(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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

qs14([]).
append23([], T33, T34, cons(T33, T34)).
append23(cons(T45, T46), T47, T48, cons(T45, T50)) :- append23(T46, T47, T48, T50).
less36(0, s(T78)).
less36(s(T83), s(T84)) :- less36(T83, T84).
split46([], T97, [], []).
split46(cons(T110, T111), T112, cons(T110, X163), X164) :- less36(T110, T112).
split46(cons(T110, T111), T112, cons(T110, X163), X164) :- ','(less36(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)) :- ','(geq62(T125, T127), split46(T126, T127, X195, X196)).
geq62(T136, T136).
geq62(s(T141), 0).
geq62(s(T146), s(T147)) :- geq62(T146, T147).
qs79([], []).
qs79(cons(T162, T163), X265) :- split46(T163, T162, X261, X262).
qs79(cons(T162, T163), X265) :- ','(split46(T163, T162, T166, T167), qs79(T166, X263)).
qs79(cons(T162, T163), X265) :- ','(split46(T163, T162, T166, T167), ','(qs79(T166, T170), qs79(T167, X264))).
qs79(cons(T162, T163), X265) :- ','(split46(T163, T162, T166, T167), ','(qs79(T166, T170), ','(qs79(T167, T171), append94(T170, T162, T171, X265)))).
append94([], T184, T185, cons(T184, T185)).
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) :- ','(qs79(T90, T157), append23(T154, T69, T157, T9)).
qs1([], []).
qs1(cons(T14, []), T9) :- qs14(X12).
qs1(cons(T14, []), T9) :- ','(qs14(T17), qs14(X13)).
qs1(cons(T14, []), T9) :- ','(qs14(T17), ','(qs14(T20), append23(T17, T14, T20, T9))).
qs1(cons(T69, cons(T67, T68)), T9) :- less36(T67, T69).
qs1(cons(T69, cons(T67, T68)), T9) :- ','(less36(T67, T69), split46(T68, T69, X93, X94)).
qs1(cons(T69, cons(T67, T68)), T9) :- ','(less36(T67, T69), ','(split46(T68, T69, T89, T90), qs1(cons(T67, T89), X12))).
qs1(cons(T69, cons(T67, T68)), T9) :- ','(less36(T67, T69), ','(split46(T68, T69, T89, T90), ','(qs1(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) :- ','(geq62(T212, T214), split46(T213, T214, X323, X324)).
qs1(cons(T214, cons(T212, T213)), T9) :- ','(geq62(T212, T214), ','(split46(T213, T214, T221, T222), qs79(T221, X12))).
qs1(cons(T214, cons(T212, T213)), T9) :- ','(geq62(T212, T214), ','(split46(T213, T214, T221, T222), ','(qs79(T221, T225), p78(cons(T212, T222), X13, T225, T214, T9)))).

Queries:

qs1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
qs1_in: (b,f) (f,f)
append23_in: (b,b,b,f) (b,f,b,f) (f,f,f,f) (f,b,f,f)
less36_in: (b,b) (f,f)
split46_in: (b,b,f,f) (f,f,f,f)
geq62_in: (b,b) (f,f)
qs79_in: (f,f)
append94_in: (f,f,f,f)
p78_in: (f,f,f,f,f) (f,f,f,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
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)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
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)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_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)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_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)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
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)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
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)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_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)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_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)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

QS1_IN_GA(cons(T14, []), T9) → U19_GA(T14, T9, qs14_in_a(X12))
QS1_IN_GA(cons(T14, []), T9) → QS14_IN_A(X12)
QS1_IN_GA(cons(T14, []), T9) → U20_GA(T14, T9, qs14_in_a(T17))
U20_GA(T14, T9, qs14_out_a(T17)) → U21_GA(T14, T9, qs14_in_a(X13))
U20_GA(T14, T9, qs14_out_a(T17)) → QS14_IN_A(X13)
U20_GA(T14, T9, qs14_out_a(T17)) → U22_GA(T14, T9, T17, qs14_in_a(T20))
U22_GA(T14, T9, T17, qs14_out_a(T20)) → U23_GA(T14, T9, append23_in_ggga(T17, T14, T20, T9))
U22_GA(T14, T9, T17, qs14_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)
U24_GA(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_GA(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
U24_GA(T69, T67, T68, T9, less36_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)
U3_GGAA(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_GGAA(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
U3_GGAA(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → SPLIT46_IN_GGAA(T111, T112, X163, X164)
SPLIT46_IN_GGAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_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)) → U7_GG(T146, T147, geq62_in_gg(T146, T147))
GEQ62_IN_GG(s(T146), s(T147)) → GEQ62_IN_GG(T146, T147)
U5_GGAA(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_GGAA(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U5_GGAA(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → SPLIT46_IN_GGAA(T126, T127, X195, X196)
U24_GA(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_GA(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_GA(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_GA(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
U26_GA(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → QS1_IN_AA(cons(T67, T89), X12)
QS1_IN_AA(cons(T14, []), T9) → U19_AA(T14, T9, qs14_in_a(X12))
QS1_IN_AA(cons(T14, []), T9) → QS14_IN_A(X12)
QS1_IN_AA(cons(T14, []), T9) → U20_AA(T14, T9, qs14_in_a(T17))
U20_AA(T14, T9, qs14_out_a(T17)) → U21_AA(T14, T9, qs14_in_a(X13))
U20_AA(T14, T9, qs14_out_a(T17)) → QS14_IN_A(X13)
U20_AA(T14, T9, qs14_out_a(T17)) → U22_AA(T14, T9, T17, qs14_in_a(T20))
U22_AA(T14, T9, T17, qs14_out_a(T20)) → U23_AA(T14, T9, append23_in_gaga(T17, T14, T20, T9))
U22_AA(T14, T9, T17, qs14_out_a(T20)) → APPEND23_IN_GAGA(T17, T14, T20, T9)
APPEND23_IN_GAGA(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_GAGA(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
APPEND23_IN_GAGA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)
QS1_IN_AA(cons(T69, cons(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, less36_in_aa(T67, T69))
QS1_IN_AA(cons(T69, cons(T67, T68)), T9) → LESS36_IN_AA(T67, T69)
LESS36_IN_AA(s(T83), s(T84)) → U2_AA(T83, T84, less36_in_aa(T83, T84))
LESS36_IN_AA(s(T83), s(T84)) → LESS36_IN_AA(T83, T84)
U24_AA(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_AA(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
U24_AA(T69, T67, T68, T9, less36_out_aa(T67, T69)) → SPLIT46_IN_AAAA(T68, T69, X93, X94)
SPLIT46_IN_AAAA(cons(T110, T111), T112, cons(T110, X163), X164) → U3_AAAA(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
SPLIT46_IN_AAAA(cons(T110, T111), T112, cons(T110, X163), X164) → LESS36_IN_AA(T110, T112)
U3_AAAA(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_AAAA(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
U3_AAAA(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → SPLIT46_IN_AAAA(T111, T112, X163, X164)
SPLIT46_IN_AAAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_AAAA(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
SPLIT46_IN_AAAA(cons(T125, T126), T127, X195, cons(T125, X196)) → GEQ62_IN_AA(T125, T127)
GEQ62_IN_AA(s(T146), s(T147)) → U7_AA(T146, T147, geq62_in_aa(T146, T147))
GEQ62_IN_AA(s(T146), s(T147)) → GEQ62_IN_AA(T146, T147)
U5_AAAA(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_AAAA(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U5_AAAA(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → SPLIT46_IN_AAAA(T126, T127, X195, X196)
U24_AA(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_AA(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_AA(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
U26_AA(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → QS1_IN_AA(cons(T67, T89), X12)
QS1_IN_AA(cons(T214, cons(T212, T213)), T9) → U30_AA(T214, T212, T213, T9, geq62_in_aa(T212, T214))
QS1_IN_AA(cons(T214, cons(T212, T213)), T9) → GEQ62_IN_AA(T212, T214)
U30_AA(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_AA(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U30_AA(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → SPLIT46_IN_AAAA(T213, T214, X323, X324)
U30_AA(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_AA(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_AA(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_AA(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U32_AA(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → QS79_IN_AA(T221, X12)
QS79_IN_AA(cons(T162, T163), X265) → U8_AA(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
QS79_IN_AA(cons(T162, T163), X265) → SPLIT46_IN_AAAA(T163, T162, X261, X262)
QS79_IN_AA(cons(T162, T163), X265) → U9_AA(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_AA(T162, T163, X265, qs79_in_aa(T166, X263))
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → QS79_IN_AA(T166, X263)
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_AA(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_AA(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_AA(T162, T163, X265, qs79_in_aa(T167, X264))
U11_AA(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → QS79_IN_AA(T167, X264)
U11_AA(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_AA(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_AA(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_AA(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
U13_AA(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → APPEND94_IN_AAAA(T170, T162, T171, X265)
APPEND94_IN_AAAA(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_AAAA(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
APPEND94_IN_AAAA(cons(T194, T195), T196, T197, cons(T194, X295)) → APPEND94_IN_AAAA(T195, T196, T197, X295)
U32_AA(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_AA(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_AA(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_AA(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
U34_AA(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → P78_IN_AAAAA(cons(T212, T222), X13, T225, T214, T9)
P78_IN_AAAAA(T90, X13, T154, T69, T9) → U16_AAAAA(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
P78_IN_AAAAA(T90, X13, T154, T69, T9) → QS79_IN_AA(T90, X13)
P78_IN_AAAAA(T90, T157, T154, T69, T9) → U17_AAAAA(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_AAAAA(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_AAAAA(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
U17_AAAAA(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → APPEND23_IN_AAAA(T154, T69, T157, T9)
APPEND23_IN_AAAA(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_AAAA(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
APPEND23_IN_AAAA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)
U26_AA(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_AA(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_AA(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_AA(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U28_AA(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → P78_IN_AAAAA(T90, X13, T154, T69, T9)
U26_GA(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_GA(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_GA(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_GA(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
U28_GA(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → P78_IN_AAAGA(T90, X13, T154, T69, T9)
P78_IN_AAAGA(T90, X13, T154, T69, T9) → U16_AAAGA(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
P78_IN_AAAGA(T90, X13, T154, T69, T9) → QS79_IN_AA(T90, X13)
P78_IN_AAAGA(T90, T157, T154, T69, T9) → U17_AAAGA(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_AAAGA(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_AAAGA(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
U17_AAAGA(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → APPEND23_IN_AGAA(T154, T69, T157, T9)
APPEND23_IN_AGAA(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_AGAA(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
APPEND23_IN_AGAA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)
QS1_IN_GA(cons(T214, cons(T212, T213)), T9) → U30_GA(T214, T212, T213, T9, geq62_in_gg(T212, T214))
QS1_IN_GA(cons(T214, cons(T212, T213)), T9) → GEQ62_IN_GG(T212, T214)
U30_GA(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_GA(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U30_GA(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → SPLIT46_IN_GGAA(T213, T214, X323, X324)
U30_GA(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_GA(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_GA(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_GA(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U32_GA(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → QS79_IN_AA(T221, X12)
U32_GA(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_GA(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_GA(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_GA(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U34_GA(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → P78_IN_AAAGA(cons(T212, T222), X13, T225, T214, T9)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
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)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
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)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_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)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_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)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
QS1_IN_GA(x1, x2)  =  QS1_IN_GA(x1)
U19_GA(x1, x2, x3)  =  U19_GA(x1, x3)
QS14_IN_A(x1)  =  QS14_IN_A
U20_GA(x1, x2, x3)  =  U20_GA(x1, x3)
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)
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)
GEQ62_IN_GG(x1, x2)  =  GEQ62_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)
U26_GA(x1, x2, x3, x4, x5)  =  U26_GA(x1, x2, x3, x5)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x3, x5)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U19_AA(x1, x2, x3)  =  U19_AA(x3)
U20_AA(x1, x2, x3)  =  U20_AA(x3)
U21_AA(x1, x2, x3)  =  U21_AA(x3)
U22_AA(x1, x2, x3, x4)  =  U22_AA(x3, x4)
U23_AA(x1, x2, x3)  =  U23_AA(x3)
APPEND23_IN_GAGA(x1, x2, x3, x4)  =  APPEND23_IN_GAGA(x1, x3)
U1_GAGA(x1, x2, x3, x4, x5, x6)  =  U1_GAGA(x1, x2, x4, x6)
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
LESS36_IN_AA(x1, x2)  =  LESS36_IN_AA
U2_AA(x1, x2, x3)  =  U2_AA(x3)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
SPLIT46_IN_AAAA(x1, x2, x3, x4)  =  SPLIT46_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)
U4_AAAA(x1, x2, x3, x4, x5, x6)  =  U4_AAAA(x6)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)
GEQ62_IN_AA(x1, x2)  =  GEQ62_IN_AA
U7_AA(x1, x2, x3)  =  U7_AA(x3)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x6)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)
U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x5)
U30_AA(x1, x2, x3, x4, x5)  =  U30_AA(x5)
U31_AA(x1, x2, x3, x4, x5)  =  U31_AA(x5)
U32_AA(x1, x2, x3, x4, x5)  =  U32_AA(x5)
U33_AA(x1, x2, x3, x4, x5)  =  U33_AA(x5)
QS79_IN_AA(x1, x2)  =  QS79_IN_AA
U8_AA(x1, x2, x3, x4)  =  U8_AA(x4)
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U10_AA(x1, x2, x3, x4)  =  U10_AA(x4)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)
U12_AA(x1, x2, x3, x4)  =  U12_AA(x4)
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4)  =  U14_AA(x4)
APPEND94_IN_AAAA(x1, x2, x3, x4)  =  APPEND94_IN_AAAA
U15_AAAA(x1, x2, x3, x4, x5, x6)  =  U15_AAAA(x6)
U34_AA(x1, x2, x3, x4, x5, x6)  =  U34_AA(x6)
U35_AA(x1, x2, x3, x4, x5)  =  U35_AA(x5)
P78_IN_AAAAA(x1, x2, x3, x4, x5)  =  P78_IN_AAAAA
U16_AAAAA(x1, x2, x3, x4, x5, x6)  =  U16_AAAAA(x6)
U17_AAAAA(x1, x2, x3, x4, x5, x6)  =  U17_AAAAA(x6)
U18_AAAAA(x1, x2, x3, x4, x5, x6)  =  U18_AAAAA(x6)
APPEND23_IN_AAAA(x1, x2, x3, x4)  =  APPEND23_IN_AAAA
U1_AAAA(x1, x2, x3, x4, x5, x6)  =  U1_AAAA(x6)
U28_AA(x1, x2, x3, x4, x5, x6)  =  U28_AA(x6)
U29_AA(x1, x2, x3, x4, x5)  =  U29_AA(x5)
U28_GA(x1, x2, x3, x4, x5, x6)  =  U28_GA(x1, x2, x3, x6)
U29_GA(x1, x2, x3, x4, x5)  =  U29_GA(x1, x2, x3, x5)
P78_IN_AAAGA(x1, x2, x3, x4, x5)  =  P78_IN_AAAGA(x4)
U16_AAAGA(x1, x2, x3, x4, x5, x6)  =  U16_AAAGA(x4, x6)
U17_AAAGA(x1, x2, x3, x4, x5, x6)  =  U17_AAAGA(x4, x6)
U18_AAAGA(x1, x2, x3, x4, x5, x6)  =  U18_AAAGA(x4, x6)
APPEND23_IN_AGAA(x1, x2, x3, x4)  =  APPEND23_IN_AGAA(x2)
U1_AGAA(x1, x2, x3, x4, x5, x6)  =  U1_AGAA(x3, x6)
U30_GA(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_GA(x1, x2, x3, x6)
U35_GA(x1, x2, x3, x4, x5)  =  U35_GA(x1, x2, x3, x5)

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

(6) Obligation:

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

QS1_IN_GA(cons(T14, []), T9) → U19_GA(T14, T9, qs14_in_a(X12))
QS1_IN_GA(cons(T14, []), T9) → QS14_IN_A(X12)
QS1_IN_GA(cons(T14, []), T9) → U20_GA(T14, T9, qs14_in_a(T17))
U20_GA(T14, T9, qs14_out_a(T17)) → U21_GA(T14, T9, qs14_in_a(X13))
U20_GA(T14, T9, qs14_out_a(T17)) → QS14_IN_A(X13)
U20_GA(T14, T9, qs14_out_a(T17)) → U22_GA(T14, T9, T17, qs14_in_a(T20))
U22_GA(T14, T9, T17, qs14_out_a(T20)) → U23_GA(T14, T9, append23_in_ggga(T17, T14, T20, T9))
U22_GA(T14, T9, T17, qs14_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)
U24_GA(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_GA(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
U24_GA(T69, T67, T68, T9, less36_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)
U3_GGAA(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_GGAA(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
U3_GGAA(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → SPLIT46_IN_GGAA(T111, T112, X163, X164)
SPLIT46_IN_GGAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_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)) → U7_GG(T146, T147, geq62_in_gg(T146, T147))
GEQ62_IN_GG(s(T146), s(T147)) → GEQ62_IN_GG(T146, T147)
U5_GGAA(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_GGAA(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U5_GGAA(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → SPLIT46_IN_GGAA(T126, T127, X195, X196)
U24_GA(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_GA(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_GA(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_GA(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
U26_GA(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → QS1_IN_AA(cons(T67, T89), X12)
QS1_IN_AA(cons(T14, []), T9) → U19_AA(T14, T9, qs14_in_a(X12))
QS1_IN_AA(cons(T14, []), T9) → QS14_IN_A(X12)
QS1_IN_AA(cons(T14, []), T9) → U20_AA(T14, T9, qs14_in_a(T17))
U20_AA(T14, T9, qs14_out_a(T17)) → U21_AA(T14, T9, qs14_in_a(X13))
U20_AA(T14, T9, qs14_out_a(T17)) → QS14_IN_A(X13)
U20_AA(T14, T9, qs14_out_a(T17)) → U22_AA(T14, T9, T17, qs14_in_a(T20))
U22_AA(T14, T9, T17, qs14_out_a(T20)) → U23_AA(T14, T9, append23_in_gaga(T17, T14, T20, T9))
U22_AA(T14, T9, T17, qs14_out_a(T20)) → APPEND23_IN_GAGA(T17, T14, T20, T9)
APPEND23_IN_GAGA(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_GAGA(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
APPEND23_IN_GAGA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)
QS1_IN_AA(cons(T69, cons(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, less36_in_aa(T67, T69))
QS1_IN_AA(cons(T69, cons(T67, T68)), T9) → LESS36_IN_AA(T67, T69)
LESS36_IN_AA(s(T83), s(T84)) → U2_AA(T83, T84, less36_in_aa(T83, T84))
LESS36_IN_AA(s(T83), s(T84)) → LESS36_IN_AA(T83, T84)
U24_AA(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_AA(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
U24_AA(T69, T67, T68, T9, less36_out_aa(T67, T69)) → SPLIT46_IN_AAAA(T68, T69, X93, X94)
SPLIT46_IN_AAAA(cons(T110, T111), T112, cons(T110, X163), X164) → U3_AAAA(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
SPLIT46_IN_AAAA(cons(T110, T111), T112, cons(T110, X163), X164) → LESS36_IN_AA(T110, T112)
U3_AAAA(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_AAAA(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
U3_AAAA(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → SPLIT46_IN_AAAA(T111, T112, X163, X164)
SPLIT46_IN_AAAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_AAAA(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
SPLIT46_IN_AAAA(cons(T125, T126), T127, X195, cons(T125, X196)) → GEQ62_IN_AA(T125, T127)
GEQ62_IN_AA(s(T146), s(T147)) → U7_AA(T146, T147, geq62_in_aa(T146, T147))
GEQ62_IN_AA(s(T146), s(T147)) → GEQ62_IN_AA(T146, T147)
U5_AAAA(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_AAAA(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U5_AAAA(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → SPLIT46_IN_AAAA(T126, T127, X195, X196)
U24_AA(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_AA(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_AA(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
U26_AA(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → QS1_IN_AA(cons(T67, T89), X12)
QS1_IN_AA(cons(T214, cons(T212, T213)), T9) → U30_AA(T214, T212, T213, T9, geq62_in_aa(T212, T214))
QS1_IN_AA(cons(T214, cons(T212, T213)), T9) → GEQ62_IN_AA(T212, T214)
U30_AA(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_AA(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U30_AA(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → SPLIT46_IN_AAAA(T213, T214, X323, X324)
U30_AA(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_AA(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_AA(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_AA(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U32_AA(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → QS79_IN_AA(T221, X12)
QS79_IN_AA(cons(T162, T163), X265) → U8_AA(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
QS79_IN_AA(cons(T162, T163), X265) → SPLIT46_IN_AAAA(T163, T162, X261, X262)
QS79_IN_AA(cons(T162, T163), X265) → U9_AA(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_AA(T162, T163, X265, qs79_in_aa(T166, X263))
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → QS79_IN_AA(T166, X263)
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_AA(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_AA(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_AA(T162, T163, X265, qs79_in_aa(T167, X264))
U11_AA(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → QS79_IN_AA(T167, X264)
U11_AA(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_AA(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_AA(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_AA(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
U13_AA(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → APPEND94_IN_AAAA(T170, T162, T171, X265)
APPEND94_IN_AAAA(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_AAAA(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
APPEND94_IN_AAAA(cons(T194, T195), T196, T197, cons(T194, X295)) → APPEND94_IN_AAAA(T195, T196, T197, X295)
U32_AA(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_AA(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_AA(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_AA(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
U34_AA(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → P78_IN_AAAAA(cons(T212, T222), X13, T225, T214, T9)
P78_IN_AAAAA(T90, X13, T154, T69, T9) → U16_AAAAA(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
P78_IN_AAAAA(T90, X13, T154, T69, T9) → QS79_IN_AA(T90, X13)
P78_IN_AAAAA(T90, T157, T154, T69, T9) → U17_AAAAA(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_AAAAA(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_AAAAA(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
U17_AAAAA(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → APPEND23_IN_AAAA(T154, T69, T157, T9)
APPEND23_IN_AAAA(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_AAAA(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
APPEND23_IN_AAAA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)
U26_AA(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_AA(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_AA(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_AA(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U28_AA(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → P78_IN_AAAAA(T90, X13, T154, T69, T9)
U26_GA(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_GA(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_GA(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_GA(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
U28_GA(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → P78_IN_AAAGA(T90, X13, T154, T69, T9)
P78_IN_AAAGA(T90, X13, T154, T69, T9) → U16_AAAGA(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
P78_IN_AAAGA(T90, X13, T154, T69, T9) → QS79_IN_AA(T90, X13)
P78_IN_AAAGA(T90, T157, T154, T69, T9) → U17_AAAGA(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_AAAGA(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_AAAGA(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
U17_AAAGA(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → APPEND23_IN_AGAA(T154, T69, T157, T9)
APPEND23_IN_AGAA(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_AGAA(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
APPEND23_IN_AGAA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)
QS1_IN_GA(cons(T214, cons(T212, T213)), T9) → U30_GA(T214, T212, T213, T9, geq62_in_gg(T212, T214))
QS1_IN_GA(cons(T214, cons(T212, T213)), T9) → GEQ62_IN_GG(T212, T214)
U30_GA(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_GA(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U30_GA(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → SPLIT46_IN_GGAA(T213, T214, X323, X324)
U30_GA(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_GA(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_GA(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_GA(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U32_GA(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → QS79_IN_AA(T221, X12)
U32_GA(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_GA(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_GA(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_GA(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U34_GA(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → P78_IN_AAAGA(cons(T212, T222), X13, T225, T214, T9)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
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)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
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)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_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)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_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)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
QS1_IN_GA(x1, x2)  =  QS1_IN_GA(x1)
U19_GA(x1, x2, x3)  =  U19_GA(x1, x3)
QS14_IN_A(x1)  =  QS14_IN_A
U20_GA(x1, x2, x3)  =  U20_GA(x1, x3)
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)
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)
GEQ62_IN_GG(x1, x2)  =  GEQ62_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)
U26_GA(x1, x2, x3, x4, x5)  =  U26_GA(x1, x2, x3, x5)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x3, x5)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U19_AA(x1, x2, x3)  =  U19_AA(x3)
U20_AA(x1, x2, x3)  =  U20_AA(x3)
U21_AA(x1, x2, x3)  =  U21_AA(x3)
U22_AA(x1, x2, x3, x4)  =  U22_AA(x3, x4)
U23_AA(x1, x2, x3)  =  U23_AA(x3)
APPEND23_IN_GAGA(x1, x2, x3, x4)  =  APPEND23_IN_GAGA(x1, x3)
U1_GAGA(x1, x2, x3, x4, x5, x6)  =  U1_GAGA(x1, x2, x4, x6)
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
LESS36_IN_AA(x1, x2)  =  LESS36_IN_AA
U2_AA(x1, x2, x3)  =  U2_AA(x3)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
SPLIT46_IN_AAAA(x1, x2, x3, x4)  =  SPLIT46_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)
U4_AAAA(x1, x2, x3, x4, x5, x6)  =  U4_AAAA(x6)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)
GEQ62_IN_AA(x1, x2)  =  GEQ62_IN_AA
U7_AA(x1, x2, x3)  =  U7_AA(x3)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x6)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)
U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x5)
U30_AA(x1, x2, x3, x4, x5)  =  U30_AA(x5)
U31_AA(x1, x2, x3, x4, x5)  =  U31_AA(x5)
U32_AA(x1, x2, x3, x4, x5)  =  U32_AA(x5)
U33_AA(x1, x2, x3, x4, x5)  =  U33_AA(x5)
QS79_IN_AA(x1, x2)  =  QS79_IN_AA
U8_AA(x1, x2, x3, x4)  =  U8_AA(x4)
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U10_AA(x1, x2, x3, x4)  =  U10_AA(x4)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)
U12_AA(x1, x2, x3, x4)  =  U12_AA(x4)
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4)  =  U14_AA(x4)
APPEND94_IN_AAAA(x1, x2, x3, x4)  =  APPEND94_IN_AAAA
U15_AAAA(x1, x2, x3, x4, x5, x6)  =  U15_AAAA(x6)
U34_AA(x1, x2, x3, x4, x5, x6)  =  U34_AA(x6)
U35_AA(x1, x2, x3, x4, x5)  =  U35_AA(x5)
P78_IN_AAAAA(x1, x2, x3, x4, x5)  =  P78_IN_AAAAA
U16_AAAAA(x1, x2, x3, x4, x5, x6)  =  U16_AAAAA(x6)
U17_AAAAA(x1, x2, x3, x4, x5, x6)  =  U17_AAAAA(x6)
U18_AAAAA(x1, x2, x3, x4, x5, x6)  =  U18_AAAAA(x6)
APPEND23_IN_AAAA(x1, x2, x3, x4)  =  APPEND23_IN_AAAA
U1_AAAA(x1, x2, x3, x4, x5, x6)  =  U1_AAAA(x6)
U28_AA(x1, x2, x3, x4, x5, x6)  =  U28_AA(x6)
U29_AA(x1, x2, x3, x4, x5)  =  U29_AA(x5)
U28_GA(x1, x2, x3, x4, x5, x6)  =  U28_GA(x1, x2, x3, x6)
U29_GA(x1, x2, x3, x4, x5)  =  U29_GA(x1, x2, x3, x5)
P78_IN_AAAGA(x1, x2, x3, x4, x5)  =  P78_IN_AAAGA(x4)
U16_AAAGA(x1, x2, x3, x4, x5, x6)  =  U16_AAAGA(x4, x6)
U17_AAAGA(x1, x2, x3, x4, x5, x6)  =  U17_AAAGA(x4, x6)
U18_AAAGA(x1, x2, x3, x4, x5, x6)  =  U18_AAAGA(x4, x6)
APPEND23_IN_AGAA(x1, x2, x3, x4)  =  APPEND23_IN_AGAA(x2)
U1_AGAA(x1, x2, x3, x4, x5, x6)  =  U1_AGAA(x3, x6)
U30_GA(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_GA(x1, x2, x3, x6)
U35_GA(x1, x2, x3, x4, x5)  =  U35_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 13 SCCs with 87 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APPEND23_IN_AGAA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
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)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
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)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_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)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_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)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
APPEND23_IN_AGAA(x1, x2, x3, x4)  =  APPEND23_IN_AGAA(x2)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

APPEND23_IN_AGAA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
APPEND23_IN_AGAA(x1, x2, x3, x4)  =  APPEND23_IN_AGAA(x2)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

APPEND23_IN_AGAA(T47) → APPEND23_IN_AGAA(T47)

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

(14) NonTerminationProof (EQUIVALENT transformation)

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

s = APPEND23_IN_AGAA(T47) evaluates to t =APPEND23_IN_AGAA(T47)

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND23_IN_AGAA(T47) to APPEND23_IN_AGAA(T47).



(15) NO

(16) Obligation:

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

APPEND23_IN_AAAA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
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)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
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)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_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)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_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)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
APPEND23_IN_AAAA(x1, x2, x3, x4)  =  APPEND23_IN_AAAA

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:

APPEND23_IN_AAAA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
APPEND23_IN_AAAA(x1, x2, x3, x4)  =  APPEND23_IN_AAAA

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

APPEND23_IN_AAAAAPPEND23_IN_AAAA

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

(21) NonTerminationProof (EQUIVALENT transformation)

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

s = APPEND23_IN_AAAA evaluates to t =APPEND23_IN_AAAA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND23_IN_AAAA to APPEND23_IN_AAAA.



(22) NO

(23) Obligation:

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

APPEND94_IN_AAAA(cons(T194, T195), T196, T197, cons(T194, X295)) → APPEND94_IN_AAAA(T195, T196, T197, X295)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
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)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
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)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_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)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_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)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
APPEND94_IN_AAAA(x1, x2, x3, x4)  =  APPEND94_IN_AAAA

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:

APPEND94_IN_AAAA(cons(T194, T195), T196, T197, cons(T194, X295)) → APPEND94_IN_AAAA(T195, T196, T197, X295)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
APPEND94_IN_AAAA(x1, x2, x3, x4)  =  APPEND94_IN_AAAA

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

APPEND94_IN_AAAAAPPEND94_IN_AAAA

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

(28) NonTerminationProof (EQUIVALENT transformation)

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

s = APPEND94_IN_AAAA evaluates to t =APPEND94_IN_AAAA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND94_IN_AAAA to APPEND94_IN_AAAA.



(29) NO

(30) Obligation:

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

GEQ62_IN_AA(s(T146), s(T147)) → GEQ62_IN_AA(T146, T147)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
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)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
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)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_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)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_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)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
GEQ62_IN_AA(x1, x2)  =  GEQ62_IN_AA

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:

GEQ62_IN_AA(s(T146), s(T147)) → GEQ62_IN_AA(T146, T147)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GEQ62_IN_AA(x1, x2)  =  GEQ62_IN_AA

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:

GEQ62_IN_AAGEQ62_IN_AA

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

(35) NonTerminationProof (EQUIVALENT transformation)

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

s = GEQ62_IN_AA evaluates to t =GEQ62_IN_AA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from GEQ62_IN_AA to GEQ62_IN_AA.



(36) NO

(37) Obligation:

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

LESS36_IN_AA(s(T83), s(T84)) → LESS36_IN_AA(T83, T84)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
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)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
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)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_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)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_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)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
LESS36_IN_AA(x1, x2)  =  LESS36_IN_AA

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:

LESS36_IN_AA(s(T83), s(T84)) → LESS36_IN_AA(T83, T84)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS36_IN_AA(x1, x2)  =  LESS36_IN_AA

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:

LESS36_IN_AALESS36_IN_AA

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

(42) NonTerminationProof (EQUIVALENT transformation)

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

s = LESS36_IN_AA evaluates to t =LESS36_IN_AA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LESS36_IN_AA to LESS36_IN_AA.



