(0) Obligation:

Clauses:

qs([], []).
qs(cons(X, L), S) :- ','(split(L, X, L1, L2), ','(qs(L1, S1), ','(qs(L2, S2), append(S1, cons(X, S2), S)))).
append([], L, L).
append(cons(X, L1), L2, cons(X, L3)) :- append(L1, L2, L3).
split([], X, [], []).
split(cons(X, L), Y, cons(X, L1), L2) :- ','(less(X, Y), split(L, Y, L1, L2)).
split(cons(X, L), Y, L1, cons(X, L2)) :- ','(geq(X, Y), split(L, Y, L1, L2)).
less(0, s(X)).
less(s(X), s(Y)) :- less(X, Y).
geq(X, X).
geq(s(X), 0).
geq(s(X), s(Y)) :- geq(X, Y).

Queries:

qs(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

append23(cons(T45, T46), T47, T48, cons(T45, T50)) :- append23(T46, T47, T48, T50).
less36(s(T83), s(T84)) :- less36(T83, T84).
split46(cons(T110, T111), T112, cons(T110, X163), X164) :- less36(T110, T112).
split46(cons(T110, T111), T112, cons(T110, X163), X164) :- ','(lessc36(T110, T112), split46(T111, T112, X163, X164)).
split46(cons(T125, T126), T127, X195, cons(T125, X196)) :- geq62(T125, T127).
split46(cons(T125, T126), T127, X195, cons(T125, X196)) :- ','(geqc62(T125, T127), split46(T126, T127, X195, X196)).
geq62(s(T146), s(T147)) :- geq62(T146, T147).
qs79(cons(T162, T163), X265) :- split46(T163, T162, X261, X262).
qs79(cons(T162, T163), X265) :- ','(splitc46(T163, T162, T166, T167), qs79(T166, X263)).
qs79(cons(T162, T163), X265) :- ','(splitc46(T163, T162, T166, T167), ','(qsc79(T166, T170), qs79(T167, X264))).
qs79(cons(T162, T163), X265) :- ','(splitc46(T163, T162, T166, T167), ','(qsc79(T166, T170), ','(qsc79(T167, T171), append94(T170, T162, T171, X265)))).
append94(cons(T194, T195), T196, T197, cons(T194, X295)) :- append94(T195, T196, T197, X295).
p78(T90, X13, T154, T69, T9) :- qs79(T90, X13).
p78(T90, T157, T154, T69, T9) :- ','(qsc79(T90, T157), append23(T154, T69, T157, T9)).
qs1(cons(T14, []), T9) :- qs14(X12).
qs1(cons(T14, []), T9) :- ','(qsc14(T17), qs14(X13)).
qs1(cons(T14, []), T9) :- ','(qsc14(T17), ','(qsc14(T20), append23(T17, T14, T20, T9))).
qs1(cons(T69, cons(T67, T68)), T9) :- less36(T67, T69).
qs1(cons(T69, cons(T67, T68)), T9) :- ','(lessc36(T67, T69), split46(T68, T69, X93, X94)).
qs1(cons(T69, cons(T67, T68)), T9) :- ','(lessc36(T67, T69), ','(splitc46(T68, T69, T89, T90), qs1(cons(T67, T89), X12))).
qs1(cons(T69, cons(T67, T68)), T9) :- ','(lessc36(T67, T69), ','(splitc46(T68, T69, T89, T90), ','(qsc1(cons(T67, T89), T154), p78(T90, X13, T154, T69, T9)))).
qs1(cons(T214, cons(T212, T213)), T9) :- geq62(T212, T214).
qs1(cons(T214, cons(T212, T213)), T9) :- ','(geqc62(T212, T214), split46(T213, T214, X323, X324)).
qs1(cons(T214, cons(T212, T213)), T9) :- ','(geqc62(T212, T214), ','(splitc46(T213, T214, T221, T222), qs79(T221, X12))).
qs1(cons(T214, cons(T212, T213)), T9) :- ','(geqc62(T212, T214), ','(splitc46(T213, T214, T221, T222), ','(qsc79(T221, T225), p78(cons(T212, T222), X13, T225, T214, T9)))).

Clauses:

qsc14([]).
appendc23([], T33, T34, cons(T33, T34)).
appendc23(cons(T45, T46), T47, T48, cons(T45, T50)) :- appendc23(T46, T47, T48, T50).
lessc36(0, s(T78)).
lessc36(s(T83), s(T84)) :- lessc36(T83, T84).
splitc46([], T97, [], []).
splitc46(cons(T110, T111), T112, cons(T110, X163), X164) :- ','(lessc36(T110, T112), splitc46(T111, T112, X163, X164)).
splitc46(cons(T125, T126), T127, X195, cons(T125, X196)) :- ','(geqc62(T125, T127), splitc46(T126, T127, X195, X196)).
geqc62(T136, T136).
geqc62(s(T141), 0).
geqc62(s(T146), s(T147)) :- geqc62(T146, T147).
qsc1([], []).
qsc1(cons(T14, []), T9) :- ','(qsc14(T17), ','(qsc14(T20), appendc23(T17, T14, T20, T9))).
qsc1(cons(T69, cons(T67, T68)), T9) :- ','(lessc36(T67, T69), ','(splitc46(T68, T69, T89, T90), ','(qsc1(cons(T67, T89), T154), qc78(T90, X13, T154, T69, T9)))).
qsc1(cons(T214, cons(T212, T213)), T9) :- ','(geqc62(T212, T214), ','(splitc46(T213, T214, T221, T222), ','(qsc79(T221, T225), qc78(cons(T212, T222), X13, T225, T214, T9)))).
qsc79([], []).
qsc79(cons(T162, T163), X265) :- ','(splitc46(T163, T162, T166, T167), ','(qsc79(T166, T170), ','(qsc79(T167, T171), appendc94(T170, T162, T171, X265)))).
appendc94([], T184, T185, cons(T184, T185)).
appendc94(cons(T194, T195), T196, T197, cons(T194, X295)) :- appendc94(T195, T196, T197, X295).
qc78(T90, T157, T154, T69, T9) :- ','(qsc79(T90, T157), appendc23(T154, T69, T157, T9)).

Afs:

qs1(x1, x2)  =  qs1(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

append23(cons(T45, T46), T47, T48, cons(T45, T50)) :- append23(T46, T47, T48, T50).
less36(s(T83), s(T84)) :- less36(T83, T84).
split46(cons(T110, T111), T112, cons(T110, X163), X164) :- less36(T110, T112).
split46(cons(T110, T111), T112, cons(T110, X163), X164) :- ','(lessc36(T110, T112), split46(T111, T112, X163, X164)).
split46(cons(T125, T126), T127, X195, cons(T125, X196)) :- geq62(T125, T127).
split46(cons(T125, T126), T127, X195, cons(T125, X196)) :- ','(geqc62(T125, T127), split46(T126, T127, X195, X196)).
geq62(s(T146), s(T147)) :- geq62(T146, T147).
qs79(cons(T162, T163), X265) :- split46(T163, T162, X261, X262).
qs79(cons(T162, T163), X265) :- ','(splitc46(T163, T162, T166, T167), qs79(T166, X263)).
qs79(cons(T162, T163), X265) :- ','(splitc46(T163, T162, T166, T167), ','(qsc79(T166, T170), qs79(T167, X264))).
qs79(cons(T162, T163), X265) :- ','(splitc46(T163, T162, T166, T167), ','(qsc79(T166, T170), ','(qsc79(T167, T171), append94(T170, T162, T171, X265)))).
append94(cons(T194, T195), T196, T197, cons(T194, X295)) :- append94(T195, T196, T197, X295).
p78(T90, X13, T154, T69, T9) :- qs79(T90, X13).
p78(T90, T157, T154, T69, T9) :- ','(qsc79(T90, T157), append23(T154, T69, T157, T9)).
qs1(cons(T14, []), T9) :- ','(qsc14(T17), ','(qsc14(T20), append23(T17, T14, T20, T9))).
qs1(cons(T69, cons(T67, T68)), T9) :- less36(T67, T69).
qs1(cons(T69, cons(T67, T68)), T9) :- ','(lessc36(T67, T69), split46(T68, T69, X93, X94)).
qs1(cons(T69, cons(T67, T68)), T9) :- ','(lessc36(T67, T69), ','(splitc46(T68, T69, T89, T90), qs1(cons(T67, T89), X12))).
qs1(cons(T69, cons(T67, T68)), T9) :- ','(lessc36(T67, T69), ','(splitc46(T68, T69, T89, T90), ','(qsc1(cons(T67, T89), T154), p78(T90, X13, T154, T69, T9)))).
qs1(cons(T214, cons(T212, T213)), T9) :- geq62(T212, T214).
qs1(cons(T214, cons(T212, T213)), T9) :- ','(geqc62(T212, T214), split46(T213, T214, X323, X324)).
qs1(cons(T214, cons(T212, T213)), T9) :- ','(geqc62(T212, T214), ','(splitc46(T213, T214, T221, T222), qs79(T221, X12))).
qs1(cons(T214, cons(T212, T213)), T9) :- ','(geqc62(T212, T214), ','(splitc46(T213, T214, T221, T222), ','(qsc79(T221, T225), p78(cons(T212, T222), X13, T225, T214, T9)))).

Clauses:

qsc14([]).
appendc23([], T33, T34, cons(T33, T34)).
appendc23(cons(T45, T46), T47, T48, cons(T45, T50)) :- appendc23(T46, T47, T48, T50).
lessc36(0, s(T78)).
lessc36(s(T83), s(T84)) :- lessc36(T83, T84).
splitc46([], T97, [], []).
splitc46(cons(T110, T111), T112, cons(T110, X163), X164) :- ','(lessc36(T110, T112), splitc46(T111, T112, X163, X164)).
splitc46(cons(T125, T126), T127, X195, cons(T125, X196)) :- ','(geqc62(T125, T127), splitc46(T126, T127, X195, X196)).
geqc62(T136, T136).
geqc62(s(T141), 0).
geqc62(s(T146), s(T147)) :- geqc62(T146, T147).
qsc1([], []).
qsc1(cons(T14, []), T9) :- ','(qsc14(T17), ','(qsc14(T20), appendc23(T17, T14, T20, T9))).
qsc1(cons(T69, cons(T67, T68)), T9) :- ','(lessc36(T67, T69), ','(splitc46(T68, T69, T89, T90), ','(qsc1(cons(T67, T89), T154), qc78(T90, X13, T154, T69, T9)))).
qsc1(cons(T214, cons(T212, T213)), T9) :- ','(geqc62(T212, T214), ','(splitc46(T213, T214, T221, T222), ','(qsc79(T221, T225), qc78(cons(T212, T222), X13, T225, T214, T9)))).
qsc79([], []).
qsc79(cons(T162, T163), X265) :- ','(splitc46(T163, T162, T166, T167), ','(qsc79(T166, T170), ','(qsc79(T167, T171), appendc94(T170, T162, T171, X265)))).
appendc94([], T184, T185, cons(T184, T185)).
appendc94(cons(T194, T195), T196, T197, cons(T194, X295)) :- appendc94(T195, T196, T197, X295).
qc78(T90, T157, T154, T69, T9) :- ','(qsc79(T90, T157), appendc23(T154, T69, T157, T9)).

Afs:

qs1(x1, x2)  =  qs1(x1)

(5) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
qs1_in: (b,f)
append23_in: (b,b,b,f)
less36_in: (b,b)
lessc36_in: (b,b)
split46_in: (b,b,f,f)
geq62_in: (b,b)
geqc62_in: (b,b)
splitc46_in: (b,b,f,f)
qs79_in: (b,f)
qsc79_in: (b,f)
appendc94_in: (b,b,b,f)
append94_in: (b,b,b,f)
p78_in: (b,f,b,b,f)
qsc1_in: (b,f)
appendc23_in: (b,b,b,f)
qc78_in: (b,f,b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

QS1_IN_GA(cons(T14, []), T9) → U21_GA(T14, T9, qsc14_in_a(T17))
U21_GA(T14, T9, qsc14_out_a(T17)) → U22_GA(T14, T9, T17, qsc14_in_a(T20))
U22_GA(T14, T9, T17, qsc14_out_a(T20)) → U23_GA(T14, T9, append23_in_ggga(T17, T14, T20, T9))
U22_GA(T14, T9, T17, qsc14_out_a(T20)) → APPEND23_IN_GGGA(T17, T14, T20, T9)
APPEND23_IN_GGGA(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_GGGA(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
APPEND23_IN_GGGA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_GGGA(T46, T47, T48, T50)
QS1_IN_GA(cons(T69, cons(T67, T68)), T9) → U24_GA(T69, T67, T68, T9, less36_in_gg(T67, T69))
QS1_IN_GA(cons(T69, cons(T67, T68)), T9) → LESS36_IN_GG(T67, T69)
LESS36_IN_GG(s(T83), s(T84)) → U2_GG(T83, T84, less36_in_gg(T83, T84))
LESS36_IN_GG(s(T83), s(T84)) → LESS36_IN_GG(T83, T84)
QS1_IN_GA(cons(T69, cons(T67, T68)), T9) → U25_GA(T69, T67, T68, T9, lessc36_in_gg(T67, T69))
U25_GA(T69, T67, T68, T9, lessc36_out_gg(T67, T69)) → U26_GA(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
U25_GA(T69, T67, T68, T9, lessc36_out_gg(T67, T69)) → SPLIT46_IN_GGAA(T68, T69, X93, X94)
SPLIT46_IN_GGAA(cons(T110, T111), T112, cons(T110, X163), X164) → U3_GGAA(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
SPLIT46_IN_GGAA(cons(T110, T111), T112, cons(T110, X163), X164) → LESS36_IN_GG(T110, T112)
SPLIT46_IN_GGAA(cons(T110, T111), T112, cons(T110, X163), X164) → U4_GGAA(T110, T111, T112, X163, X164, lessc36_in_gg(T110, T112))
U4_GGAA(T110, T111, T112, X163, X164, lessc36_out_gg(T110, T112)) → U5_GGAA(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
U4_GGAA(T110, T111, T112, X163, X164, lessc36_out_gg(T110, T112)) → SPLIT46_IN_GGAA(T111, T112, X163, X164)
SPLIT46_IN_GGAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U6_GGAA(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
SPLIT46_IN_GGAA(cons(T125, T126), T127, X195, cons(T125, X196)) → GEQ62_IN_GG(T125, T127)
GEQ62_IN_GG(s(T146), s(T147)) → U9_GG(T146, T147, geq62_in_gg(T146, T147))
GEQ62_IN_GG(s(T146), s(T147)) → GEQ62_IN_GG(T146, T147)
SPLIT46_IN_GGAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U7_GGAA(T125, T126, T127, X195, X196, geqc62_in_gg(T125, T127))
U7_GGAA(T125, T126, T127, X195, X196, geqc62_out_gg(T125, T127)) → U8_GGAA(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U7_GGAA(T125, T126, T127, X195, X196, geqc62_out_gg(T125, T127)) → SPLIT46_IN_GGAA(T126, T127, X195, X196)
U25_GA(T69, T67, T68, T9, lessc36_out_gg(T67, T69)) → U27_GA(T69, T67, T68, T9, splitc46_in_ggaa(T68, T69, T89, T90))
U27_GA(T69, T67, T68, T9, splitc46_out_ggaa(T68, T69, T89, T90)) → U28_GA(T69, T67, T68, T9, qs1_in_ga(cons(T67, T89), X12))
U27_GA(T69, T67, T68, T9, splitc46_out_ggaa(T68, T69, T89, T90)) → QS1_IN_GA(cons(T67, T89), X12)
QS1_IN_GA(cons(T214, cons(T212, T213)), T9) → U31_GA(T214, T212, T213, T9, geq62_in_gg(T212, T214))
QS1_IN_GA(cons(T214, cons(T212, T213)), T9) → GEQ62_IN_GG(T212, T214)
QS1_IN_GA(cons(T214, cons(T212, T213)), T9) → U32_GA(T214, T212, T213, T9, geqc62_in_gg(T212, T214))
U32_GA(T214, T212, T213, T9, geqc62_out_gg(T212, T214)) → U33_GA(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U32_GA(T214, T212, T213, T9, geqc62_out_gg(T212, T214)) → SPLIT46_IN_GGAA(T213, T214, X323, X324)
U32_GA(T214, T212, T213, T9, geqc62_out_gg(T212, T214)) → U34_GA(T214, T212, T213, T9, splitc46_in_ggaa(T213, T214, T221, T222))
U34_GA(T214, T212, T213, T9, splitc46_out_ggaa(T213, T214, T221, T222)) → U35_GA(T214, T212, T213, T9, qs79_in_ga(T221, X12))
U34_GA(T214, T212, T213, T9, splitc46_out_ggaa(T213, T214, T221, T222)) → QS79_IN_GA(T221, X12)
QS79_IN_GA(cons(T162, T163), X265) → U10_GA(T162, T163, X265, split46_in_ggaa(T163, T162, X261, X262))
QS79_IN_GA(cons(T162, T163), X265) → SPLIT46_IN_GGAA(T163, T162, X261, X262)
QS79_IN_GA(cons(T162, T163), X265) → U11_GA(T162, T163, X265, splitc46_in_ggaa(T163, T162, T166, T167))
U11_GA(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → U12_GA(T162, T163, X265, qs79_in_ga(T166, X263))
U11_GA(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → QS79_IN_GA(T166, X263)
U11_GA(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → U13_GA(T162, T163, X265, T167, qsc79_in_ga(T166, T170))
U13_GA(T162, T163, X265, T167, qsc79_out_ga(T166, T170)) → U14_GA(T162, T163, X265, qs79_in_ga(T167, X264))
U13_GA(T162, T163, X265, T167, qsc79_out_ga(T166, T170)) → QS79_IN_GA(T167, X264)
U13_GA(T162, T163, X265, T167, qsc79_out_ga(T166, T170)) → U15_GA(T162, T163, X265, T170, qsc79_in_ga(T167, T171))
U15_GA(T162, T163, X265, T170, qsc79_out_ga(T167, T171)) → U16_GA(T162, T163, X265, append94_in_ggga(T170, T162, T171, X265))
U15_GA(T162, T163, X265, T170, qsc79_out_ga(T167, T171)) → APPEND94_IN_GGGA(T170, T162, T171, X265)
APPEND94_IN_GGGA(cons(T194, T195), T196, T197, cons(T194, X295)) → U17_GGGA(T194, T195, T196, T197, X295, append94_in_ggga(T195, T196, T197, X295))
APPEND94_IN_GGGA(cons(T194, T195), T196, T197, cons(T194, X295)) → APPEND94_IN_GGGA(T195, T196, T197, X295)
U34_GA(T214, T212, T213, T9, splitc46_out_ggaa(T213, T214, T221, T222)) → U36_GA(T214, T212, T213, T9, T222, qsc79_in_ga(T221, T225))
U36_GA(T214, T212, T213, T9, T222, qsc79_out_ga(T221, T225)) → U37_GA(T214, T212, T213, T9, p78_in_gagga(cons(T212, T222), X13, T225, T214, T9))
U36_GA(T214, T212, T213, T9, T222, qsc79_out_ga(T221, T225)) → P78_IN_GAGGA(cons(T212, T222), X13, T225, T214, T9)
P78_IN_GAGGA(T90, X13, T154, T69, T9) → U18_GAGGA(T90, X13, T154, T69, T9, qs79_in_ga(T90, X13))
P78_IN_GAGGA(T90, X13, T154, T69, T9) → QS79_IN_GA(T90, X13)
P78_IN_GAGGA(T90, T157, T154, T69, T9) → U19_GAGGA(T90, T157, T154, T69, T9, qsc79_in_ga(T90, T157))
U19_GAGGA(T90, T157, T154, T69, T9, qsc79_out_ga(T90, T157)) → U20_GAGGA(T90, T157, T154, T69, T9, append23_in_ggga(T154, T69, T157, T9))
U19_GAGGA(T90, T157, T154, T69, T9, qsc79_out_ga(T90, T157)) → APPEND23_IN_GGGA(T154, T69, T157, T9)
U27_GA(T69, T67, T68, T9, splitc46_out_ggaa(T68, T69, T89, T90)) → U29_GA(T69, T67, T68, T9, T90, qsc1_in_ga(cons(T67, T89), T154))
U29_GA(T69, T67, T68, T9, T90, qsc1_out_ga(cons(T67, T89), T154)) → U30_GA(T69, T67, T68, T9, p78_in_gagga(T90, X13, T154, T69, T9))
U29_GA(T69, T67, T68, T9, T90, qsc1_out_ga(cons(T67, T89), T154)) → P78_IN_GAGGA(T90, X13, T154, T69, T9)

