x2)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
tappend5_in_aaa(x1, x2, x3)  =  tappend5_in_aaa
tappend5_out_aaa(x1, x2, x3)  =  tappend5_out_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
U5_aaa(x1, x2, x3, x4, x5, x6)  =  U5_aaa(x6)
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
U7_aaa(x1, x2)  =  U7_aaa(x2)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U20_g(x1, x2)  =  U20_g(x1, x2)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
U23_g(x1, x2)  =  U23_g(x1, x2)
U24_g(x1, x2)  =  U24_g(x1, x2)
U25_g(x1, x2)  =  U25_g(x1, x2)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U10_G(x1, x2)  =  U10_G(x1, x2)
S2T4_IN_GA(x1, x2)  =  S2T4_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U11_G(x1, x2)  =  U11_G(x1, x2)
S2T4_IN_GG(x1, x2)  =  S2T4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x1, x2, x3, x4)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x1, x2, x3, x4)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x1, x2, x3, x4)
U12_G(x1, x2)  =  U12_G(x1, x2)
TAPPEND5_IN_GAA(x1, x2, x3)  =  TAPPEND5_IN_GAA(x1)
U4_GAA(x1, x2, x3)  =  U4_GAA(x3)
U5_GAA(x1, x2, x3, x4, x5, x6)  =  U5_GAA(x1, x2, x3, x6)
U6_GAA(x1, x2, x3, x4, x5, x6)  =  U6_GAA(x1, x2, x3, x6)
U7_GAA(x1, x2)  =  U7_GAA(x2)
U8_GAA(x1, x2, x3, x4, x5)  =  U8_GAA(x1, x2, x3, x5)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x1, x2, x3, x5)
U13_G(x1, x2)  =  U13_G(x1, x2)
U14_G(x1, x2)  =  U14_G(x1, x2)
U15_G(x1, x2)  =  U15_G(x1, x2)
U16_G(x1, x2)  =  U16_G(x1, x2)
U17_G(x1, x2)  =  U17_G(x1, x2)
U18_G(x1, x2)  =  U18_G(x1, x2)
U19_G(x1, x2)  =  U19_G(x1, x2)
TAPPEND5_IN_AAA(x1, x2, x3)  =  TAPPEND5_IN_AAA
U4_AAA(x1, x2, x3)  =  U4_AAA(x3)
U5_AAA(x1, x2, x3, x4, x5, x6)  =  U5_AAA(x6)
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(x6)
U7_AAA(x1, x2)  =  U7_AAA(x2)
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
U20_G(x1, x2)  =  U20_G(x1, x2)
U21_G(x1, x2)  =  U21_G(x1, x2)
U22_G(x1, x2)  =  U22_G(x1, x2)
U23_G(x1, x2)  =  U23_G(x1, x2)
U24_G(x1, x2)  =  U24_G(x1, x2)
U25_G(x1, x2)  =  U25_G(x1, x2)

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

(47) Obligation:

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

GOAL1_IN_G(T5) → U10_G(T5, s2t4_in_ga(T5, X15))
GOAL1_IN_G(T5) → S2T4_IN_GA(T5, X15)
S2T4_IN_GA(s(T8), node(X34, X35, X34)) → U1_GA(T8, X34, X35, s2t4_in_ga(T8, X34))
S2T4_IN_GA(s(T8), node(X34, X35, X34)) → S2T4_IN_GA(T8, X34)
S2T4_IN_GA(s(T10), node(nil, X48, X49)) → U2_GA(T10, X48, X49, s2t4_in_ga(T10, X49))
S2T4_IN_GA(s(T10), node(nil, X48, X49)) → S2T4_IN_GA(T10, X49)
S2T4_IN_GA(s(T12), node(X62, X63, nil)) → U3_GA(T12, X62, X63, s2t4_in_ga(T12, X62))
S2T4_IN_GA(s(T12), node(X62, X63, nil)) → S2T4_IN_GA(T12, X62)
GOAL1_IN_G(T5) → U11_G(T5, s2t4_in_gg(T5, nil))
GOAL1_IN_G(T5) → S2T4_IN_GG(T5, nil)
S2T4_IN_GG(s(T8), node(X34, X35, X34)) → U1_GG(T8, X34, X35, s2t4_in_gg(T8, X34))
S2T4_IN_GG(s(T8), node(X34, X35, X34)) → S2T4_IN_GG(T8, X34)
S2T4_IN_GG(s(T10), node(nil, X48, X49)) → U2_GG(T10, X48, X49, s2t4_in_gg(T10, X49))
S2T4_IN_GG(s(T10), node(nil, X48, X49)) → S2T4_IN_GG(T10, X49)
S2T4_IN_GG(s(T12), node(X62, X63, nil)) → U3_GG(T12, X62, X63, s2t4_in_gg(T12, X62))
S2T4_IN_GG(s(T12), node(X62, X63, nil)) → S2T4_IN_GG(T12, X62)
U11_G(T5, s2t4_out_gg(T5, nil)) → U12_G(T5, tappend5_in_gaa(nil, X210, X211))
U11_G(T5, s2t4_out_gg(T5, nil)) → TAPPEND5_IN_GAA(nil, X210, X211)
TAPPEND5_IN_GAA(nil, X210, node(X211, nil, nil)) → U4_GAA(X210, X211, tappend5_in_gaa(nil, X210, X211))
TAPPEND5_IN_GAA(nil, X210, node(X211, nil, nil)) → TAPPEND5_IN_GAA(nil, X210, X211)
TAPPEND5_IN_GAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_GAA(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
TAPPEND5_IN_GAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → TAPPEND5_IN_GAA(T83, X210, X211)
TAPPEND5_IN_GAA(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_GAA(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
TAPPEND5_IN_GAA(nil, nil, node(nil, nil, X268)) → U7_GAA(X268, tappend5_in_gaa(nil, X265, X268))
TAPPEND5_IN_GAA(nil, nil, node(nil, nil, X268)) → TAPPEND5_IN_GAA(nil, X265, X268)
TAPPEND5_IN_GAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_GAA(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
TAPPEND5_IN_GAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → TAPPEND5_IN_GAA(T117, X265, X268)
TAPPEND5_IN_GAA(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_GAA(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U11_G(T5, s2t4_out_gg(T5, nil)) → U13_G(T5, tappend5_in_gaa(nil, X265, X268))
GOAL1_IN_G(T5) → U14_G(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
GOAL1_IN_G(T5) → S2T4_IN_GA(T5, node(nil, T25, T26))
GOAL1_IN_G(T5) → U15_G(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
GOAL1_IN_G(T5) → U16_G(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
GOAL1_IN_G(T5) → S2T4_IN_GA(T5, node(T51, T52, nil))
GOAL1_IN_G(T5) → U17_G(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
GOAL1_IN_G(T5) → U18_G(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
GOAL1_IN_G(T5) → S2T4_IN_GA(T5, node(T83, T81, T82))
U18_G(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_G(T5, tappend5_in_aaa(T83, X210, X211))
U18_G(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → TAPPEND5_IN_AAA(T83, X210, X211)
TAPPEND5_IN_AAA(nil, X210, node(X211, nil, nil)) → U4_AAA(X210, X211, tappend5_in_gaa(nil, X210, X211))
TAPPEND5_IN_AAA(nil, X210, node(X211, nil, nil)) → TAPPEND5_IN_GAA(nil, X210, X211)
TAPPEND5_IN_AAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_AAA(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
TAPPEND5_IN_AAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → TAPPEND5_IN_AAA(T83, X210, X211)
TAPPEND5_IN_AAA(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_AAA(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
TAPPEND5_IN_AAA(nil, nil, node(nil, nil, X268)) → U7_AAA(X268, tappend5_in_gaa(nil, X265, X268))
TAPPEND5_IN_AAA(nil, nil, node(nil, nil, X268)) → TAPPEND5_IN_GAA(nil, X265, X268)
TAPPEND5_IN_AAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_AAA(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
TAPPEND5_IN_AAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → TAPPEND5_IN_AAA(T117, X265, X268)
TAPPEND5_IN_AAA(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_AAA(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
GOAL1_IN_G(T5) → U20_G(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_G(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_G(T5, tappend5_in_aaa(T99, X210, X211))
U20_G(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → TAPPEND5_IN_AAA(T99, X210, X211)
GOAL1_IN_G(T5) → U22_G(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_G(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_G(T5, tappend5_in_aaa(T117, X265, X268))
U22_G(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → TAPPEND5_IN_AAA(T117, X265, X268)
GOAL1_IN_G(T5) → U24_G(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_G(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_G(T5, tappend5_in_aaa(T133, X265, X268))
U24_G(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → TAPPEND5_IN_AAA(T133, X265, X268)