(43) NO

(44) Obligation:

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

SPLIT46_IN_AAAA(cons(T110, T111), T112, cons(T110, X163), X164) → U3_AAAA(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_AAAA(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → SPLIT46_IN_AAAA(T111, T112, X163, X164)
SPLIT46_IN_AAAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_AAAA(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
U5_AAAA(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → SPLIT46_IN_AAAA(T126, T127, X195, X196)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
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)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
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)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_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)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_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)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
SPLIT46_IN_AAAA(x1, x2, x3, x4)  =  SPLIT46_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)

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

(45) UsableRulesProof (EQUIVALENT transformation)

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

(46) Obligation:

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

SPLIT46_IN_AAAA(cons(T110, T111), T112, cons(T110, X163), X164) → U3_AAAA(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_AAAA(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → SPLIT46_IN_AAAA(T111, T112, X163, X164)
SPLIT46_IN_AAAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_AAAA(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
U5_AAAA(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → SPLIT46_IN_AAAA(T126, T127, X195, X196)

The TRS R consists of the following rules:

less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
SPLIT46_IN_AAAA(x1, x2, x3, x4)  =  SPLIT46_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)

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

(47) PiDPToQDPProof (SOUND transformation)

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

(48) Obligation:

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

SPLIT46_IN_AAAAU3_AAAA(less36_in_aa)
U3_AAAA(less36_out_aa(T110)) → SPLIT46_IN_AAAA
SPLIT46_IN_AAAAU5_AAAA(geq62_in_aa)
U5_AAAA(geq62_out_aa) → SPLIT46_IN_AAAA

The TRS R consists of the following rules:

less36_in_aaless36_out_aa(0)
less36_in_aaU2_aa(less36_in_aa)
geq62_in_aageq62_out_aa
geq62_in_aaU7_aa(geq62_in_aa)
U2_aa(less36_out_aa(T83)) → less36_out_aa(s(T83))
U7_aa(geq62_out_aa) → geq62_out_aa

The set Q consists of the following terms:

less36_in_aa
geq62_in_aa
U2_aa(x0)
U7_aa(x0)

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

(49) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SPLIT46_IN_AAAAU3_AAAA(less36_in_aa) at position [0] we obtained the following new rules [LPAR04]:

SPLIT46_IN_AAAAU3_AAAA(less36_out_aa(0))
SPLIT46_IN_AAAAU3_AAAA(U2_aa(less36_in_aa))

(50) Obligation:

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

U3_AAAA(less36_out_aa(T110)) → SPLIT46_IN_AAAA
SPLIT46_IN_AAAAU5_AAAA(geq62_in_aa)
U5_AAAA(geq62_out_aa) → SPLIT46_IN_AAAA
SPLIT46_IN_AAAAU3_AAAA(less36_out_aa(0))
SPLIT46_IN_AAAAU3_AAAA(U2_aa(less36_in_aa))

The TRS R consists of the following rules:

less36_in_aaless36_out_aa(0)
less36_in_aaU2_aa(less36_in_aa)
geq62_in_aageq62_out_aa
geq62_in_aaU7_aa(geq62_in_aa)
U2_aa(less36_out_aa(T83)) → less36_out_aa(s(T83))
U7_aa(geq62_out_aa) → geq62_out_aa

The set Q consists of the following terms:

less36_in_aa
geq62_in_aa
U2_aa(x0)
U7_aa(x0)

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

(51) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SPLIT46_IN_AAAAU5_AAAA(geq62_in_aa) at position [0] we obtained the following new rules [LPAR04]:

SPLIT46_IN_AAAAU5_AAAA(geq62_out_aa)
SPLIT46_IN_AAAAU5_AAAA(U7_aa(geq62_in_aa))

(52) Obligation:

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

U3_AAAA(less36_out_aa(T110)) → SPLIT46_IN_AAAA
U5_AAAA(geq62_out_aa) → SPLIT46_IN_AAAA
SPLIT46_IN_AAAAU3_AAAA(less36_out_aa(0))
SPLIT46_IN_AAAAU3_AAAA(U2_aa(less36_in_aa))
SPLIT46_IN_AAAAU5_AAAA(geq62_out_aa)
SPLIT46_IN_AAAAU5_AAAA(U7_aa(geq62_in_aa))

The TRS R consists of the following rules:

less36_in_aaless36_out_aa(0)
less36_in_aaU2_aa(less36_in_aa)
geq62_in_aageq62_out_aa
geq62_in_aaU7_aa(geq62_in_aa)
U2_aa(less36_out_aa(T83)) → less36_out_aa(s(T83))
U7_aa(geq62_out_aa) → geq62_out_aa

The set Q consists of the following terms:

less36_in_aa
geq62_in_aa
U2_aa(x0)
U7_aa(x0)

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

(53) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = SPLIT46_IN_AAAA evaluates to t =SPLIT46_IN_AAAA

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




Rewriting sequence

SPLIT46_IN_AAAAU3_AAAA(less36_out_aa(0))
with rule SPLIT46_IN_AAAAU3_AAAA(less36_out_aa(0)) at position [] and matcher [ ]

U3_AAAA(less36_out_aa(0))SPLIT46_IN_AAAA
with rule U3_AAAA(less36_out_aa(T110)) → SPLIT46_IN_AAAA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.



(54) NO

(55) Obligation:

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

QS79_IN_AA(cons(T162, T163), X265) → U9_AA(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → QS79_IN_AA(T166, X263)
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_AA(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_AA(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → QS79_IN_AA(T167, X264)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
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)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
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)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_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)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_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)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
QS79_IN_AA(x1, x2)  =  QS79_IN_AA
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)

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

(56) UsableRulesProof (EQUIVALENT transformation)

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

(57) Obligation:

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

QS79_IN_AA(cons(T162, T163), X265) → U9_AA(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → QS79_IN_AA(T166, X263)
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_AA(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_AA(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → QS79_IN_AA(T167, X264)

The TRS R consists of the following rules:

split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))

The argument filtering Pi contains the following mapping:
[]  =  []
cons(x1, x2)  =  cons(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
QS79_IN_AA(x1, x2)  =  QS79_IN_AA
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)

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

(58) PiDPToQDPProof (SOUND transformation)

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

(59) Obligation:

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

QS79_IN_AAU9_AA(split46_in_aaaa)
U9_AA(split46_out_aaaa) → QS79_IN_AA
U9_AA(split46_out_aaaa) → U11_AA(qs79_in_aa)
U11_AA(qs79_out_aa) → QS79_IN_AA

The TRS R consists of the following rules:

split46_in_aaaasplit46_out_aaaa
split46_in_aaaaU3_aaaa(less36_in_aa)
split46_in_aaaaU5_aaaa(geq62_in_aa)
qs79_in_aaqs79_out_aa
qs79_in_aaU8_aa(split46_in_aaaa)
qs79_in_aaU9_aa(split46_in_aaaa)
U3_aaaa(less36_out_aa(T110)) → split46_out_aaaa
U3_aaaa(less36_out_aa(T110)) → U4_aaaa(split46_in_aaaa)
U5_aaaa(geq62_out_aa) → split46_out_aaaa
U5_aaaa(geq62_out_aa) → U6_aaaa(split46_in_aaaa)
U8_aa(split46_out_aaaa) → qs79_out_aa
U9_aa(split46_out_aaaa) → U10_aa(qs79_in_aa)
U9_aa(split46_out_aaaa) → U11_aa(qs79_in_aa)
less36_in_aaless36_out_aa(0)
less36_in_aaU2_aa(less36_in_aa)
U4_aaaa(split46_out_aaaa) → split46_out_aaaa
geq62_in_aageq62_out_aa
geq62_in_aaU7_aa(geq62_in_aa)
U6_aaaa(split46_out_aaaa) → split46_out_aaaa
U10_aa(qs79_out_aa) → qs79_out_aa
U11_aa(qs79_out_aa) → U12_aa(qs79_in_aa)
U11_aa(qs79_out_aa) → U13_aa(qs79_in_aa)
U2_aa(less36_out_aa(T83)) → less36_out_aa(s(T83))
U7_aa(geq62_out_aa) → geq62_out_aa
U12_aa(qs79_out_aa) → qs79_out_aa
U13_aa(qs79_out_aa) → U14_aa(append94_in_aaaa)
U14_aa(append94_out_aaaa) → qs79_out_aa
append94_in_aaaaappend94_out_aaaa
append94_in_aaaaU15_aaaa(append94_in_aaaa)
U15_aaaa(append94_out_aaaa) → append94_out_aaaa

The set Q consists of the following terms:

split46_in_aaaa
qs79_in_aa
U3_aaaa(x0)
U5_aaaa(x0)
U8_aa(x0)
U9_aa(x0)
less36_in_aa
U4_aaaa(x0)
geq62_in_aa
U6_aaaa(x0)
U10_aa(x0)
U11_aa(x0)
U2_aa(x0)
U7_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
append94_in_aaaa
U15_aaaa(x0)

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

(60) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS79_IN_AAU9_AA(split46_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

QS79_IN_AAU9_AA(split46_out_aaaa)
QS79_IN_AAU9_AA(U3_aaaa(less36_in_aa))
QS79_IN_AAU9_AA(U5_aaaa(geq62_in_aa))

(61) Obligation:

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

U9_AA(split46_out_aaaa) → QS79_IN_AA
U9_AA(split46_out_aaaa) → U11_AA(qs79_in_aa)
U11_AA(qs79_out_aa) → QS79_IN_AA
QS79_IN_AAU9_AA(split46_out_aaaa)
QS79_IN_AAU9_AA(U3_aaaa(less36_in_aa))
QS79_IN_AAU9_AA(U5_aaaa(geq62_in_aa))

The TRS R consists of the following rules:

split46_in_aaaasplit46_out_aaaa
split46_in_aaaaU3_aaaa(less36_in_aa)
split46_in_aaaaU5_aaaa(geq62_in_aa)
qs79_in_aaqs79_out_aa
qs79_in_aaU8_aa(split46_in_aaaa)
qs79_in_aaU9_aa(split46_in_aaaa)
U3_aaaa(less36_out_aa(T110)) → split46_out_aaaa
U3_aaaa(less36_out_aa(T110)) → U4_aaaa(split46_in_aaaa)
U5_aaaa(geq62_out_aa) → split46_out_aaaa
U5_aaaa(geq62_out_aa) → U6_aaaa(split46_in_aaaa)
U8_aa(split46_out_aaaa) → qs79_out_aa
U9_aa(split46_out_aaaa) → U10_aa(qs79_in_aa)
U9_aa(split46_out_aaaa) → U11_aa(qs79_in_aa)
less36_in_aaless36_out_aa(0)
less36_in_aaU2_aa(less36_in_aa)
U4_aaaa(split46_out_aaaa) → split46_out_aaaa
geq62_in_aageq62_out_aa
geq62_in_aaU7_aa(geq62_in_aa)
U6_aaaa(split46_out_aaaa) → split46_out_aaaa
U10_aa(qs79_out_aa) → qs79_out_aa
U11_aa(qs79_out_aa) → U12_aa(qs79_in_aa)
U11_aa(qs79_out_aa) → U13_aa(qs79_in_aa)
U2_aa(less36_out_aa(T83)) → less36_out_aa(s(T83))
U7_aa(geq62_out_aa) → geq62_out_aa
U12_aa(qs79_out_aa) → qs79_out_aa
U13_aa(qs79_out_aa) → U14_aa(append94_in_aaaa)
U14_aa(append94_out_aaaa) → qs79_out_aa
append94_in_aaaaappend94_out_aaaa
append94_in_aaaaU15_aaaa(append94_in_aaaa)
U15_aaaa(append94_out_aaaa) → append94_out_aaaa

The set Q consists of the following terms:

split46_in_aaaa
qs79_in_aa
U3_aaaa(x0)
U5_aaaa(x0)
U8_aa(x0)
U9_aa(x0)
less36_in_aa
U4_aaaa(x0)
geq62_in_aa
U6_aaaa(x0)
U10_aa(x0)
U11_aa(x0)
U2_aa(x0)
U7_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
append94_in_aaaa
U15_aaaa(x0)

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

(62) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U9_AA(split46_out_aaaa) → U11_AA(qs79_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U9_AA(split46_out_aaaa) → U11_AA(qs79_out_aa)
U9_AA(split46_out_aaaa) → U11_AA(U8_aa(split46_in_aaaa))
U9_AA(split46_out_aaaa) → U11_AA(U9_aa(split46_in_aaaa))

(63) Obligation:

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

U9_AA(split46_out_aaaa) → QS79_IN_AA
U11_AA(qs79_out_aa) → QS79_IN_AA
QS79_IN_AAU9_AA(split46_out_aaaa)
QS79_IN_AAU9_AA(U3_aaaa(less36_in_aa))
QS79_IN_AAU9_AA(U5_aaaa(geq62_in_aa))
U9_AA(split46_out_aaaa) → U11_AA(qs79_out_aa)
U9_AA(split46_out_aaaa) → U11_AA(U8_aa(split46_in_aaaa))
U9_AA(split46_out_aaaa) → U11_AA(U9_aa(split46_in_aaaa))

The TRS R consists of the following rules:

split46_in_aaaasplit46_out_aaaa
split46_in_aaaaU3_aaaa(less36_in_aa)
split46_in_aaaaU5_aaaa(geq62_in_aa)
qs79_in_aaqs79_out_aa
qs79_in_aaU8_aa(split46_in_aaaa)
qs79_in_aaU9_aa(split46_in_aaaa)
U3_aaaa(less36_out_aa(T110)) → split46_out_aaaa
U3_aaaa(less36_out_aa(T110)) → U4_aaaa(split46_in_aaaa)
U5_aaaa(geq62_out_aa) → split46_out_aaaa
U5_aaaa(geq62_out_aa) → U6_aaaa(split46_in_aaaa)
U8_aa(split46_out_aaaa) → qs79_out_aa
U9_aa(split46_out_aaaa) → U10_aa(qs79_in_aa)
U9_aa(split46_out_aaaa) → U11_aa(qs79_in_aa)
less36_in_aaless36_out_aa(0)
less36_in_aaU2_aa(less36_in_aa)
U4_aaaa(split46_out_aaaa) → split46_out_aaaa
geq62_in_aageq62_out_aa
geq62_in_aaU7_aa(geq62_in_aa)
U6_aaaa(split46_out_aaaa) → split46_out_aaaa
U10_aa(qs79_out_aa) → qs79_out_aa
U11_aa(qs79_out_aa) → U12_aa(qs79_in_aa)
U11_aa(qs79_out_aa) → U13_aa(qs79_in_aa)
U2_aa(less36_out_aa(T83)) → less36_out_aa(s(T83))
U7_aa(geq62_out_aa) → geq62_out_aa
U12_aa(qs79_out_aa) → qs79_out_aa
U13_aa(qs79_out_aa) → U14_aa(append94_in_aaaa)
U14_aa(append94_out_aaaa) → qs79_out_aa
append94_in_aaaaappend94_out_aaaa
append94_in_aaaaU15_aaaa(append94_in_aaaa)
U15_aaaa(append94_out_aaaa) → append94_out_aaaa

The set Q consists of the following terms:

split46_in_aaaa
qs79_in_aa
U3_aaaa(x0)
U5_aaaa(x0)
U8_aa(x0)
U9_aa(x0)
less36_in_aa
U4_aaaa(x0)
geq62_in_aa
U6_aaaa(x0)
U10_aa(x0)
U11_aa(x0)
U2_aa(x0)
U7_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
append94_in_aaaa
U15_aaaa(x0)

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

(64) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = QS79_IN_AA evaluates to t =QS79_IN_AA

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




Rewriting sequence

QS79_IN_AAU9_AA(split46_out_aaaa)
with rule QS79_IN_AAU9_AA(split46_out_aaaa) at position [] and matcher [ ]

U9_AA(split46_out_aaaa)QS79_IN_AA
with rule U9_AA(split46_out_aaaa) → QS79_IN_AA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.