The TRS R consists of the following rules:

qsc14_in_a([]) → qsc14_out_a([])
lessc36_in_gg(0, s(T78)) → lessc36_out_gg(0, s(T78))
lessc36_in_gg(s(T83), s(T84)) → U40_gg(T83, T84, lessc36_in_gg(T83, T84))
U40_gg(T83, T84, lessc36_out_gg(T83, T84)) → lessc36_out_gg(s(T83), s(T84))
geqc62_in_gg(T136, T136) → geqc62_out_gg(T136, T136)
geqc62_in_gg(s(T141), 0) → geqc62_out_gg(s(T141), 0)
geqc62_in_gg(s(T146), s(T147)) → U45_gg(T146, T147, geqc62_in_gg(T146, T147))
U45_gg(T146, T147, geqc62_out_gg(T146, T147)) → geqc62_out_gg(s(T146), s(T147))
splitc46_in_ggaa([], T97, [], []) → splitc46_out_ggaa([], T97, [], [])
splitc46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U41_ggaa(T110, T111, T112, X163, X164, lessc36_in_gg(T110, T112))
U41_ggaa(T110, T111, T112, X163, X164, lessc36_out_gg(T110, T112)) → U42_ggaa(T110, T111, T112, X163, X164, splitc46_in_ggaa(T111, T112, X163, X164))
splitc46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U43_ggaa(T125, T126, T127, X195, X196, geqc62_in_gg(T125, T127))
U43_ggaa(T125, T126, T127, X195, X196, geqc62_out_gg(T125, T127)) → U44_ggaa(T125, T126, T127, X195, X196, splitc46_in_ggaa(T126, T127, X195, X196))
U44_ggaa(T125, T126, T127, X195, X196, splitc46_out_ggaa(T126, T127, X195, X196)) → splitc46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U42_ggaa(T110, T111, T112, X163, X164, splitc46_out_ggaa(T111, T112, X163, X164)) → splitc46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
qsc79_in_ga([], []) → qsc79_out_ga([], [])
qsc79_in_ga(cons(T162, T163), X265) → U57_ga(T162, T163, X265, splitc46_in_ggaa(T163, T162, T166, T167))
U57_ga(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → U58_ga(T162, T163, X265, T167, qsc79_in_ga(T166, T170))
U58_ga(T162, T163, X265, T167, qsc79_out_ga(T166, T170)) → U59_ga(T162, T163, X265, T170, qsc79_in_ga(T167, T171))
U59_ga(T162, T163, X265, T170, qsc79_out_ga(T167, T171)) → U60_ga(T162, T163, X265, appendc94_in_ggga(T170, T162, T171, X265))
appendc94_in_ggga([], T184, T185, cons(T184, T185)) → appendc94_out_ggga([], T184, T185, cons(T184, T185))
appendc94_in_ggga(cons(T194, T195), T196, T197, cons(T194, X295)) → U61_ggga(T194, T195, T196, T197, X295, appendc94_in_ggga(T195, T196, T197, X295))
U61_ggga(T194, T195, T196, T197, X295, appendc94_out_ggga(T195, T196, T197, X295)) → appendc94_out_ggga(cons(T194, T195), T196, T197, cons(T194, X295))
U60_ga(T162, T163, X265, appendc94_out_ggga(T170, T162, T171, X265)) → qsc79_out_ga(cons(T162, T163), X265)
qsc1_in_ga([], []) → qsc1_out_ga([], [])
qsc1_in_ga(cons(T14, []), T9) → U46_ga(T14, T9, qsc14_in_a(T17))
U46_ga(T14, T9, qsc14_out_a(T17)) → U47_ga(T14, T9, T17, qsc14_in_a(T20))
U47_ga(T14, T9, T17, qsc14_out_a(T20)) → U48_ga(T14, T9, appendc23_in_ggga(T17, T14, T20, T9))
appendc23_in_ggga([], T33, T34, cons(T33, T34)) → appendc23_out_ggga([], T33, T34, cons(T33, T34))
appendc23_in_ggga(cons(T45, T46), T47, T48, cons(T45, T50)) → U39_ggga(T45, T46, T47, T48, T50, appendc23_in_ggga(T46, T47, T48, T50))
U39_ggga(T45, T46, T47, T48, T50, appendc23_out_ggga(T46, T47, T48, T50)) → appendc23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U48_ga(T14, T9, appendc23_out_ggga(T17, T14, T20, T9)) → qsc1_out_ga(cons(T14, []), T9)
qsc1_in_ga(cons(T69, cons(T67, T68)), T9) → U49_ga(T69, T67, T68, T9, lessc36_in_gg(T67, T69))
U49_ga(T69, T67, T68, T9, lessc36_out_gg(T67, T69)) → U50_ga(T69, T67, T68, T9, splitc46_in_ggaa(T68, T69, T89, T90))
U50_ga(T69, T67, T68, T9, splitc46_out_ggaa(T68, T69, T89, T90)) → U51_ga(T69, T67, T68, T9, T90, qsc1_in_ga(cons(T67, T89), T154))
qsc1_in_ga(cons(T214, cons(T212, T213)), T9) → U53_ga(T214, T212, T213, T9, geqc62_in_gg(T212, T214))
U53_ga(T214, T212, T213, T9, geqc62_out_gg(T212, T214)) → U54_ga(T214, T212, T213, T9, splitc46_in_ggaa(T213, T214, T221, T222))
U54_ga(T214, T212, T213, T9, splitc46_out_ggaa(T213, T214, T221, T222)) → U55_ga(T214, T212, T213, T9, T222, qsc79_in_ga(T221, T225))
U55_ga(T214, T212, T213, T9, T222, qsc79_out_ga(T221, T225)) → U56_ga(T214, T212, T213, T9, qc78_in_gagga(cons(T212, T222), X13, T225, T214, T9))
qc78_in_gagga(T90, T157, T154, T69, T9) → U62_gagga(T90, T157, T154, T69, T9, qsc79_in_ga(T90, T157))
U62_gagga(T90, T157, T154, T69, T9, qsc79_out_ga(T90, T157)) → U63_gagga(T90, T157, T154, T69, T9, appendc23_in_ggga(T154, T69, T157, T9))
U63_gagga(T90, T157, T154, T69, T9, appendc23_out_ggga(T154, T69, T157, T9)) → qc78_out_gagga(T90, T157, T154, T69, T9)
U56_ga(T214, T212, T213, T9, qc78_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → qsc1_out_ga(cons(T214, cons(T212, T213)), T9)
U51_ga(T69, T67, T68, T9, T90, qsc1_out_ga(cons(T67, T89), T154)) → U52_ga(T69, T67, T68, T9, qc78_in_gagga(T90, X13, T154, T69, T9))
U52_ga(T69, T67, T68, T9, qc78_out_gagga(T90, X13, T154, T69, T9)) → qsc1_out_ga(cons(T69, cons(T67, T68)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
[]  =  []
qsc14_in_a(x1)  =  qsc14_in_a
qsc14_out_a(x1)  =  qsc14_out_a(x1)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
s(x1)  =  s(x1)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
0  =  0
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geqc62_in_gg(x1, x2)  =  geqc62_in_gg(x1, x2)
geqc62_out_gg(x1, x2)  =  geqc62_out_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
splitc46_in_ggaa(x1, x2, x3, x4)  =  splitc46_in_ggaa(x1, x2)
splitc46_out_ggaa(x1, x2, x3, x4)  =  splitc46_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qs79_in_ga(x1, x2)  =  qs79_in_ga(x1)
qsc79_in_ga(x1, x2)  =  qsc79_in_ga(x1)
qsc79_out_ga(x1, x2)  =  qsc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
append94_in_ggga(x1, x2, x3, x4)  =  append94_in_ggga(x1, x2, x3)
p78_in_gagga(x1, x2, x3, x4, x5)  =  p78_in_gagga(x1, x3, x4)
qsc1_in_ga(x1, x2)  =  qsc1_in_ga(x1)
qsc1_out_ga(x1, x2)  =  qsc1_out_ga(x1, x2)
U46_ga(x1, x2, x3)  =  U46_ga(x1, x3)
U47_ga(x1, x2, x3, x4)  =  U47_ga(x1, x3, x4)
U48_ga(x1, x2, x3)  =  U48_ga(x1, x3)
appendc23_in_ggga(x1, x2, x3, x4)  =  appendc23_in_ggga(x1, x2, x3)
appendc23_out_ggga(x1, x2, x3, x4)  =  appendc23_out_ggga(x1, x2, x3, x4)
U39_ggga(x1, x2, x3, x4, x5, x6)  =  U39_ggga(x1, x2, x3, x4, x6)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6)  =  U51_ga(x1, x2, x3, x5, x6)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x2, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x1, x2, x3, x5)
U55_ga(x1, x2, x3, x4, x5, x6)  =  U55_ga(x1, x2, x3, x5, x6)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x1, x2, x3, x5)
qc78_in_gagga(x1, x2, x3, x4, x5)  =  qc78_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6)  =  U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6)  =  U63_gagga(x1, x2, x3, x4, x6)
qc78_out_gagga(x1, x2, x3, x4, x5)  =  qc78_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x1, x2, x3, x5)
QS1_IN_GA(x1, x2)  =  QS1_IN_GA(x1)
U21_GA(x1, x2, x3)  =  U21_GA(x1, x3)
U22_GA(x1, x2, x3, x4)  =  U22_GA(x1, x3, x4)
U23_GA(x1, x2, x3)  =  U23_GA(x1, x3)
APPEND23_IN_GGGA(x1, x2, x3, x4)  =  APPEND23_IN_GGGA(x1, x2, x3)
U1_GGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGA(x1, x2, x3, x4, x6)
U24_GA(x1, x2, x3, x4, x5)  =  U24_GA(x1, x2, x3, x5)
LESS36_IN_GG(x1, x2)  =  LESS36_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x1, x2, x3, x5)
U26_GA(x1, x2, x3, x4, x5)  =  U26_GA(x1, x2, x3, x5)
SPLIT46_IN_GGAA(x1, x2, x3, x4)  =  SPLIT46_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x1, x2, x3, x6)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x1, x2, x3, x6)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)
GEQ62_IN_GG(x1, x2)  =  GEQ62_IN_GG(x1, x2)
U9_GG(x1, x2, x3)  =  U9_GG(x1, x2, x3)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)
U8_GGAA(x1, x2, x3, x4, x5, x6)  =  U8_GGAA(x1, x2, x3, x6)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x3, x5)
U28_GA(x1, x2, x3, x4, x5)  =  U28_GA(x1, x2, x3, x5)
U31_GA(x1, x2, x3, x4, x5)  =  U31_GA(x1, x2, x3, x5)
U32_GA(x1, x2, x3, x4, x5)  =  U32_GA(x1, x2, x3, x5)
U33_GA(x1, x2, x3, x4, x5)  =  U33_GA(x1, x2, x3, x5)
U34_GA(x1, x2, x3, x4, x5)  =  U34_GA(x1, x2, x3, x5)
U35_GA(x1, x2, x3, x4, x5)  =  U35_GA(x1, x2, x3, x5)
QS79_IN_GA(x1, x2)  =  QS79_IN_GA(x1)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x1, x2, x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x1, x2, x4)
U12_GA(x1, x2, x3, x4)  =  U12_GA(x1, x2, x4)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x1, x2, x4, x5)
U14_GA(x1, x2, x3, x4)  =  U14_GA(x1, x2, x4)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x2, x4, x5)
U16_GA(x1, x2, x3, x4)  =  U16_GA(x1, x2, x4)
APPEND94_IN_GGGA(x1, x2, x3, x4)  =  APPEND94_IN_GGGA(x1, x2, x3)
U17_GGGA(x1, x2, x3, x4, x5, x6)  =  U17_GGGA(x1, x2, x3, x4, x6)
U36_GA(x1, x2, x3, x4, x5, x6)  =  U36_GA(x1, x2, x3, x5, x6)
U37_GA(x1, x2, x3, x4, x5)  =  U37_GA(x1, x2, x3, x5)
P78_IN_GAGGA(x1, x2, x3, x4, x5)  =  P78_IN_GAGGA(x1, x3, x4)
U18_GAGGA(x1, x2, x3, x4, x5, x6)  =  U18_GAGGA(x1, x3, x4, x6)
U19_GAGGA(x1, x2, x3, x4, x5, x6)  =  U19_GAGGA(x1, x3, x4, x6)
U20_GAGGA(x1, x2, x3, x4, x5, x6)  =  U20_GAGGA(x1, x3, x4, x6)
U29_GA(x1, x2, x3, x4, x5, x6)  =  U29_GA(x1, x2, x3, x5, x6)
U30_GA(x1, x2, x3, x4, x5)  =  U30_GA(x1, x2, x3, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(6) Obligation:

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

QS1_IN_GA(cons(T14, []), T9) → U21_GA(T14, T9, qsc14_in_a(T17))
U21_GA(T14, T9, qsc14_out_a(T17)) → U22_GA(T14, T9, T17, qsc14_in_a(T20))
U22_GA(T14, T9, T17, qsc14_out_a(T20)) → U23_GA(T14, T9, append23_in_ggga(T17, T14, T20, T9))
U22_GA(T14, T9, T17, qsc14_out_a(T20)) → APPEND23_IN_GGGA(T17, T14, T20, T9)
APPEND23_IN_GGGA(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_GGGA(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
APPEND23_IN_GGGA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_GGGA(T46, T47, T48, T50)
QS1_IN_GA(cons(T69, cons(T67, T68)), T9) → U24_GA(T69, T67, T68, T9, less36_in_gg(T67, T69))
QS1_IN_GA(cons(T69, cons(T67, T68)), T9) → LESS36_IN_GG(T67, T69)
LESS36_IN_GG(s(T83), s(T84)) → U2_GG(T83, T84, less36_in_gg(T83, T84))
LESS36_IN_GG(s(T83), s(T84)) → LESS36_IN_GG(T83, T84)
QS1_IN_GA(cons(T69, cons(T67, T68)), T9) → U25_GA(T69, T67, T68, T9, lessc36_in_gg(T67, T69))
U25_GA(T69, T67, T68, T9, lessc36_out_gg(T67, T69)) → U26_GA(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
U25_GA(T69, T67, T68, T9, lessc36_out_gg(T67, T69)) → SPLIT46_IN_GGAA(T68, T69, X93, X94)
SPLIT46_IN_GGAA(cons(T110, T111), T112, cons(T110, X163), X164) → U3_GGAA(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
SPLIT46_IN_GGAA(cons(T110, T111), T112, cons(T110, X163), X164) → LESS36_IN_GG(T110, T112)
SPLIT46_IN_GGAA(cons(T110, T111), T112, cons(T110, X163), X164) → U4_GGAA(T110, T111, T112, X163, X164, lessc36_in_gg(T110, T112))
U4_GGAA(T110, T111, T112, X163, X164, lessc36_out_gg(T110, T112)) → U5_GGAA(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
U4_GGAA(T110, T111, T112, X163, X164, lessc36_out_gg(T110, T112)) → SPLIT46_IN_GGAA(T111, T112, X163, X164)
SPLIT46_IN_GGAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U6_GGAA(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
SPLIT46_IN_GGAA(cons(T125, T126), T127, X195, cons(T125, X196)) → GEQ62_IN_GG(T125, T127)
GEQ62_IN_GG(s(T146), s(T147)) → U9_GG(T146, T147, geq62_in_gg(T146, T147))
GEQ62_IN_GG(s(T146), s(T147)) → GEQ62_IN_GG(T146, T147)
SPLIT46_IN_GGAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U7_GGAA(T125, T126, T127, X195, X196, geqc62_in_gg(T125, T127))
U7_GGAA(T125, T126, T127, X195, X196, geqc62_out_gg(T125, T127)) → U8_GGAA(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U7_GGAA(T125, T126, T127, X195, X196, geqc62_out_gg(T125, T127)) → SPLIT46_IN_GGAA(T126, T127, X195, X196)
U25_GA(T69, T67, T68, T9, lessc36_out_gg(T67, T69)) → U27_GA(T69, T67, T68, T9, splitc46_in_ggaa(T68, T69, T89, T90))
U27_GA(T69, T67, T68, T9, splitc46_out_ggaa(T68, T69, T89, T90)) → U28_GA(T69, T67, T68, T9, qs1_in_ga(cons(T67, T89), X12))
U27_GA(T69, T67, T68, T9, splitc46_out_ggaa(T68, T69, T89, T90)) → QS1_IN_GA(cons(T67, T89), X12)
QS1_IN_GA(cons(T214, cons(T212, T213)), T9) → U31_GA(T214, T212, T213, T9, geq62_in_gg(T212, T214))
QS1_IN_GA(cons(T214, cons(T212, T213)), T9) → GEQ62_IN_GG(T212, T214)
QS1_IN_GA(cons(T214, cons(T212, T213)), T9) → U32_GA(T214, T212, T213, T9, geqc62_in_gg(T212, T214))
U32_GA(T214, T212, T213, T9, geqc62_out_gg(T212, T214)) → U33_GA(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U32_GA(T214, T212, T213, T9, geqc62_out_gg(T212, T214)) → SPLIT46_IN_GGAA(T213, T214, X323, X324)
U32_GA(T214, T212, T213, T9, geqc62_out_gg(T212, T214)) → U34_GA(T214, T212, T213, T9, splitc46_in_ggaa(T213, T214, T221, T222))
U34_GA(T214, T212, T213, T9, splitc46_out_ggaa(T213, T214, T221, T222)) → U35_GA(T214, T212, T213, T9, qs79_in_ga(T221, X12))
U34_GA(T214, T212, T213, T9, splitc46_out_ggaa(T213, T214, T221, T222)) → QS79_IN_GA(T221, X12)
QS79_IN_GA(cons(T162, T163), X265) → U10_GA(T162, T163, X265, split46_in_ggaa(T163, T162, X261, X262))
QS79_IN_GA(cons(T162, T163), X265) → SPLIT46_IN_GGAA(T163, T162, X261, X262)
QS79_IN_GA(cons(T162, T163), X265) → U11_GA(T162, T163, X265, splitc46_in_ggaa(T163, T162, T166, T167))
U11_GA(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → U12_GA(T162, T163, X265, qs79_in_ga(T166, X263))
U11_GA(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → QS79_IN_GA(T166, X263)
U11_GA(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → U13_GA(T162, T163, X265, T167, qsc79_in_ga(T166, T170))
U13_GA(T162, T163, X265, T167, qsc79_out_ga(T166, T170)) → U14_GA(T162, T163, X265, qs79_in_ga(T167, X264))
U13_GA(T162, T163, X265, T167, qsc79_out_ga(T166, T170)) → QS79_IN_GA(T167, X264)
U13_GA(T162, T163, X265, T167, qsc79_out_ga(T166, T170)) → U15_GA(T162, T163, X265, T170, qsc79_in_ga(T167, T171))
U15_GA(T162, T163, X265, T170, qsc79_out_ga(T167, T171)) → U16_GA(T162, T163, X265, append94_in_ggga(T170, T162, T171, X265))
U15_GA(T162, T163, X265, T170, qsc79_out_ga(T167, T171)) → APPEND94_IN_GGGA(T170, T162, T171, X265)
APPEND94_IN_GGGA(cons(T194, T195), T196, T197, cons(T194, X295)) → U17_GGGA(T194, T195, T196, T197, X295, append94_in_ggga(T195, T196, T197, X295))
APPEND94_IN_GGGA(cons(T194, T195), T196, T197, cons(T194, X295)) → APPEND94_IN_GGGA(T195, T196, T197, X295)
U34_GA(T214, T212, T213, T9, splitc46_out_ggaa(T213, T214, T221, T222)) → U36_GA(T214, T212, T213, T9, T222, qsc79_in_ga(T221, T225))
U36_GA(T214, T212, T213, T9, T222, qsc79_out_ga(T221, T225)) → U37_GA(T214, T212, T213, T9, p78_in_gagga(cons(T212, T222), X13, T225, T214, T9))
U36_GA(T214, T212, T213, T9, T222, qsc79_out_ga(T221, T225)) → P78_IN_GAGGA(cons(T212, T222), X13, T225, T214, T9)
P78_IN_GAGGA(T90, X13, T154, T69, T9) → U18_GAGGA(T90, X13, T154, T69, T9, qs79_in_ga(T90, X13))
P78_IN_GAGGA(T90, X13, T154, T69, T9) → QS79_IN_GA(T90, X13)
P78_IN_GAGGA(T90, T157, T154, T69, T9) → U19_GAGGA(T90, T157, T154, T69, T9, qsc79_in_ga(T90, T157))
U19_GAGGA(T90, T157, T154, T69, T9, qsc79_out_ga(T90, T157)) → U20_GAGGA(T90, T157, T154, T69, T9, append23_in_ggga(T154, T69, T157, T9))
U19_GAGGA(T90, T157, T154, T69, T9, qsc79_out_ga(T90, T157)) → APPEND23_IN_GGGA(T154, T69, T157, T9)
U27_GA(T69, T67, T68, T9, splitc46_out_ggaa(T68, T69, T89, T90)) → U29_GA(T69, T67, T68, T9, T90, qsc1_in_ga(cons(T67, T89), T154))
U29_GA(T69, T67, T68, T9, T90, qsc1_out_ga(cons(T67, T89), T154)) → U30_GA(T69, T67, T68, T9, p78_in_gagga(T90, X13, T154, T69, T9))
U29_GA(T69, T67, T68, T9, T90, qsc1_out_ga(cons(T67, T89), T154)) → P78_IN_GAGGA(T90, X13, T154, T69, T9)