The TRS R consists of the following rules:

goal1_in_g(T5) → U10_g(T5, s2t4_in_ga(T5, X15))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T8), node(X34, X35, X34)) → U1_ga(T8, X34, X35, s2t4_in_ga(T8, X34))
s2t4_in_ga(s(T10), node(nil, X48, X49)) → U2_ga(T10, X48, X49, s2t4_in_ga(T10, X49))
s2t4_in_ga(s(T12), node(X62, X63, nil)) → U3_ga(T12, X62, X63, s2t4_in_ga(T12, X62))
s2t4_in_ga(T13, node(nil, X70, nil)) → s2t4_out_ga(T13, node(nil, X70, nil))
U3_ga(T12, X62, X63, s2t4_out_ga(T12, X62)) → s2t4_out_ga(s(T12), node(X62, X63, nil))
U2_ga(T10, X48, X49, s2t4_out_ga(T10, X49)) → s2t4_out_ga(s(T10), node(nil, X48, X49))
U1_ga(T8, X34, X35, s2t4_out_ga(T8, X34)) → s2t4_out_ga(s(T8), node(X34, X35, X34))
U10_g(T5, s2t4_out_ga(T5, X15)) → goal1_out_g(T5)
goal1_in_g(T5) → U11_g(T5, s2t4_in_gg(T5, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T8), node(X34, X35, X34)) → U1_gg(T8, X34, X35, s2t4_in_gg(T8, X34))
s2t4_in_gg(s(T10), node(nil, X48, X49)) → U2_gg(T10, X48, X49, s2t4_in_gg(T10, X49))
s2t4_in_gg(s(T12), node(X62, X63, nil)) → U3_gg(T12, X62, X63, s2t4_in_gg(T12, X62))
s2t4_in_gg(T13, node(nil, X70, nil)) → s2t4_out_gg(T13, node(nil, X70, nil))
U3_gg(T12, X62, X63, s2t4_out_gg(T12, X62)) → s2t4_out_gg(s(T12), node(X62, X63, nil))
U2_gg(T10, X48, X49, s2t4_out_gg(T10, X49)) → s2t4_out_gg(s(T10), node(nil, X48, X49))
U1_gg(T8, X34, X35, s2t4_out_gg(T8, X34)) → s2t4_out_gg(s(T8), node(X34, X35, X34))
U11_g(T5, s2t4_out_gg(T5, nil)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U12_g(T5, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(nil, X79, X79) → tappend5_out_gaa(nil, X79, X79)
tappend5_in_gaa(nil, X92, node(X92, nil, nil)) → tappend5_out_gaa(nil, X92, node(X92, nil, nil))
tappend5_in_gaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_gaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_gaa(nil, X149, node(nil, nil, X149)) → tappend5_out_gaa(nil, X149, node(nil, nil, X149))
tappend5_in_gaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_gaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_gaa(nil, X210, node(X211, nil, nil)) → U4_gaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_gaa(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
tappend5_in_gaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_gaa(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
tappend5_in_gaa(nil, nil, node(nil, nil, X268)) → U7_gaa(X268, tappend5_in_gaa(nil, X265, X268))
tappend5_in_gaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_gaa(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
tappend5_in_gaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_gaa(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U9_gaa(T130, T131, T133, X268, tappend5_out_gaa(T133, X265, X268)) → tappend5_out_gaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_gaa(T114, T115, T117, X268, tappend5_out_gaa(T117, X265, X268)) → tappend5_out_gaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U7_gaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_gaa(nil, nil, node(nil, nil, X268))
U6_gaa(T99, T97, T98, X210, X211, tappend5_out_gaa(T99, X210, X211)) → tappend5_out_gaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_gaa(T83, T81, T82, X210, X211, tappend5_out_gaa(T83, X210, X211)) → tappend5_out_gaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U4_gaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_gaa(nil, X210, node(X211, nil, nil))
U12_g(T5, tappend5_out_gaa(nil, X210, X211)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U13_g(T5, tappend5_in_gaa(nil, X265, X268))
U13_g(T5, tappend5_out_gaa(nil, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U14_g(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
U14_g(T5, s2t4_out_ga(T5, node(nil, T25, T26))) → goal1_out_g(T5)
goal1_in_g(T5) → U15_g(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
U15_g(T5, s2t4_out_ga(T5, node(nil, T36, T37))) → goal1_out_g(T5)
goal1_in_g(T5) → U16_g(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
U16_g(T5, s2t4_out_ga(T5, node(T51, T52, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U17_g(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
U17_g(T5, s2t4_out_ga(T5, node(T64, T65, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U18_g(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
U18_g(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_g(T5, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(nil, X79, X79) → tappend5_out_aaa(nil, X79, X79)
tappend5_in_aaa(nil, X92, node(X92, nil, nil)) → tappend5_out_aaa(nil, X92, node(X92, nil, nil))
tappend5_in_aaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_aaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_aaa(nil, X149, node(nil, nil, X149)) → tappend5_out_aaa(nil, X149, node(nil, nil, X149))
tappend5_in_aaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_aaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_aaa(nil, X210, node(X211, nil, nil)) → U4_aaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
U4_aaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_aaa(nil, X210, node(X211, nil, nil))
tappend5_in_aaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_aaa(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_aaa(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
tappend5_in_aaa(nil, nil, node(nil, nil, X268)) → U7_aaa(X268, tappend5_in_gaa(nil, X265, X268))
U7_aaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_aaa(nil, nil, node(nil, nil, X268))
tappend5_in_aaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_aaa(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
tappend5_in_aaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_aaa(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
U9_aaa(T130, T131, T133, X268, tappend5_out_aaa(T133, X265, X268)) → tappend5_out_aaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_aaa(T114, T115, T117, X268, tappend5_out_aaa(T117, X265, X268)) → tappend5_out_aaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U6_aaa(T99, T97, T98, X210, X211, tappend5_out_aaa(T99, X210, X211)) → tappend5_out_aaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_aaa(T83, T81, T82, X210, X211, tappend5_out_aaa(T83, X210, X211)) → tappend5_out_aaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U19_g(T5, tappend5_out_aaa(T83, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U20_g(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_g(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_g(T5, tappend5_in_aaa(T99, X210, X211))
U21_g(T5, tappend5_out_aaa(T99, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U22_g(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_g(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_g(T5, tappend5_in_aaa(T117, X265, X268))
U23_g(T5, tappend5_out_aaa(T117, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U24_g(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_g(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_g(T5, tappend5_in_aaa(T133, X265, X268))
U25_g(T5, tappend5_out_aaa(T133, X265, X268)) → goal1_out_g(T5)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
goal1_out_g(x1)  =  goal1_out_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg(x1, x2)
node(x1, x2, x3)  =  node(x1, x2, x3)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x3, x4)
U12_g(x1, x2)  =  U12_g(x1, x2)
tappend5_in_gaa(x1, x2, x3)  =  tappend5_in_gaa(x1)
tappend5_out_gaa(x1, x2, x3)  =  tappend5_out_gaa(x1)
U4_gaa(x1, x2, x3)  =  U4_gaa(x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x1, x2, x3, x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x2, x3, x6)
U7_gaa(x1, x2)  =  U7_gaa(x2)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x1, x2, x3, x5)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x3, x5)
U13_g(x1, x2)  =  U13_g(x1, x2)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
tappend5_in_aaa(x1, x2, x3)  =  tappend5_in_aaa
tappend5_out_aaa(x1, x2, x3)  =  tappend5_out_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
U5_aaa(x1, x2, x3, x4, x5, x6)  =  U5_aaa(x6)
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
U7_aaa(x1, x2)  =  U7_aaa(x2)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U20_g(x1, x2)  =  U20_g(x1, x2)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
U23_g(x1, x2)  =  U23_g(x1, x2)
U24_g(x1, x2)  =  U24_g(x1, x2)
U25_g(x1, x2)  =  U25_g(x1, x2)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U10_G(x1, x2)  =  U10_G(x1, x2)
S2T4_IN_GA(x1, x2)  =  S2T4_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U11_G(x1, x2)  =  U11_G(x1, x2)
S2T4_IN_GG(x1, x2)  =  S2T4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x1, x2, x3, x4)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x1, x2, x3, x4)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x1, x2, x3, x4)
U12_G(x1, x2)  =  U12_G(x1, x2)
TAPPEND5_IN_GAA(x1, x2, x3)  =  TAPPEND5_IN_GAA(x1)
U4_GAA(x1, x2, x3)  =  U4_GAA(x3)
U5_GAA(x1, x2, x3, x4, x5, x6)  =  U5_GAA(x1, x2, x3, x6)
U6_GAA(x1, x2, x3, x4, x5, x6)  =  U6_GAA(x1, x2, x3, x6)
U7_GAA(x1, x2)  =  U7_GAA(x2)
U8_GAA(x1, x2, x3, x4, x5)  =  U8_GAA(x1, x2, x3, x5)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x1, x2, x3, x5)
U13_G(x1, x2)  =  U13_G(x1, x2)
U14_G(x1, x2)  =  U14_G(x1, x2)
U15_G(x1, x2)  =  U15_G(x1, x2)
U16_G(x1, x2)  =  U16_G(x1, x2)
U17_G(x1, x2)  =  U17_G(x1, x2)
U18_G(x1, x2)  =  U18_G(x1, x2)
U19_G(x1, x2)  =  U19_G(x1, x2)
TAPPEND5_IN_AAA(x1, x2, x3)  =  TAPPEND5_IN_AAA
U4_AAA(x1, x2, x3)  =  U4_AAA(x3)
U5_AAA(x1, x2, x3, x4, x5, x6)  =  U5_AAA(x6)
U6_AAA(x1, x2, x3, x4, x5, x6)  =  U6_AAA(x6)
U7_AAA(x1, x2)  =  U7_AAA(x2)
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
U20_G(x1, x2)  =  U20_G(x1, x2)
U21_G(x1, x2)  =  U21_G(x1, x2)
U22_G(x1, x2)  =  U22_G(x1, x2)
U23_G(x1, x2)  =  U23_G(x1, x2)
U24_G(x1, x2)  =  U24_G(x1, x2)
U25_G(x1, x2)  =  U25_G(x1, x2)