(65) NO

(66) Obligation:

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

APPEND23_IN_GAGA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
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)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
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)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_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)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_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)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
APPEND23_IN_GAGA(x1, x2, x3, x4)  =  APPEND23_IN_GAGA(x1, x3)

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

(67) UsableRulesProof (EQUIVALENT transformation)

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

(68) Obligation:

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

APPEND23_IN_GAGA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
APPEND23_IN_GAGA(x1, x2, x3, x4)  =  APPEND23_IN_GAGA(x1, x3)

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

(69) PiDPToQDPProof (SOUND transformation)

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

(70) Obligation:

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

APPEND23_IN_GAGA(cons(T45, T46), T48) → APPEND23_IN_GAGA(T46, T48)

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

(71) 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_GAGA(cons(T45, T46), T48) → APPEND23_IN_GAGA(T46, T48)
    The graph contains the following edges 1 > 1, 2 >= 2

(72) YES

(73) Obligation:

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

QS1_IN_AA(cons(T69, cons(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, less36_in_aa(T67, T69))
U24_AA(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_AA(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → QS1_IN_AA(cons(T67, T89), X12)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
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)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
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)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_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)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_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)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)

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

(74) UsableRulesProof (EQUIVALENT transformation)

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

(75) Obligation:

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

QS1_IN_AA(cons(T69, cons(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, less36_in_aa(T67, T69))
U24_AA(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_AA(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → QS1_IN_AA(cons(T67, T89), X12)

The TRS R consists of the following rules:

less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))

The argument filtering Pi contains the following mapping:
[]  =  []
cons(x1, x2)  =  cons(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)

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

(76) PiDPToQDPProof (SOUND transformation)

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

(77) Obligation:

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

QS1_IN_AAU24_AA(less36_in_aa)
U24_AA(less36_out_aa(T67)) → U26_AA(split46_in_aaaa)
U26_AA(split46_out_aaaa) → QS1_IN_AA

The TRS R consists of the following rules:

less36_in_aaless36_out_aa(0)
less36_in_aaU2_aa(less36_in_aa)
split46_in_aaaasplit46_out_aaaa
split46_in_aaaaU3_aaaa(less36_in_aa)
split46_in_aaaaU5_aaaa(geq62_in_aa)
U2_aa(less36_out_aa(T83)) → less36_out_aa(s(T83))
U3_aaaa(less36_out_aa(T110)) → split46_out_aaaa
U3_aaaa(less36_out_aa(T110)) → U4_aaaa(split46_in_aaaa)
U5_aaaa(geq62_out_aa) → split46_out_aaaa
U5_aaaa(geq62_out_aa) → U6_aaaa(split46_in_aaaa)
U4_aaaa(split46_out_aaaa) → split46_out_aaaa
geq62_in_aageq62_out_aa
geq62_in_aaU7_aa(geq62_in_aa)
U6_aaaa(split46_out_aaaa) → split46_out_aaaa
U7_aa(geq62_out_aa) → geq62_out_aa

The set Q consists of the following terms:

less36_in_aa
split46_in_aaaa
U2_aa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U4_aaaa(x0)
geq62_in_aa
U6_aaaa(x0)
U7_aa(x0)

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

(78) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS1_IN_AAU24_AA(less36_in_aa) at position [0] we obtained the following new rules [LPAR04]:

QS1_IN_AAU24_AA(less36_out_aa(0))
QS1_IN_AAU24_AA(U2_aa(less36_in_aa))

(79) Obligation:

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

U24_AA(less36_out_aa(T67)) → U26_AA(split46_in_aaaa)
U26_AA(split46_out_aaaa) → QS1_IN_AA
QS1_IN_AAU24_AA(less36_out_aa(0))
QS1_IN_AAU24_AA(U2_aa(less36_in_aa))

The TRS R consists of the following rules:

less36_in_aaless36_out_aa(0)
less36_in_aaU2_aa(less36_in_aa)
split46_in_aaaasplit46_out_aaaa
split46_in_aaaaU3_aaaa(less36_in_aa)
split46_in_aaaaU5_aaaa(geq62_in_aa)
U2_aa(less36_out_aa(T83)) → less36_out_aa(s(T83))
U3_aaaa(less36_out_aa(T110)) → split46_out_aaaa
U3_aaaa(less36_out_aa(T110)) → U4_aaaa(split46_in_aaaa)
U5_aaaa(geq62_out_aa) → split46_out_aaaa
U5_aaaa(geq62_out_aa) → U6_aaaa(split46_in_aaaa)
U4_aaaa(split46_out_aaaa) → split46_out_aaaa
geq62_in_aageq62_out_aa
geq62_in_aaU7_aa(geq62_in_aa)
U6_aaaa(split46_out_aaaa) → split46_out_aaaa
U7_aa(geq62_out_aa) → geq62_out_aa

The set Q consists of the following terms:

less36_in_aa
split46_in_aaaa
U2_aa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U4_aaaa(x0)
geq62_in_aa
U6_aaaa(x0)
U7_aa(x0)

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

(80) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_AA(less36_out_aa(T67)) → U26_AA(split46_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

U24_AA(less36_out_aa(y0)) → U26_AA(split46_out_aaaa)
U24_AA(less36_out_aa(y0)) → U26_AA(U3_aaaa(less36_in_aa))
U24_AA(less36_out_aa(y0)) → U26_AA(U5_aaaa(geq62_in_aa))

(81) Obligation:

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

U26_AA(split46_out_aaaa) → QS1_IN_AA
QS1_IN_AAU24_AA(less36_out_aa(0))
QS1_IN_AAU24_AA(U2_aa(less36_in_aa))
U24_AA(less36_out_aa(y0)) → U26_AA(split46_out_aaaa)
U24_AA(less36_out_aa(y0)) → U26_AA(U3_aaaa(less36_in_aa))
U24_AA(less36_out_aa(y0)) → U26_AA(U5_aaaa(geq62_in_aa))

The TRS R consists of the following rules:

less36_in_aaless36_out_aa(0)
less36_in_aaU2_aa(less36_in_aa)
split46_in_aaaasplit46_out_aaaa
split46_in_aaaaU3_aaaa(less36_in_aa)
split46_in_aaaaU5_aaaa(geq62_in_aa)
U2_aa(less36_out_aa(T83)) → less36_out_aa(s(T83))
U3_aaaa(less36_out_aa(T110)) → split46_out_aaaa
U3_aaaa(less36_out_aa(T110)) → U4_aaaa(split46_in_aaaa)
U5_aaaa(geq62_out_aa) → split46_out_aaaa
U5_aaaa(geq62_out_aa) → U6_aaaa(split46_in_aaaa)
U4_aaaa(split46_out_aaaa) → split46_out_aaaa
geq62_in_aageq62_out_aa
geq62_in_aaU7_aa(geq62_in_aa)
U6_aaaa(split46_out_aaaa) → split46_out_aaaa
U7_aa(geq62_out_aa) → geq62_out_aa

The set Q consists of the following terms:

less36_in_aa
split46_in_aaaa
U2_aa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U4_aaaa(x0)
geq62_in_aa
U6_aaaa(x0)
U7_aa(x0)

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

(82) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = QS1_IN_AA evaluates to t =QS1_IN_AA

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




Rewriting sequence

QS1_IN_AAU24_AA(less36_out_aa(0))
with rule QS1_IN_AAU24_AA(less36_out_aa(0)) at position [] and matcher [ ]

U24_AA(less36_out_aa(0))U26_AA(split46_out_aaaa)
with rule U24_AA(less36_out_aa(y0)) → U26_AA(split46_out_aaaa) at position [] and matcher [y0 / 0]

U26_AA(split46_out_aaaa)QS1_IN_AA
with rule U26_AA(split46_out_aaaa) → QS1_IN_AA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.



(83) NO

(84) 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:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
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)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
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)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_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)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_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)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
GEQ62_IN_GG(x1, x2)  =  GEQ62_IN_GG(x1, x2)

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

(85) UsableRulesProof (EQUIVALENT transformation)

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

(86) 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

(87) PiDPToQDPProof (EQUIVALENT transformation)

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

(88) 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.

(89) 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

(90) YES

(91) 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:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
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)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
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)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_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)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_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)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x1, x2, x3, x5)
LESS36_IN_GG(x1, x2)  =  LESS36_IN_GG(x1, x2)

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

(92) UsableRulesProof (EQUIVALENT transformation)

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

(93) 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

(94) PiDPToQDPProof (EQUIVALENT transformation)

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

(95) 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.

(96) 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

(97) YES

(98) Obligation:

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

SPLIT46_IN_GGAA(cons(T110, T111), T112, cons(T110, X163), X164) → U3_GGAA(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_GGAA(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → SPLIT46_IN_GGAA(T111, T112, X163, X164)
SPLIT46_IN_GGAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_GGAA(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
U5_GGAA(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → SPLIT46_IN_GGAA(T126, T127, X195, X196)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
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)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
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)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_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)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_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)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_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)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

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

(99) UsableRulesProof (EQUIVALENT transformation)

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

(100) Obligation:

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

SPLIT46_IN_GGAA(cons(T110, T111), T112, cons(T110, X163), X164) → U3_GGAA(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_GGAA(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → SPLIT46_IN_GGAA(T111, T112, X163, X164)
SPLIT46_IN_GGAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_GGAA(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
U5_GGAA(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → SPLIT46_IN_GGAA(T126, T127, X195, X196)

The TRS R consists of the following rules:

less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg(x1, x2)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
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)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

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

(101) PiDPToQDPProof (SOUND transformation)

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

(102) Obligation:

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

SPLIT46_IN_GGAA(cons(T110, T111), T112) → U3_GGAA(T110, T111, T112, less36_in_gg(T110, T112))
U3_GGAA(T110, T111, T112, less36_out_gg(T110, T112)) → SPLIT46_IN_GGAA(T111, T112)
SPLIT46_IN_GGAA(cons(T125, T126), T127) → U5_GGAA(T125, T126, T127, geq62_in_gg(T125, T127))
U5_GGAA(T125, T126, T127, geq62_out_gg(T125, T127)) → SPLIT46_IN_GGAA(T126, T127)

The TRS R consists of the following rules:

less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))

The set Q consists of the following terms:

less36_in_gg(x0, x1)
geq62_in_gg(x0, x1)
U2_gg(x0, x1, x2)
U7_gg(x0, x1, x2)

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

(103) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • U3_GGAA(T110, T111, T112, less36_out_gg(T110, T112)) → SPLIT46_IN_GGAA(T111, T112)
    The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 2

  • U5_GGAA(T125, T126, T127, geq62_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) → U3_GGAA(T110, T111, T112, less36_in_gg(T110, T112))
    The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3

  • SPLIT46_IN_GGAA(cons(T125, T126), T127) → U5_GGAA(T125, T126, T127, geq62_in_gg(T125, T127))
    The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3

(104) YES

(105) 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:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
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)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x1, x2, x3, x4)
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)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_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)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_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)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x2, x3, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x1, x2, x3, x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga(x1, x3)
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x1, x2, x4, x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x2, x3, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x4, x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga(x4)
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x4, x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x3, x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_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, x6)  =  U34_ga(x1, x2, x3, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_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

(106) UsableRulesProof (EQUIVALENT transformation)

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

(107) 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

(108) PiDPToQDPProof (SOUND transformation)

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

(109) 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.

(110) 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

(111) YES

(112) PrologToPiTRSProof (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) (f,f)
append23_in: (b,b,b,f) (b,f,b,f) (f,f,f,f) (f,b,f,f)
less36_in: (b,b) (f,f)
split46_in: (b,b,f,f) (f,f,f,f)
geq62_in: (b,b) (f,f)
qs79_in: (f,f)
append94_in: (f,f,f,f)
p78_in: (f,f,f,f,f) (f,f,f,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(113) Obligation:

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

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)

(114) DependencyPairsProof (EQUIVALENT transformation)