The TRS R consists of the following rules:

qsc14_in_a([]) → qsc14_out_a([])
lessc36_in_gg(0, s(T78)) → lessc36_out_gg(0, s(T78))
lessc36_in_gg(s(T83), s(T84)) → U40_gg(T83, T84, lessc36_in_gg(T83, T84))
U40_gg(T83, T84, lessc36_out_gg(T83, T84)) → lessc36_out_gg(s(T83), s(T84))
geqc62_in_gg(T136, T136) → geqc62_out_gg(T136, T136)
geqc62_in_gg(s(T141), 0) → geqc62_out_gg(s(T141), 0)
geqc62_in_gg(s(T146), s(T147)) → U45_gg(T146, T147, geqc62_in_gg(T146, T147))
U45_gg(T146, T147, geqc62_out_gg(T146, T147)) → geqc62_out_gg(s(T146), s(T147))
splitc46_in_ggaa([], T97, [], []) → splitc46_out_ggaa([], T97, [], [])
splitc46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U41_ggaa(T110, T111, T112, X163, X164, lessc36_in_gg(T110, T112))
U41_ggaa(T110, T111, T112, X163, X164, lessc36_out_gg(T110, T112)) → U42_ggaa(T110, T111, T112, X163, X164, splitc46_in_ggaa(T111, T112, X163, X164))
splitc46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U43_ggaa(T125, T126, T127, X195, X196, geqc62_in_gg(T125, T127))
U43_ggaa(T125, T126, T127, X195, X196, geqc62_out_gg(T125, T127)) → U44_ggaa(T125, T126, T127, X195, X196, splitc46_in_ggaa(T126, T127, X195, X196))
U44_ggaa(T125, T126, T127, X195, X196, splitc46_out_ggaa(T126, T127, X195, X196)) → splitc46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U42_ggaa(T110, T111, T112, X163, X164, splitc46_out_ggaa(T111, T112, X163, X164)) → splitc46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
qsc79_in_ga([], []) → qsc79_out_ga([], [])
qsc79_in_ga(cons(T162, T163), X265) → U57_ga(T162, T163, X265, splitc46_in_ggaa(T163, T162, T166, T167))
U57_ga(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → U58_ga(T162, T163, X265, T167, qsc79_in_ga(T166, T170))
U58_ga(T162, T163, X265, T167, qsc79_out_ga(T166, T170)) → U59_ga(T162, T163, X265, T170, qsc79_in_ga(T167, T171))
U59_ga(T162, T163, X265, T170, qsc79_out_ga(T167, T171)) → U60_ga(T162, T163, X265, appendc94_in_ggga(T170, T162, T171, X265))
appendc94_in_ggga([], T184, T185, cons(T184, T185)) → appendc94_out_ggga([], T184, T185, cons(T184, T185))
appendc94_in_ggga(cons(T194, T195), T196, T197, cons(T194, X295)) → U61_ggga(T194, T195, T196, T197, X295, appendc94_in_ggga(T195, T196, T197, X295))
U61_ggga(T194, T195, T196, T197, X295, appendc94_out_ggga(T195, T196, T197, X295)) → appendc94_out_ggga(cons(T194, T195), T196, T197, cons(T194, X295))
U60_ga(T162, T163, X265, appendc94_out_ggga(T170, T162, T171, X265)) → qsc79_out_ga(cons(T162, T163), X265)
qsc1_in_ga([], []) → qsc1_out_ga([], [])
qsc1_in_ga(cons(T14, []), T9) → U46_ga(T14, T9, qsc14_in_a(T17))
U46_ga(T14, T9, qsc14_out_a(T17)) → U47_ga(T14, T9, T17, qsc14_in_a(T20))
U47_ga(T14, T9, T17, qsc14_out_a(T20)) → U48_ga(T14, T9, appendc23_in_ggga(T17, T14, T20, T9))
appendc23_in_ggga([], T33, T34, cons(T33, T34)) → appendc23_out_ggga([], T33, T34, cons(T33, T34))
appendc23_in_ggga(cons(T45, T46), T47, T48, cons(T45, T50)) → U39_ggga(T45, T46, T47, T48, T50, appendc23_in_ggga(T46, T47, T48, T50))
U39_ggga(T45, T46, T47, T48, T50, appendc23_out_ggga(T46, T47, T48, T50)) → appendc23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U48_ga(T14, T9, appendc23_out_ggga(T17, T14, T20, T9)) → qsc1_out_ga(cons(T14, []), T9)
qsc1_in_ga(cons(T69, cons(T67, T68)), T9) → U49_ga(T69, T67, T68, T9, lessc36_in_gg(T67, T69))
U49_ga(T69, T67, T68, T9, lessc36_out_gg(T67, T69)) → U50_ga(T69, T67, T68, T9, splitc46_in_ggaa(T68, T69, T89, T90))
U50_ga(T69, T67, T68, T9, splitc46_out_ggaa(T68, T69, T89, T90)) → U51_ga(T69, T67, T68, T9, T90, qsc1_in_ga(cons(T67, T89), T154))
qsc1_in_ga(cons(T214, cons(T212, T213)), T9) → U53_ga(T214, T212, T213, T9, geqc62_in_gg(T212, T214))
U53_ga(T214, T212, T213, T9, geqc62_out_gg(T212, T214)) → U54_ga(T214, T212, T213, T9, splitc46_in_ggaa(T213, T214, T221, T222))
U54_ga(T214, T212, T213, T9, splitc46_out_ggaa(T213, T214, T221, T222)) → U55_ga(T214, T212, T213, T9, T222, qsc79_in_ga(T221, T225))
U55_ga(T214, T212, T213, T9, T222, qsc79_out_ga(T221, T225)) → U56_ga(T214, T212, T213, T9, qc78_in_gagga(cons(T212, T222), X13, T225, T214, T9))
qc78_in_gagga(T90, T157, T154, T69, T9) → U62_gagga(T90, T157, T154, T69, T9, qsc79_in_ga(T90, T157))
U62_gagga(T90, T157, T154, T69, T9, qsc79_out_ga(T90, T157)) → U63_gagga(T90, T157, T154, T69, T9, appendc23_in_ggga(T154, T69, T157, T9))
U63_gagga(T90, T157, T154, T69, T9, appendc23_out_ggga(T154, T69, T157, T9)) → qc78_out_gagga(T90, T157, T154, T69, T9)
U56_ga(T214, T212, T213, T9, qc78_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → qsc1_out_ga(cons(T214, cons(T212, T213)), T9)
U51_ga(T69, T67, T68, T9, T90, qsc1_out_ga(cons(T67, T89), T154)) → U52_ga(T69, T67, T68, T9, qc78_in_gagga(T90, X13, T154, T69, T9))
U52_ga(T69, T67, T68, T9, qc78_out_gagga(T90, X13, T154, T69, T9)) → qsc1_out_ga(cons(T69, cons(T67, T68)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
[]  =  []
qsc14_in_a(x1)  =  qsc14_in_a
qsc14_out_a(x1)  =  qsc14_out_a(x1)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
s(x1)  =  s(x1)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
0  =  0
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geqc62_in_gg(x1, x2)  =  geqc62_in_gg(x1, x2)
geqc62_out_gg(x1, x2)  =  geqc62_out_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
splitc46_in_ggaa(x1, x2, x3, x4)  =  splitc46_in_ggaa(x1, x2)
splitc46_out_ggaa(x1, x2, x3, x4)  =  splitc46_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qs79_in_ga(x1, x2)  =  qs79_in_ga(x1)
qsc79_in_ga(x1, x2)  =  qsc79_in_ga(x1)
qsc79_out_ga(x1, x2)  =  qsc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
append94_in_ggga(x1, x2, x3, x4)  =  append94_in_ggga(x1, x2, x3)
p78_in_gagga(x1, x2, x3, x4, x5)  =  p78_in_gagga(x1, x3, x4)
qsc1_in_ga(x1, x2)  =  qsc1_in_ga(x1)
qsc1_out_ga(x1, x2)  =  qsc1_out_ga(x1, x2)
U46_ga(x1, x2, x3)  =  U46_ga(x1, x3)
U47_ga(x1, x2, x3, x4)  =  U47_ga(x1, x3, x4)
U48_ga(x1, x2, x3)  =  U48_ga(x1, x3)
appendc23_in_ggga(x1, x2, x3, x4)  =  appendc23_in_ggga(x1, x2, x3)
appendc23_out_ggga(x1, x2, x3, x4)  =  appendc23_out_ggga(x1, x2, x3, x4)
U39_ggga(x1, x2, x3, x4, x5, x6)  =  U39_ggga(x1, x2, x3, x4, x6)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6)  =  U51_ga(x1, x2, x3, x5, x6)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x2, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x1, x2, x3, x5)
U55_ga(x1, x2, x3, x4, x5, x6)  =  U55_ga(x1, x2, x3, x5, x6)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x1, x2, x3, x5)
qc78_in_gagga(x1, x2, x3, x4, x5)  =  qc78_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6)  =  U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6)  =  U63_gagga(x1, x2, x3, x4, x6)
qc78_out_gagga(x1, x2, x3, x4, x5)  =  qc78_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x1, x2, x3, x5)
QS1_IN_GA(x1, x2)  =  QS1_IN_GA(x1)
U21_GA(x1, x2, x3)  =  U21_GA(x1, x3)
U22_GA(x1, x2, x3, x4)  =  U22_GA(x1, x3, x4)
U23_GA(x1, x2, x3)  =  U23_GA(x1, x3)
APPEND23_IN_GGGA(x1, x2, x3, x4)  =  APPEND23_IN_GGGA(x1, x2, x3)
U1_GGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGA(x1, x2, x3, x4, x6)
U24_GA(x1, x2, x3, x4, x5)  =  U24_GA(x1, x2, x3, x5)
LESS36_IN_GG(x1, x2)  =  LESS36_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x1, x2, x3, x5)
U26_GA(x1, x2, x3, x4, x5)  =  U26_GA(x1, x2, x3, x5)
SPLIT46_IN_GGAA(x1, x2, x3, x4)  =  SPLIT46_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x1, x2, x3, x6)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x1, x2, x3, x6)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)
GEQ62_IN_GG(x1, x2)  =  GEQ62_IN_GG(x1, x2)
U9_GG(x1, x2, x3)  =  U9_GG(x1, x2, x3)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)
U8_GGAA(x1, x2, x3, x4, x5, x6)  =  U8_GGAA(x1, x2, x3, x6)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x3, x5)
U28_GA(x1, x2, x3, x4, x5)  =  U28_GA(x1, x2, x3, x5)
U31_GA(x1, x2, x3, x4, x5)  =  U31_GA(x1, x2, x3, x5)
U32_GA(x1, x2, x3, x4, x5)  =  U32_GA(x1, x2, x3, x5)
U33_GA(x1, x2, x3, x4, x5)  =  U33_GA(x1, x2, x3, x5)
U34_GA(x1, x2, x3, x4, x5)  =  U34_GA(x1, x2, x3, x5)
U35_GA(x1, x2, x3, x4, x5)  =  U35_GA(x1, x2, x3, x5)
QS79_IN_GA(x1, x2)  =  QS79_IN_GA(x1)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x1, x2, x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x1, x2, x4)
U12_GA(x1, x2, x3, x4)  =  U12_GA(x1, x2, x4)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x1, x2, x4, x5)
U14_GA(x1, x2, x3, x4)  =  U14_GA(x1, x2, x4)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x2, x4, x5)
U16_GA(x1, x2, x3, x4)  =  U16_GA(x1, x2, x4)
APPEND94_IN_GGGA(x1, x2, x3, x4)  =  APPEND94_IN_GGGA(x1, x2, x3)
U17_GGGA(x1, x2, x3, x4, x5, x6)  =  U17_GGGA(x1, x2, x3, x4, x6)
U36_GA(x1, x2, x3, x4, x5, x6)  =  U36_GA(x1, x2, x3, x5, x6)
U37_GA(x1, x2, x3, x4, x5)  =  U37_GA(x1, x2, x3, x5)
P78_IN_GAGGA(x1, x2, x3, x4, x5)  =  P78_IN_GAGGA(x1, x3, x4)
U18_GAGGA(x1, x2, x3, x4, x5, x6)  =  U18_GAGGA(x1, x3, x4, x6)
U19_GAGGA(x1, x2, x3, x4, x5, x6)  =  U19_GAGGA(x1, x3, x4, x6)
U20_GAGGA(x1, x2, x3, x4, x5, x6)  =  U20_GAGGA(x1, x3, x4, x6)
U29_GA(x1, x2, x3, x4, x5, x6)  =  U29_GA(x1, x2, x3, x5, x6)
U30_GA(x1, x2, x3, x4, x5)  =  U30_GA(x1, x2, x3, x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 7 SCCs with 45 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APPEND94_IN_GGGA(cons(T194, T195), T196, T197, cons(T194, X295)) → APPEND94_IN_GGGA(T195, T196, T197, X295)

The TRS R consists of the following rules:

qsc14_in_a([]) → qsc14_out_a([])
lessc36_in_gg(0, s(T78)) → lessc36_out_gg(0, s(T78))
lessc36_in_gg(s(T83), s(T84)) → U40_gg(T83, T84, lessc36_in_gg(T83, T84))
U40_gg(T83, T84, lessc36_out_gg(T83, T84)) → lessc36_out_gg(s(T83), s(T84))
geqc62_in_gg(T136, T136) → geqc62_out_gg(T136, T136)
geqc62_in_gg(s(T141), 0) → geqc62_out_gg(s(T141), 0)
geqc62_in_gg(s(T146), s(T147)) → U45_gg(T146, T147, geqc62_in_gg(T146, T147))
U45_gg(T146, T147, geqc62_out_gg(T146, T147)) → geqc62_out_gg(s(T146), s(T147))
splitc46_in_ggaa([], T97, [], []) → splitc46_out_ggaa([], T97, [], [])
splitc46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U41_ggaa(T110, T111, T112, X163, X164, lessc36_in_gg(T110, T112))
U41_ggaa(T110, T111, T112, X163, X164, lessc36_out_gg(T110, T112)) → U42_ggaa(T110, T111, T112, X163, X164, splitc46_in_ggaa(T111, T112, X163, X164))
splitc46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U43_ggaa(T125, T126, T127, X195, X196, geqc62_in_gg(T125, T127))
U43_ggaa(T125, T126, T127, X195, X196, geqc62_out_gg(T125, T127)) → U44_ggaa(T125, T126, T127, X195, X196, splitc46_in_ggaa(T126, T127, X195, X196))
U44_ggaa(T125, T126, T127, X195, X196, splitc46_out_ggaa(T126, T127, X195, X196)) → splitc46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U42_ggaa(T110, T111, T112, X163, X164, splitc46_out_ggaa(T111, T112, X163, X164)) → splitc46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
qsc79_in_ga([], []) → qsc79_out_ga([], [])
qsc79_in_ga(cons(T162, T163), X265) → U57_ga(T162, T163, X265, splitc46_in_ggaa(T163, T162, T166, T167))
U57_ga(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → U58_ga(T162, T163, X265, T167, qsc79_in_ga(T166, T170))
U58_ga(T162, T163, X265, T167, qsc79_out_ga(T166, T170)) → U59_ga(T162, T163, X265, T170, qsc79_in_ga(T167, T171))
U59_ga(T162, T163, X265, T170, qsc79_out_ga(T167, T171)) → U60_ga(T162, T163, X265, appendc94_in_ggga(T170, T162, T171, X265))
appendc94_in_ggga([], T184, T185, cons(T184, T185)) → appendc94_out_ggga([], T184, T185, cons(T184, T185))
appendc94_in_ggga(cons(T194, T195), T196, T197, cons(T194, X295)) → U61_ggga(T194, T195, T196, T197, X295, appendc94_in_ggga(T195, T196, T197, X295))
U61_ggga(T194, T195, T196, T197, X295, appendc94_out_ggga(T195, T196, T197, X295)) → appendc94_out_ggga(cons(T194, T195), T196, T197, cons(T194, X295))
U60_ga(T162, T163, X265, appendc94_out_ggga(T170, T162, T171, X265)) → qsc79_out_ga(cons(T162, T163), X265)
qsc1_in_ga([], []) → qsc1_out_ga([], [])
qsc1_in_ga(cons(T14, []), T9) → U46_ga(T14, T9, qsc14_in_a(T17))
U46_ga(T14, T9, qsc14_out_a(T17)) → U47_ga(T14, T9, T17, qsc14_in_a(T20))
U47_ga(T14, T9, T17, qsc14_out_a(T20)) → U48_ga(T14, T9, appendc23_in_ggga(T17, T14, T20, T9))
appendc23_in_ggga([], T33, T34, cons(T33, T34)) → appendc23_out_ggga([], T33, T34, cons(T33, T34))
appendc23_in_ggga(cons(T45, T46), T47, T48, cons(T45, T50)) → U39_ggga(T45, T46, T47, T48, T50, appendc23_in_ggga(T46, T47, T48, T50))
U39_ggga(T45, T46, T47, T48, T50, appendc23_out_ggga(T46, T47, T48, T50)) → appendc23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U48_ga(T14, T9, appendc23_out_ggga(T17, T14, T20, T9)) → qsc1_out_ga(cons(T14, []), T9)
qsc1_in_ga(cons(T69, cons(T67, T68)), T9) → U49_ga(T69, T67, T68, T9, lessc36_in_gg(T67, T69))
U49_ga(T69, T67, T68, T9, lessc36_out_gg(T67, T69)) → U50_ga(T69, T67, T68, T9, splitc46_in_ggaa(T68, T69, T89, T90))
U50_ga(T69, T67, T68, T9, splitc46_out_ggaa(T68, T69, T89, T90)) → U51_ga(T69, T67, T68, T9, T90, qsc1_in_ga(cons(T67, T89), T154))
qsc1_in_ga(cons(T214, cons(T212, T213)), T9) → U53_ga(T214, T212, T213, T9, geqc62_in_gg(T212, T214))
U53_ga(T214, T212, T213, T9, geqc62_out_gg(T212, T214)) → U54_ga(T214, T212, T213, T9, splitc46_in_ggaa(T213, T214, T221, T222))
U54_ga(T214, T212, T213, T9, splitc46_out_ggaa(T213, T214, T221, T222)) → U55_ga(T214, T212, T213, T9, T222, qsc79_in_ga(T221, T225))
U55_ga(T214, T212, T213, T9, T222, qsc79_out_ga(T221, T225)) → U56_ga(T214, T212, T213, T9, qc78_in_gagga(cons(T212, T222), X13, T225, T214, T9))
qc78_in_gagga(T90, T157, T154, T69, T9) → U62_gagga(T90, T157, T154, T69, T9, qsc79_in_ga(T90, T157))
U62_gagga(T90, T157, T154, T69, T9, qsc79_out_ga(T90, T157)) → U63_gagga(T90, T157, T154, T69, T9, appendc23_in_ggga(T154, T69, T157, T9))
U63_gagga(T90, T157, T154, T69, T9, appendc23_out_ggga(T154, T69, T157, T9)) → qc78_out_gagga(T90, T157, T154, T69, T9)
U56_ga(T214, T212, T213, T9, qc78_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → qsc1_out_ga(cons(T214, cons(T212, T213)), T9)
U51_ga(T69, T67, T68, T9, T90, qsc1_out_ga(cons(T67, T89), T154)) → U52_ga(T69, T67, T68, T9, qc78_in_gagga(T90, X13, T154, T69, T9))
U52_ga(T69, T67, T68, T9, qc78_out_gagga(T90, X13, T154, T69, T9)) → qsc1_out_ga(cons(T69, cons(T67, T68)), T9)

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
[]  =  []
qsc14_in_a(x1)  =  qsc14_in_a
qsc14_out_a(x1)  =  qsc14_out_a(x1)
s(x1)  =  s(x1)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
0  =  0
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
geqc62_in_gg(x1, x2)  =  geqc62_in_gg(x1, x2)
geqc62_out_gg(x1, x2)  =  geqc62_out_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
splitc46_in_ggaa(x1, x2, x3, x4)  =  splitc46_in_ggaa(x1, x2)
splitc46_out_ggaa(x1, x2, x3, x4)  =  splitc46_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qsc79_in_ga(x1, x2)  =  qsc79_in_ga(x1)
qsc79_out_ga(x1, x2)  =  qsc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
qsc1_in_ga(x1, x2)  =  qsc1_in_ga(x1)
qsc1_out_ga(x1, x2)  =  qsc1_out_ga(x1, x2)
U46_ga(x1, x2, x3)  =  U46_ga(x1, x3)
U47_ga(x1, x2, x3, x4)  =  U47_ga(x1, x3, x4)
U48_ga(x1, x2, x3)  =  U48_ga(x1, x3)
appendc23_in_ggga(x1, x2, x3, x4)  =  appendc23_in_ggga(x1, x2, x3)
appendc23_out_ggga(x1, x2, x3, x4)  =  appendc23_out_ggga(x1, x2, x3, x4)
U39_ggga(x1, x2, x3, x4, x5, x6)  =  U39_ggga(x1, x2, x3, x4, x6)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6)  =  U51_ga(x1, x2, x3, x5, x6)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x2, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x1, x2, x3, x5)
U55_ga(x1, x2, x3, x4, x5, x6)  =  U55_ga(x1, x2, x3, x5, x6)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x1, x2, x3, x5)
qc78_in_gagga(x1, x2, x3, x4, x5)  =  qc78_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6)  =  U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6)  =  U63_gagga(x1, x2, x3, x4, x6)
qc78_out_gagga(x1, x2, x3, x4, x5)  =  qc78_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x1, x2, x3, x5)
APPEND94_IN_GGGA(x1, x2, x3, x4)  =  APPEND94_IN_GGGA(x1, x2, x3)

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:

APPEND94_IN_GGGA(cons(T194, T195), T196, T197, cons(T194, X295)) → APPEND94_IN_GGGA(T195, T196, T197, X295)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
APPEND94_IN_GGGA(x1, x2, x3, x4)  =  APPEND94_IN_GGGA(x1, x2, x3)

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:

APPEND94_IN_GGGA(cons(T194, T195), T196, T197) → APPEND94_IN_GGGA(T195, T196, T197)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • APPEND94_IN_GGGA(cons(T194, T195), T196, T197) → APPEND94_IN_GGGA(T195, T196, T197)
    The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

(15) YES

(16) Obligation:

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

GEQ62_IN_GG(s(T146), s(T147)) → GEQ62_IN_GG(T146, T147)

The TRS R consists of the following rules:

qsc14_in_a([]) → qsc14_out_a([])
lessc36_in_gg(0, s(T78)) → lessc36_out_gg(0, s(T78))
lessc36_in_gg(s(T83), s(T84)) → U40_gg(T83, T84, lessc36_in_gg(T83, T84))
U40_gg(T83, T84, lessc36_out_gg(T83, T84)) → lessc36_out_gg(s(T83), s(T84))
geqc62_in_gg(T136, T136) → geqc62_out_gg(T136, T136)
geqc62_in_gg(s(T141), 0) → geqc62_out_gg(s(T141), 0)
geqc62_in_gg(s(T146), s(T147)) → U45_gg(T146, T147, geqc62_in_gg(T146, T147))
U45_gg(T146, T147, geqc62_out_gg(T146, T147)) → geqc62_out_gg(s(T146), s(T147))
splitc46_in_ggaa([], T97, [], []) → splitc46_out_ggaa([], T97, [], [])
splitc46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U41_ggaa(T110, T111, T112, X163, X164, lessc36_in_gg(T110, T112))
U41_ggaa(T110, T111, T112, X163, X164, lessc36_out_gg(T110, T112)) → U42_ggaa(T110, T111, T112, X163, X164, splitc46_in_ggaa(T111, T112, X163, X164))
splitc46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U43_ggaa(T125, T126, T127, X195, X196, geqc62_in_gg(T125, T127))
U43_ggaa(T125, T126, T127, X195, X196, geqc62_out_gg(T125, T127)) → U44_ggaa(T125, T126, T127, X195, X196, splitc46_in_ggaa(T126, T127, X195, X196))
U44_ggaa(T125, T126, T127, X195, X196, splitc46_out_ggaa(T126, T127, X195, X196)) → splitc46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U42_ggaa(T110, T111, T112, X163, X164, splitc46_out_ggaa(T111, T112, X163, X164)) → splitc46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
qsc79_in_ga([], []) → qsc79_out_ga([], [])
qsc79_in_ga(cons(T162, T163), X265) → U57_ga(T162, T163, X265, splitc46_in_ggaa(T163, T162, T166, T167))
U57_ga(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → U58_ga(T162, T163, X265, T167, qsc79_in_ga(T166, T170))
U58_ga(T162, T163, X265, T167, qsc79_out_ga(T166, T170)) → U59_ga(T162, T163, X265, T170, qsc79_in_ga(T167, T171))
U59_ga(T162, T163, X265, T170, qsc79_out_ga(T167, T171)) → U60_ga(T162, T163, X265, appendc94_in_ggga(T170, T162, T171, X265))
appendc94_in_ggga([], T184, T185, cons(T184, T185)) → appendc94_out_ggga([], T184, T185, cons(T184, T185))
appendc94_in_ggga(cons(T194, T195), T196, T197, cons(T194, X295)) → U61_ggga(T194, T195, T196, T197, X295, appendc94_in_ggga(T195, T196, T197, X295))
U61_ggga(T194, T195, T196, T197, X295, appendc94_out_ggga(T195, T196, T197, X295)) → appendc94_out_ggga(cons(T194, T195), T196, T197, cons(T194, X295))
U60_ga(T162, T163, X265, appendc94_out_ggga(T170, T162, T171, X265)) → qsc79_out_ga(cons(T162, T163), X265)
qsc1_in_ga([], []) → qsc1_out_ga([], [])
qsc1_in_ga(cons(T14, []), T9) → U46_ga(T14, T9, qsc14_in_a(T17))
U46_ga(T14, T9, qsc14_out_a(T17)) → U47_ga(T14, T9, T17, qsc14_in_a(T20))
U47_ga(T14, T9, T17, qsc14_out_a(T20)) → U48_ga(T14, T9, appendc23_in_ggga(T17, T14, T20, T9))
appendc23_in_ggga([], T33, T34, cons(T33, T34)) → appendc23_out_ggga([], T33, T34, cons(T33, T34))
appendc23_in_ggga(cons(T45, T46), T47, T48, cons(T45, T50)) → U39_ggga(T45, T46, T47, T48, T50, appendc23_in_ggga(T46, T47, T48, T50))
U39_ggga(T45, T46, T47, T48, T50, appendc23_out_ggga(T46, T47, T48, T50)) → appendc23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U48_ga(T14, T9, appendc23_out_ggga(T17, T14, T20, T9)) → qsc1_out_ga(cons(T14, []), T9)
qsc1_in_ga(cons(T69, cons(T67, T68)), T9) → U49_ga(T69, T67, T68, T9, lessc36_in_gg(T67, T69))
U49_ga(T69, T67, T68, T9, lessc36_out_gg(T67, T69)) → U50_ga(T69, T67, T68, T9, splitc46_in_ggaa(T68, T69, T89, T90))
U50_ga(T69, T67, T68, T9, splitc46_out_ggaa(T68, T69, T89, T90)) → U51_ga(T69, T67, T68, T9, T90, qsc1_in_ga(cons(T67, T89), T154))
qsc1_in_ga(cons(T214, cons(T212, T213)), T9) → U53_ga(T214, T212, T213, T9, geqc62_in_gg(T212, T214))
U53_ga(T214, T212, T213, T9, geqc62_out_gg(T212, T214)) → U54_ga(T214, T212, T213, T9, splitc46_in_ggaa(T213, T214, T221, T222))
U54_ga(T214, T212, T213, T9, splitc46_out_ggaa(T213, T214, T221, T222)) → U55_ga(T214, T212, T213, T9, T222, qsc79_in_ga(T221, T225))
U55_ga(T214, T212, T213, T9, T222, qsc79_out_ga(T221, T225)) → U56_ga(T214, T212, T213, T9, qc78_in_gagga(cons(T212, T222), X13, T225, T214, T9))
qc78_in_gagga(T90, T157, T154, T69, T9) → U62_gagga(T90, T157, T154, T69, T9, qsc79_in_ga(T90, T157))
U62_gagga(T90, T157, T154, T69, T9, qsc79_out_ga(T90, T157)) → U63_gagga(T90, T157, T154, T69, T9, appendc23_in_ggga(T154, T69, T157, T9))
U63_gagga(T90, T157, T154, T69, T9, appendc23_out_ggga(T154, T69, T157, T9)) → qc78_out_gagga(T90, T157, T154, T69, T9)
U56_ga(T214, T212, T213, T9, qc78_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → qsc1_out_ga(cons(T214, cons(T212, T213)), T9)
U51_ga(T69, T67, T68, T9, T90, qsc1_out_ga(cons(T67, T89), T154)) → U52_ga(T69, T67, T68, T9, qc78_in_gagga(T90, X13, T154, T69, T9))
U52_ga(T69, T67, T68, T9, qc78_out_gagga(T90, X13, T154, T69, T9)) → qsc1_out_ga(cons(T69, cons(T67, T68)), T9)

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
[]  =  []
qsc14_in_a(x1)  =  qsc14_in_a
qsc14_out_a(x1)  =  qsc14_out_a(x1)
s(x1)  =  s(x1)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
0  =  0
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
geqc62_in_gg(x1, x2)  =  geqc62_in_gg(x1, x2)
geqc62_out_gg(x1, x2)  =  geqc62_out_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
splitc46_in_ggaa(x1, x2, x3, x4)  =  splitc46_in_ggaa(x1, x2)
splitc46_out_ggaa(x1, x2, x3, x4)  =  splitc46_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qsc79_in_ga(x1, x2)  =  qsc79_in_ga(x1)
qsc79_out_ga(x1, x2)  =  qsc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
qsc1_in_ga(x1, x2)  =  qsc1_in_ga(x1)
qsc1_out_ga(x1, x2)  =  qsc1_out_ga(x1, x2)
U46_ga(x1, x2, x3)  =  U46_ga(x1, x3)
U47_ga(x1, x2, x3, x4)  =  U47_ga(x1, x3, x4)
U48_ga(x1, x2, x3)  =  U48_ga(x1, x3)
appendc23_in_ggga(x1, x2, x3, x4)  =  appendc23_in_ggga(x1, x2, x3)
appendc23_out_ggga(x1, x2, x3, x4)  =  appendc23_out_ggga(x1, x2, x3, x4)
U39_ggga(x1, x2, x3, x4, x5, x6)  =  U39_ggga(x1, x2, x3, x4, x6)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6)  =  U51_ga(x1, x2, x3, x5, x6)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x2, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x1, x2, x3, x5)
U55_ga(x1, x2, x3, x4, x5, x6)  =  U55_ga(x1, x2, x3, x5, x6)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x1, x2, x3, x5)
qc78_in_gagga(x1, x2, x3, x4, x5)  =  qc78_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6)  =  U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6)  =  U63_gagga(x1, x2, x3, x4, x6)
qc78_out_gagga(x1, x2, x3, x4, x5)  =  qc78_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x1, x2, x3, x5)
GEQ62_IN_GG(x1, x2)  =  GEQ62_IN_GG(x1, x2)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

GEQ62_IN_GG(s(T146), s(T147)) → GEQ62_IN_GG(T146, T147)

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

(19) PiDPToQDPProof (EQUIVALENT 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:

GEQ62_IN_GG(s(T146), s(T147)) → GEQ62_IN_GG(T146, T147)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • GEQ62_IN_GG(s(T146), s(T147)) → GEQ62_IN_GG(T146, T147)
    The graph contains the following edges 1 > 1, 2 > 2

(22) YES

(23) Obligation:

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

LESS36_IN_GG(s(T83), s(T84)) → LESS36_IN_GG(T83, T84)

The TRS R consists of the following rules:

qsc14_in_a([]) → qsc14_out_a([])
lessc36_in_gg(0, s(T78)) → lessc36_out_gg(0, s(T78))
lessc36_in_gg(s(T83), s(T84)) → U40_gg(T83, T84, lessc36_in_gg(T83, T84))
U40_gg(T83, T84, lessc36_out_gg(T83, T84)) → lessc36_out_gg(s(T83), s(T84))
geqc62_in_gg(T136, T136) → geqc62_out_gg(T136, T136)
geqc62_in_gg(s(T141), 0) → geqc62_out_gg(s(T141), 0)
geqc62_in_gg(s(T146), s(T147)) → U45_gg(T146, T147, geqc62_in_gg(T146, T147))
U45_gg(T146, T147, geqc62_out_gg(T146, T147)) → geqc62_out_gg(s(T146), s(T147))
splitc46_in_ggaa([], T97, [], []) → splitc46_out_ggaa([], T97, [], [])
splitc46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U41_ggaa(T110, T111, T112, X163, X164, lessc36_in_gg(T110, T112))
U41_ggaa(T110, T111, T112, X163, X164, lessc36_out_gg(T110, T112)) → U42_ggaa(T110, T111, T112, X163, X164, splitc46_in_ggaa(T111, T112, X163, X164))
splitc46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U43_ggaa(T125, T126, T127, X195, X196, geqc62_in_gg(T125, T127))
U43_ggaa(T125, T126, T127, X195, X196, geqc62_out_gg(T125, T127)) → U44_ggaa(T125, T126, T127, X195, X196, splitc46_in_ggaa(T126, T127, X195, X196))
U44_ggaa(T125, T126, T127, X195, X196, splitc46_out_ggaa(T126, T127, X195, X196)) → splitc46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U42_ggaa(T110, T111, T112, X163, X164, splitc46_out_ggaa(T111, T112, X163, X164)) → splitc46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
qsc79_in_ga([], []) → qsc79_out_ga([], [])
qsc79_in_ga(cons(T162, T163), X265) → U57_ga(T162, T163, X265, splitc46_in_ggaa(T163, T162, T166, T167))
U57_ga(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → U58_ga(T162, T163, X265, T167, qsc79_in_ga(T166, T170))
U58_ga(T162, T163, X265, T167, qsc79_out_ga(T166, T170)) → U59_ga(T162, T163, X265, T170, qsc79_in_ga(T167, T171))
U59_ga(T162, T163, X265, T170, qsc79_out_ga(T167, T171)) → U60_ga(T162, T163, X265, appendc94_in_ggga(T170, T162, T171, X265))
appendc94_in_ggga([], T184, T185, cons(T184, T185)) → appendc94_out_ggga([], T184, T185, cons(T184, T185))
appendc94_in_ggga(cons(T194, T195), T196, T197, cons(T194, X295)) → U61_ggga(T194, T195, T196, T197, X295, appendc94_in_ggga(T195, T196, T197, X295))
U61_ggga(T194, T195, T196, T197, X295, appendc94_out_ggga(T195, T196, T197, X295)) → appendc94_out_ggga(cons(T194, T195), T196, T197, cons(T194, X295))
U60_ga(T162, T163, X265, appendc94_out_ggga(T170, T162, T171, X265)) → qsc79_out_ga(cons(T162, T163), X265)
qsc1_in_ga([], []) → qsc1_out_ga([], [])
qsc1_in_ga(cons(T14, []), T9) → U46_ga(T14, T9, qsc14_in_a(T17))
U46_ga(T14, T9, qsc14_out_a(T17)) → U47_ga(T14, T9, T17, qsc14_in_a(T20))
U47_ga(T14, T9, T17, qsc14_out_a(T20)) → U48_ga(T14, T9, appendc23_in_ggga(T17, T14, T20, T9))
appendc23_in_ggga([], T33, T34, cons(T33, T34)) → appendc23_out_ggga([], T33, T34, cons(T33, T34))
appendc23_in_ggga(cons(T45, T46), T47, T48, cons(T45, T50)) → U39_ggga(T45, T46, T47, T48, T50, appendc23_in_ggga(T46, T47, T48, T50))
U39_ggga(T45, T46, T47, T48, T50, appendc23_out_ggga(T46, T47, T48, T50)) → appendc23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U48_ga(T14, T9, appendc23_out_ggga(T17, T14, T20, T9)) → qsc1_out_ga(cons(T14, []), T9)
qsc1_in_ga(cons(T69, cons(T67, T68)), T9) → U49_ga(T69, T67, T68, T9, lessc36_in_gg(T67, T69))
U49_ga(T69, T67, T68, T9, lessc36_out_gg(T67, T69)) → U50_ga(T69, T67, T68, T9, splitc46_in_ggaa(T68, T69, T89, T90))
U50_ga(T69, T67, T68, T9, splitc46_out_ggaa(T68, T69, T89, T90)) → U51_ga(T69, T67, T68, T9, T90, qsc1_in_ga(cons(T67, T89), T154))
qsc1_in_ga(cons(T214, cons(T212, T213)), T9) → U53_ga(T214, T212, T213, T9, geqc62_in_gg(T212, T214))
U53_ga(T214, T212, T213, T9, geqc62_out_gg(T212, T214)) → U54_ga(T214, T212, T213, T9, splitc46_in_ggaa(T213, T214, T221, T222))
U54_ga(T214, T212, T213, T9, splitc46_out_ggaa(T213, T214, T221, T222)) → U55_ga(T214, T212, T213, T9, T222, qsc79_in_ga(T221, T225))
U55_ga(T214, T212, T213, T9, T222, qsc79_out_ga(T221, T225)) → U56_ga(T214, T212, T213, T9, qc78_in_gagga(cons(T212, T222), X13, T225, T214, T9))
qc78_in_gagga(T90, T157, T154, T69, T9) → U62_gagga(T90, T157, T154, T69, T9, qsc79_in_ga(T90, T157))
U62_gagga(T90, T157, T154, T69, T9, qsc79_out_ga(T90, T157)) → U63_gagga(T90, T157, T154, T69, T9, appendc23_in_ggga(T154, T69, T157, T9))
U63_gagga(T90, T157, T154, T69, T9, appendc23_out_ggga(T154, T69, T157, T9)) → qc78_out_gagga(T90, T157, T154, T69, T9)
U56_ga(T214, T212, T213, T9, qc78_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → qsc1_out_ga(cons(T214, cons(T212, T213)), T9)
U51_ga(T69, T67, T68, T9, T90, qsc1_out_ga(cons(T67, T89), T154)) → U52_ga(T69, T67, T68, T9, qc78_in_gagga(T90, X13, T154, T69, T9))
U52_ga(T69, T67, T68, T9, qc78_out_gagga(T90, X13, T154, T69, T9)) → qsc1_out_ga(cons(T69, cons(T67, T68)), T9)