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

(48) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 46 less nodes.

(49) Complex Obligation (AND)

(50) Obligation:

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

TAPPEND5_IN_GAA(nil, nil, node(nil, nil, X268)) → TAPPEND5_IN_GAA(nil, X265, X268)
TAPPEND5_IN_GAA(nil, X210, node(X211, nil, nil)) → TAPPEND5_IN_GAA(nil, X210, X211)

The TRS R consists of the following rules:

goal1_in_g(T5) → U10_g(T5, s2t4_in_ga(T5, X15))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T8), node(X34, X35, X34)) → U1_ga(T8, X34, X35, s2t4_in_ga(T8, X34))
s2t4_in_ga(s(T10), node(nil, X48, X49)) → U2_ga(T10, X48, X49, s2t4_in_ga(T10, X49))
s2t4_in_ga(s(T12), node(X62, X63, nil)) → U3_ga(T12, X62, X63, s2t4_in_ga(T12, X62))
s2t4_in_ga(T13, node(nil, X70, nil)) → s2t4_out_ga(T13, node(nil, X70, nil))
U3_ga(T12, X62, X63, s2t4_out_ga(T12, X62)) → s2t4_out_ga(s(T12), node(X62, X63, nil))
U2_ga(T10, X48, X49, s2t4_out_ga(T10, X49)) → s2t4_out_ga(s(T10), node(nil, X48, X49))
U1_ga(T8, X34, X35, s2t4_out_ga(T8, X34)) → s2t4_out_ga(s(T8), node(X34, X35, X34))
U10_g(T5, s2t4_out_ga(T5, X15)) → goal1_out_g(T5)
goal1_in_g(T5) → U11_g(T5, s2t4_in_gg(T5, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T8), node(X34, X35, X34)) → U1_gg(T8, X34, X35, s2t4_in_gg(T8, X34))
s2t4_in_gg(s(T10), node(nil, X48, X49)) → U2_gg(T10, X48, X49, s2t4_in_gg(T10, X49))
s2t4_in_gg(s(T12), node(X62, X63, nil)) → U3_gg(T12, X62, X63, s2t4_in_gg(T12, X62))
s2t4_in_gg(T13, node(nil, X70, nil)) → s2t4_out_gg(T13, node(nil, X70, nil))
U3_gg(T12, X62, X63, s2t4_out_gg(T12, X62)) → s2t4_out_gg(s(T12), node(X62, X63, nil))
U2_gg(T10, X48, X49, s2t4_out_gg(T10, X49)) → s2t4_out_gg(s(T10), node(nil, X48, X49))
U1_gg(T8, X34, X35, s2t4_out_gg(T8, X34)) → s2t4_out_gg(s(T8), node(X34, X35, X34))
U11_g(T5, s2t4_out_gg(T5, nil)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U12_g(T5, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(nil, X79, X79) → tappend5_out_gaa(nil, X79, X79)
tappend5_in_gaa(nil, X92, node(X92, nil, nil)) → tappend5_out_gaa(nil, X92, node(X92, nil, nil))
tappend5_in_gaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_gaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_gaa(nil, X149, node(nil, nil, X149)) → tappend5_out_gaa(nil, X149, node(nil, nil, X149))
tappend5_in_gaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_gaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_gaa(nil, X210, node(X211, nil, nil)) → U4_gaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_gaa(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
tappend5_in_gaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_gaa(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
tappend5_in_gaa(nil, nil, node(nil, nil, X268)) → U7_gaa(X268, tappend5_in_gaa(nil, X265, X268))
tappend5_in_gaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_gaa(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
tappend5_in_gaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_gaa(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U9_gaa(T130, T131, T133, X268, tappend5_out_gaa(T133, X265, X268)) → tappend5_out_gaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_gaa(T114, T115, T117, X268, tappend5_out_gaa(T117, X265, X268)) → tappend5_out_gaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U7_gaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_gaa(nil, nil, node(nil, nil, X268))
U6_gaa(T99, T97, T98, X210, X211, tappend5_out_gaa(T99, X210, X211)) → tappend5_out_gaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_gaa(T83, T81, T82, X210, X211, tappend5_out_gaa(T83, X210, X211)) → tappend5_out_gaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U4_gaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_gaa(nil, X210, node(X211, nil, nil))
U12_g(T5, tappend5_out_gaa(nil, X210, X211)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U13_g(T5, tappend5_in_gaa(nil, X265, X268))
U13_g(T5, tappend5_out_gaa(nil, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U14_g(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
U14_g(T5, s2t4_out_ga(T5, node(nil, T25, T26))) → goal1_out_g(T5)
goal1_in_g(T5) → U15_g(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
U15_g(T5, s2t4_out_ga(T5, node(nil, T36, T37))) → goal1_out_g(T5)
goal1_in_g(T5) → U16_g(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
U16_g(T5, s2t4_out_ga(T5, node(T51, T52, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U17_g(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
U17_g(T5, s2t4_out_ga(T5, node(T64, T65, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U18_g(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
U18_g(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_g(T5, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(nil, X79, X79) → tappend5_out_aaa(nil, X79, X79)
tappend5_in_aaa(nil, X92, node(X92, nil, nil)) → tappend5_out_aaa(nil, X92, node(X92, nil, nil))
tappend5_in_aaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_aaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_aaa(nil, X149, node(nil, nil, X149)) → tappend5_out_aaa(nil, X149, node(nil, nil, X149))
tappend5_in_aaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_aaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_aaa(nil, X210, node(X211, nil, nil)) → U4_aaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
U4_aaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_aaa(nil, X210, node(X211, nil, nil))
tappend5_in_aaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_aaa(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_aaa(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
tappend5_in_aaa(nil, nil, node(nil, nil, X268)) → U7_aaa(X268, tappend5_in_gaa(nil, X265, X268))
U7_aaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_aaa(nil, nil, node(nil, nil, X268))
tappend5_in_aaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_aaa(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
tappend5_in_aaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_aaa(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
U9_aaa(T130, T131, T133, X268, tappend5_out_aaa(T133, X265, X268)) → tappend5_out_aaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_aaa(T114, T115, T117, X268, tappend5_out_aaa(T117, X265, X268)) → tappend5_out_aaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U6_aaa(T99, T97, T98, X210, X211, tappend5_out_aaa(T99, X210, X211)) → tappend5_out_aaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_aaa(T83, T81, T82, X210, X211, tappend5_out_aaa(T83, X210, X211)) → tappend5_out_aaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U19_g(T5, tappend5_out_aaa(T83, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U20_g(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_g(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_g(T5, tappend5_in_aaa(T99, X210, X211))
U21_g(T5, tappend5_out_aaa(T99, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U22_g(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_g(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_g(T5, tappend5_in_aaa(T117, X265, X268))
U23_g(T5, tappend5_out_aaa(T117, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U24_g(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_g(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_g(T5, tappend5_in_aaa(T133, X265, X268))
U25_g(T5, tappend5_out_aaa(T133, X265, X268)) → goal1_out_g(T5)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
goal1_out_g(x1)  =  goal1_out_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg(x1, x2)
node(x1, x2, x3)  =  node(x1, x2, x3)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x3, x4)
U12_g(x1, x2)  =  U12_g(x1, x2)
tappend5_in_gaa(x1, x2, x3)  =  tappend5_in_gaa(x1)
tappend5_out_gaa(x1, x2, x3)  =  tappend5_out_gaa(x1)
U4_gaa(x1, x2, x3)  =  U4_gaa(x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x1, x2, x3, x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x2, x3, x6)
U7_gaa(x1, x2)  =  U7_gaa(x2)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x1, x2, x3, x5)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x3, x5)
U13_g(x1, x2)  =  U13_g(x1, x2)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
tappend5_in_aaa(x1, x2, x3)  =  tappend5_in_aaa
tappend5_out_aaa(x1, x2, x3)  =  tappend5_out_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
U5_aaa(x1, x2, x3, x4, x5, x6)  =  U5_aaa(x6)
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
U7_aaa(x1, x2)  =  U7_aaa(x2)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U20_g(x1, x2)  =  U20_g(x1, x2)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
U23_g(x1, x2)  =  U23_g(x1, x2)
U24_g(x1, x2)  =  U24_g(x1, x2)
U25_g(x1, x2)  =  U25_g(x1, x2)
TAPPEND5_IN_GAA(x1, x2, x3)  =  TAPPEND5_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:

TAPPEND5_IN_GAA(nil, nil, node(nil, nil, X268)) → TAPPEND5_IN_GAA(nil, X265, X268)
TAPPEND5_IN_GAA(nil, X210, node(X211, nil, nil)) → TAPPEND5_IN_GAA(nil, X210, X211)

R is empty.
The argument filtering Pi contains the following mapping:
nil  =  nil
node(x1, x2, x3)  =  node(x1, x2, x3)
TAPPEND5_IN_GAA(x1, x2, x3)  =  TAPPEND5_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:

TAPPEND5_IN_GAA(nil) → TAPPEND5_IN_GAA(nil)

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

(55) 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 = TAPPEND5_IN_GAA(nil) evaluates to t =TAPPEND5_IN_GAA(nil)

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 TAPPEND5_IN_GAA(nil) to TAPPEND5_IN_GAA(nil).



(56) FALSE

(57) Obligation:

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

TAPPEND5_IN_AAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → TAPPEND5_IN_AAA(T117, X265, X268)
TAPPEND5_IN_AAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → TAPPEND5_IN_AAA(T83, X210, X211)

The TRS R consists of the following rules:

goal1_in_g(T5) → U10_g(T5, s2t4_in_ga(T5, X15))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T8), node(X34, X35, X34)) → U1_ga(T8, X34, X35, s2t4_in_ga(T8, X34))
s2t4_in_ga(s(T10), node(nil, X48, X49)) → U2_ga(T10, X48, X49, s2t4_in_ga(T10, X49))
s2t4_in_ga(s(T12), node(X62, X63, nil)) → U3_ga(T12, X62, X63, s2t4_in_ga(T12, X62))
s2t4_in_ga(T13, node(nil, X70, nil)) → s2t4_out_ga(T13, node(nil, X70, nil))
U3_ga(T12, X62, X63, s2t4_out_ga(T12, X62)) → s2t4_out_ga(s(T12), node(X62, X63, nil))
U2_ga(T10, X48, X49, s2t4_out_ga(T10, X49)) → s2t4_out_ga(s(T10), node(nil, X48, X49))
U1_ga(T8, X34, X35, s2t4_out_ga(T8, X34)) → s2t4_out_ga(s(T8), node(X34, X35, X34))
U10_g(T5, s2t4_out_ga(T5, X15)) → goal1_out_g(T5)
goal1_in_g(T5) → U11_g(T5, s2t4_in_gg(T5, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T8), node(X34, X35, X34)) → U1_gg(T8, X34, X35, s2t4_in_gg(T8, X34))
s2t4_in_gg(s(T10), node(nil, X48, X49)) → U2_gg(T10, X48, X49, s2t4_in_gg(T10, X49))
s2t4_in_gg(s(T12), node(X62, X63, nil)) → U3_gg(T12, X62, X63, s2t4_in_gg(T12, X62))
s2t4_in_gg(T13, node(nil, X70, nil)) → s2t4_out_gg(T13, node(nil, X70, nil))
U3_gg(T12, X62, X63, s2t4_out_gg(T12, X62)) → s2t4_out_gg(s(T12), node(X62, X63, nil))
U2_gg(T10, X48, X49, s2t4_out_gg(T10, X49)) → s2t4_out_gg(s(T10), node(nil, X48, X49))
U1_gg(T8, X34, X35, s2t4_out_gg(T8, X34)) → s2t4_out_gg(s(T8), node(X34, X35, X34))
U11_g(T5, s2t4_out_gg(T5, nil)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U12_g(T5, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(nil, X79, X79) → tappend5_out_gaa(nil, X79, X79)
tappend5_in_gaa(nil, X92, node(X92, nil, nil)) → tappend5_out_gaa(nil, X92, node(X92, nil, nil))
tappend5_in_gaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_gaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_gaa(nil, X149, node(nil, nil, X149)) → tappend5_out_gaa(nil, X149, node(nil, nil, X149))
tappend5_in_gaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_gaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_gaa(nil, X210, node(X211, nil, nil)) → U4_gaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_gaa(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
tappend5_in_gaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_gaa(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
tappend5_in_gaa(nil, nil, node(nil, nil, X268)) → U7_gaa(X268, tappend5_in_gaa(nil, X265, X268))
tappend5_in_gaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_gaa(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
tappend5_in_gaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_gaa(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U9_gaa(T130, T131, T133, X268, tappend5_out_gaa(T133, X265, X268)) → tappend5_out_gaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_gaa(T114, T115, T117, X268, tappend5_out_gaa(T117, X265, X268)) → tappend5_out_gaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U7_gaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_gaa(nil, nil, node(nil, nil, X268))
U6_gaa(T99, T97, T98, X210, X211, tappend5_out_gaa(T99, X210, X211)) → tappend5_out_gaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_gaa(T83, T81, T82, X210, X211, tappend5_out_gaa(T83, X210, X211)) → tappend5_out_gaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U4_gaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_gaa(nil, X210, node(X211, nil, nil))
U12_g(T5, tappend5_out_gaa(nil, X210, X211)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U13_g(T5, tappend5_in_gaa(nil, X265, X268))
U13_g(T5, tappend5_out_gaa(nil, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U14_g(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
U14_g(T5, s2t4_out_ga(T5, node(nil, T25, T26))) → goal1_out_g(T5)
goal1_in_g(T5) → U15_g(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
U15_g(T5, s2t4_out_ga(T5, node(nil, T36, T37))) → goal1_out_g(T5)
goal1_in_g(T5) → U16_g(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
U16_g(T5, s2t4_out_ga(T5, node(T51, T52, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U17_g(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
U17_g(T5, s2t4_out_ga(T5, node(T64, T65, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U18_g(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
U18_g(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_g(T5, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(nil, X79, X79) → tappend5_out_aaa(nil, X79, X79)
tappend5_in_aaa(nil, X92, node(X92, nil, nil)) → tappend5_out_aaa(nil, X92, node(X92, nil, nil))
tappend5_in_aaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_aaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_aaa(nil, X149, node(nil, nil, X149)) → tappend5_out_aaa(nil, X149, node(nil, nil, X149))
tappend5_in_aaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_aaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_aaa(nil, X210, node(X211, nil, nil)) → U4_aaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
U4_aaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_aaa(nil, X210, node(X211, nil, nil))
tappend5_in_aaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_aaa(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_aaa(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
tappend5_in_aaa(nil, nil, node(nil, nil, X268)) → U7_aaa(X268, tappend5_in_gaa(nil, X265, X268))
U7_aaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_aaa(nil, nil, node(nil, nil, X268))
tappend5_in_aaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_aaa(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
tappend5_in_aaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_aaa(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
U9_aaa(T130, T131, T133, X268, tappend5_out_aaa(T133, X265, X268)) → tappend5_out_aaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_aaa(T114, T115, T117, X268, tappend5_out_aaa(T117, X265, X268)) → tappend5_out_aaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U6_aaa(T99, T97, T98, X210, X211, tappend5_out_aaa(T99, X210, X211)) → tappend5_out_aaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_aaa(T83, T81, T82, X210, X211, tappend5_out_aaa(T83, X210, X211)) → tappend5_out_aaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U19_g(T5, tappend5_out_aaa(T83, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U20_g(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_g(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_g(T5, tappend5_in_aaa(T99, X210, X211))
U21_g(T5, tappend5_out_aaa(T99, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U22_g(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_g(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_g(T5, tappend5_in_aaa(T117, X265, X268))
U23_g(T5, tappend5_out_aaa(T117, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U24_g(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_g(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_g(T5, tappend5_in_aaa(T133, X265, X268))
U25_g(T5, tappend5_out_aaa(T133, X265, X268)) → goal1_out_g(T5)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
goal1_out_g(x1)  =  goal1_out_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg(x1, x2)
node(x1, x2, x3)  =  node(x1, x2, x3)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x3, x4)
U12_g(x1, x2)  =  U12_g(x1, x2)
tappend5_in_gaa(x1, x2, x3)  =  tappend5_in_gaa(x1)
tappend5_out_gaa(x1, x2, x3)  =  tappend5_out_gaa(x1)
U4_gaa(x1, x2, x3)  =  U4_gaa(x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x1, x2, x3, x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x2, x3, x6)
U7_gaa(x1, x2)  =  U7_gaa(x2)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x1, x2, x3, x5)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x3, x5)
U13_g(x1, x2)  =  U13_g(x1, x2)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
tappend5_in_aaa(x1, x2, x3)  =  tappend5_in_aaa
tappend5_out_aaa(x1, x2, x3)  =  tappend5_out_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
U5_aaa(x1, x2, x3, x4, x5, x6)  =  U5_aaa(x6)
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
U7_aaa(x1, x2)  =  U7_aaa(x2)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U20_g(x1, x2)  =  U20_g(x1, x2)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
U23_g(x1, x2)  =  U23_g(x1, x2)
U24_g(x1, x2)  =  U24_g(x1, x2)
U25_g(x1, x2)  =  U25_g(x1, x2)
TAPPEND5_IN_AAA(x1, x2, x3)  =  TAPPEND5_IN_AAA