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

QS1_IN_GA(cons(T14, []), T9) → U19_GA(T14, T9, qs14_in_a(X12))
QS1_IN_GA(cons(T14, []), T9) → QS14_IN_A(X12)
QS1_IN_GA(cons(T14, []), T9) → U20_GA(T14, T9, qs14_in_a(T17))
U20_GA(T14, T9, qs14_out_a(T17)) → U21_GA(T14, T9, qs14_in_a(X13))
U20_GA(T14, T9, qs14_out_a(T17)) → QS14_IN_A(X13)
U20_GA(T14, T9, qs14_out_a(T17)) → U22_GA(T14, T9, T17, qs14_in_a(T20))
U22_GA(T14, T9, T17, qs14_out_a(T20)) → U23_GA(T14, T9, append23_in_ggga(T17, T14, T20, T9))
U22_GA(T14, T9, T17, qs14_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)
U24_GA(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_GA(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
U24_GA(T69, T67, T68, T9, less36_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)
U3_GGAA(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_GGAA(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
U3_GGAA(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → SPLIT46_IN_GGAA(T111, T112, X163, X164)
SPLIT46_IN_GGAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_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)) → U7_GG(T146, T147, geq62_in_gg(T146, T147))
GEQ62_IN_GG(s(T146), s(T147)) → GEQ62_IN_GG(T146, T147)
U5_GGAA(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_GGAA(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U5_GGAA(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → SPLIT46_IN_GGAA(T126, T127, X195, X196)
U24_GA(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_GA(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_GA(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_GA(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
U26_GA(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → QS1_IN_AA(cons(T67, T89), X12)
QS1_IN_AA(cons(T14, []), T9) → U19_AA(T14, T9, qs14_in_a(X12))
QS1_IN_AA(cons(T14, []), T9) → QS14_IN_A(X12)
QS1_IN_AA(cons(T14, []), T9) → U20_AA(T14, T9, qs14_in_a(T17))
U20_AA(T14, T9, qs14_out_a(T17)) → U21_AA(T14, T9, qs14_in_a(X13))
U20_AA(T14, T9, qs14_out_a(T17)) → QS14_IN_A(X13)
U20_AA(T14, T9, qs14_out_a(T17)) → U22_AA(T14, T9, T17, qs14_in_a(T20))
U22_AA(T14, T9, T17, qs14_out_a(T20)) → U23_AA(T14, T9, append23_in_gaga(T17, T14, T20, T9))
U22_AA(T14, T9, T17, qs14_out_a(T20)) → APPEND23_IN_GAGA(T17, T14, T20, T9)
APPEND23_IN_GAGA(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_GAGA(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
APPEND23_IN_GAGA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)
QS1_IN_AA(cons(T69, cons(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, less36_in_aa(T67, T69))
QS1_IN_AA(cons(T69, cons(T67, T68)), T9) → LESS36_IN_AA(T67, T69)
LESS36_IN_AA(s(T83), s(T84)) → U2_AA(T83, T84, less36_in_aa(T83, T84))
LESS36_IN_AA(s(T83), s(T84)) → LESS36_IN_AA(T83, T84)
U24_AA(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_AA(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
U24_AA(T69, T67, T68, T9, less36_out_aa(T67, T69)) → SPLIT46_IN_AAAA(T68, T69, X93, X94)
SPLIT46_IN_AAAA(cons(T110, T111), T112, cons(T110, X163), X164) → U3_AAAA(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
SPLIT46_IN_AAAA(cons(T110, T111), T112, cons(T110, X163), X164) → LESS36_IN_AA(T110, T112)
U3_AAAA(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_AAAA(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
U3_AAAA(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → SPLIT46_IN_AAAA(T111, T112, X163, X164)
SPLIT46_IN_AAAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_AAAA(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
SPLIT46_IN_AAAA(cons(T125, T126), T127, X195, cons(T125, X196)) → GEQ62_IN_AA(T125, T127)
GEQ62_IN_AA(s(T146), s(T147)) → U7_AA(T146, T147, geq62_in_aa(T146, T147))
GEQ62_IN_AA(s(T146), s(T147)) → GEQ62_IN_AA(T146, T147)
U5_AAAA(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_AAAA(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U5_AAAA(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → SPLIT46_IN_AAAA(T126, T127, X195, X196)
U24_AA(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_AA(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_AA(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
U26_AA(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → QS1_IN_AA(cons(T67, T89), X12)
QS1_IN_AA(cons(T214, cons(T212, T213)), T9) → U30_AA(T214, T212, T213, T9, geq62_in_aa(T212, T214))
QS1_IN_AA(cons(T214, cons(T212, T213)), T9) → GEQ62_IN_AA(T212, T214)
U30_AA(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_AA(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U30_AA(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → SPLIT46_IN_AAAA(T213, T214, X323, X324)
U30_AA(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_AA(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_AA(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_AA(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U32_AA(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → QS79_IN_AA(T221, X12)
QS79_IN_AA(cons(T162, T163), X265) → U8_AA(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
QS79_IN_AA(cons(T162, T163), X265) → SPLIT46_IN_AAAA(T163, T162, X261, X262)
QS79_IN_AA(cons(T162, T163), X265) → U9_AA(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_AA(T162, T163, X265, qs79_in_aa(T166, X263))
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → QS79_IN_AA(T166, X263)
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_AA(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_AA(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_AA(T162, T163, X265, qs79_in_aa(T167, X264))
U11_AA(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → QS79_IN_AA(T167, X264)
U11_AA(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_AA(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_AA(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_AA(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
U13_AA(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → APPEND94_IN_AAAA(T170, T162, T171, X265)
APPEND94_IN_AAAA(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_AAAA(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
APPEND94_IN_AAAA(cons(T194, T195), T196, T197, cons(T194, X295)) → APPEND94_IN_AAAA(T195, T196, T197, X295)
U32_AA(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_AA(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_AA(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_AA(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
U34_AA(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → P78_IN_AAAAA(cons(T212, T222), X13, T225, T214, T9)
P78_IN_AAAAA(T90, X13, T154, T69, T9) → U16_AAAAA(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
P78_IN_AAAAA(T90, X13, T154, T69, T9) → QS79_IN_AA(T90, X13)
P78_IN_AAAAA(T90, T157, T154, T69, T9) → U17_AAAAA(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_AAAAA(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_AAAAA(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
U17_AAAAA(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → APPEND23_IN_AAAA(T154, T69, T157, T9)
APPEND23_IN_AAAA(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_AAAA(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
APPEND23_IN_AAAA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)
U26_AA(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_AA(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_AA(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_AA(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U28_AA(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → P78_IN_AAAAA(T90, X13, T154, T69, T9)
U26_GA(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_GA(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_GA(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_GA(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
U28_GA(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → P78_IN_AAAGA(T90, X13, T154, T69, T9)
P78_IN_AAAGA(T90, X13, T154, T69, T9) → U16_AAAGA(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
P78_IN_AAAGA(T90, X13, T154, T69, T9) → QS79_IN_AA(T90, X13)
P78_IN_AAAGA(T90, T157, T154, T69, T9) → U17_AAAGA(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_AAAGA(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_AAAGA(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
U17_AAAGA(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → APPEND23_IN_AGAA(T154, T69, T157, T9)
APPEND23_IN_AGAA(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_AGAA(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
APPEND23_IN_AGAA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)
QS1_IN_GA(cons(T214, cons(T212, T213)), T9) → U30_GA(T214, T212, T213, T9, geq62_in_gg(T212, T214))
QS1_IN_GA(cons(T214, cons(T212, T213)), T9) → GEQ62_IN_GG(T212, T214)
U30_GA(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_GA(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U30_GA(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → SPLIT46_IN_GGAA(T213, T214, X323, X324)
U30_GA(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_GA(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_GA(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_GA(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U32_GA(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → QS79_IN_AA(T221, X12)
U32_GA(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_GA(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_GA(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_GA(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U34_GA(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → P78_IN_AAAGA(cons(T212, T222), X13, T225, T214, T9)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
QS1_IN_GA(x1, x2)  =  QS1_IN_GA(x1)
U19_GA(x1, x2, x3)  =  U19_GA(x3)
QS14_IN_A(x1)  =  QS14_IN_A
U20_GA(x1, x2, x3)  =  U20_GA(x1, x3)
U21_GA(x1, x2, x3)  =  U21_GA(x3)
U22_GA(x1, x2, x3, x4)  =  U22_GA(x1, x3, x4)
U23_GA(x1, x2, x3)  =  U23_GA(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, x6)
U24_GA(x1, x2, x3, x4, x5)  =  U24_GA(x1, x3, x5)
LESS36_IN_GG(x1, x2)  =  LESS36_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x3)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x5)
SPLIT46_IN_GGAA(x1, x2, x3, x4)  =  SPLIT46_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x2, x3, x6)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x6)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x2, x3, x6)
GEQ62_IN_GG(x1, x2)  =  GEQ62_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x3)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x6)
U26_GA(x1, x2, x3, x4, x5)  =  U26_GA(x1, x5)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x5)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U19_AA(x1, x2, x3)  =  U19_AA(x3)
U20_AA(x1, x2, x3)  =  U20_AA(x3)
U21_AA(x1, x2, x3)  =  U21_AA(x3)
U22_AA(x1, x2, x3, x4)  =  U22_AA(x3, x4)
U23_AA(x1, x2, x3)  =  U23_AA(x3)
APPEND23_IN_GAGA(x1, x2, x3, x4)  =  APPEND23_IN_GAGA(x1, x3)
U1_GAGA(x1, x2, x3, x4, x5, x6)  =  U1_GAGA(x6)
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
LESS36_IN_AA(x1, x2)  =  LESS36_IN_AA
U2_AA(x1, x2, x3)  =  U2_AA(x3)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
SPLIT46_IN_AAAA(x1, x2, x3, x4)  =  SPLIT46_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)
U4_AAAA(x1, x2, x3, x4, x5, x6)  =  U4_AAAA(x6)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)
GEQ62_IN_AA(x1, x2)  =  GEQ62_IN_AA
U7_AA(x1, x2, x3)  =  U7_AA(x3)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x6)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)
U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x5)
U30_AA(x1, x2, x3, x4, x5)  =  U30_AA(x5)
U31_AA(x1, x2, x3, x4, x5)  =  U31_AA(x5)
U32_AA(x1, x2, x3, x4, x5)  =  U32_AA(x5)
U33_AA(x1, x2, x3, x4, x5)  =  U33_AA(x5)
QS79_IN_AA(x1, x2)  =  QS79_IN_AA
U8_AA(x1, x2, x3, x4)  =  U8_AA(x4)
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U10_AA(x1, x2, x3, x4)  =  U10_AA(x4)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)
U12_AA(x1, x2, x3, x4)  =  U12_AA(x4)
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4)  =  U14_AA(x4)
APPEND94_IN_AAAA(x1, x2, x3, x4)  =  APPEND94_IN_AAAA
U15_AAAA(x1, x2, x3, x4, x5, x6)  =  U15_AAAA(x6)
U34_AA(x1, x2, x3, x4, x5, x6)  =  U34_AA(x6)
U35_AA(x1, x2, x3, x4, x5)  =  U35_AA(x5)
P78_IN_AAAAA(x1, x2, x3, x4, x5)  =  P78_IN_AAAAA
U16_AAAAA(x1, x2, x3, x4, x5, x6)  =  U16_AAAAA(x6)
U17_AAAAA(x1, x2, x3, x4, x5, x6)  =  U17_AAAAA(x6)
U18_AAAAA(x1, x2, x3, x4, x5, x6)  =  U18_AAAAA(x6)
APPEND23_IN_AAAA(x1, x2, x3, x4)  =  APPEND23_IN_AAAA
U1_AAAA(x1, x2, x3, x4, x5, x6)  =  U1_AAAA(x6)
U28_AA(x1, x2, x3, x4, x5, x6)  =  U28_AA(x6)
U29_AA(x1, x2, x3, x4, x5)  =  U29_AA(x5)
U28_GA(x1, x2, x3, x4, x5, x6)  =  U28_GA(x1, x6)
U29_GA(x1, x2, x3, x4, x5)  =  U29_GA(x5)
P78_IN_AAAGA(x1, x2, x3, x4, x5)  =  P78_IN_AAAGA(x4)
U16_AAAGA(x1, x2, x3, x4, x5, x6)  =  U16_AAAGA(x6)
U17_AAAGA(x1, x2, x3, x4, x5, x6)  =  U17_AAAGA(x4, x6)
U18_AAAGA(x1, x2, x3, x4, x5, x6)  =  U18_AAAGA(x6)
APPEND23_IN_AGAA(x1, x2, x3, x4)  =  APPEND23_IN_AGAA(x2)
U1_AGAA(x1, x2, x3, x4, x5, x6)  =  U1_AGAA(x6)
U30_GA(x1, x2, x3, x4, x5)  =  U30_GA(x1, x3, x5)
U31_GA(x1, x2, x3, x4, x5)  =  U31_GA(x5)
U32_GA(x1, x2, x3, x4, x5)  =  U32_GA(x1, x5)
U33_GA(x1, x2, x3, x4, x5)  =  U33_GA(x5)
U34_GA(x1, x2, x3, x4, x5, x6)  =  U34_GA(x1, x6)
U35_GA(x1, x2, x3, x4, x5)  =  U35_GA(x5)

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

(115) Obligation:

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

QS1_IN_GA(cons(T14, []), T9) → U19_GA(T14, T9, qs14_in_a(X12))
QS1_IN_GA(cons(T14, []), T9) → QS14_IN_A(X12)
QS1_IN_GA(cons(T14, []), T9) → U20_GA(T14, T9, qs14_in_a(T17))
U20_GA(T14, T9, qs14_out_a(T17)) → U21_GA(T14, T9, qs14_in_a(X13))
U20_GA(T14, T9, qs14_out_a(T17)) → QS14_IN_A(X13)
U20_GA(T14, T9, qs14_out_a(T17)) → U22_GA(T14, T9, T17, qs14_in_a(T20))
U22_GA(T14, T9, T17, qs14_out_a(T20)) → U23_GA(T14, T9, append23_in_ggga(T17, T14, T20, T9))
U22_GA(T14, T9, T17, qs14_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)
U24_GA(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_GA(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
U24_GA(T69, T67, T68, T9, less36_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)
U3_GGAA(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_GGAA(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
U3_GGAA(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → SPLIT46_IN_GGAA(T111, T112, X163, X164)
SPLIT46_IN_GGAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_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)) → U7_GG(T146, T147, geq62_in_gg(T146, T147))
GEQ62_IN_GG(s(T146), s(T147)) → GEQ62_IN_GG(T146, T147)
U5_GGAA(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_GGAA(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U5_GGAA(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → SPLIT46_IN_GGAA(T126, T127, X195, X196)
U24_GA(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_GA(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_GA(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_GA(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
U26_GA(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → QS1_IN_AA(cons(T67, T89), X12)
QS1_IN_AA(cons(T14, []), T9) → U19_AA(T14, T9, qs14_in_a(X12))
QS1_IN_AA(cons(T14, []), T9) → QS14_IN_A(X12)
QS1_IN_AA(cons(T14, []), T9) → U20_AA(T14, T9, qs14_in_a(T17))
U20_AA(T14, T9, qs14_out_a(T17)) → U21_AA(T14, T9, qs14_in_a(X13))
U20_AA(T14, T9, qs14_out_a(T17)) → QS14_IN_A(X13)
U20_AA(T14, T9, qs14_out_a(T17)) → U22_AA(T14, T9, T17, qs14_in_a(T20))
U22_AA(T14, T9, T17, qs14_out_a(T20)) → U23_AA(T14, T9, append23_in_gaga(T17, T14, T20, T9))
U22_AA(T14, T9, T17, qs14_out_a(T20)) → APPEND23_IN_GAGA(T17, T14, T20, T9)
APPEND23_IN_GAGA(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_GAGA(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
APPEND23_IN_GAGA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)
QS1_IN_AA(cons(T69, cons(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, less36_in_aa(T67, T69))
QS1_IN_AA(cons(T69, cons(T67, T68)), T9) → LESS36_IN_AA(T67, T69)
LESS36_IN_AA(s(T83), s(T84)) → U2_AA(T83, T84, less36_in_aa(T83, T84))
LESS36_IN_AA(s(T83), s(T84)) → LESS36_IN_AA(T83, T84)
U24_AA(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_AA(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
U24_AA(T69, T67, T68, T9, less36_out_aa(T67, T69)) → SPLIT46_IN_AAAA(T68, T69, X93, X94)
SPLIT46_IN_AAAA(cons(T110, T111), T112, cons(T110, X163), X164) → U3_AAAA(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
SPLIT46_IN_AAAA(cons(T110, T111), T112, cons(T110, X163), X164) → LESS36_IN_AA(T110, T112)
U3_AAAA(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_AAAA(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
U3_AAAA(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → SPLIT46_IN_AAAA(T111, T112, X163, X164)
SPLIT46_IN_AAAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_AAAA(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
SPLIT46_IN_AAAA(cons(T125, T126), T127, X195, cons(T125, X196)) → GEQ62_IN_AA(T125, T127)
GEQ62_IN_AA(s(T146), s(T147)) → U7_AA(T146, T147, geq62_in_aa(T146, T147))
GEQ62_IN_AA(s(T146), s(T147)) → GEQ62_IN_AA(T146, T147)
U5_AAAA(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_AAAA(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U5_AAAA(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → SPLIT46_IN_AAAA(T126, T127, X195, X196)
U24_AA(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_AA(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_AA(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
U26_AA(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → QS1_IN_AA(cons(T67, T89), X12)
QS1_IN_AA(cons(T214, cons(T212, T213)), T9) → U30_AA(T214, T212, T213, T9, geq62_in_aa(T212, T214))
QS1_IN_AA(cons(T214, cons(T212, T213)), T9) → GEQ62_IN_AA(T212, T214)
U30_AA(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_AA(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U30_AA(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → SPLIT46_IN_AAAA(T213, T214, X323, X324)
U30_AA(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_AA(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_AA(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_AA(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U32_AA(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → QS79_IN_AA(T221, X12)
QS79_IN_AA(cons(T162, T163), X265) → U8_AA(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
QS79_IN_AA(cons(T162, T163), X265) → SPLIT46_IN_AAAA(T163, T162, X261, X262)
QS79_IN_AA(cons(T162, T163), X265) → U9_AA(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_AA(T162, T163, X265, qs79_in_aa(T166, X263))
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → QS79_IN_AA(T166, X263)
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_AA(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_AA(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_AA(T162, T163, X265, qs79_in_aa(T167, X264))
U11_AA(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → QS79_IN_AA(T167, X264)
U11_AA(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_AA(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_AA(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_AA(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
U13_AA(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → APPEND94_IN_AAAA(T170, T162, T171, X265)
APPEND94_IN_AAAA(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_AAAA(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
APPEND94_IN_AAAA(cons(T194, T195), T196, T197, cons(T194, X295)) → APPEND94_IN_AAAA(T195, T196, T197, X295)
U32_AA(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_AA(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_AA(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_AA(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
U34_AA(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → P78_IN_AAAAA(cons(T212, T222), X13, T225, T214, T9)
P78_IN_AAAAA(T90, X13, T154, T69, T9) → U16_AAAAA(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
P78_IN_AAAAA(T90, X13, T154, T69, T9) → QS79_IN_AA(T90, X13)
P78_IN_AAAAA(T90, T157, T154, T69, T9) → U17_AAAAA(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_AAAAA(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_AAAAA(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
U17_AAAAA(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → APPEND23_IN_AAAA(T154, T69, T157, T9)
APPEND23_IN_AAAA(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_AAAA(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
APPEND23_IN_AAAA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)
U26_AA(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_AA(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_AA(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_AA(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U28_AA(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → P78_IN_AAAAA(T90, X13, T154, T69, T9)
U26_GA(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_GA(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_GA(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_GA(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
U28_GA(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → P78_IN_AAAGA(T90, X13, T154, T69, T9)
P78_IN_AAAGA(T90, X13, T154, T69, T9) → U16_AAAGA(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
P78_IN_AAAGA(T90, X13, T154, T69, T9) → QS79_IN_AA(T90, X13)
P78_IN_AAAGA(T90, T157, T154, T69, T9) → U17_AAAGA(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_AAAGA(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_AAAGA(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
U17_AAAGA(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → APPEND23_IN_AGAA(T154, T69, T157, T9)
APPEND23_IN_AGAA(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_AGAA(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
APPEND23_IN_AGAA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)
QS1_IN_GA(cons(T214, cons(T212, T213)), T9) → U30_GA(T214, T212, T213, T9, geq62_in_gg(T212, T214))
QS1_IN_GA(cons(T214, cons(T212, T213)), T9) → GEQ62_IN_GG(T212, T214)
U30_GA(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_GA(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U30_GA(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → SPLIT46_IN_GGAA(T213, T214, X323, X324)
U30_GA(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_GA(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_GA(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_GA(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U32_GA(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → QS79_IN_AA(T221, X12)
U32_GA(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_GA(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_GA(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_GA(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U34_GA(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → P78_IN_AAAGA(cons(T212, T222), X13, T225, T214, T9)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
QS1_IN_GA(x1, x2)  =  QS1_IN_GA(x1)
U19_GA(x1, x2, x3)  =  U19_GA(x3)
QS14_IN_A(x1)  =  QS14_IN_A
U20_GA(x1, x2, x3)  =  U20_GA(x1, x3)
U21_GA(x1, x2, x3)  =  U21_GA(x3)
U22_GA(x1, x2, x3, x4)  =  U22_GA(x1, x3, x4)
U23_GA(x1, x2, x3)  =  U23_GA(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, x6)
U24_GA(x1, x2, x3, x4, x5)  =  U24_GA(x1, x3, x5)
LESS36_IN_GG(x1, x2)  =  LESS36_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x3)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x5)
SPLIT46_IN_GGAA(x1, x2, x3, x4)  =  SPLIT46_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x2, x3, x6)
U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x6)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x2, x3, x6)
GEQ62_IN_GG(x1, x2)  =  GEQ62_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x3)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x6)
U26_GA(x1, x2, x3, x4, x5)  =  U26_GA(x1, x5)
U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x5)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U19_AA(x1, x2, x3)  =  U19_AA(x3)
U20_AA(x1, x2, x3)  =  U20_AA(x3)
U21_AA(x1, x2, x3)  =  U21_AA(x3)
U22_AA(x1, x2, x3, x4)  =  U22_AA(x3, x4)
U23_AA(x1, x2, x3)  =  U23_AA(x3)
APPEND23_IN_GAGA(x1, x2, x3, x4)  =  APPEND23_IN_GAGA(x1, x3)
U1_GAGA(x1, x2, x3, x4, x5, x6)  =  U1_GAGA(x6)
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
LESS36_IN_AA(x1, x2)  =  LESS36_IN_AA
U2_AA(x1, x2, x3)  =  U2_AA(x3)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
SPLIT46_IN_AAAA(x1, x2, x3, x4)  =  SPLIT46_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)
U4_AAAA(x1, x2, x3, x4, x5, x6)  =  U4_AAAA(x6)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)
GEQ62_IN_AA(x1, x2)  =  GEQ62_IN_AA
U7_AA(x1, x2, x3)  =  U7_AA(x3)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x6)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)
U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x5)
U30_AA(x1, x2, x3, x4, x5)  =  U30_AA(x5)
U31_AA(x1, x2, x3, x4, x5)  =  U31_AA(x5)
U32_AA(x1, x2, x3, x4, x5)  =  U32_AA(x5)
U33_AA(x1, x2, x3, x4, x5)  =  U33_AA(x5)
QS79_IN_AA(x1, x2)  =  QS79_IN_AA
U8_AA(x1, x2, x3, x4)  =  U8_AA(x4)
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U10_AA(x1, x2, x3, x4)  =  U10_AA(x4)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)
U12_AA(x1, x2, x3, x4)  =  U12_AA(x4)
U13_AA(x1, x2, x3, x4, x5)  =  U13_AA(x5)
U14_AA(x1, x2, x3, x4)  =  U14_AA(x4)
APPEND94_IN_AAAA(x1, x2, x3, x4)  =  APPEND94_IN_AAAA
U15_AAAA(x1, x2, x3, x4, x5, x6)  =  U15_AAAA(x6)
U34_AA(x1, x2, x3, x4, x5, x6)  =  U34_AA(x6)
U35_AA(x1, x2, x3, x4, x5)  =  U35_AA(x5)
P78_IN_AAAAA(x1, x2, x3, x4, x5)  =  P78_IN_AAAAA
U16_AAAAA(x1, x2, x3, x4, x5, x6)  =  U16_AAAAA(x6)
U17_AAAAA(x1, x2, x3, x4, x5, x6)  =  U17_AAAAA(x6)
U18_AAAAA(x1, x2, x3, x4, x5, x6)  =  U18_AAAAA(x6)
APPEND23_IN_AAAA(x1, x2, x3, x4)  =  APPEND23_IN_AAAA
U1_AAAA(x1, x2, x3, x4, x5, x6)  =  U1_AAAA(x6)
U28_AA(x1, x2, x3, x4, x5, x6)  =  U28_AA(x6)
U29_AA(x1, x2, x3, x4, x5)  =  U29_AA(x5)
U28_GA(x1, x2, x3, x4, x5, x6)  =  U28_GA(x1, x6)
U29_GA(x1, x2, x3, x4, x5)  =  U29_GA(x5)
P78_IN_AAAGA(x1, x2, x3, x4, x5)  =  P78_IN_AAAGA(x4)
U16_AAAGA(x1, x2, x3, x4, x5, x6)  =  U16_AAAGA(x6)
U17_AAAGA(x1, x2, x3, x4, x5, x6)  =  U17_AAAGA(x4, x6)
U18_AAAGA(x1, x2, x3, x4, x5, x6)  =  U18_AAAGA(x6)
APPEND23_IN_AGAA(x1, x2, x3, x4)  =  APPEND23_IN_AGAA(x2)
U1_AGAA(x1, x2, x3, x4, x5, x6)  =  U1_AGAA(x6)
U30_GA(x1, x2, x3, x4, x5)  =  U30_GA(x1, x3, x5)
U31_GA(x1, x2, x3, x4, x5)  =  U31_GA(x5)
U32_GA(x1, x2, x3, x4, x5)  =  U32_GA(x1, x5)
U33_GA(x1, x2, x3, x4, x5)  =  U33_GA(x5)
U34_GA(x1, x2, x3, x4, x5, x6)  =  U34_GA(x1, x6)
U35_GA(x1, x2, x3, x4, x5)  =  U35_GA(x5)