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
[]  =  []
qsc14_in_a(x1)  =  qsc14_in_a
qsc14_out_a(x1)  =  qsc14_out_a(x1)
s(x1)  =  s(x1)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
0  =  0
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
geqc62_in_gg(x1, x2)  =  geqc62_in_gg(x1, x2)
geqc62_out_gg(x1, x2)  =  geqc62_out_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
splitc46_in_ggaa(x1, x2, x3, x4)  =  splitc46_in_ggaa(x1, x2)
splitc46_out_ggaa(x1, x2, x3, x4)  =  splitc46_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qsc79_in_ga(x1, x2)  =  qsc79_in_ga(x1)
qsc79_out_ga(x1, x2)  =  qsc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
qsc1_in_ga(x1, x2)  =  qsc1_in_ga(x1)
qsc1_out_ga(x1, x2)  =  qsc1_out_ga(x1, x2)
U46_ga(x1, x2, x3)  =  U46_ga(x1, x3)
U47_ga(x1, x2, x3, x4)  =  U47_ga(x1, x3, x4)
U48_ga(x1, x2, x3)  =  U48_ga(x1, x3)
appendc23_in_ggga(x1, x2, x3, x4)  =  appendc23_in_ggga(x1, x2, x3)
appendc23_out_ggga(x1, x2, x3, x4)  =  appendc23_out_ggga(x1, x2, x3, x4)
U39_ggga(x1, x2, x3, x4, x5, x6)  =  U39_ggga(x1, x2, x3, x4, x6)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6)  =  U51_ga(x1, x2, x3, x5, x6)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x2, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x1, x2, x3, x5)
U55_ga(x1, x2, x3, x4, x5, x6)  =  U55_ga(x1, x2, x3, x5, x6)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x1, x2, x3, x5)
qc78_in_gagga(x1, x2, x3, x4, x5)  =  qc78_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6)  =  U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6)  =  U63_gagga(x1, x2, x3, x4, x6)
qc78_out_gagga(x1, x2, x3, x4, x5)  =  qc78_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x1, x2, x3, x5)
LESS36_IN_GG(x1, x2)  =  LESS36_IN_GG(x1, x2)

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:

LESS36_IN_GG(s(T83), s(T84)) → LESS36_IN_GG(T83, T84)

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

(26) PiDPToQDPProof (EQUIVALENT 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:

LESS36_IN_GG(s(T83), s(T84)) → LESS36_IN_GG(T83, T84)

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:

  • LESS36_IN_GG(s(T83), s(T84)) → LESS36_IN_GG(T83, T84)
    The graph contains the following edges 1 > 1, 2 > 2

(29) YES

(30) Obligation:

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

SPLIT46_IN_GGAA(cons(T110, T111), T112, cons(T110, X163), X164) → U4_GGAA(T110, T111, T112, X163, X164, lessc36_in_gg(T110, T112))
U4_GGAA(T110, T111, T112, X163, X164, lessc36_out_gg(T110, T112)) → SPLIT46_IN_GGAA(T111, T112, X163, X164)
SPLIT46_IN_GGAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U7_GGAA(T125, T126, T127, X195, X196, geqc62_in_gg(T125, T127))
U7_GGAA(T125, T126, T127, X195, X196, geqc62_out_gg(T125, T127)) → SPLIT46_IN_GGAA(T126, T127, X195, X196)

The TRS R consists of the following rules:

qsc14_in_a([]) → qsc14_out_a([])
lessc36_in_gg(0, s(T78)) → lessc36_out_gg(0, s(T78))
lessc36_in_gg(s(T83), s(T84)) → U40_gg(T83, T84, lessc36_in_gg(T83, T84))
U40_gg(T83, T84, lessc36_out_gg(T83, T84)) → lessc36_out_gg(s(T83), s(T84))
geqc62_in_gg(T136, T136) → geqc62_out_gg(T136, T136)
geqc62_in_gg(s(T141), 0) → geqc62_out_gg(s(T141), 0)
geqc62_in_gg(s(T146), s(T147)) → U45_gg(T146, T147, geqc62_in_gg(T146, T147))
U45_gg(T146, T147, geqc62_out_gg(T146, T147)) → geqc62_out_gg(s(T146), s(T147))
splitc46_in_ggaa([], T97, [], []) → splitc46_out_ggaa([], T97, [], [])
splitc46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U41_ggaa(T110, T111, T112, X163, X164, lessc36_in_gg(T110, T112))
U41_ggaa(T110, T111, T112, X163, X164, lessc36_out_gg(T110, T112)) → U42_ggaa(T110, T111, T112, X163, X164, splitc46_in_ggaa(T111, T112, X163, X164))
splitc46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U43_ggaa(T125, T126, T127, X195, X196, geqc62_in_gg(T125, T127))
U43_ggaa(T125, T126, T127, X195, X196, geqc62_out_gg(T125, T127)) → U44_ggaa(T125, T126, T127, X195, X196, splitc46_in_ggaa(T126, T127, X195, X196))
U44_ggaa(T125, T126, T127, X195, X196, splitc46_out_ggaa(T126, T127, X195, X196)) → splitc46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U42_ggaa(T110, T111, T112, X163, X164, splitc46_out_ggaa(T111, T112, X163, X164)) → splitc46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
qsc79_in_ga([], []) → qsc79_out_ga([], [])
qsc79_in_ga(cons(T162, T163), X265) → U57_ga(T162, T163, X265, splitc46_in_ggaa(T163, T162, T166, T167))
U57_ga(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → U58_ga(T162, T163, X265, T167, qsc79_in_ga(T166, T170))
U58_ga(T162, T163, X265, T167, qsc79_out_ga(T166, T170)) → U59_ga(T162, T163, X265, T170, qsc79_in_ga(T167, T171))
U59_ga(T162, T163, X265, T170, qsc79_out_ga(T167, T171)) → U60_ga(T162, T163, X265, appendc94_in_ggga(T170, T162, T171, X265))
appendc94_in_ggga([], T184, T185, cons(T184, T185)) → appendc94_out_ggga([], T184, T185, cons(T184, T185))
appendc94_in_ggga(cons(T194, T195), T196, T197, cons(T194, X295)) → U61_ggga(T194, T195, T196, T197, X295, appendc94_in_ggga(T195, T196, T197, X295))
U61_ggga(T194, T195, T196, T197, X295, appendc94_out_ggga(T195, T196, T197, X295)) → appendc94_out_ggga(cons(T194, T195), T196, T197, cons(T194, X295))
U60_ga(T162, T163, X265, appendc94_out_ggga(T170, T162, T171, X265)) → qsc79_out_ga(cons(T162, T163), X265)
qsc1_in_ga([], []) → qsc1_out_ga([], [])
qsc1_in_ga(cons(T14, []), T9) → U46_ga(T14, T9, qsc14_in_a(T17))
U46_ga(T14, T9, qsc14_out_a(T17)) → U47_ga(T14, T9, T17, qsc14_in_a(T20))
U47_ga(T14, T9, T17, qsc14_out_a(T20)) → U48_ga(T14, T9, appendc23_in_ggga(T17, T14, T20, T9))
appendc23_in_ggga([], T33, T34, cons(T33, T34)) → appendc23_out_ggga([], T33, T34, cons(T33, T34))
appendc23_in_ggga(cons(T45, T46), T47, T48, cons(T45, T50)) → U39_ggga(T45, T46, T47, T48, T50, appendc23_in_ggga(T46, T47, T48, T50))
U39_ggga(T45, T46, T47, T48, T50, appendc23_out_ggga(T46, T47, T48, T50)) → appendc23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U48_ga(T14, T9, appendc23_out_ggga(T17, T14, T20, T9)) → qsc1_out_ga(cons(T14, []), T9)
qsc1_in_ga(cons(T69, cons(T67, T68)), T9) → U49_ga(T69, T67, T68, T9, lessc36_in_gg(T67, T69))
U49_ga(T69, T67, T68, T9, lessc36_out_gg(T67, T69)) → U50_ga(T69, T67, T68, T9, splitc46_in_ggaa(T68, T69, T89, T90))
U50_ga(T69, T67, T68, T9, splitc46_out_ggaa(T68, T69, T89, T90)) → U51_ga(T69, T67, T68, T9, T90, qsc1_in_ga(cons(T67, T89), T154))
qsc1_in_ga(cons(T214, cons(T212, T213)), T9) → U53_ga(T214, T212, T213, T9, geqc62_in_gg(T212, T214))
U53_ga(T214, T212, T213, T9, geqc62_out_gg(T212, T214)) → U54_ga(T214, T212, T213, T9, splitc46_in_ggaa(T213, T214, T221, T222))
U54_ga(T214, T212, T213, T9, splitc46_out_ggaa(T213, T214, T221, T222)) → U55_ga(T214, T212, T213, T9, T222, qsc79_in_ga(T221, T225))
U55_ga(T214, T212, T213, T9, T222, qsc79_out_ga(T221, T225)) → U56_ga(T214, T212, T213, T9, qc78_in_gagga(cons(T212, T222), X13, T225, T214, T9))
qc78_in_gagga(T90, T157, T154, T69, T9) → U62_gagga(T90, T157, T154, T69, T9, qsc79_in_ga(T90, T157))
U62_gagga(T90, T157, T154, T69, T9, qsc79_out_ga(T90, T157)) → U63_gagga(T90, T157, T154, T69, T9, appendc23_in_ggga(T154, T69, T157, T9))
U63_gagga(T90, T157, T154, T69, T9, appendc23_out_ggga(T154, T69, T157, T9)) → qc78_out_gagga(T90, T157, T154, T69, T9)
U56_ga(T214, T212, T213, T9, qc78_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → qsc1_out_ga(cons(T214, cons(T212, T213)), T9)
U51_ga(T69, T67, T68, T9, T90, qsc1_out_ga(cons(T67, T89), T154)) → U52_ga(T69, T67, T68, T9, qc78_in_gagga(T90, X13, T154, T69, T9))
U52_ga(T69, T67, T68, T9, qc78_out_gagga(T90, X13, T154, T69, T9)) → qsc1_out_ga(cons(T69, cons(T67, T68)), T9)

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
[]  =  []
qsc14_in_a(x1)  =  qsc14_in_a
qsc14_out_a(x1)  =  qsc14_out_a(x1)
s(x1)  =  s(x1)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
0  =  0
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
geqc62_in_gg(x1, x2)  =  geqc62_in_gg(x1, x2)
geqc62_out_gg(x1, x2)  =  geqc62_out_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
splitc46_in_ggaa(x1, x2, x3, x4)  =  splitc46_in_ggaa(x1, x2)
splitc46_out_ggaa(x1, x2, x3, x4)  =  splitc46_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qsc79_in_ga(x1, x2)  =  qsc79_in_ga(x1)
qsc79_out_ga(x1, x2)  =  qsc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
qsc1_in_ga(x1, x2)  =  qsc1_in_ga(x1)
qsc1_out_ga(x1, x2)  =  qsc1_out_ga(x1, x2)
U46_ga(x1, x2, x3)  =  U46_ga(x1, x3)
U47_ga(x1, x2, x3, x4)  =  U47_ga(x1, x3, x4)
U48_ga(x1, x2, x3)  =  U48_ga(x1, x3)
appendc23_in_ggga(x1, x2, x3, x4)  =  appendc23_in_ggga(x1, x2, x3)
appendc23_out_ggga(x1, x2, x3, x4)  =  appendc23_out_ggga(x1, x2, x3, x4)
U39_ggga(x1, x2, x3, x4, x5, x6)  =  U39_ggga(x1, x2, x3, x4, x6)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6)  =  U51_ga(x1, x2, x3, x5, x6)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x2, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x1, x2, x3, x5)
U55_ga(x1, x2, x3, x4, x5, x6)  =  U55_ga(x1, x2, x3, x5, x6)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x1, x2, x3, x5)
qc78_in_gagga(x1, x2, x3, x4, x5)  =  qc78_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6)  =  U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6)  =  U63_gagga(x1, x2, x3, x4, x6)
qc78_out_gagga(x1, x2, x3, x4, x5)  =  qc78_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x1, x2, x3, x5)
SPLIT46_IN_GGAA(x1, x2, x3, x4)  =  SPLIT46_IN_GGAA(x1, x2)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x1, x2, x3, x6)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)

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:

SPLIT46_IN_GGAA(cons(T110, T111), T112, cons(T110, X163), X164) → U4_GGAA(T110, T111, T112, X163, X164, lessc36_in_gg(T110, T112))
U4_GGAA(T110, T111, T112, X163, X164, lessc36_out_gg(T110, T112)) → SPLIT46_IN_GGAA(T111, T112, X163, X164)
SPLIT46_IN_GGAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U7_GGAA(T125, T126, T127, X195, X196, geqc62_in_gg(T125, T127))
U7_GGAA(T125, T126, T127, X195, X196, geqc62_out_gg(T125, T127)) → SPLIT46_IN_GGAA(T126, T127, X195, X196)

The TRS R consists of the following rules:

lessc36_in_gg(0, s(T78)) → lessc36_out_gg(0, s(T78))
lessc36_in_gg(s(T83), s(T84)) → U40_gg(T83, T84, lessc36_in_gg(T83, T84))
geqc62_in_gg(T136, T136) → geqc62_out_gg(T136, T136)
geqc62_in_gg(s(T141), 0) → geqc62_out_gg(s(T141), 0)
geqc62_in_gg(s(T146), s(T147)) → U45_gg(T146, T147, geqc62_in_gg(T146, T147))
U40_gg(T83, T84, lessc36_out_gg(T83, T84)) → lessc36_out_gg(s(T83), s(T84))
U45_gg(T146, T147, geqc62_out_gg(T146, T147)) → geqc62_out_gg(s(T146), s(T147))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
s(x1)  =  s(x1)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
0  =  0
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
geqc62_in_gg(x1, x2)  =  geqc62_in_gg(x1, x2)
geqc62_out_gg(x1, x2)  =  geqc62_out_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
SPLIT46_IN_GGAA(x1, x2, x3, x4)  =  SPLIT46_IN_GGAA(x1, x2)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x1, x2, x3, x6)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

SPLIT46_IN_GGAA(cons(T110, T111), T112) → U4_GGAA(T110, T111, T112, lessc36_in_gg(T110, T112))
U4_GGAA(T110, T111, T112, lessc36_out_gg(T110, T112)) → SPLIT46_IN_GGAA(T111, T112)
SPLIT46_IN_GGAA(cons(T125, T126), T127) → U7_GGAA(T125, T126, T127, geqc62_in_gg(T125, T127))
U7_GGAA(T125, T126, T127, geqc62_out_gg(T125, T127)) → SPLIT46_IN_GGAA(T126, T127)

The TRS R consists of the following rules:

lessc36_in_gg(0, s(T78)) → lessc36_out_gg(0, s(T78))
lessc36_in_gg(s(T83), s(T84)) → U40_gg(T83, T84, lessc36_in_gg(T83, T84))
geqc62_in_gg(T136, T136) → geqc62_out_gg(T136, T136)
geqc62_in_gg(s(T141), 0) → geqc62_out_gg(s(T141), 0)
geqc62_in_gg(s(T146), s(T147)) → U45_gg(T146, T147, geqc62_in_gg(T146, T147))
U40_gg(T83, T84, lessc36_out_gg(T83, T84)) → lessc36_out_gg(s(T83), s(T84))
U45_gg(T146, T147, geqc62_out_gg(T146, T147)) → geqc62_out_gg(s(T146), s(T147))

The set Q consists of the following terms:

lessc36_in_gg(x0, x1)
geqc62_in_gg(x0, x1)
U40_gg(x0, x1, x2)
U45_gg(x0, x1, x2)

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:

  • U4_GGAA(T110, T111, T112, lessc36_out_gg(T110, T112)) → SPLIT46_IN_GGAA(T111, T112)
    The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 2

  • U7_GGAA(T125, T126, T127, geqc62_out_gg(T125, T127)) → SPLIT46_IN_GGAA(T126, T127)
    The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 2

  • SPLIT46_IN_GGAA(cons(T110, T111), T112) → U4_GGAA(T110, T111, T112, lessc36_in_gg(T110, T112))
    The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3

  • SPLIT46_IN_GGAA(cons(T125, T126), T127) → U7_GGAA(T125, T126, T127, geqc62_in_gg(T125, T127))
    The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3

(36) YES

(37) Obligation:

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

QS79_IN_GA(cons(T162, T163), X265) → U11_GA(T162, T163, X265, splitc46_in_ggaa(T163, T162, T166, T167))
U11_GA(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → QS79_IN_GA(T166, X263)
U11_GA(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → U13_GA(T162, T163, X265, T167, qsc79_in_ga(T166, T170))
U13_GA(T162, T163, X265, T167, qsc79_out_ga(T166, T170)) → QS79_IN_GA(T167, X264)

The TRS R consists of the following rules:

qsc14_in_a([]) → qsc14_out_a([])
lessc36_in_gg(0, s(T78)) → lessc36_out_gg(0, s(T78))
lessc36_in_gg(s(T83), s(T84)) → U40_gg(T83, T84, lessc36_in_gg(T83, T84))
U40_gg(T83, T84, lessc36_out_gg(T83, T84)) → lessc36_out_gg(s(T83), s(T84))
geqc62_in_gg(T136, T136) → geqc62_out_gg(T136, T136)
geqc62_in_gg(s(T141), 0) → geqc62_out_gg(s(T141), 0)
geqc62_in_gg(s(T146), s(T147)) → U45_gg(T146, T147, geqc62_in_gg(T146, T147))
U45_gg(T146, T147, geqc62_out_gg(T146, T147)) → geqc62_out_gg(s(T146), s(T147))
splitc46_in_ggaa([], T97, [], []) → splitc46_out_ggaa([], T97, [], [])
splitc46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U41_ggaa(T110, T111, T112, X163, X164, lessc36_in_gg(T110, T112))
U41_ggaa(T110, T111, T112, X163, X164, lessc36_out_gg(T110, T112)) → U42_ggaa(T110, T111, T112, X163, X164, splitc46_in_ggaa(T111, T112, X163, X164))
splitc46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U43_ggaa(T125, T126, T127, X195, X196, geqc62_in_gg(T125, T127))
U43_ggaa(T125, T126, T127, X195, X196, geqc62_out_gg(T125, T127)) → U44_ggaa(T125, T126, T127, X195, X196, splitc46_in_ggaa(T126, T127, X195, X196))
U44_ggaa(T125, T126, T127, X195, X196, splitc46_out_ggaa(T126, T127, X195, X196)) → splitc46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U42_ggaa(T110, T111, T112, X163, X164, splitc46_out_ggaa(T111, T112, X163, X164)) → splitc46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
qsc79_in_ga([], []) → qsc79_out_ga([], [])
qsc79_in_ga(cons(T162, T163), X265) → U57_ga(T162, T163, X265, splitc46_in_ggaa(T163, T162, T166, T167))
U57_ga(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → U58_ga(T162, T163, X265, T167, qsc79_in_ga(T166, T170))
U58_ga(T162, T163, X265, T167, qsc79_out_ga(T166, T170)) → U59_ga(T162, T163, X265, T170, qsc79_in_ga(T167, T171))
U59_ga(T162, T163, X265, T170, qsc79_out_ga(T167, T171)) → U60_ga(T162, T163, X265, appendc94_in_ggga(T170, T162, T171, X265))
appendc94_in_ggga([], T184, T185, cons(T184, T185)) → appendc94_out_ggga([], T184, T185, cons(T184, T185))
appendc94_in_ggga(cons(T194, T195), T196, T197, cons(T194, X295)) → U61_ggga(T194, T195, T196, T197, X295, appendc94_in_ggga(T195, T196, T197, X295))
U61_ggga(T194, T195, T196, T197, X295, appendc94_out_ggga(T195, T196, T197, X295)) → appendc94_out_ggga(cons(T194, T195), T196, T197, cons(T194, X295))
U60_ga(T162, T163, X265, appendc94_out_ggga(T170, T162, T171, X265)) → qsc79_out_ga(cons(T162, T163), X265)
qsc1_in_ga([], []) → qsc1_out_ga([], [])
qsc1_in_ga(cons(T14, []), T9) → U46_ga(T14, T9, qsc14_in_a(T17))
U46_ga(T14, T9, qsc14_out_a(T17)) → U47_ga(T14, T9, T17, qsc14_in_a(T20))
U47_ga(T14, T9, T17, qsc14_out_a(T20)) → U48_ga(T14, T9, appendc23_in_ggga(T17, T14, T20, T9))
appendc23_in_ggga([], T33, T34, cons(T33, T34)) → appendc23_out_ggga([], T33, T34, cons(T33, T34))
appendc23_in_ggga(cons(T45, T46), T47, T48, cons(T45, T50)) → U39_ggga(T45, T46, T47, T48, T50, appendc23_in_ggga(T46, T47, T48, T50))
U39_ggga(T45, T46, T47, T48, T50, appendc23_out_ggga(T46, T47, T48, T50)) → appendc23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U48_ga(T14, T9, appendc23_out_ggga(T17, T14, T20, T9)) → qsc1_out_ga(cons(T14, []), T9)
qsc1_in_ga(cons(T69, cons(T67, T68)), T9) → U49_ga(T69, T67, T68, T9, lessc36_in_gg(T67, T69))
U49_ga(T69, T67, T68, T9, lessc36_out_gg(T67, T69)) → U50_ga(T69, T67, T68, T9, splitc46_in_ggaa(T68, T69, T89, T90))
U50_ga(T69, T67, T68, T9, splitc46_out_ggaa(T68, T69, T89, T90)) → U51_ga(T69, T67, T68, T9, T90, qsc1_in_ga(cons(T67, T89), T154))
qsc1_in_ga(cons(T214, cons(T212, T213)), T9) → U53_ga(T214, T212, T213, T9, geqc62_in_gg(T212, T214))
U53_ga(T214, T212, T213, T9, geqc62_out_gg(T212, T214)) → U54_ga(T214, T212, T213, T9, splitc46_in_ggaa(T213, T214, T221, T222))
U54_ga(T214, T212, T213, T9, splitc46_out_ggaa(T213, T214, T221, T222)) → U55_ga(T214, T212, T213, T9, T222, qsc79_in_ga(T221, T225))
U55_ga(T214, T212, T213, T9, T222, qsc79_out_ga(T221, T225)) → U56_ga(T214, T212, T213, T9, qc78_in_gagga(cons(T212, T222), X13, T225, T214, T9))
qc78_in_gagga(T90, T157, T154, T69, T9) → U62_gagga(T90, T157, T154, T69, T9, qsc79_in_ga(T90, T157))
U62_gagga(T90, T157, T154, T69, T9, qsc79_out_ga(T90, T157)) → U63_gagga(T90, T157, T154, T69, T9, appendc23_in_ggga(T154, T69, T157, T9))
U63_gagga(T90, T157, T154, T69, T9, appendc23_out_ggga(T154, T69, T157, T9)) → qc78_out_gagga(T90, T157, T154, T69, T9)
U56_ga(T214, T212, T213, T9, qc78_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → qsc1_out_ga(cons(T214, cons(T212, T213)), T9)
U51_ga(T69, T67, T68, T9, T90, qsc1_out_ga(cons(T67, T89), T154)) → U52_ga(T69, T67, T68, T9, qc78_in_gagga(T90, X13, T154, T69, T9))
U52_ga(T69, T67, T68, T9, qc78_out_gagga(T90, X13, T154, T69, T9)) → qsc1_out_ga(cons(T69, cons(T67, T68)), T9)

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
[]  =  []
qsc14_in_a(x1)  =  qsc14_in_a
qsc14_out_a(x1)  =  qsc14_out_a(x1)
s(x1)  =  s(x1)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
0  =  0
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
geqc62_in_gg(x1, x2)  =  geqc62_in_gg(x1, x2)
geqc62_out_gg(x1, x2)  =  geqc62_out_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
splitc46_in_ggaa(x1, x2, x3, x4)  =  splitc46_in_ggaa(x1, x2)
splitc46_out_ggaa(x1, x2, x3, x4)  =  splitc46_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qsc79_in_ga(x1, x2)  =  qsc79_in_ga(x1)
qsc79_out_ga(x1, x2)  =  qsc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
qsc1_in_ga(x1, x2)  =  qsc1_in_ga(x1)
qsc1_out_ga(x1, x2)  =  qsc1_out_ga(x1, x2)
U46_ga(x1, x2, x3)  =  U46_ga(x1, x3)
U47_ga(x1, x2, x3, x4)  =  U47_ga(x1, x3, x4)
U48_ga(x1, x2, x3)  =  U48_ga(x1, x3)
appendc23_in_ggga(x1, x2, x3, x4)  =  appendc23_in_ggga(x1, x2, x3)
appendc23_out_ggga(x1, x2, x3, x4)  =  appendc23_out_ggga(x1, x2, x3, x4)
U39_ggga(x1, x2, x3, x4, x5, x6)  =  U39_ggga(x1, x2, x3, x4, x6)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6)  =  U51_ga(x1, x2, x3, x5, x6)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x2, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x1, x2, x3, x5)
U55_ga(x1, x2, x3, x4, x5, x6)  =  U55_ga(x1, x2, x3, x5, x6)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x1, x2, x3, x5)
qc78_in_gagga(x1, x2, x3, x4, x5)  =  qc78_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6)  =  U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6)  =  U63_gagga(x1, x2, x3, x4, x6)
qc78_out_gagga(x1, x2, x3, x4, x5)  =  qc78_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x1, x2, x3, x5)
QS79_IN_GA(x1, x2)  =  QS79_IN_GA(x1)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x1, x2, x4)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x1, x2, x4, x5)

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:

QS79_IN_GA(cons(T162, T163), X265) → U11_GA(T162, T163, X265, splitc46_in_ggaa(T163, T162, T166, T167))
U11_GA(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → QS79_IN_GA(T166, X263)
U11_GA(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → U13_GA(T162, T163, X265, T167, qsc79_in_ga(T166, T170))
U13_GA(T162, T163, X265, T167, qsc79_out_ga(T166, T170)) → QS79_IN_GA(T167, X264)

The TRS R consists of the following rules:

splitc46_in_ggaa([], T97, [], []) → splitc46_out_ggaa([], T97, [], [])
splitc46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U41_ggaa(T110, T111, T112, X163, X164, lessc36_in_gg(T110, T112))
splitc46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U43_ggaa(T125, T126, T127, X195, X196, geqc62_in_gg(T125, T127))
qsc79_in_ga([], []) → qsc79_out_ga([], [])
qsc79_in_ga(cons(T162, T163), X265) → U57_ga(T162, T163, X265, splitc46_in_ggaa(T163, T162, T166, T167))
U41_ggaa(T110, T111, T112, X163, X164, lessc36_out_gg(T110, T112)) → U42_ggaa(T110, T111, T112, X163, X164, splitc46_in_ggaa(T111, T112, X163, X164))
U43_ggaa(T125, T126, T127, X195, X196, geqc62_out_gg(T125, T127)) → U44_ggaa(T125, T126, T127, X195, X196, splitc46_in_ggaa(T126, T127, X195, X196))
U57_ga(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → U58_ga(T162, T163, X265, T167, qsc79_in_ga(T166, T170))
lessc36_in_gg(0, s(T78)) → lessc36_out_gg(0, s(T78))
lessc36_in_gg(s(T83), s(T84)) → U40_gg(T83, T84, lessc36_in_gg(T83, T84))
U42_ggaa(T110, T111, T112, X163, X164, splitc46_out_ggaa(T111, T112, X163, X164)) → splitc46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
geqc62_in_gg(T136, T136) → geqc62_out_gg(T136, T136)
geqc62_in_gg(s(T141), 0) → geqc62_out_gg(s(T141), 0)
geqc62_in_gg(s(T146), s(T147)) → U45_gg(T146, T147, geqc62_in_gg(T146, T147))
U44_ggaa(T125, T126, T127, X195, X196, splitc46_out_ggaa(T126, T127, X195, X196)) → splitc46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U58_ga(T162, T163, X265, T167, qsc79_out_ga(T166, T170)) → U59_ga(T162, T163, X265, T170, qsc79_in_ga(T167, T171))
U40_gg(T83, T84, lessc36_out_gg(T83, T84)) → lessc36_out_gg(s(T83), s(T84))
U45_gg(T146, T147, geqc62_out_gg(T146, T147)) → geqc62_out_gg(s(T146), s(T147))
U59_ga(T162, T163, X265, T170, qsc79_out_ga(T167, T171)) → U60_ga(T162, T163, X265, appendc94_in_ggga(T170, T162, T171, X265))
U60_ga(T162, T163, X265, appendc94_out_ggga(T170, T162, T171, X265)) → qsc79_out_ga(cons(T162, T163), X265)
appendc94_in_ggga([], T184, T185, cons(T184, T185)) → appendc94_out_ggga([], T184, T185, cons(T184, T185))
appendc94_in_ggga(cons(T194, T195), T196, T197, cons(T194, X295)) → U61_ggga(T194, T195, T196, T197, X295, appendc94_in_ggga(T195, T196, T197, X295))
U61_ggga(T194, T195, T196, T197, X295, appendc94_out_ggga(T195, T196, T197, X295)) → appendc94_out_ggga(cons(T194, T195), T196, T197, cons(T194, X295))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
[]  =  []
s(x1)  =  s(x1)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
0  =  0
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
geqc62_in_gg(x1, x2)  =  geqc62_in_gg(x1, x2)
geqc62_out_gg(x1, x2)  =  geqc62_out_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
splitc46_in_ggaa(x1, x2, x3, x4)  =  splitc46_in_ggaa(x1, x2)
splitc46_out_ggaa(x1, x2, x3, x4)  =  splitc46_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qsc79_in_ga(x1, x2)  =  qsc79_in_ga(x1)
qsc79_out_ga(x1, x2)  =  qsc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
QS79_IN_GA(x1, x2)  =  QS79_IN_GA(x1)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x1, x2, x4)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x1, x2, x4, x5)

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:

QS79_IN_GA(cons(T162, T163)) → U11_GA(T162, T163, splitc46_in_ggaa(T163, T162))
U11_GA(T162, T163, splitc46_out_ggaa(T163, T162, T166, T167)) → QS79_IN_GA(T166)
U11_GA(T162, T163, splitc46_out_ggaa(T163, T162, T166, T167)) → U13_GA(T162, T163, T167, qsc79_in_ga(T166))
U13_GA(T162, T163, T167, qsc79_out_ga(T166, T170)) → QS79_IN_GA(T167)

The TRS R consists of the following rules:

splitc46_in_ggaa([], T97) → splitc46_out_ggaa([], T97, [], [])
splitc46_in_ggaa(cons(T110, T111), T112) → U41_ggaa(T110, T111, T112, lessc36_in_gg(T110, T112))
splitc46_in_ggaa(cons(T125, T126), T127) → U43_ggaa(T125, T126, T127, geqc62_in_gg(T125, T127))
qsc79_in_ga([]) → qsc79_out_ga([], [])
qsc79_in_ga(cons(T162, T163)) → U57_ga(T162, T163, splitc46_in_ggaa(T163, T162))
U41_ggaa(T110, T111, T112, lessc36_out_gg(T110, T112)) → U42_ggaa(T110, T111, T112, splitc46_in_ggaa(T111, T112))
U43_ggaa(T125, T126, T127, geqc62_out_gg(T125, T127)) → U44_ggaa(T125, T126, T127, splitc46_in_ggaa(T126, T127))
U57_ga(T162, T163, splitc46_out_ggaa(T163, T162, T166, T167)) → U58_ga(T162, T163, T167, qsc79_in_ga(T166))
lessc36_in_gg(0, s(T78)) → lessc36_out_gg(0, s(T78))
lessc36_in_gg(s(T83), s(T84)) → U40_gg(T83, T84, lessc36_in_gg(T83, T84))
U42_ggaa(T110, T111, T112, splitc46_out_ggaa(T111, T112, X163, X164)) → splitc46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
geqc62_in_gg(T136, T136) → geqc62_out_gg(T136, T136)
geqc62_in_gg(s(T141), 0) → geqc62_out_gg(s(T141), 0)
geqc62_in_gg(s(T146), s(T147)) → U45_gg(T146, T147, geqc62_in_gg(T146, T147))
U44_ggaa(T125, T126, T127, splitc46_out_ggaa(T126, T127, X195, X196)) → splitc46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U58_ga(T162, T163, T167, qsc79_out_ga(T166, T170)) → U59_ga(T162, T163, T170, qsc79_in_ga(T167))
U40_gg(T83, T84, lessc36_out_gg(T83, T84)) → lessc36_out_gg(s(T83), s(T84))
U45_gg(T146, T147, geqc62_out_gg(T146, T147)) → geqc62_out_gg(s(T146), s(T147))
U59_ga(T162, T163, T170, qsc79_out_ga(T167, T171)) → U60_ga(T162, T163, appendc94_in_ggga(T170, T162, T171))
U60_ga(T162, T163, appendc94_out_ggga(T170, T162, T171, X265)) → qsc79_out_ga(cons(T162, T163), X265)
appendc94_in_ggga([], T184, T185) → appendc94_out_ggga([], T184, T185, cons(T184, T185))
appendc94_in_ggga(cons(T194, T195), T196, T197) → U61_ggga(T194, T195, T196, T197, appendc94_in_ggga(T195, T196, T197))
U61_ggga(T194, T195, T196, T197, appendc94_out_ggga(T195, T196, T197, X295)) → appendc94_out_ggga(cons(T194, T195), T196, T197, cons(T194, X295))

The set Q consists of the following terms:

splitc46_in_ggaa(x0, x1)
qsc79_in_ga(x0)
U41_ggaa(x0, x1, x2, x3)
U43_ggaa(x0, x1, x2, x3)
U57_ga(x0, x1, x2)
lessc36_in_gg(x0, x1)
U42_ggaa(x0, x1, x2, x3)
geqc62_in_gg(x0, x1)
U44_ggaa(x0, x1, x2, x3)
U58_ga(x0, x1, x2, x3)
U40_gg(x0, x1, x2)
U45_gg(x0, x1, x2)
U59_ga(x0, x1, x2, x3)
U60_ga(x0, x1, x2)
appendc94_in_ggga(x0, x1, x2)
U61_ggga(x0, x1, x2, x3, x4)

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

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


QS79_IN_GA(cons(T162, T163)) → U11_GA(T162, T163, splitc46_in_ggaa(T163, T162))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(QS79_IN_GA(x1)) = 1 + x1   
POL(U11_GA(x1, x2, x3)) = 1 + x3   
POL(U13_GA(x1, x2, x3, x4)) = 1 + x3   
POL(U40_gg(x1, x2, x3)) = x2   
POL(U41_ggaa(x1, x2, x3, x4)) = 1 + x2   
POL(U42_ggaa(x1, x2, x3, x4)) = 1 + x4   
POL(U43_ggaa(x1, x2, x3, x4)) = 1 + x2   
POL(U44_ggaa(x1, x2, x3, x4)) = 1 + x4   
POL(U45_gg(x1, x2, x3)) = 0   
POL(U57_ga(x1, x2, x3)) = 0   
POL(U58_ga(x1, x2, x3, x4)) = 0   
POL(U59_ga(x1, x2, x3, x4)) = 0   
POL(U60_ga(x1, x2, x3)) = 0   
POL(U61_ggga(x1, x2, x3, x4, x5)) = 0   
POL([]) = 0   
POL(appendc94_in_ggga(x1, x2, x3)) = 0   
POL(appendc94_out_ggga(x1, x2, x3, x4)) = 0   
POL(cons(x1, x2)) = 1 + x2   
POL(geqc62_in_gg(x1, x2)) = 0   
POL(geqc62_out_gg(x1, x2)) = 0   
POL(lessc36_in_gg(x1, x2)) = x2   
POL(lessc36_out_gg(x1, x2)) = 0   
POL(qsc79_in_ga(x1)) = 0   
POL(qsc79_out_ga(x1, x2)) = 0   
POL(s(x1)) = 1 + x1   
POL(splitc46_in_ggaa(x1, x2)) = x1   
POL(splitc46_out_ggaa(x1, x2, x3, x4)) = x3 + x4   