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:

TAPPEND5_IN_AAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → TAPPEND5_IN_AAA(T117, X265, X268)
TAPPEND5_IN_AAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → TAPPEND5_IN_AAA(T83, X210, X211)

R is empty.
The argument filtering Pi contains the following mapping:
node(x1, x2, x3)  =  node(x1, x2, x3)
TAPPEND5_IN_AAA(x1, x2, x3)  =  TAPPEND5_IN_AAA

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:

TAPPEND5_IN_AAATAPPEND5_IN_AAA

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 = TAPPEND5_IN_AAA evaluates to t =TAPPEND5_IN_AAA

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 TAPPEND5_IN_AAA to TAPPEND5_IN_AAA.



(63) FALSE

(64) Obligation:

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

TAPPEND5_IN_GAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → TAPPEND5_IN_GAA(T117, X265, X268)
TAPPEND5_IN_GAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → TAPPEND5_IN_GAA(T83, X210, X211)

The TRS R consists of the following rules:

goal1_in_g(T5) → U10_g(T5, s2t4_in_ga(T5, X15))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T8), node(X34, X35, X34)) → U1_ga(T8, X34, X35, s2t4_in_ga(T8, X34))
s2t4_in_ga(s(T10), node(nil, X48, X49)) → U2_ga(T10, X48, X49, s2t4_in_ga(T10, X49))
s2t4_in_ga(s(T12), node(X62, X63, nil)) → U3_ga(T12, X62, X63, s2t4_in_ga(T12, X62))
s2t4_in_ga(T13, node(nil, X70, nil)) → s2t4_out_ga(T13, node(nil, X70, nil))
U3_ga(T12, X62, X63, s2t4_out_ga(T12, X62)) → s2t4_out_ga(s(T12), node(X62, X63, nil))
U2_ga(T10, X48, X49, s2t4_out_ga(T10, X49)) → s2t4_out_ga(s(T10), node(nil, X48, X49))
U1_ga(T8, X34, X35, s2t4_out_ga(T8, X34)) → s2t4_out_ga(s(T8), node(X34, X35, X34))
U10_g(T5, s2t4_out_ga(T5, X15)) → goal1_out_g(T5)
goal1_in_g(T5) → U11_g(T5, s2t4_in_gg(T5, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T8), node(X34, X35, X34)) → U1_gg(T8, X34, X35, s2t4_in_gg(T8, X34))
s2t4_in_gg(s(T10), node(nil, X48, X49)) → U2_gg(T10, X48, X49, s2t4_in_gg(T10, X49))
s2t4_in_gg(s(T12), node(X62, X63, nil)) → U3_gg(T12, X62, X63, s2t4_in_gg(T12, X62))
s2t4_in_gg(T13, node(nil, X70, nil)) → s2t4_out_gg(T13, node(nil, X70, nil))
U3_gg(T12, X62, X63, s2t4_out_gg(T12, X62)) → s2t4_out_gg(s(T12), node(X62, X63, nil))
U2_gg(T10, X48, X49, s2t4_out_gg(T10, X49)) → s2t4_out_gg(s(T10), node(nil, X48, X49))
U1_gg(T8, X34, X35, s2t4_out_gg(T8, X34)) → s2t4_out_gg(s(T8), node(X34, X35, X34))
U11_g(T5, s2t4_out_gg(T5, nil)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U12_g(T5, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(nil, X79, X79) → tappend5_out_gaa(nil, X79, X79)
tappend5_in_gaa(nil, X92, node(X92, nil, nil)) → tappend5_out_gaa(nil, X92, node(X92, nil, nil))
tappend5_in_gaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_gaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_gaa(nil, X149, node(nil, nil, X149)) → tappend5_out_gaa(nil, X149, node(nil, nil, X149))
tappend5_in_gaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_gaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_gaa(nil, X210, node(X211, nil, nil)) → U4_gaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_gaa(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
tappend5_in_gaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_gaa(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
tappend5_in_gaa(nil, nil, node(nil, nil, X268)) → U7_gaa(X268, tappend5_in_gaa(nil, X265, X268))
tappend5_in_gaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_gaa(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
tappend5_in_gaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_gaa(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U9_gaa(T130, T131, T133, X268, tappend5_out_gaa(T133, X265, X268)) → tappend5_out_gaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_gaa(T114, T115, T117, X268, tappend5_out_gaa(T117, X265, X268)) → tappend5_out_gaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U7_gaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_gaa(nil, nil, node(nil, nil, X268))
U6_gaa(T99, T97, T98, X210, X211, tappend5_out_gaa(T99, X210, X211)) → tappend5_out_gaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_gaa(T83, T81, T82, X210, X211, tappend5_out_gaa(T83, X210, X211)) → tappend5_out_gaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U4_gaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_gaa(nil, X210, node(X211, nil, nil))
U12_g(T5, tappend5_out_gaa(nil, X210, X211)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U13_g(T5, tappend5_in_gaa(nil, X265, X268))
U13_g(T5, tappend5_out_gaa(nil, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U14_g(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
U14_g(T5, s2t4_out_ga(T5, node(nil, T25, T26))) → goal1_out_g(T5)
goal1_in_g(T5) → U15_g(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
U15_g(T5, s2t4_out_ga(T5, node(nil, T36, T37))) → goal1_out_g(T5)
goal1_in_g(T5) → U16_g(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
U16_g(T5, s2t4_out_ga(T5, node(T51, T52, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U17_g(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
U17_g(T5, s2t4_out_ga(T5, node(T64, T65, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U18_g(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
U18_g(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_g(T5, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(nil, X79, X79) → tappend5_out_aaa(nil, X79, X79)
tappend5_in_aaa(nil, X92, node(X92, nil, nil)) → tappend5_out_aaa(nil, X92, node(X92, nil, nil))
tappend5_in_aaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_aaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_aaa(nil, X149, node(nil, nil, X149)) → tappend5_out_aaa(nil, X149, node(nil, nil, X149))
tappend5_in_aaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_aaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_aaa(nil, X210, node(X211, nil, nil)) → U4_aaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
U4_aaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_aaa(nil, X210, node(X211, nil, nil))
tappend5_in_aaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_aaa(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_aaa(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
tappend5_in_aaa(nil, nil, node(nil, nil, X268)) → U7_aaa(X268, tappend5_in_gaa(nil, X265, X268))
U7_aaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_aaa(nil, nil, node(nil, nil, X268))
tappend5_in_aaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_aaa(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
tappend5_in_aaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_aaa(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
U9_aaa(T130, T131, T133, X268, tappend5_out_aaa(T133, X265, X268)) → tappend5_out_aaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_aaa(T114, T115, T117, X268, tappend5_out_aaa(T117, X265, X268)) → tappend5_out_aaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U6_aaa(T99, T97, T98, X210, X211, tappend5_out_aaa(T99, X210, X211)) → tappend5_out_aaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_aaa(T83, T81, T82, X210, X211, tappend5_out_aaa(T83, X210, X211)) → tappend5_out_aaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U19_g(T5, tappend5_out_aaa(T83, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U20_g(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_g(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_g(T5, tappend5_in_aaa(T99, X210, X211))
U21_g(T5, tappend5_out_aaa(T99, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U22_g(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_g(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_g(T5, tappend5_in_aaa(T117, X265, X268))
U23_g(T5, tappend5_out_aaa(T117, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U24_g(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_g(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_g(T5, tappend5_in_aaa(T133, X265, X268))
U25_g(T5, tappend5_out_aaa(T133, X265, X268)) → goal1_out_g(T5)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
goal1_out_g(x1)  =  goal1_out_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg(x1, x2)
node(x1, x2, x3)  =  node(x1, x2, x3)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x3, x4)
U12_g(x1, x2)  =  U12_g(x1, x2)
tappend5_in_gaa(x1, x2, x3)  =  tappend5_in_gaa(x1)
tappend5_out_gaa(x1, x2, x3)  =  tappend5_out_gaa(x1)
U4_gaa(x1, x2, x3)  =  U4_gaa(x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x1, x2, x3, x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x2, x3, x6)
U7_gaa(x1, x2)  =  U7_gaa(x2)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x1, x2, x3, x5)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x3, x5)
U13_g(x1, x2)  =  U13_g(x1, x2)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
tappend5_in_aaa(x1, x2, x3)  =  tappend5_in_aaa
tappend5_out_aaa(x1, x2, x3)  =  tappend5_out_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
U5_aaa(x1, x2, x3, x4, x5, x6)  =  U5_aaa(x6)
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
U7_aaa(x1, x2)  =  U7_aaa(x2)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U20_g(x1, x2)  =  U20_g(x1, x2)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
U23_g(x1, x2)  =  U23_g(x1, x2)
U24_g(x1, x2)  =  U24_g(x1, x2)
U25_g(x1, x2)  =  U25_g(x1, x2)
TAPPEND5_IN_GAA(x1, x2, x3)  =  TAPPEND5_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:

TAPPEND5_IN_GAA(node(T114, T115, T117), T114, node(T114, T115, X268)) → TAPPEND5_IN_GAA(T117, X265, X268)
TAPPEND5_IN_GAA(node(T83, T81, T82), X210, node(X211, T81, T82)) → TAPPEND5_IN_GAA(T83, X210, X211)

R is empty.
The argument filtering Pi contains the following mapping:
node(x1, x2, x3)  =  node(x1, x2, x3)
TAPPEND5_IN_GAA(x1, x2, x3)  =  TAPPEND5_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:

TAPPEND5_IN_GAA(node(T114, T115, T117)) → TAPPEND5_IN_GAA(T117)
TAPPEND5_IN_GAA(node(T83, T81, T82)) → TAPPEND5_IN_GAA(T83)

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:

  • TAPPEND5_IN_GAA(node(T114, T115, T117)) → TAPPEND5_IN_GAA(T117)
    The graph contains the following edges 1 > 1

  • TAPPEND5_IN_GAA(node(T83, T81, T82)) → TAPPEND5_IN_GAA(T83)
    The graph contains the following edges 1 > 1

(70) TRUE

(71) Obligation:

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

S2T4_IN_GG(s(T10), node(nil, X48, X49)) → S2T4_IN_GG(T10, X49)
S2T4_IN_GG(s(T8), node(X34, X35, X34)) → S2T4_IN_GG(T8, X34)
S2T4_IN_GG(s(T12), node(X62, X63, nil)) → S2T4_IN_GG(T12, X62)

The TRS R consists of the following rules:

goal1_in_g(T5) → U10_g(T5, s2t4_in_ga(T5, X15))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T8), node(X34, X35, X34)) → U1_ga(T8, X34, X35, s2t4_in_ga(T8, X34))
s2t4_in_ga(s(T10), node(nil, X48, X49)) → U2_ga(T10, X48, X49, s2t4_in_ga(T10, X49))
s2t4_in_ga(s(T12), node(X62, X63, nil)) → U3_ga(T12, X62, X63, s2t4_in_ga(T12, X62))
s2t4_in_ga(T13, node(nil, X70, nil)) → s2t4_out_ga(T13, node(nil, X70, nil))
U3_ga(T12, X62, X63, s2t4_out_ga(T12, X62)) → s2t4_out_ga(s(T12), node(X62, X63, nil))
U2_ga(T10, X48, X49, s2t4_out_ga(T10, X49)) → s2t4_out_ga(s(T10), node(nil, X48, X49))
U1_ga(T8, X34, X35, s2t4_out_ga(T8, X34)) → s2t4_out_ga(s(T8), node(X34, X35, X34))
U10_g(T5, s2t4_out_ga(T5, X15)) → goal1_out_g(T5)
goal1_in_g(T5) → U11_g(T5, s2t4_in_gg(T5, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T8), node(X34, X35, X34)) → U1_gg(T8, X34, X35, s2t4_in_gg(T8, X34))
s2t4_in_gg(s(T10), node(nil, X48, X49)) → U2_gg(T10, X48, X49, s2t4_in_gg(T10, X49))
s2t4_in_gg(s(T12), node(X62, X63, nil)) → U3_gg(T12, X62, X63, s2t4_in_gg(T12, X62))
s2t4_in_gg(T13, node(nil, X70, nil)) → s2t4_out_gg(T13, node(nil, X70, nil))
U3_gg(T12, X62, X63, s2t4_out_gg(T12, X62)) → s2t4_out_gg(s(T12), node(X62, X63, nil))
U2_gg(T10, X48, X49, s2t4_out_gg(T10, X49)) → s2t4_out_gg(s(T10), node(nil, X48, X49))
U1_gg(T8, X34, X35, s2t4_out_gg(T8, X34)) → s2t4_out_gg(s(T8), node(X34, X35, X34))
U11_g(T5, s2t4_out_gg(T5, nil)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U12_g(T5, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(nil, X79, X79) → tappend5_out_gaa(nil, X79, X79)
tappend5_in_gaa(nil, X92, node(X92, nil, nil)) → tappend5_out_gaa(nil, X92, node(X92, nil, nil))
tappend5_in_gaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_gaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_gaa(nil, X149, node(nil, nil, X149)) → tappend5_out_gaa(nil, X149, node(nil, nil, X149))
tappend5_in_gaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_gaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_gaa(nil, X210, node(X211, nil, nil)) → U4_gaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_gaa(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
tappend5_in_gaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_gaa(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
tappend5_in_gaa(nil, nil, node(nil, nil, X268)) → U7_gaa(X268, tappend5_in_gaa(nil, X265, X268))
tappend5_in_gaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_gaa(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
tappend5_in_gaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_gaa(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U9_gaa(T130, T131, T133, X268, tappend5_out_gaa(T133, X265, X268)) → tappend5_out_gaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_gaa(T114, T115, T117, X268, tappend5_out_gaa(T117, X265, X268)) → tappend5_out_gaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U7_gaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_gaa(nil, nil, node(nil, nil, X268))
U6_gaa(T99, T97, T98, X210, X211, tappend5_out_gaa(T99, X210, X211)) → tappend5_out_gaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_gaa(T83, T81, T82, X210, X211, tappend5_out_gaa(T83, X210, X211)) → tappend5_out_gaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U4_gaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_gaa(nil, X210, node(X211, nil, nil))
U12_g(T5, tappend5_out_gaa(nil, X210, X211)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U13_g(T5, tappend5_in_gaa(nil, X265, X268))
U13_g(T5, tappend5_out_gaa(nil, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U14_g(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
U14_g(T5, s2t4_out_ga(T5, node(nil, T25, T26))) → goal1_out_g(T5)
goal1_in_g(T5) → U15_g(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
U15_g(T5, s2t4_out_ga(T5, node(nil, T36, T37))) → goal1_out_g(T5)
goal1_in_g(T5) → U16_g(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
U16_g(T5, s2t4_out_ga(T5, node(T51, T52, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U17_g(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
U17_g(T5, s2t4_out_ga(T5, node(T64, T65, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U18_g(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
U18_g(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_g(T5, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(nil, X79, X79) → tappend5_out_aaa(nil, X79, X79)
tappend5_in_aaa(nil, X92, node(X92, nil, nil)) → tappend5_out_aaa(nil, X92, node(X92, nil, nil))
tappend5_in_aaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_aaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_aaa(nil, X149, node(nil, nil, X149)) → tappend5_out_aaa(nil, X149, node(nil, nil, X149))
tappend5_in_aaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_aaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_aaa(nil, X210, node(X211, nil, nil)) → U4_aaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
U4_aaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_aaa(nil, X210, node(X211, nil, nil))
tappend5_in_aaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_aaa(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_aaa(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
tappend5_in_aaa(nil, nil, node(nil, nil, X268)) → U7_aaa(X268, tappend5_in_gaa(nil, X265, X268))
U7_aaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_aaa(nil, nil, node(nil, nil, X268))
tappend5_in_aaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_aaa(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
tappend5_in_aaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_aaa(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
U9_aaa(T130, T131, T133, X268, tappend5_out_aaa(T133, X265, X268)) → tappend5_out_aaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_aaa(T114, T115, T117, X268, tappend5_out_aaa(T117, X265, X268)) → tappend5_out_aaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U6_aaa(T99, T97, T98, X210, X211, tappend5_out_aaa(T99, X210, X211)) → tappend5_out_aaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_aaa(T83, T81, T82, X210, X211, tappend5_out_aaa(T83, X210, X211)) → tappend5_out_aaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U19_g(T5, tappend5_out_aaa(T83, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U20_g(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_g(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_g(T5, tappend5_in_aaa(T99, X210, X211))
U21_g(T5, tappend5_out_aaa(T99, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U22_g(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_g(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_g(T5, tappend5_in_aaa(T117, X265, X268))
U23_g(T5, tappend5_out_aaa(T117, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U24_g(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_g(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_g(T5, tappend5_in_aaa(T133, X265, X268))
U25_g(T5, tappend5_out_aaa(T133, X265, X268)) → goal1_out_g(T5)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
goal1_out_g(x1)  =  goal1_out_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg(x1, x2)
node(x1, x2, x3)  =  node(x1, x2, x3)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x3, x4)
U12_g(x1, x2)  =  U12_g(x1, x2)
tappend5_in_gaa(x1, x2, x3)  =  tappend5_in_gaa(x1)
tappend5_out_gaa(x1, x2, x3)  =  tappend5_out_gaa(x1)
U4_gaa(x1, x2, x3)  =  U4_gaa(x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x1, x2, x3, x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x2, x3, x6)
U7_gaa(x1, x2)  =  U7_gaa(x2)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x1, x2, x3, x5)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x3, x5)
U13_g(x1, x2)  =  U13_g(x1, x2)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
tappend5_in_aaa(x1, x2, x3)  =  tappend5_in_aaa
tappend5_out_aaa(x1, x2, x3)  =  tappend5_out_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
U5_aaa(x1, x2, x3, x4, x5, x6)  =  U5_aaa(x6)
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
U7_aaa(x1, x2)  =  U7_aaa(x2)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U20_g(x1, x2)  =  U20_g(x1, x2)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
U23_g(x1, x2)  =  U23_g(x1, x2)
U24_g(x1, x2)  =  U24_g(x1, x2)
U25_g(x1, x2)  =  U25_g(x1, x2)
S2T4_IN_GG(x1, x2)  =  S2T4_IN_GG(x1, 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:

S2T4_IN_GG(s(T10), node(nil, X48, X49)) → S2T4_IN_GG(T10, X49)
S2T4_IN_GG(s(T8), node(X34, X35, X34)) → S2T4_IN_GG(T8, X34)
S2T4_IN_GG(s(T12), node(X62, X63, nil)) → S2T4_IN_GG(T12, X62)

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

(74) PiDPToQDPProof (EQUIVALENT 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:

S2T4_IN_GG(s(T10), node(nil, X48, X49)) → S2T4_IN_GG(T10, X49)
S2T4_IN_GG(s(T8), node(X34, X35, X34)) → S2T4_IN_GG(T8, X34)
S2T4_IN_GG(s(T12), node(X62, X63, nil)) → S2T4_IN_GG(T12, X62)

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

(76) 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:

  • S2T4_IN_GG(s(T10), node(nil, X48, X49)) → S2T4_IN_GG(T10, X49)
    The graph contains the following edges 1 > 1, 2 > 2

  • S2T4_IN_GG(s(T8), node(X34, X35, X34)) → S2T4_IN_GG(T8, X34)
    The graph contains the following edges 1 > 1, 2 > 2

  • S2T4_IN_GG(s(T12), node(X62, X63, nil)) → S2T4_IN_GG(T12, X62)
    The graph contains the following edges 1 > 1, 2 > 2

(77) TRUE

(78) Obligation:

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

S2T4_IN_GA(s(T10), node(nil, X48, X49)) → S2T4_IN_GA(T10, X49)
S2T4_IN_GA(s(T8), node(X34, X35, X34)) → S2T4_IN_GA(T8, X34)
S2T4_IN_GA(s(T12), node(X62, X63, nil)) → S2T4_IN_GA(T12, X62)

The TRS R consists of the following rules:

goal1_in_g(T5) → U10_g(T5, s2t4_in_ga(T5, X15))
s2t4_in_ga(0, nil) → s2t4_out_ga(0, nil)
s2t4_in_ga(s(T8), node(X34, X35, X34)) → U1_ga(T8, X34, X35, s2t4_in_ga(T8, X34))
s2t4_in_ga(s(T10), node(nil, X48, X49)) → U2_ga(T10, X48, X49, s2t4_in_ga(T10, X49))
s2t4_in_ga(s(T12), node(X62, X63, nil)) → U3_ga(T12, X62, X63, s2t4_in_ga(T12, X62))
s2t4_in_ga(T13, node(nil, X70, nil)) → s2t4_out_ga(T13, node(nil, X70, nil))
U3_ga(T12, X62, X63, s2t4_out_ga(T12, X62)) → s2t4_out_ga(s(T12), node(X62, X63, nil))
U2_ga(T10, X48, X49, s2t4_out_ga(T10, X49)) → s2t4_out_ga(s(T10), node(nil, X48, X49))
U1_ga(T8, X34, X35, s2t4_out_ga(T8, X34)) → s2t4_out_ga(s(T8), node(X34, X35, X34))
U10_g(T5, s2t4_out_ga(T5, X15)) → goal1_out_g(T5)
goal1_in_g(T5) → U11_g(T5, s2t4_in_gg(T5, nil))
s2t4_in_gg(0, nil) → s2t4_out_gg(0, nil)
s2t4_in_gg(s(T8), node(X34, X35, X34)) → U1_gg(T8, X34, X35, s2t4_in_gg(T8, X34))
s2t4_in_gg(s(T10), node(nil, X48, X49)) → U2_gg(T10, X48, X49, s2t4_in_gg(T10, X49))
s2t4_in_gg(s(T12), node(X62, X63, nil)) → U3_gg(T12, X62, X63, s2t4_in_gg(T12, X62))
s2t4_in_gg(T13, node(nil, X70, nil)) → s2t4_out_gg(T13, node(nil, X70, nil))
U3_gg(T12, X62, X63, s2t4_out_gg(T12, X62)) → s2t4_out_gg(s(T12), node(X62, X63, nil))
U2_gg(T10, X48, X49, s2t4_out_gg(T10, X49)) → s2t4_out_gg(s(T10), node(nil, X48, X49))
U1_gg(T8, X34, X35, s2t4_out_gg(T8, X34)) → s2t4_out_gg(s(T8), node(X34, X35, X34))
U11_g(T5, s2t4_out_gg(T5, nil)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U12_g(T5, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(nil, X79, X79) → tappend5_out_gaa(nil, X79, X79)
tappend5_in_gaa(nil, X92, node(X92, nil, nil)) → tappend5_out_gaa(nil, X92, node(X92, nil, nil))
tappend5_in_gaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_gaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_gaa(nil, X149, node(nil, nil, X149)) → tappend5_out_gaa(nil, X149, node(nil, nil, X149))
tappend5_in_gaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_gaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_gaa(nil, X210, node(X211, nil, nil)) → U4_gaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
tappend5_in_gaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_gaa(T83, T81, T82, X210, X211, tappend5_in_gaa(T83, X210, X211))
tappend5_in_gaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_gaa(T99, T97, T98, X210, X211, tappend5_in_gaa(T99, X210, X211))
tappend5_in_gaa(nil, nil, node(nil, nil, X268)) → U7_gaa(X268, tappend5_in_gaa(nil, X265, X268))
tappend5_in_gaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_gaa(T114, T115, T117, X268, tappend5_in_gaa(T117, X265, X268))
tappend5_in_gaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_gaa(T130, T131, T133, X268, tappend5_in_gaa(T133, X265, X268))
U9_gaa(T130, T131, T133, X268, tappend5_out_gaa(T133, X265, X268)) → tappend5_out_gaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_gaa(T114, T115, T117, X268, tappend5_out_gaa(T117, X265, X268)) → tappend5_out_gaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U7_gaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_gaa(nil, nil, node(nil, nil, X268))
U6_gaa(T99, T97, T98, X210, X211, tappend5_out_gaa(T99, X210, X211)) → tappend5_out_gaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_gaa(T83, T81, T82, X210, X211, tappend5_out_gaa(T83, X210, X211)) → tappend5_out_gaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U4_gaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_gaa(nil, X210, node(X211, nil, nil))
U12_g(T5, tappend5_out_gaa(nil, X210, X211)) → goal1_out_g(T5)
U11_g(T5, s2t4_out_gg(T5, nil)) → U13_g(T5, tappend5_in_gaa(nil, X265, X268))
U13_g(T5, tappend5_out_gaa(nil, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U14_g(T5, s2t4_in_ga(T5, node(nil, T25, T26)))
U14_g(T5, s2t4_out_ga(T5, node(nil, T25, T26))) → goal1_out_g(T5)
goal1_in_g(T5) → U15_g(T5, s2t4_in_ga(T5, node(nil, T36, T37)))
U15_g(T5, s2t4_out_ga(T5, node(nil, T36, T37))) → goal1_out_g(T5)
goal1_in_g(T5) → U16_g(T5, s2t4_in_ga(T5, node(T51, T52, nil)))
U16_g(T5, s2t4_out_ga(T5, node(T51, T52, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U17_g(T5, s2t4_in_ga(T5, node(T64, T65, nil)))
U17_g(T5, s2t4_out_ga(T5, node(T64, T65, nil))) → goal1_out_g(T5)
goal1_in_g(T5) → U18_g(T5, s2t4_in_ga(T5, node(T83, T81, T82)))
U18_g(T5, s2t4_out_ga(T5, node(T83, T81, T82))) → U19_g(T5, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(nil, X79, X79) → tappend5_out_aaa(nil, X79, X79)
tappend5_in_aaa(nil, X92, node(X92, nil, nil)) → tappend5_out_aaa(nil, X92, node(X92, nil, nil))
tappend5_in_aaa(node(nil, T25, T26), X92, node(X92, T25, T26)) → tappend5_out_aaa(node(nil, T25, T26), X92, node(X92, T25, T26))
tappend5_in_aaa(nil, X149, node(nil, nil, X149)) → tappend5_out_aaa(nil, X149, node(nil, nil, X149))
tappend5_in_aaa(node(T51, T52, nil), X149, node(T51, T52, X149)) → tappend5_out_aaa(node(T51, T52, nil), X149, node(T51, T52, X149))
tappend5_in_aaa(nil, X210, node(X211, nil, nil)) → U4_aaa(X210, X211, tappend5_in_gaa(nil, X210, X211))
U4_aaa(X210, X211, tappend5_out_gaa(nil, X210, X211)) → tappend5_out_aaa(nil, X210, node(X211, nil, nil))
tappend5_in_aaa(node(T83, T81, T82), X210, node(X211, T81, T82)) → U5_aaa(T83, T81, T82, X210, X211, tappend5_in_aaa(T83, X210, X211))
tappend5_in_aaa(node(T99, T97, T98), X210, node(X211, T97, T98)) → U6_aaa(T99, T97, T98, X210, X211, tappend5_in_aaa(T99, X210, X211))
tappend5_in_aaa(nil, nil, node(nil, nil, X268)) → U7_aaa(X268, tappend5_in_gaa(nil, X265, X268))
U7_aaa(X268, tappend5_out_gaa(nil, X265, X268)) → tappend5_out_aaa(nil, nil, node(nil, nil, X268))
tappend5_in_aaa(node(T114, T115, T117), T114, node(T114, T115, X268)) → U8_aaa(T114, T115, T117, X268, tappend5_in_aaa(T117, X265, X268))
tappend5_in_aaa(node(T130, T131, T133), T130, node(T130, T131, X268)) → U9_aaa(T130, T131, T133, X268, tappend5_in_aaa(T133, X265, X268))
U9_aaa(T130, T131, T133, X268, tappend5_out_aaa(T133, X265, X268)) → tappend5_out_aaa(node(T130, T131, T133), T130, node(T130, T131, X268))
U8_aaa(T114, T115, T117, X268, tappend5_out_aaa(T117, X265, X268)) → tappend5_out_aaa(node(T114, T115, T117), T114, node(T114, T115, X268))
U6_aaa(T99, T97, T98, X210, X211, tappend5_out_aaa(T99, X210, X211)) → tappend5_out_aaa(node(T99, T97, T98), X210, node(X211, T97, T98))
U5_aaa(T83, T81, T82, X210, X211, tappend5_out_aaa(T83, X210, X211)) → tappend5_out_aaa(node(T83, T81, T82), X210, node(X211, T81, T82))
U19_g(T5, tappend5_out_aaa(T83, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U20_g(T5, s2t4_in_ga(T5, node(T99, T97, T98)))
U20_g(T5, s2t4_out_ga(T5, node(T99, T97, T98))) → U21_g(T5, tappend5_in_aaa(T99, X210, X211))
U21_g(T5, tappend5_out_aaa(T99, X210, X211)) → goal1_out_g(T5)
goal1_in_g(T5) → U22_g(T5, s2t4_in_ga(T5, node(T114, T115, T117)))
U22_g(T5, s2t4_out_ga(T5, node(T114, T115, T117))) → U23_g(T5, tappend5_in_aaa(T117, X265, X268))
U23_g(T5, tappend5_out_aaa(T117, X265, X268)) → goal1_out_g(T5)
goal1_in_g(T5) → U24_g(T5, s2t4_in_ga(T5, node(T130, T131, T133)))
U24_g(T5, s2t4_out_ga(T5, node(T130, T131, T133))) → U25_g(T5, tappend5_in_aaa(T133, X265, X268))
U25_g(T5, tappend5_out_aaa(T133, X265, X268)) → goal1_out_g(T5)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U10_g(x1, x2)  =  U10_g(x1, x2)
s2t4_in_ga(x1, x2)  =  s2t4_in_ga(x1)
0  =  0
s2t4_out_ga(x1, x2)  =  s2t4_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
goal1_out_g(x1)  =  goal1_out_g(x1)
U11_g(x1, x2)  =  U11_g(x1, x2)
s2t4_in_gg(x1, x2)  =  s2t4_in_gg(x1, x2)
nil  =  nil
s2t4_out_gg(x1, x2)  =  s2t4_out_gg(x1, x2)
node(x1, x2, x3)  =  node(x1, x2, x3)
U1_gg(x1, x2, x3, x4)  =  U1_gg(x1, x2, x3, x4)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x2, x3, x4)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x3, x4)
U12_g(x1, x2)  =  U12_g(x1, x2)
tappend5_in_gaa(x1, x2, x3)  =  tappend5_in_gaa(x1)
tappend5_out_gaa(x1, x2, x3)  =  tappend5_out_gaa(x1)
U4_gaa(x1, x2, x3)  =  U4_gaa(x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x1, x2, x3, x6)
U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x2, x3, x6)
U7_gaa(x1, x2)  =  U7_gaa(x2)
U8_gaa(x1, x2, x3, x4, x5)  =  U8_gaa(x1, x2, x3, x5)
U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x3, x5)
U13_g(x1, x2)  =  U13_g(x1, x2)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
tappend5_in_aaa(x1, x2, x3)  =  tappend5_in_aaa
tappend5_out_aaa(x1, x2, x3)  =  tappend5_out_aaa
U4_aaa(x1, x2, x3)  =  U4_aaa(x3)
U5_aaa(x1, x2, x3, x4, x5, x6)  =  U5_aaa(x6)
U6_aaa(x1, x2, x3, x4, x5, x6)  =  U6_aaa(x6)
U7_aaa(x1, x2)  =  U7_aaa(x2)
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U20_g(x1, x2)  =  U20_g(x1, x2)
U21_g(x1, x2)  =  U21_g(x1, x2)
U22_g(x1, x2)  =  U22_g(x1, x2)
U23_g(x1, x2)  =  U23_g(x1, x2)
U24_g(x1, x2)  =  U24_g(x1, x2)
U25_g(x1, x2)  =  U25_g(x1, x2)
S2T4_IN_GA(x1, x2)  =  S2T4_IN_GA(x1)

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:

S2T4_IN_GA(s(T10), node(nil, X48, X49)) → S2T4_IN_GA(T10, X49)
S2T4_IN_GA(s(T8), node(X34, X35, X34)) → S2T4_IN_GA(T8, X34)
S2T4_IN_GA(s(T12), node(X62, X63, nil)) → S2T4_IN_GA(T12, X62)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
nil  =  nil
node(x1, x2, x3)  =  node(x1, x2, x3)
S2T4_IN_GA(x1, x2)  =  S2T4_IN_GA(x1)

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:

S2T4_IN_GA(s(T10)) → S2T4_IN_GA(T10)

R is empty.
Q is empty.
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:

  • S2T4_IN_GA(s(T10)) → S2T4_IN_GA(T10)
    The graph contains the following edges 1 > 1

(84) TRUE