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

(116) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 13 SCCs with 87 less nodes.

(117) Complex Obligation (AND)

(118) Obligation:

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

APPEND23_IN_AGAA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
APPEND23_IN_AGAA(x1, x2, x3, x4)  =  APPEND23_IN_AGAA(x2)

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

(119) UsableRulesProof (EQUIVALENT transformation)

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

(120) Obligation:

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

APPEND23_IN_AGAA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
APPEND23_IN_AGAA(x1, x2, x3, x4)  =  APPEND23_IN_AGAA(x2)

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

(121) PiDPToQDPProof (SOUND transformation)

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

(122) Obligation:

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

APPEND23_IN_AGAA(T47) → APPEND23_IN_AGAA(T47)

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

(123) NonTerminationProof (EQUIVALENT transformation)

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

s = APPEND23_IN_AGAA(T47) evaluates to t =APPEND23_IN_AGAA(T47)

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND23_IN_AGAA(T47) to APPEND23_IN_AGAA(T47).



(124) NO

(125) Obligation:

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

APPEND23_IN_AAAA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
APPEND23_IN_AAAA(x1, x2, x3, x4)  =  APPEND23_IN_AAAA

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

(126) UsableRulesProof (EQUIVALENT transformation)

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

(127) Obligation:

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

APPEND23_IN_AAAA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
APPEND23_IN_AAAA(x1, x2, x3, x4)  =  APPEND23_IN_AAAA

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

(128) PiDPToQDPProof (SOUND transformation)

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

(129) Obligation:

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

APPEND23_IN_AAAAAPPEND23_IN_AAAA

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

(130) NonTerminationProof (EQUIVALENT transformation)

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

s = APPEND23_IN_AAAA evaluates to t =APPEND23_IN_AAAA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND23_IN_AAAA to APPEND23_IN_AAAA.



(131) NO

(132) Obligation:

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

APPEND94_IN_AAAA(cons(T194, T195), T196, T197, cons(T194, X295)) → APPEND94_IN_AAAA(T195, T196, T197, X295)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
APPEND94_IN_AAAA(x1, x2, x3, x4)  =  APPEND94_IN_AAAA

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

(133) UsableRulesProof (EQUIVALENT transformation)

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

(134) Obligation:

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

APPEND94_IN_AAAA(cons(T194, T195), T196, T197, cons(T194, X295)) → APPEND94_IN_AAAA(T195, T196, T197, X295)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
APPEND94_IN_AAAA(x1, x2, x3, x4)  =  APPEND94_IN_AAAA

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

(135) PiDPToQDPProof (SOUND transformation)

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

(136) Obligation:

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

APPEND94_IN_AAAAAPPEND94_IN_AAAA

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

(137) NonTerminationProof (EQUIVALENT transformation)

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

s = APPEND94_IN_AAAA evaluates to t =APPEND94_IN_AAAA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND94_IN_AAAA to APPEND94_IN_AAAA.



(138) NO

(139) Obligation:

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

GEQ62_IN_AA(s(T146), s(T147)) → GEQ62_IN_AA(T146, T147)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
GEQ62_IN_AA(x1, x2)  =  GEQ62_IN_AA

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

(140) UsableRulesProof (EQUIVALENT transformation)

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

(141) Obligation:

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

GEQ62_IN_AA(s(T146), s(T147)) → GEQ62_IN_AA(T146, T147)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GEQ62_IN_AA(x1, x2)  =  GEQ62_IN_AA

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

(142) PiDPToQDPProof (SOUND transformation)

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

(143) Obligation:

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

GEQ62_IN_AAGEQ62_IN_AA

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

(144) NonTerminationProof (EQUIVALENT transformation)

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

s = GEQ62_IN_AA evaluates to t =GEQ62_IN_AA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from GEQ62_IN_AA to GEQ62_IN_AA.



(145) NO

(146) Obligation:

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

LESS36_IN_AA(s(T83), s(T84)) → LESS36_IN_AA(T83, T84)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
LESS36_IN_AA(x1, x2)  =  LESS36_IN_AA

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

(147) UsableRulesProof (EQUIVALENT transformation)

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

(148) Obligation:

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

LESS36_IN_AA(s(T83), s(T84)) → LESS36_IN_AA(T83, T84)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS36_IN_AA(x1, x2)  =  LESS36_IN_AA

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

(149) PiDPToQDPProof (SOUND transformation)

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

(150) Obligation:

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

LESS36_IN_AALESS36_IN_AA

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

(151) NonTerminationProof (EQUIVALENT transformation)

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

s = LESS36_IN_AA evaluates to t =LESS36_IN_AA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LESS36_IN_AA to LESS36_IN_AA.



(152) NO

(153) Obligation:

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

SPLIT46_IN_AAAA(cons(T110, T111), T112, cons(T110, X163), X164) → U3_AAAA(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_AAAA(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → SPLIT46_IN_AAAA(T111, T112, X163, X164)
SPLIT46_IN_AAAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_AAAA(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
U5_AAAA(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → SPLIT46_IN_AAAA(T126, T127, X195, X196)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
SPLIT46_IN_AAAA(x1, x2, x3, x4)  =  SPLIT46_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)

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

(154) UsableRulesProof (EQUIVALENT transformation)

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

(155) Obligation:

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

SPLIT46_IN_AAAA(cons(T110, T111), T112, cons(T110, X163), X164) → U3_AAAA(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_AAAA(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → SPLIT46_IN_AAAA(T111, T112, X163, X164)
SPLIT46_IN_AAAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_AAAA(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
U5_AAAA(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → SPLIT46_IN_AAAA(T126, T127, X195, X196)

The TRS R consists of the following rules:

less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
SPLIT46_IN_AAAA(x1, x2, x3, x4)  =  SPLIT46_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)
U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)

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

(156) PiDPToQDPProof (SOUND transformation)

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

(157) Obligation:

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

SPLIT46_IN_AAAAU3_AAAA(less36_in_aa)
U3_AAAA(less36_out_aa(T110)) → SPLIT46_IN_AAAA
SPLIT46_IN_AAAAU5_AAAA(geq62_in_aa)
U5_AAAA(geq62_out_aa) → SPLIT46_IN_AAAA

The TRS R consists of the following rules:

less36_in_aaless36_out_aa(0)
less36_in_aaU2_aa(less36_in_aa)
geq62_in_aageq62_out_aa
geq62_in_aaU7_aa(geq62_in_aa)
U2_aa(less36_out_aa(T83)) → less36_out_aa(s(T83))
U7_aa(geq62_out_aa) → geq62_out_aa

The set Q consists of the following terms:

less36_in_aa
geq62_in_aa
U2_aa(x0)
U7_aa(x0)

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

(158) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SPLIT46_IN_AAAAU3_AAAA(less36_in_aa) at position [0] we obtained the following new rules [LPAR04]:

SPLIT46_IN_AAAAU3_AAAA(less36_out_aa(0))
SPLIT46_IN_AAAAU3_AAAA(U2_aa(less36_in_aa))

(159) Obligation:

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

U3_AAAA(less36_out_aa(T110)) → SPLIT46_IN_AAAA
SPLIT46_IN_AAAAU5_AAAA(geq62_in_aa)
U5_AAAA(geq62_out_aa) → SPLIT46_IN_AAAA
SPLIT46_IN_AAAAU3_AAAA(less36_out_aa(0))
SPLIT46_IN_AAAAU3_AAAA(U2_aa(less36_in_aa))

The TRS R consists of the following rules:

less36_in_aaless36_out_aa(0)
less36_in_aaU2_aa(less36_in_aa)
geq62_in_aageq62_out_aa
geq62_in_aaU7_aa(geq62_in_aa)
U2_aa(less36_out_aa(T83)) → less36_out_aa(s(T83))
U7_aa(geq62_out_aa) → geq62_out_aa

The set Q consists of the following terms:

less36_in_aa
geq62_in_aa
U2_aa(x0)
U7_aa(x0)

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

(160) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SPLIT46_IN_AAAAU5_AAAA(geq62_in_aa) at position [0] we obtained the following new rules [LPAR04]:

SPLIT46_IN_AAAAU5_AAAA(geq62_out_aa)
SPLIT46_IN_AAAAU5_AAAA(U7_aa(geq62_in_aa))

(161) Obligation:

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

U3_AAAA(less36_out_aa(T110)) → SPLIT46_IN_AAAA
U5_AAAA(geq62_out_aa) → SPLIT46_IN_AAAA
SPLIT46_IN_AAAAU3_AAAA(less36_out_aa(0))
SPLIT46_IN_AAAAU3_AAAA(U2_aa(less36_in_aa))
SPLIT46_IN_AAAAU5_AAAA(geq62_out_aa)
SPLIT46_IN_AAAAU5_AAAA(U7_aa(geq62_in_aa))

The TRS R consists of the following rules:

less36_in_aaless36_out_aa(0)
less36_in_aaU2_aa(less36_in_aa)
geq62_in_aageq62_out_aa
geq62_in_aaU7_aa(geq62_in_aa)
U2_aa(less36_out_aa(T83)) → less36_out_aa(s(T83))
U7_aa(geq62_out_aa) → geq62_out_aa

The set Q consists of the following terms:

less36_in_aa
geq62_in_aa
U2_aa(x0)
U7_aa(x0)

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

(162) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = SPLIT46_IN_AAAA evaluates to t =SPLIT46_IN_AAAA

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




Rewriting sequence

SPLIT46_IN_AAAAU3_AAAA(less36_out_aa(0))
with rule SPLIT46_IN_AAAAU3_AAAA(less36_out_aa(0)) at position [] and matcher [ ]

U3_AAAA(less36_out_aa(0))SPLIT46_IN_AAAA
with rule U3_AAAA(less36_out_aa(T110)) → SPLIT46_IN_AAAA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.