The following usable rules [FROCOS05] were oriented:

splitc46_in_ggaa([], T97) → splitc46_out_ggaa([], T97, [], [])
splitc46_in_ggaa(cons(T110, T111), T112) → U41_ggaa(T110, T111, T112, lessc36_in_gg(T110, T112))
splitc46_in_ggaa(cons(T125, T126), T127) → U43_ggaa(T125, T126, T127, geqc62_in_gg(T125, T127))
U41_ggaa(T110, T111, T112, lessc36_out_gg(T110, T112)) → U42_ggaa(T110, T111, T112, splitc46_in_ggaa(T111, T112))
U42_ggaa(T110, T111, T112, splitc46_out_ggaa(T111, T112, X163, X164)) → splitc46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U43_ggaa(T125, T126, T127, geqc62_out_gg(T125, T127)) → U44_ggaa(T125, T126, T127, splitc46_in_ggaa(T126, T127))
U44_ggaa(T125, T126, T127, splitc46_out_ggaa(T126, T127, X195, X196)) → splitc46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))

(43) Obligation:

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

U11_GA(T162, T163, splitc46_out_ggaa(T163, T162, T166, T167)) → QS79_IN_GA(T166)
U11_GA(T162, T163, splitc46_out_ggaa(T163, T162, T166, T167)) → U13_GA(T162, T163, T167, qsc79_in_ga(T166))
U13_GA(T162, T163, T167, qsc79_out_ga(T166, T170)) → QS79_IN_GA(T167)

The TRS R consists of the following rules:

splitc46_in_ggaa([], T97) → splitc46_out_ggaa([], T97, [], [])
splitc46_in_ggaa(cons(T110, T111), T112) → U41_ggaa(T110, T111, T112, lessc36_in_gg(T110, T112))
splitc46_in_ggaa(cons(T125, T126), T127) → U43_ggaa(T125, T126, T127, geqc62_in_gg(T125, T127))
qsc79_in_ga([]) → qsc79_out_ga([], [])
qsc79_in_ga(cons(T162, T163)) → U57_ga(T162, T163, splitc46_in_ggaa(T163, T162))
U41_ggaa(T110, T111, T112, lessc36_out_gg(T110, T112)) → U42_ggaa(T110, T111, T112, splitc46_in_ggaa(T111, T112))
U43_ggaa(T125, T126, T127, geqc62_out_gg(T125, T127)) → U44_ggaa(T125, T126, T127, splitc46_in_ggaa(T126, T127))
U57_ga(T162, T163, splitc46_out_ggaa(T163, T162, T166, T167)) → U58_ga(T162, T163, T167, qsc79_in_ga(T166))
lessc36_in_gg(0, s(T78)) → lessc36_out_gg(0, s(T78))
lessc36_in_gg(s(T83), s(T84)) → U40_gg(T83, T84, lessc36_in_gg(T83, T84))
U42_ggaa(T110, T111, T112, splitc46_out_ggaa(T111, T112, X163, X164)) → splitc46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
geqc62_in_gg(T136, T136) → geqc62_out_gg(T136, T136)
geqc62_in_gg(s(T141), 0) → geqc62_out_gg(s(T141), 0)
geqc62_in_gg(s(T146), s(T147)) → U45_gg(T146, T147, geqc62_in_gg(T146, T147))
U44_ggaa(T125, T126, T127, splitc46_out_ggaa(T126, T127, X195, X196)) → splitc46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U58_ga(T162, T163, T167, qsc79_out_ga(T166, T170)) → U59_ga(T162, T163, T170, qsc79_in_ga(T167))
U40_gg(T83, T84, lessc36_out_gg(T83, T84)) → lessc36_out_gg(s(T83), s(T84))
U45_gg(T146, T147, geqc62_out_gg(T146, T147)) → geqc62_out_gg(s(T146), s(T147))
U59_ga(T162, T163, T170, qsc79_out_ga(T167, T171)) → U60_ga(T162, T163, appendc94_in_ggga(T170, T162, T171))
U60_ga(T162, T163, appendc94_out_ggga(T170, T162, T171, X265)) → qsc79_out_ga(cons(T162, T163), X265)
appendc94_in_ggga([], T184, T185) → appendc94_out_ggga([], T184, T185, cons(T184, T185))
appendc94_in_ggga(cons(T194, T195), T196, T197) → U61_ggga(T194, T195, T196, T197, appendc94_in_ggga(T195, T196, T197))
U61_ggga(T194, T195, T196, T197, appendc94_out_ggga(T195, T196, T197, X295)) → appendc94_out_ggga(cons(T194, T195), T196, T197, cons(T194, X295))

The set Q consists of the following terms:

splitc46_in_ggaa(x0, x1)
qsc79_in_ga(x0)
U41_ggaa(x0, x1, x2, x3)
U43_ggaa(x0, x1, x2, x3)
U57_ga(x0, x1, x2)
lessc36_in_gg(x0, x1)
U42_ggaa(x0, x1, x2, x3)
geqc62_in_gg(x0, x1)
U44_ggaa(x0, x1, x2, x3)
U58_ga(x0, x1, x2, x3)
U40_gg(x0, x1, x2)
U45_gg(x0, x1, x2)
U59_ga(x0, x1, x2, x3)
U60_ga(x0, x1, x2)
appendc94_in_ggga(x0, x1, x2)
U61_ggga(x0, x1, x2, x3, x4)

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

(44) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(45) TRUE

(46) Obligation:

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

APPEND23_IN_GGGA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_GGGA(T46, T47, T48, T50)

The TRS R consists of the following rules:

qsc14_in_a([]) → qsc14_out_a([])
lessc36_in_gg(0, s(T78)) → lessc36_out_gg(0, s(T78))
lessc36_in_gg(s(T83), s(T84)) → U40_gg(T83, T84, lessc36_in_gg(T83, T84))
U40_gg(T83, T84, lessc36_out_gg(T83, T84)) → lessc36_out_gg(s(T83), s(T84))
geqc62_in_gg(T136, T136) → geqc62_out_gg(T136, T136)
geqc62_in_gg(s(T141), 0) → geqc62_out_gg(s(T141), 0)
geqc62_in_gg(s(T146), s(T147)) → U45_gg(T146, T147, geqc62_in_gg(T146, T147))
U45_gg(T146, T147, geqc62_out_gg(T146, T147)) → geqc62_out_gg(s(T146), s(T147))
splitc46_in_ggaa([], T97, [], []) → splitc46_out_ggaa([], T97, [], [])
splitc46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U41_ggaa(T110, T111, T112, X163, X164, lessc36_in_gg(T110, T112))
U41_ggaa(T110, T111, T112, X163, X164, lessc36_out_gg(T110, T112)) → U42_ggaa(T110, T111, T112, X163, X164, splitc46_in_ggaa(T111, T112, X163, X164))
splitc46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U43_ggaa(T125, T126, T127, X195, X196, geqc62_in_gg(T125, T127))
U43_ggaa(T125, T126, T127, X195, X196, geqc62_out_gg(T125, T127)) → U44_ggaa(T125, T126, T127, X195, X196, splitc46_in_ggaa(T126, T127, X195, X196))
U44_ggaa(T125, T126, T127, X195, X196, splitc46_out_ggaa(T126, T127, X195, X196)) → splitc46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U42_ggaa(T110, T111, T112, X163, X164, splitc46_out_ggaa(T111, T112, X163, X164)) → splitc46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
qsc79_in_ga([], []) → qsc79_out_ga([], [])
qsc79_in_ga(cons(T162, T163), X265) → U57_ga(T162, T163, X265, splitc46_in_ggaa(T163, T162, T166, T167))
U57_ga(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → U58_ga(T162, T163, X265, T167, qsc79_in_ga(T166, T170))
U58_ga(T162, T163, X265, T167, qsc79_out_ga(T166, T170)) → U59_ga(T162, T163, X265, T170, qsc79_in_ga(T167, T171))
U59_ga(T162, T163, X265, T170, qsc79_out_ga(T167, T171)) → U60_ga(T162, T163, X265, appendc94_in_ggga(T170, T162, T171, X265))
appendc94_in_ggga([], T184, T185, cons(T184, T185)) → appendc94_out_ggga([], T184, T185, cons(T184, T185))
appendc94_in_ggga(cons(T194, T195), T196, T197, cons(T194, X295)) → U61_ggga(T194, T195, T196, T197, X295, appendc94_in_ggga(T195, T196, T197, X295))
U61_ggga(T194, T195, T196, T197, X295, appendc94_out_ggga(T195, T196, T197, X295)) → appendc94_out_ggga(cons(T194, T195), T196, T197, cons(T194, X295))
U60_ga(T162, T163, X265, appendc94_out_ggga(T170, T162, T171, X265)) → qsc79_out_ga(cons(T162, T163), X265)
qsc1_in_ga([], []) → qsc1_out_ga([], [])
qsc1_in_ga(cons(T14, []), T9) → U46_ga(T14, T9, qsc14_in_a(T17))
U46_ga(T14, T9, qsc14_out_a(T17)) → U47_ga(T14, T9, T17, qsc14_in_a(T20))
U47_ga(T14, T9, T17, qsc14_out_a(T20)) → U48_ga(T14, T9, appendc23_in_ggga(T17, T14, T20, T9))
appendc23_in_ggga([], T33, T34, cons(T33, T34)) → appendc23_out_ggga([], T33, T34, cons(T33, T34))
appendc23_in_ggga(cons(T45, T46), T47, T48, cons(T45, T50)) → U39_ggga(T45, T46, T47, T48, T50, appendc23_in_ggga(T46, T47, T48, T50))
U39_ggga(T45, T46, T47, T48, T50, appendc23_out_ggga(T46, T47, T48, T50)) → appendc23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U48_ga(T14, T9, appendc23_out_ggga(T17, T14, T20, T9)) → qsc1_out_ga(cons(T14, []), T9)
qsc1_in_ga(cons(T69, cons(T67, T68)), T9) → U49_ga(T69, T67, T68, T9, lessc36_in_gg(T67, T69))
U49_ga(T69, T67, T68, T9, lessc36_out_gg(T67, T69)) → U50_ga(T69, T67, T68, T9, splitc46_in_ggaa(T68, T69, T89, T90))
U50_ga(T69, T67, T68, T9, splitc46_out_ggaa(T68, T69, T89, T90)) → U51_ga(T69, T67, T68, T9, T90, qsc1_in_ga(cons(T67, T89), T154))
qsc1_in_ga(cons(T214, cons(T212, T213)), T9) → U53_ga(T214, T212, T213, T9, geqc62_in_gg(T212, T214))
U53_ga(T214, T212, T213, T9, geqc62_out_gg(T212, T214)) → U54_ga(T214, T212, T213, T9, splitc46_in_ggaa(T213, T214, T221, T222))
U54_ga(T214, T212, T213, T9, splitc46_out_ggaa(T213, T214, T221, T222)) → U55_ga(T214, T212, T213, T9, T222, qsc79_in_ga(T221, T225))
U55_ga(T214, T212, T213, T9, T222, qsc79_out_ga(T221, T225)) → U56_ga(T214, T212, T213, T9, qc78_in_gagga(cons(T212, T222), X13, T225, T214, T9))
qc78_in_gagga(T90, T157, T154, T69, T9) → U62_gagga(T90, T157, T154, T69, T9, qsc79_in_ga(T90, T157))
U62_gagga(T90, T157, T154, T69, T9, qsc79_out_ga(T90, T157)) → U63_gagga(T90, T157, T154, T69, T9, appendc23_in_ggga(T154, T69, T157, T9))
U63_gagga(T90, T157, T154, T69, T9, appendc23_out_ggga(T154, T69, T157, T9)) → qc78_out_gagga(T90, T157, T154, T69, T9)
U56_ga(T214, T212, T213, T9, qc78_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → qsc1_out_ga(cons(T214, cons(T212, T213)), T9)
U51_ga(T69, T67, T68, T9, T90, qsc1_out_ga(cons(T67, T89), T154)) → U52_ga(T69, T67, T68, T9, qc78_in_gagga(T90, X13, T154, T69, T9))
U52_ga(T69, T67, T68, T9, qc78_out_gagga(T90, X13, T154, T69, T9)) → qsc1_out_ga(cons(T69, cons(T67, T68)), T9)

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
[]  =  []
qsc14_in_a(x1)  =  qsc14_in_a
qsc14_out_a(x1)  =  qsc14_out_a(x1)
s(x1)  =  s(x1)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
0  =  0
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
geqc62_in_gg(x1, x2)  =  geqc62_in_gg(x1, x2)
geqc62_out_gg(x1, x2)  =  geqc62_out_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
splitc46_in_ggaa(x1, x2, x3, x4)  =  splitc46_in_ggaa(x1, x2)
splitc46_out_ggaa(x1, x2, x3, x4)  =  splitc46_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qsc79_in_ga(x1, x2)  =  qsc79_in_ga(x1)
qsc79_out_ga(x1, x2)  =  qsc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
qsc1_in_ga(x1, x2)  =  qsc1_in_ga(x1)
qsc1_out_ga(x1, x2)  =  qsc1_out_ga(x1, x2)
U46_ga(x1, x2, x3)  =  U46_ga(x1, x3)
U47_ga(x1, x2, x3, x4)  =  U47_ga(x1, x3, x4)
U48_ga(x1, x2, x3)  =  U48_ga(x1, x3)
appendc23_in_ggga(x1, x2, x3, x4)  =  appendc23_in_ggga(x1, x2, x3)
appendc23_out_ggga(x1, x2, x3, x4)  =  appendc23_out_ggga(x1, x2, x3, x4)
U39_ggga(x1, x2, x3, x4, x5, x6)  =  U39_ggga(x1, x2, x3, x4, x6)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6)  =  U51_ga(x1, x2, x3, x5, x6)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x2, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x1, x2, x3, x5)
U55_ga(x1, x2, x3, x4, x5, x6)  =  U55_ga(x1, x2, x3, x5, x6)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x1, x2, x3, x5)
qc78_in_gagga(x1, x2, x3, x4, x5)  =  qc78_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6)  =  U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6)  =  U63_gagga(x1, x2, x3, x4, x6)
qc78_out_gagga(x1, x2, x3, x4, x5)  =  qc78_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x1, x2, x3, x5)
APPEND23_IN_GGGA(x1, x2, x3, x4)  =  APPEND23_IN_GGGA(x1, x2, x3)

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

(47) UsableRulesProof (EQUIVALENT transformation)

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

(48) Obligation:

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

APPEND23_IN_GGGA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_GGGA(T46, T47, T48, T50)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
APPEND23_IN_GGGA(x1, x2, x3, x4)  =  APPEND23_IN_GGGA(x1, x2, x3)

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

(49) PiDPToQDPProof (SOUND transformation)

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

(50) Obligation:

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

APPEND23_IN_GGGA(cons(T45, T46), T47, T48) → APPEND23_IN_GGGA(T46, T47, T48)

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

(51) 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:

  • APPEND23_IN_GGGA(cons(T45, T46), T47, T48) → APPEND23_IN_GGGA(T46, T47, T48)
    The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

(52) YES

(53) Obligation:

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

QS1_IN_GA(cons(T69, cons(T67, T68)), T9) → U25_GA(T69, T67, T68, T9, lessc36_in_gg(T67, T69))
U25_GA(T69, T67, T68, T9, lessc36_out_gg(T67, T69)) → U27_GA(T69, T67, T68, T9, splitc46_in_ggaa(T68, T69, T89, T90))
U27_GA(T69, T67, T68, T9, splitc46_out_ggaa(T68, T69, T89, T90)) → QS1_IN_GA(cons(T67, T89), X12)

The TRS R consists of the following rules:

qsc14_in_a([]) → qsc14_out_a([])
lessc36_in_gg(0, s(T78)) → lessc36_out_gg(0, s(T78))
lessc36_in_gg(s(T83), s(T84)) → U40_gg(T83, T84, lessc36_in_gg(T83, T84))
U40_gg(T83, T84, lessc36_out_gg(T83, T84)) → lessc36_out_gg(s(T83), s(T84))
geqc62_in_gg(T136, T136) → geqc62_out_gg(T136, T136)
geqc62_in_gg(s(T141), 0) → geqc62_out_gg(s(T141), 0)
geqc62_in_gg(s(T146), s(T147)) → U45_gg(T146, T147, geqc62_in_gg(T146, T147))
U45_gg(T146, T147, geqc62_out_gg(T146, T147)) → geqc62_out_gg(s(T146), s(T147))
splitc46_in_ggaa([], T97, [], []) → splitc46_out_ggaa([], T97, [], [])
splitc46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U41_ggaa(T110, T111, T112, X163, X164, lessc36_in_gg(T110, T112))
U41_ggaa(T110, T111, T112, X163, X164, lessc36_out_gg(T110, T112)) → U42_ggaa(T110, T111, T112, X163, X164, splitc46_in_ggaa(T111, T112, X163, X164))
splitc46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U43_ggaa(T125, T126, T127, X195, X196, geqc62_in_gg(T125, T127))
U43_ggaa(T125, T126, T127, X195, X196, geqc62_out_gg(T125, T127)) → U44_ggaa(T125, T126, T127, X195, X196, splitc46_in_ggaa(T126, T127, X195, X196))
U44_ggaa(T125, T126, T127, X195, X196, splitc46_out_ggaa(T126, T127, X195, X196)) → splitc46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U42_ggaa(T110, T111, T112, X163, X164, splitc46_out_ggaa(T111, T112, X163, X164)) → splitc46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
qsc79_in_ga([], []) → qsc79_out_ga([], [])
qsc79_in_ga(cons(T162, T163), X265) → U57_ga(T162, T163, X265, splitc46_in_ggaa(T163, T162, T166, T167))
U57_ga(T162, T163, X265, splitc46_out_ggaa(T163, T162, T166, T167)) → U58_ga(T162, T163, X265, T167, qsc79_in_ga(T166, T170))
U58_ga(T162, T163, X265, T167, qsc79_out_ga(T166, T170)) → U59_ga(T162, T163, X265, T170, qsc79_in_ga(T167, T171))
U59_ga(T162, T163, X265, T170, qsc79_out_ga(T167, T171)) → U60_ga(T162, T163, X265, appendc94_in_ggga(T170, T162, T171, X265))
appendc94_in_ggga([], T184, T185, cons(T184, T185)) → appendc94_out_ggga([], T184, T185, cons(T184, T185))
appendc94_in_ggga(cons(T194, T195), T196, T197, cons(T194, X295)) → U61_ggga(T194, T195, T196, T197, X295, appendc94_in_ggga(T195, T196, T197, X295))
U61_ggga(T194, T195, T196, T197, X295, appendc94_out_ggga(T195, T196, T197, X295)) → appendc94_out_ggga(cons(T194, T195), T196, T197, cons(T194, X295))
U60_ga(T162, T163, X265, appendc94_out_ggga(T170, T162, T171, X265)) → qsc79_out_ga(cons(T162, T163), X265)
qsc1_in_ga([], []) → qsc1_out_ga([], [])
qsc1_in_ga(cons(T14, []), T9) → U46_ga(T14, T9, qsc14_in_a(T17))
U46_ga(T14, T9, qsc14_out_a(T17)) → U47_ga(T14, T9, T17, qsc14_in_a(T20))
U47_ga(T14, T9, T17, qsc14_out_a(T20)) → U48_ga(T14, T9, appendc23_in_ggga(T17, T14, T20, T9))
appendc23_in_ggga([], T33, T34, cons(T33, T34)) → appendc23_out_ggga([], T33, T34, cons(T33, T34))
appendc23_in_ggga(cons(T45, T46), T47, T48, cons(T45, T50)) → U39_ggga(T45, T46, T47, T48, T50, appendc23_in_ggga(T46, T47, T48, T50))
U39_ggga(T45, T46, T47, T48, T50, appendc23_out_ggga(T46, T47, T48, T50)) → appendc23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U48_ga(T14, T9, appendc23_out_ggga(T17, T14, T20, T9)) → qsc1_out_ga(cons(T14, []), T9)
qsc1_in_ga(cons(T69, cons(T67, T68)), T9) → U49_ga(T69, T67, T68, T9, lessc36_in_gg(T67, T69))
U49_ga(T69, T67, T68, T9, lessc36_out_gg(T67, T69)) → U50_ga(T69, T67, T68, T9, splitc46_in_ggaa(T68, T69, T89, T90))
U50_ga(T69, T67, T68, T9, splitc46_out_ggaa(T68, T69, T89, T90)) → U51_ga(T69, T67, T68, T9, T90, qsc1_in_ga(cons(T67, T89), T154))
qsc1_in_ga(cons(T214, cons(T212, T213)), T9) → U53_ga(T214, T212, T213, T9, geqc62_in_gg(T212, T214))
U53_ga(T214, T212, T213, T9, geqc62_out_gg(T212, T214)) → U54_ga(T214, T212, T213, T9, splitc46_in_ggaa(T213, T214, T221, T222))
U54_ga(T214, T212, T213, T9, splitc46_out_ggaa(T213, T214, T221, T222)) → U55_ga(T214, T212, T213, T9, T222, qsc79_in_ga(T221, T225))
U55_ga(T214, T212, T213, T9, T222, qsc79_out_ga(T221, T225)) → U56_ga(T214, T212, T213, T9, qc78_in_gagga(cons(T212, T222), X13, T225, T214, T9))
qc78_in_gagga(T90, T157, T154, T69, T9) → U62_gagga(T90, T157, T154, T69, T9, qsc79_in_ga(T90, T157))
U62_gagga(T90, T157, T154, T69, T9, qsc79_out_ga(T90, T157)) → U63_gagga(T90, T157, T154, T69, T9, appendc23_in_ggga(T154, T69, T157, T9))
U63_gagga(T90, T157, T154, T69, T9, appendc23_out_ggga(T154, T69, T157, T9)) → qc78_out_gagga(T90, T157, T154, T69, T9)
U56_ga(T214, T212, T213, T9, qc78_out_gagga(cons(T212, T222), X13, T225, T214, T9)) → qsc1_out_ga(cons(T214, cons(T212, T213)), T9)
U51_ga(T69, T67, T68, T9, T90, qsc1_out_ga(cons(T67, T89), T154)) → U52_ga(T69, T67, T68, T9, qc78_in_gagga(T90, X13, T154, T69, T9))
U52_ga(T69, T67, T68, T9, qc78_out_gagga(T90, X13, T154, T69, T9)) → qsc1_out_ga(cons(T69, cons(T67, T68)), T9)