(163) NO

(164) Obligation:

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

QS79_IN_AA(cons(T162, T163), X265) → U9_AA(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → QS79_IN_AA(T166, X263)
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_AA(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_AA(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → QS79_IN_AA(T167, X264)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
QS79_IN_AA(x1, x2)  =  QS79_IN_AA
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)

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

(165) UsableRulesProof (EQUIVALENT transformation)

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

(166) Obligation:

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

QS79_IN_AA(cons(T162, T163), X265) → U9_AA(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → QS79_IN_AA(T166, X263)
U9_AA(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_AA(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_AA(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → QS79_IN_AA(T167, X264)

The TRS R consists of the following rules:

split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))

The argument filtering Pi contains the following mapping:
[]  =  []
cons(x1, x2)  =  cons(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
QS79_IN_AA(x1, x2)  =  QS79_IN_AA
U9_AA(x1, x2, x3, x4)  =  U9_AA(x4)
U11_AA(x1, x2, x3, x4, x5)  =  U11_AA(x5)

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

(167) PiDPToQDPProof (SOUND transformation)

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

(168) Obligation:

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

QS79_IN_AAU9_AA(split46_in_aaaa)
U9_AA(split46_out_aaaa) → QS79_IN_AA
U9_AA(split46_out_aaaa) → U11_AA(qs79_in_aa)
U11_AA(qs79_out_aa) → QS79_IN_AA

The TRS R consists of the following rules:

split46_in_aaaasplit46_out_aaaa
split46_in_aaaaU3_aaaa(less36_in_aa)
split46_in_aaaaU5_aaaa(geq62_in_aa)
qs79_in_aaqs79_out_aa
qs79_in_aaU8_aa(split46_in_aaaa)
qs79_in_aaU9_aa(split46_in_aaaa)
U3_aaaa(less36_out_aa(T110)) → split46_out_aaaa
U3_aaaa(less36_out_aa(T110)) → U4_aaaa(split46_in_aaaa)
U5_aaaa(geq62_out_aa) → split46_out_aaaa
U5_aaaa(geq62_out_aa) → U6_aaaa(split46_in_aaaa)
U8_aa(split46_out_aaaa) → qs79_out_aa
U9_aa(split46_out_aaaa) → U10_aa(qs79_in_aa)
U9_aa(split46_out_aaaa) → U11_aa(qs79_in_aa)
less36_in_aaless36_out_aa(0)
less36_in_aaU2_aa(less36_in_aa)
U4_aaaa(split46_out_aaaa) → split46_out_aaaa
geq62_in_aageq62_out_aa
geq62_in_aaU7_aa(geq62_in_aa)
U6_aaaa(split46_out_aaaa) → split46_out_aaaa
U10_aa(qs79_out_aa) → qs79_out_aa
U11_aa(qs79_out_aa) → U12_aa(qs79_in_aa)
U11_aa(qs79_out_aa) → U13_aa(qs79_in_aa)
U2_aa(less36_out_aa(T83)) → less36_out_aa(s(T83))
U7_aa(geq62_out_aa) → geq62_out_aa
U12_aa(qs79_out_aa) → qs79_out_aa
U13_aa(qs79_out_aa) → U14_aa(append94_in_aaaa)
U14_aa(append94_out_aaaa) → qs79_out_aa
append94_in_aaaaappend94_out_aaaa
append94_in_aaaaU15_aaaa(append94_in_aaaa)
U15_aaaa(append94_out_aaaa) → append94_out_aaaa

The set Q consists of the following terms:

split46_in_aaaa
qs79_in_aa
U3_aaaa(x0)
U5_aaaa(x0)
U8_aa(x0)
U9_aa(x0)
less36_in_aa
U4_aaaa(x0)
geq62_in_aa
U6_aaaa(x0)
U10_aa(x0)
U11_aa(x0)
U2_aa(x0)
U7_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
append94_in_aaaa
U15_aaaa(x0)

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

(169) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS79_IN_AAU9_AA(split46_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

QS79_IN_AAU9_AA(split46_out_aaaa)
QS79_IN_AAU9_AA(U3_aaaa(less36_in_aa))
QS79_IN_AAU9_AA(U5_aaaa(geq62_in_aa))

(170) Obligation:

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

U9_AA(split46_out_aaaa) → QS79_IN_AA
U9_AA(split46_out_aaaa) → U11_AA(qs79_in_aa)
U11_AA(qs79_out_aa) → QS79_IN_AA
QS79_IN_AAU9_AA(split46_out_aaaa)
QS79_IN_AAU9_AA(U3_aaaa(less36_in_aa))
QS79_IN_AAU9_AA(U5_aaaa(geq62_in_aa))

The TRS R consists of the following rules:

split46_in_aaaasplit46_out_aaaa
split46_in_aaaaU3_aaaa(less36_in_aa)
split46_in_aaaaU5_aaaa(geq62_in_aa)
qs79_in_aaqs79_out_aa
qs79_in_aaU8_aa(split46_in_aaaa)
qs79_in_aaU9_aa(split46_in_aaaa)
U3_aaaa(less36_out_aa(T110)) → split46_out_aaaa
U3_aaaa(less36_out_aa(T110)) → U4_aaaa(split46_in_aaaa)
U5_aaaa(geq62_out_aa) → split46_out_aaaa
U5_aaaa(geq62_out_aa) → U6_aaaa(split46_in_aaaa)
U8_aa(split46_out_aaaa) → qs79_out_aa
U9_aa(split46_out_aaaa) → U10_aa(qs79_in_aa)
U9_aa(split46_out_aaaa) → U11_aa(qs79_in_aa)
less36_in_aaless36_out_aa(0)
less36_in_aaU2_aa(less36_in_aa)
U4_aaaa(split46_out_aaaa) → split46_out_aaaa
geq62_in_aageq62_out_aa
geq62_in_aaU7_aa(geq62_in_aa)
U6_aaaa(split46_out_aaaa) → split46_out_aaaa
U10_aa(qs79_out_aa) → qs79_out_aa
U11_aa(qs79_out_aa) → U12_aa(qs79_in_aa)
U11_aa(qs79_out_aa) → U13_aa(qs79_in_aa)
U2_aa(less36_out_aa(T83)) → less36_out_aa(s(T83))
U7_aa(geq62_out_aa) → geq62_out_aa
U12_aa(qs79_out_aa) → qs79_out_aa
U13_aa(qs79_out_aa) → U14_aa(append94_in_aaaa)
U14_aa(append94_out_aaaa) → qs79_out_aa
append94_in_aaaaappend94_out_aaaa
append94_in_aaaaU15_aaaa(append94_in_aaaa)
U15_aaaa(append94_out_aaaa) → append94_out_aaaa

The set Q consists of the following terms:

split46_in_aaaa
qs79_in_aa
U3_aaaa(x0)
U5_aaaa(x0)
U8_aa(x0)
U9_aa(x0)
less36_in_aa
U4_aaaa(x0)
geq62_in_aa
U6_aaaa(x0)
U10_aa(x0)
U11_aa(x0)
U2_aa(x0)
U7_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
append94_in_aaaa
U15_aaaa(x0)

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

(171) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U9_AA(split46_out_aaaa) → U11_AA(qs79_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U9_AA(split46_out_aaaa) → U11_AA(qs79_out_aa)
U9_AA(split46_out_aaaa) → U11_AA(U8_aa(split46_in_aaaa))
U9_AA(split46_out_aaaa) → U11_AA(U9_aa(split46_in_aaaa))

(172) Obligation:

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

U9_AA(split46_out_aaaa) → QS79_IN_AA
U11_AA(qs79_out_aa) → QS79_IN_AA
QS79_IN_AAU9_AA(split46_out_aaaa)
QS79_IN_AAU9_AA(U3_aaaa(less36_in_aa))
QS79_IN_AAU9_AA(U5_aaaa(geq62_in_aa))
U9_AA(split46_out_aaaa) → U11_AA(qs79_out_aa)
U9_AA(split46_out_aaaa) → U11_AA(U8_aa(split46_in_aaaa))
U9_AA(split46_out_aaaa) → U11_AA(U9_aa(split46_in_aaaa))

The TRS R consists of the following rules:

split46_in_aaaasplit46_out_aaaa
split46_in_aaaaU3_aaaa(less36_in_aa)
split46_in_aaaaU5_aaaa(geq62_in_aa)
qs79_in_aaqs79_out_aa
qs79_in_aaU8_aa(split46_in_aaaa)
qs79_in_aaU9_aa(split46_in_aaaa)
U3_aaaa(less36_out_aa(T110)) → split46_out_aaaa
U3_aaaa(less36_out_aa(T110)) → U4_aaaa(split46_in_aaaa)
U5_aaaa(geq62_out_aa) → split46_out_aaaa
U5_aaaa(geq62_out_aa) → U6_aaaa(split46_in_aaaa)
U8_aa(split46_out_aaaa) → qs79_out_aa
U9_aa(split46_out_aaaa) → U10_aa(qs79_in_aa)
U9_aa(split46_out_aaaa) → U11_aa(qs79_in_aa)
less36_in_aaless36_out_aa(0)
less36_in_aaU2_aa(less36_in_aa)
U4_aaaa(split46_out_aaaa) → split46_out_aaaa
geq62_in_aageq62_out_aa
geq62_in_aaU7_aa(geq62_in_aa)
U6_aaaa(split46_out_aaaa) → split46_out_aaaa
U10_aa(qs79_out_aa) → qs79_out_aa
U11_aa(qs79_out_aa) → U12_aa(qs79_in_aa)
U11_aa(qs79_out_aa) → U13_aa(qs79_in_aa)
U2_aa(less36_out_aa(T83)) → less36_out_aa(s(T83))
U7_aa(geq62_out_aa) → geq62_out_aa
U12_aa(qs79_out_aa) → qs79_out_aa
U13_aa(qs79_out_aa) → U14_aa(append94_in_aaaa)
U14_aa(append94_out_aaaa) → qs79_out_aa
append94_in_aaaaappend94_out_aaaa
append94_in_aaaaU15_aaaa(append94_in_aaaa)
U15_aaaa(append94_out_aaaa) → append94_out_aaaa

The set Q consists of the following terms:

split46_in_aaaa
qs79_in_aa
U3_aaaa(x0)
U5_aaaa(x0)
U8_aa(x0)
U9_aa(x0)
less36_in_aa
U4_aaaa(x0)
geq62_in_aa
U6_aaaa(x0)
U10_aa(x0)
U11_aa(x0)
U2_aa(x0)
U7_aa(x0)
U12_aa(x0)
U13_aa(x0)
U14_aa(x0)
append94_in_aaaa
U15_aaaa(x0)

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

(173) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = QS79_IN_AA evaluates to t =QS79_IN_AA

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




Rewriting sequence

QS79_IN_AAU9_AA(split46_out_aaaa)
with rule QS79_IN_AAU9_AA(split46_out_aaaa) at position [] and matcher [ ]

U9_AA(split46_out_aaaa)QS79_IN_AA
with rule U9_AA(split46_out_aaaa) → QS79_IN_AA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.



(174) NO

(175) Obligation:

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

APPEND23_IN_GAGA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
APPEND23_IN_GAGA(x1, x2, x3, x4)  =  APPEND23_IN_GAGA(x1, x3)

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

(176) UsableRulesProof (EQUIVALENT transformation)

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

(177) Obligation:

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

APPEND23_IN_GAGA(cons(T45, T46), T47, T48, cons(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
APPEND23_IN_GAGA(x1, x2, x3, x4)  =  APPEND23_IN_GAGA(x1, x3)

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

(178) PiDPToQDPProof (SOUND transformation)

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

(179) Obligation:

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

APPEND23_IN_GAGA(cons(T45, T46), T48) → APPEND23_IN_GAGA(T46, T48)

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

(180) 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_GAGA(cons(T45, T46), T48) → APPEND23_IN_GAGA(T46, T48)
    The graph contains the following edges 1 > 1, 2 >= 2

(181) YES

(182) Obligation:

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

QS1_IN_AA(cons(T69, cons(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, less36_in_aa(T67, T69))
U24_AA(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_AA(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → QS1_IN_AA(cons(T67, T89), X12)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)

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

(183) UsableRulesProof (EQUIVALENT transformation)

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

(184) Obligation:

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

QS1_IN_AA(cons(T69, cons(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, less36_in_aa(T67, T69))
U24_AA(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_AA(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → QS1_IN_AA(cons(T67, T89), X12)

The TRS R consists of the following rules:

less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))

The argument filtering Pi contains the following mapping:
[]  =  []
cons(x1, x2)  =  cons(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
QS1_IN_AA(x1, x2)  =  QS1_IN_AA
U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)
U26_AA(x1, x2, x3, x4, x5)  =  U26_AA(x5)

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

(185) PiDPToQDPProof (SOUND transformation)

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

(186) Obligation:

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

QS1_IN_AAU24_AA(less36_in_aa)
U24_AA(less36_out_aa(T67)) → U26_AA(split46_in_aaaa)
U26_AA(split46_out_aaaa) → QS1_IN_AA

The TRS R consists of the following rules:

less36_in_aaless36_out_aa(0)
less36_in_aaU2_aa(less36_in_aa)
split46_in_aaaasplit46_out_aaaa
split46_in_aaaaU3_aaaa(less36_in_aa)
split46_in_aaaaU5_aaaa(geq62_in_aa)
U2_aa(less36_out_aa(T83)) → less36_out_aa(s(T83))
U3_aaaa(less36_out_aa(T110)) → split46_out_aaaa
U3_aaaa(less36_out_aa(T110)) → U4_aaaa(split46_in_aaaa)
U5_aaaa(geq62_out_aa) → split46_out_aaaa
U5_aaaa(geq62_out_aa) → U6_aaaa(split46_in_aaaa)
U4_aaaa(split46_out_aaaa) → split46_out_aaaa
geq62_in_aageq62_out_aa
geq62_in_aaU7_aa(geq62_in_aa)
U6_aaaa(split46_out_aaaa) → split46_out_aaaa
U7_aa(geq62_out_aa) → geq62_out_aa

The set Q consists of the following terms:

less36_in_aa
split46_in_aaaa
U2_aa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U4_aaaa(x0)
geq62_in_aa
U6_aaaa(x0)
U7_aa(x0)

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

(187) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QS1_IN_AAU24_AA(less36_in_aa) at position [0] we obtained the following new rules [LPAR04]:

QS1_IN_AAU24_AA(less36_out_aa(0))
QS1_IN_AAU24_AA(U2_aa(less36_in_aa))

(188) Obligation:

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

U24_AA(less36_out_aa(T67)) → U26_AA(split46_in_aaaa)
U26_AA(split46_out_aaaa) → QS1_IN_AA
QS1_IN_AAU24_AA(less36_out_aa(0))
QS1_IN_AAU24_AA(U2_aa(less36_in_aa))

The TRS R consists of the following rules:

less36_in_aaless36_out_aa(0)
less36_in_aaU2_aa(less36_in_aa)
split46_in_aaaasplit46_out_aaaa
split46_in_aaaaU3_aaaa(less36_in_aa)
split46_in_aaaaU5_aaaa(geq62_in_aa)
U2_aa(less36_out_aa(T83)) → less36_out_aa(s(T83))
U3_aaaa(less36_out_aa(T110)) → split46_out_aaaa
U3_aaaa(less36_out_aa(T110)) → U4_aaaa(split46_in_aaaa)
U5_aaaa(geq62_out_aa) → split46_out_aaaa
U5_aaaa(geq62_out_aa) → U6_aaaa(split46_in_aaaa)
U4_aaaa(split46_out_aaaa) → split46_out_aaaa
geq62_in_aageq62_out_aa
geq62_in_aaU7_aa(geq62_in_aa)
U6_aaaa(split46_out_aaaa) → split46_out_aaaa
U7_aa(geq62_out_aa) → geq62_out_aa

The set Q consists of the following terms:

less36_in_aa
split46_in_aaaa
U2_aa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U4_aaaa(x0)
geq62_in_aa
U6_aaaa(x0)
U7_aa(x0)

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

(189) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_AA(less36_out_aa(T67)) → U26_AA(split46_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

U24_AA(less36_out_aa(y0)) → U26_AA(split46_out_aaaa)
U24_AA(less36_out_aa(y0)) → U26_AA(U3_aaaa(less36_in_aa))
U24_AA(less36_out_aa(y0)) → U26_AA(U5_aaaa(geq62_in_aa))

(190) Obligation:

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

U26_AA(split46_out_aaaa) → QS1_IN_AA
QS1_IN_AAU24_AA(less36_out_aa(0))
QS1_IN_AAU24_AA(U2_aa(less36_in_aa))
U24_AA(less36_out_aa(y0)) → U26_AA(split46_out_aaaa)
U24_AA(less36_out_aa(y0)) → U26_AA(U3_aaaa(less36_in_aa))
U24_AA(less36_out_aa(y0)) → U26_AA(U5_aaaa(geq62_in_aa))

The TRS R consists of the following rules:

less36_in_aaless36_out_aa(0)
less36_in_aaU2_aa(less36_in_aa)
split46_in_aaaasplit46_out_aaaa
split46_in_aaaaU3_aaaa(less36_in_aa)
split46_in_aaaaU5_aaaa(geq62_in_aa)
U2_aa(less36_out_aa(T83)) → less36_out_aa(s(T83))
U3_aaaa(less36_out_aa(T110)) → split46_out_aaaa
U3_aaaa(less36_out_aa(T110)) → U4_aaaa(split46_in_aaaa)
U5_aaaa(geq62_out_aa) → split46_out_aaaa
U5_aaaa(geq62_out_aa) → U6_aaaa(split46_in_aaaa)
U4_aaaa(split46_out_aaaa) → split46_out_aaaa
geq62_in_aageq62_out_aa
geq62_in_aaU7_aa(geq62_in_aa)
U6_aaaa(split46_out_aaaa) → split46_out_aaaa
U7_aa(geq62_out_aa) → geq62_out_aa

The set Q consists of the following terms:

less36_in_aa
split46_in_aaaa
U2_aa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U4_aaaa(x0)
geq62_in_aa
U6_aaaa(x0)
U7_aa(x0)

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

(191) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = QS1_IN_AA evaluates to t =QS1_IN_AA

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




Rewriting sequence

QS1_IN_AAU24_AA(less36_out_aa(0))
with rule QS1_IN_AAU24_AA(less36_out_aa(0)) at position [] and matcher [ ]

U24_AA(less36_out_aa(0))U26_AA(split46_out_aaaa)
with rule U24_AA(less36_out_aa(y0)) → U26_AA(split46_out_aaaa) at position [] and matcher [y0 / 0]

U26_AA(split46_out_aaaa)QS1_IN_AA
with rule U26_AA(split46_out_aaaa) → QS1_IN_AA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.