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
[]  =  []
qsc14_in_a(x1)  =  qsc14_in_a
qsc14_out_a(x1)  =  qsc14_out_a(x1)
s(x1)  =  s(x1)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
0  =  0
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
geqc62_in_gg(x1, x2)  =  geqc62_in_gg(x1, x2)
geqc62_out_gg(x1, x2)  =  geqc62_out_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
splitc46_in_ggaa(x1, x2, x3, x4)  =  splitc46_in_ggaa(x1, x2)
splitc46_out_ggaa(x1, x2, x3, x4)  =  splitc46_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
qsc79_in_ga(x1, x2)  =  qsc79_in_ga(x1)
qsc79_out_ga(x1, x2)  =  qsc79_out_ga(x1, x2)
U57_ga(x1, x2, x3, x4)  =  U57_ga(x1, x2, x4)
U58_ga(x1, x2, x3, x4, x5)  =  U58_ga(x1, x2, x4, x5)
U59_ga(x1, x2, x3, x4, x5)  =  U59_ga(x1, x2, x4, x5)
U60_ga(x1, x2, x3, x4)  =  U60_ga(x1, x2, x4)
appendc94_in_ggga(x1, x2, x3, x4)  =  appendc94_in_ggga(x1, x2, x3)
appendc94_out_ggga(x1, x2, x3, x4)  =  appendc94_out_ggga(x1, x2, x3, x4)
U61_ggga(x1, x2, x3, x4, x5, x6)  =  U61_ggga(x1, x2, x3, x4, x6)
qsc1_in_ga(x1, x2)  =  qsc1_in_ga(x1)
qsc1_out_ga(x1, x2)  =  qsc1_out_ga(x1, x2)
U46_ga(x1, x2, x3)  =  U46_ga(x1, x3)
U47_ga(x1, x2, x3, x4)  =  U47_ga(x1, x3, x4)
U48_ga(x1, x2, x3)  =  U48_ga(x1, x3)
appendc23_in_ggga(x1, x2, x3, x4)  =  appendc23_in_ggga(x1, x2, x3)
appendc23_out_ggga(x1, x2, x3, x4)  =  appendc23_out_ggga(x1, x2, x3, x4)
U39_ggga(x1, x2, x3, x4, x5, x6)  =  U39_ggga(x1, x2, x3, x4, x6)
U49_ga(x1, x2, x3, x4, x5)  =  U49_ga(x1, x2, x3, x5)
U50_ga(x1, x2, x3, x4, x5)  =  U50_ga(x1, x2, x3, x5)
U51_ga(x1, x2, x3, x4, x5, x6)  =  U51_ga(x1, x2, x3, x5, x6)
U53_ga(x1, x2, x3, x4, x5)  =  U53_ga(x1, x2, x3, x5)
U54_ga(x1, x2, x3, x4, x5)  =  U54_ga(x1, x2, x3, x5)
U55_ga(x1, x2, x3, x4, x5, x6)  =  U55_ga(x1, x2, x3, x5, x6)
U56_ga(x1, x2, x3, x4, x5)  =  U56_ga(x1, x2, x3, x5)
qc78_in_gagga(x1, x2, x3, x4, x5)  =  qc78_in_gagga(x1, x3, x4)
U62_gagga(x1, x2, x3, x4, x5, x6)  =  U62_gagga(x1, x3, x4, x6)
U63_gagga(x1, x2, x3, x4, x5, x6)  =  U63_gagga(x1, x2, x3, x4, x6)
qc78_out_gagga(x1, x2, x3, x4, x5)  =  qc78_out_gagga(x1, x2, x3, x4, x5)
U52_ga(x1, x2, x3, x4, x5)  =  U52_ga(x1, x2, x3, x5)
QS1_IN_GA(x1, x2)  =  QS1_IN_GA(x1)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x1, x2, x3, x5)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x3, x5)

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

(54) UsableRulesProof (EQUIVALENT transformation)

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

(55) Obligation:

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

QS1_IN_GA(cons(T69, cons(T67, T68)), T9) → U25_GA(T69, T67, T68, T9, lessc36_in_gg(T67, T69))
U25_GA(T69, T67, T68, T9, lessc36_out_gg(T67, T69)) → U27_GA(T69, T67, T68, T9, splitc46_in_ggaa(T68, T69, T89, T90))
U27_GA(T69, T67, T68, T9, splitc46_out_ggaa(T68, T69, T89, T90)) → QS1_IN_GA(cons(T67, T89), X12)

The TRS R consists of the following rules:

lessc36_in_gg(0, s(T78)) → lessc36_out_gg(0, s(T78))
lessc36_in_gg(s(T83), s(T84)) → U40_gg(T83, T84, lessc36_in_gg(T83, T84))
splitc46_in_ggaa([], T97, [], []) → splitc46_out_ggaa([], T97, [], [])
splitc46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U41_ggaa(T110, T111, T112, X163, X164, lessc36_in_gg(T110, T112))
splitc46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U43_ggaa(T125, T126, T127, X195, X196, geqc62_in_gg(T125, T127))
U40_gg(T83, T84, lessc36_out_gg(T83, T84)) → lessc36_out_gg(s(T83), s(T84))
U41_ggaa(T110, T111, T112, X163, X164, lessc36_out_gg(T110, T112)) → U42_ggaa(T110, T111, T112, X163, X164, splitc46_in_ggaa(T111, T112, X163, X164))
U43_ggaa(T125, T126, T127, X195, X196, geqc62_out_gg(T125, T127)) → U44_ggaa(T125, T126, T127, X195, X196, splitc46_in_ggaa(T126, T127, X195, X196))
U42_ggaa(T110, T111, T112, X163, X164, splitc46_out_ggaa(T111, T112, X163, X164)) → splitc46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
geqc62_in_gg(T136, T136) → geqc62_out_gg(T136, T136)
geqc62_in_gg(s(T141), 0) → geqc62_out_gg(s(T141), 0)
geqc62_in_gg(s(T146), s(T147)) → U45_gg(T146, T147, geqc62_in_gg(T146, T147))
U44_ggaa(T125, T126, T127, X195, X196, splitc46_out_ggaa(T126, T127, X195, X196)) → splitc46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U45_gg(T146, T147, geqc62_out_gg(T146, T147)) → geqc62_out_gg(s(T146), s(T147))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
[]  =  []
s(x1)  =  s(x1)
lessc36_in_gg(x1, x2)  =  lessc36_in_gg(x1, x2)
0  =  0
lessc36_out_gg(x1, x2)  =  lessc36_out_gg(x1, x2)
U40_gg(x1, x2, x3)  =  U40_gg(x1, x2, x3)
geqc62_in_gg(x1, x2)  =  geqc62_in_gg(x1, x2)
geqc62_out_gg(x1, x2)  =  geqc62_out_gg(x1, x2)
U45_gg(x1, x2, x3)  =  U45_gg(x1, x2, x3)
splitc46_in_ggaa(x1, x2, x3, x4)  =  splitc46_in_ggaa(x1, x2)
splitc46_out_ggaa(x1, x2, x3, x4)  =  splitc46_out_ggaa(x1, x2, x3, x4)
U41_ggaa(x1, x2, x3, x4, x5, x6)  =  U41_ggaa(x1, x2, x3, x6)
U42_ggaa(x1, x2, x3, x4, x5, x6)  =  U42_ggaa(x1, x2, x3, x6)
U43_ggaa(x1, x2, x3, x4, x5, x6)  =  U43_ggaa(x1, x2, x3, x6)
U44_ggaa(x1, x2, x3, x4, x5, x6)  =  U44_ggaa(x1, x2, x3, x6)
QS1_IN_GA(x1, x2)  =  QS1_IN_GA(x1)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x1, x2, x3, x5)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x3, x5)

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

(56) PiDPToQDPProof (SOUND transformation)

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

(57) Obligation:

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

QS1_IN_GA(cons(T69, cons(T67, T68))) → U25_GA(T69, T67, T68, lessc36_in_gg(T67, T69))
U25_GA(T69, T67, T68, lessc36_out_gg(T67, T69)) → U27_GA(T69, T67, T68, splitc46_in_ggaa(T68, T69))
U27_GA(T69, T67, T68, splitc46_out_ggaa(T68, T69, T89, T90)) → QS1_IN_GA(cons(T67, T89))

The TRS R consists of the following rules:

lessc36_in_gg(0, s(T78)) → lessc36_out_gg(0, s(T78))
lessc36_in_gg(s(T83), s(T84)) → U40_gg(T83, T84, lessc36_in_gg(T83, T84))
splitc46_in_ggaa([], T97) → splitc46_out_ggaa([], T97, [], [])
splitc46_in_ggaa(cons(T110, T111), T112) → U41_ggaa(T110, T111, T112, lessc36_in_gg(T110, T112))
splitc46_in_ggaa(cons(T125, T126), T127) → U43_ggaa(T125, T126, T127, geqc62_in_gg(T125, T127))
U40_gg(T83, T84, lessc36_out_gg(T83, T84)) → lessc36_out_gg(s(T83), s(T84))
U41_ggaa(T110, T111, T112, lessc36_out_gg(T110, T112)) → U42_ggaa(T110, T111, T112, splitc46_in_ggaa(T111, T112))
U43_ggaa(T125, T126, T127, geqc62_out_gg(T125, T127)) → U44_ggaa(T125, T126, T127, splitc46_in_ggaa(T126, T127))
U42_ggaa(T110, T111, T112, splitc46_out_ggaa(T111, T112, X163, X164)) → splitc46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
geqc62_in_gg(T136, T136) → geqc62_out_gg(T136, T136)
geqc62_in_gg(s(T141), 0) → geqc62_out_gg(s(T141), 0)
geqc62_in_gg(s(T146), s(T147)) → U45_gg(T146, T147, geqc62_in_gg(T146, T147))
U44_ggaa(T125, T126, T127, splitc46_out_ggaa(T126, T127, X195, X196)) → splitc46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U45_gg(T146, T147, geqc62_out_gg(T146, T147)) → geqc62_out_gg(s(T146), s(T147))

The set Q consists of the following terms:

lessc36_in_gg(x0, x1)
splitc46_in_ggaa(x0, x1)
U40_gg(x0, x1, x2)
U41_ggaa(x0, x1, x2, x3)
U43_ggaa(x0, x1, x2, x3)
U42_ggaa(x0, x1, x2, x3)
geqc62_in_gg(x0, x1)
U44_ggaa(x0, x1, x2, x3)
U45_gg(x0, x1, x2)

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


QS1_IN_GA(cons(T69, cons(T67, T68))) → U25_GA(T69, T67, T68, lessc36_in_gg(T67, T69))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(QS1_IN_GA(x1)) = 1 + x1   
POL(U25_GA(x1, x2, x3, x4)) = 1 + x1 + x3 + x4   
POL(U27_GA(x1, x2, x3, x4)) = 1 + x2 + x4   
POL(U40_gg(x1, x2, x3)) = 1   
POL(U41_ggaa(x1, x2, x3, x4)) = 1 + x1 + x2 + x4   
POL(U42_ggaa(x1, x2, x3, x4)) = 1 + x1 + x4   
POL(U43_ggaa(x1, x2, x3, x4)) = 1 + x2 + x4   
POL(U44_ggaa(x1, x2, x3, x4)) = x4   
POL(U45_gg(x1, x2, x3)) = 1   
POL([]) = 0   
POL(cons(x1, x2)) = 1 + x1 + x2   
POL(geqc62_in_gg(x1, x2)) = 1   
POL(geqc62_out_gg(x1, x2)) = 1   
POL(lessc36_in_gg(x1, x2)) = 1   
POL(lessc36_out_gg(x1, x2)) = 1 + x1   
POL(s(x1)) = 0   
POL(splitc46_in_ggaa(x1, x2)) = 1 + x1   
POL(splitc46_out_ggaa(x1, x2, x3, x4)) = 1 + x3   

The following usable rules [FROCOS05] were oriented:

lessc36_in_gg(0, s(T78)) → lessc36_out_gg(0, s(T78))
lessc36_in_gg(s(T83), s(T84)) → U40_gg(T83, T84, lessc36_in_gg(T83, T84))
splitc46_in_ggaa([], T97) → splitc46_out_ggaa([], T97, [], [])
splitc46_in_ggaa(cons(T110, T111), T112) → U41_ggaa(T110, T111, T112, lessc36_in_gg(T110, T112))
splitc46_in_ggaa(cons(T125, T126), T127) → U43_ggaa(T125, T126, T127, geqc62_in_gg(T125, T127))
U41_ggaa(T110, T111, T112, lessc36_out_gg(T110, T112)) → U42_ggaa(T110, T111, T112, splitc46_in_ggaa(T111, T112))
U42_ggaa(T110, T111, T112, splitc46_out_ggaa(T111, T112, X163, X164)) → splitc46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
geqc62_in_gg(T136, T136) → geqc62_out_gg(T136, T136)
geqc62_in_gg(s(T141), 0) → geqc62_out_gg(s(T141), 0)
geqc62_in_gg(s(T146), s(T147)) → U45_gg(T146, T147, geqc62_in_gg(T146, T147))
U43_ggaa(T125, T126, T127, geqc62_out_gg(T125, T127)) → U44_ggaa(T125, T126, T127, splitc46_in_ggaa(T126, T127))
U44_ggaa(T125, T126, T127, splitc46_out_ggaa(T126, T127, X195, X196)) → splitc46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U40_gg(T83, T84, lessc36_out_gg(T83, T84)) → lessc36_out_gg(s(T83), s(T84))
U45_gg(T146, T147, geqc62_out_gg(T146, T147)) → geqc62_out_gg(s(T146), s(T147))

(59) Obligation:

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

U25_GA(T69, T67, T68, lessc36_out_gg(T67, T69)) → U27_GA(T69, T67, T68, splitc46_in_ggaa(T68, T69))
U27_GA(T69, T67, T68, splitc46_out_ggaa(T68, T69, T89, T90)) → QS1_IN_GA(cons(T67, T89))

The TRS R consists of the following rules:

lessc36_in_gg(0, s(T78)) → lessc36_out_gg(0, s(T78))
lessc36_in_gg(s(T83), s(T84)) → U40_gg(T83, T84, lessc36_in_gg(T83, T84))
splitc46_in_ggaa([], T97) → splitc46_out_ggaa([], T97, [], [])
splitc46_in_ggaa(cons(T110, T111), T112) → U41_ggaa(T110, T111, T112, lessc36_in_gg(T110, T112))
splitc46_in_ggaa(cons(T125, T126), T127) → U43_ggaa(T125, T126, T127, geqc62_in_gg(T125, T127))
U40_gg(T83, T84, lessc36_out_gg(T83, T84)) → lessc36_out_gg(s(T83), s(T84))
U41_ggaa(T110, T111, T112, lessc36_out_gg(T110, T112)) → U42_ggaa(T110, T111, T112, splitc46_in_ggaa(T111, T112))
U43_ggaa(T125, T126, T127, geqc62_out_gg(T125, T127)) → U44_ggaa(T125, T126, T127, splitc46_in_ggaa(T126, T127))
U42_ggaa(T110, T111, T112, splitc46_out_ggaa(T111, T112, X163, X164)) → splitc46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
geqc62_in_gg(T136, T136) → geqc62_out_gg(T136, T136)
geqc62_in_gg(s(T141), 0) → geqc62_out_gg(s(T141), 0)
geqc62_in_gg(s(T146), s(T147)) → U45_gg(T146, T147, geqc62_in_gg(T146, T147))
U44_ggaa(T125, T126, T127, splitc46_out_ggaa(T126, T127, X195, X196)) → splitc46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U45_gg(T146, T147, geqc62_out_gg(T146, T147)) → geqc62_out_gg(s(T146), s(T147))

The set Q consists of the following terms:

lessc36_in_gg(x0, x1)
splitc46_in_ggaa(x0, x1)
U40_gg(x0, x1, x2)
U41_ggaa(x0, x1, x2, x3)
U43_ggaa(x0, x1, x2, x3)
U42_ggaa(x0, x1, x2, x3)
geqc62_in_gg(x0, x1)
U44_ggaa(x0, x1, x2, x3)
U45_gg(x0, x1, x2)

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

(60) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(61) TRUE