(192) NO

(193) 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:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
GEQ62_IN_GG(x1, x2)  =  GEQ62_IN_GG(x1, x2)

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

(194) UsableRulesProof (EQUIVALENT transformation)

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

(195) 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

(196) PiDPToQDPProof (EQUIVALENT transformation)

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

(197) 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.

(198) 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

(199) YES

(200) 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:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
LESS36_IN_GG(x1, x2)  =  LESS36_IN_GG(x1, x2)

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

(201) UsableRulesProof (EQUIVALENT transformation)

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

(202) 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

(203) PiDPToQDPProof (EQUIVALENT transformation)

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

(204) 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.

(205) 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

(206) YES

(207) Obligation:

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

SPLIT46_IN_GGAA(cons(T110, T111), T112, cons(T110, X163), X164) → U3_GGAA(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_GGAA(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → SPLIT46_IN_GGAA(T111, T112, X163, X164)
SPLIT46_IN_GGAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_GGAA(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
U5_GGAA(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → SPLIT46_IN_GGAA(T126, T127, X195, X196)

The TRS R consists of the following rules:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
SPLIT46_IN_GGAA(x1, x2, x3, x4)  =  SPLIT46_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x2, x3, x6)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x2, x3, x6)

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

(208) UsableRulesProof (EQUIVALENT transformation)

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

(209) Obligation:

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

SPLIT46_IN_GGAA(cons(T110, T111), T112, cons(T110, X163), X164) → U3_GGAA(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_GGAA(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → SPLIT46_IN_GGAA(T111, T112, X163, X164)
SPLIT46_IN_GGAA(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_GGAA(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
U5_GGAA(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → SPLIT46_IN_GGAA(T126, T127, X195, X196)

The TRS R consists of the following rules:

less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
SPLIT46_IN_GGAA(x1, x2, x3, x4)  =  SPLIT46_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x2, x3, x6)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x2, x3, x6)

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

(210) PiDPToQDPProof (SOUND transformation)

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

(211) Obligation:

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

SPLIT46_IN_GGAA(cons(T110, T111), T112) → U3_GGAA(T111, T112, less36_in_gg(T110, T112))
U3_GGAA(T111, T112, less36_out_gg) → SPLIT46_IN_GGAA(T111, T112)
SPLIT46_IN_GGAA(cons(T125, T126), T127) → U5_GGAA(T126, T127, geq62_in_gg(T125, T127))
U5_GGAA(T126, T127, geq62_out_gg) → SPLIT46_IN_GGAA(T126, T127)

The TRS R consists of the following rules:

less36_in_gg(0, s(T78)) → less36_out_gg
less36_in_gg(s(T83), s(T84)) → U2_gg(less36_in_gg(T83, T84))
geq62_in_gg(T136, T136) → geq62_out_gg
geq62_in_gg(s(T141), 0) → geq62_out_gg
geq62_in_gg(s(T146), s(T147)) → U7_gg(geq62_in_gg(T146, T147))
U2_gg(less36_out_gg) → less36_out_gg
U7_gg(geq62_out_gg) → geq62_out_gg

The set Q consists of the following terms:

less36_in_gg(x0, x1)
geq62_in_gg(x0, x1)
U2_gg(x0)
U7_gg(x0)

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

(212) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • U3_GGAA(T111, T112, less36_out_gg) → SPLIT46_IN_GGAA(T111, T112)
    The graph contains the following edges 1 >= 1, 2 >= 2

  • U5_GGAA(T126, T127, geq62_out_gg) → SPLIT46_IN_GGAA(T126, T127)
    The graph contains the following edges 1 >= 1, 2 >= 2

  • SPLIT46_IN_GGAA(cons(T110, T111), T112) → U3_GGAA(T111, T112, less36_in_gg(T110, T112))
    The graph contains the following edges 1 > 1, 2 >= 2

  • SPLIT46_IN_GGAA(cons(T125, T126), T127) → U5_GGAA(T126, T127, geq62_in_gg(T125, T127))
    The graph contains the following edges 1 > 1, 2 >= 2

(213) YES

(214) 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:

qs1_in_ga([], []) → qs1_out_ga([], [])
qs1_in_ga(cons(T14, []), T9) → U19_ga(T14, T9, qs14_in_a(X12))
qs14_in_a([]) → qs14_out_a([])
U19_ga(T14, T9, qs14_out_a(X12)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T14, []), T9) → U20_ga(T14, T9, qs14_in_a(T17))
U20_ga(T14, T9, qs14_out_a(T17)) → U21_ga(T14, T9, qs14_in_a(X13))
U21_ga(T14, T9, qs14_out_a(X13)) → qs1_out_ga(cons(T14, []), T9)
U20_ga(T14, T9, qs14_out_a(T17)) → U22_ga(T14, T9, T17, qs14_in_a(T20))
U22_ga(T14, T9, T17, qs14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, cons(T33, T34)) → append23_out_ggga([], T33, T34, cons(T33, T34))
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))
U1_ggga(T45, T46, T47, T48, T50, append23_out_ggga(T46, T47, T48, T50)) → append23_out_ggga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qs1_out_ga(cons(T14, []), T9)
qs1_in_ga(cons(T69, cons(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, less36_in_gg(T67, T69))
less36_in_gg(0, s(T78)) → less36_out_gg(0, s(T78))
less36_in_gg(s(T83), s(T84)) → U2_gg(T83, T84, less36_in_gg(T83, T84))
U2_gg(T83, T84, less36_out_gg(T83, T84)) → less36_out_gg(s(T83), s(T84))
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, X93, X94))
split46_in_ggaa([], T97, [], []) → split46_out_ggaa([], T97, [], [])
split46_in_ggaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_ggaa(T110, T111, T112, X163, X164, less36_in_gg(T110, T112))
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_ggaa(T110, T111, T112, X163, X164, less36_out_gg(T110, T112)) → U4_ggaa(T110, T111, T112, X163, X164, split46_in_ggaa(T111, T112, X163, X164))
split46_in_ggaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_ggaa(T125, T126, T127, X195, X196, geq62_in_gg(T125, T127))
geq62_in_gg(T136, T136) → geq62_out_gg(T136, T136)
geq62_in_gg(s(T141), 0) → geq62_out_gg(s(T141), 0)
geq62_in_gg(s(T146), s(T147)) → U7_gg(T146, T147, geq62_in_gg(T146, T147))
U7_gg(T146, T147, geq62_out_gg(T146, T147)) → geq62_out_gg(s(T146), s(T147))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_ggaa(T125, T126, T127, X195, X196, geq62_out_gg(T125, T127)) → U6_ggaa(T125, T126, T127, X195, X196, split46_in_ggaa(T126, T127, X195, X196))
U6_ggaa(T125, T126, T127, X195, X196, split46_out_ggaa(T126, T127, X195, X196)) → split46_out_ggaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_ggaa(T110, T111, T112, X163, X164, split46_out_ggaa(T111, T112, X163, X164)) → split46_out_ggaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, X93, X94)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, less36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split46_in_ggaa(T68, T69, T89, T90))
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U27_ga(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa([], []) → qs1_out_aa([], [])
qs1_in_aa(cons(T14, []), T9) → U19_aa(T14, T9, qs14_in_a(X12))
U19_aa(T14, T9, qs14_out_a(X12)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T14, []), T9) → U20_aa(T14, T9, qs14_in_a(T17))
U20_aa(T14, T9, qs14_out_a(T17)) → U21_aa(T14, T9, qs14_in_a(X13))
U21_aa(T14, T9, qs14_out_a(X13)) → qs1_out_aa(cons(T14, []), T9)
U20_aa(T14, T9, qs14_out_a(T17)) → U22_aa(T14, T9, T17, qs14_in_a(T20))
U22_aa(T14, T9, T17, qs14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, cons(T33, T34)) → append23_out_gaga([], T33, T34, cons(T33, T34))
append23_in_gaga(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_gaga(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
U1_gaga(T45, T46, T47, T48, T50, append23_out_gaga(T46, T47, T48, T50)) → append23_out_gaga(cons(T45, T46), T47, T48, cons(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qs1_out_aa(cons(T14, []), T9)
qs1_in_aa(cons(T69, cons(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, less36_in_aa(T67, T69))
less36_in_aa(0, s(T78)) → less36_out_aa(0, s(T78))
less36_in_aa(s(T83), s(T84)) → U2_aa(T83, T84, less36_in_aa(T83, T84))
U2_aa(T83, T84, less36_out_aa(T83, T84)) → less36_out_aa(s(T83), s(T84))
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, X93, X94))
split46_in_aaaa([], T97, [], []) → split46_out_aaaa([], T97, [], [])
split46_in_aaaa(cons(T110, T111), T112, cons(T110, X163), X164) → U3_aaaa(T110, T111, T112, X163, X164, less36_in_aa(T110, T112))
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U3_aaaa(T110, T111, T112, X163, X164, less36_out_aa(T110, T112)) → U4_aaaa(T110, T111, T112, X163, X164, split46_in_aaaa(T111, T112, X163, X164))
split46_in_aaaa(cons(T125, T126), T127, X195, cons(T125, X196)) → U5_aaaa(T125, T126, T127, X195, X196, geq62_in_aa(T125, T127))
geq62_in_aa(T136, T136) → geq62_out_aa(T136, T136)
geq62_in_aa(s(T141), 0) → geq62_out_aa(s(T141), 0)
geq62_in_aa(s(T146), s(T147)) → U7_aa(T146, T147, geq62_in_aa(T146, T147))
U7_aa(T146, T147, geq62_out_aa(T146, T147)) → geq62_out_aa(s(T146), s(T147))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U5_aaaa(T125, T126, T127, X195, X196, geq62_out_aa(T125, T127)) → U6_aaaa(T125, T126, T127, X195, X196, split46_in_aaaa(T126, T127, X195, X196))
U6_aaaa(T125, T126, T127, X195, X196, split46_out_aaaa(T126, T127, X195, X196)) → split46_out_aaaa(cons(T125, T126), T127, X195, cons(T125, X196))
U4_aaaa(T110, T111, T112, X163, X164, split46_out_aaaa(T111, T112, X163, X164)) → split46_out_aaaa(cons(T110, T111), T112, cons(T110, X163), X164)
U25_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, X93, X94)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, less36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split46_in_aaaa(T68, T69, T89, T90))
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U27_aa(T69, T67, T68, T9, qs1_in_aa(cons(T67, T89), X12))
qs1_in_aa(cons(T214, cons(T212, T213)), T9) → U30_aa(T214, T212, T213, T9, geq62_in_aa(T212, T214))
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U31_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, X323, X324))
U31_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, X323, X324)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U30_aa(T214, T212, T213, T9, geq62_out_aa(T212, T214)) → U32_aa(T214, T212, T213, T9, split46_in_aaaa(T213, T214, T221, T222))
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U33_aa(T214, T212, T213, T9, qs79_in_aa(T221, X12))
qs79_in_aa([], []) → qs79_out_aa([], [])
qs79_in_aa(cons(T162, T163), X265) → U8_aa(T162, T163, X265, split46_in_aaaa(T163, T162, X261, X262))
U8_aa(T162, T163, X265, split46_out_aaaa(T163, T162, X261, X262)) → qs79_out_aa(cons(T162, T163), X265)
qs79_in_aa(cons(T162, T163), X265) → U9_aa(T162, T163, X265, split46_in_aaaa(T163, T162, T166, T167))
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U10_aa(T162, T163, X265, qs79_in_aa(T166, X263))
U10_aa(T162, T163, X265, qs79_out_aa(T166, X263)) → qs79_out_aa(cons(T162, T163), X265)
U9_aa(T162, T163, X265, split46_out_aaaa(T163, T162, T166, T167)) → U11_aa(T162, T163, X265, T167, qs79_in_aa(T166, T170))
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U12_aa(T162, T163, X265, qs79_in_aa(T167, X264))
U12_aa(T162, T163, X265, qs79_out_aa(T167, X264)) → qs79_out_aa(cons(T162, T163), X265)
U11_aa(T162, T163, X265, T167, qs79_out_aa(T166, T170)) → U13_aa(T162, T163, X265, T170, qs79_in_aa(T167, T171))
U13_aa(T162, T163, X265, T170, qs79_out_aa(T167, T171)) → U14_aa(T162, T163, X265, append94_in_aaaa(T170, T162, T171, X265))
append94_in_aaaa([], T184, T185, cons(T184, T185)) → append94_out_aaaa([], T184, T185, cons(T184, T185))
append94_in_aaaa(cons(T194, T195), T196, T197, cons(T194, X295)) → U15_aaaa(T194, T195, T196, T197, X295, append94_in_aaaa(T195, T196, T197, X295))
U15_aaaa(T194, T195, T196, T197, X295, append94_out_aaaa(T195, T196, T197, X295)) → append94_out_aaaa(cons(T194, T195), T196, T197, cons(T194, X295))
U14_aa(T162, T163, X265, append94_out_aaaa(T170, T162, T171, X265)) → qs79_out_aa(cons(T162, T163), X265)
U33_aa(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U32_aa(T214, T212, T213, T9, split46_out_aaaa(T213, T214, T221, T222)) → U34_aa(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_aa(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_aa(T214, T212, T213, T9, p78_in_aaaaa(cons(T212, T222), X13, T225, T214, T9))
p78_in_aaaaa(T90, X13, T154, T69, T9) → U16_aaaaa(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaaa(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaaa(T90, X13, T154, T69, T9)
p78_in_aaaaa(T90, T157, T154, T69, T9) → U17_aaaaa(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaaa(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaaa(T90, T157, T154, T69, T9, append23_in_aaaa(T154, T69, T157, T9))
append23_in_aaaa([], T33, T34, cons(T33, T34)) → append23_out_aaaa([], T33, T34, cons(T33, T34))
append23_in_aaaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_aaaa(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
U1_aaaa(T45, T46, T47, T48, T50, append23_out_aaaa(T46, T47, T48, T50)) → append23_out_aaaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaaa(T90, T157, T154, T69, T9, append23_out_aaaa(T154, T69, T157, T9)) → p78_out_aaaaa(T90, T157, T154, T69, T9)
U35_aa(T214, T212, T213, T9, p78_out_aaaaa(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_aa(cons(T214, cons(T212, T213)), T9)
U27_aa(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split46_out_aaaa(T68, T69, T89, T90)) → U28_aa(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_aa(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T90, X13, T154, T69, T9))
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T90, X13, T154, T69, T9)) → qs1_out_aa(cons(T69, cons(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qs1_out_aa(cons(T67, T89), X12)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split46_out_ggaa(T68, T69, T89, T90)) → U28_ga(T69, T67, T68, T9, T90, qs1_in_aa(cons(T67, T89), T154))
U28_ga(T69, T67, T68, T9, T90, qs1_out_aa(cons(T67, T89), T154)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T90, X13, T154, T69, T9))
p78_in_aaaga(T90, X13, T154, T69, T9) → U16_aaaga(T90, X13, T154, T69, T9, qs79_in_aa(T90, X13))
U16_aaaga(T90, X13, T154, T69, T9, qs79_out_aa(T90, X13)) → p78_out_aaaga(T90, X13, T154, T69, T9)
p78_in_aaaga(T90, T157, T154, T69, T9) → U17_aaaga(T90, T157, T154, T69, T9, qs79_in_aa(T90, T157))
U17_aaaga(T90, T157, T154, T69, T9, qs79_out_aa(T90, T157)) → U18_aaaga(T90, T157, T154, T69, T9, append23_in_agaa(T154, T69, T157, T9))
append23_in_agaa([], T33, T34, cons(T33, T34)) → append23_out_agaa([], T33, T34, cons(T33, T34))
append23_in_agaa(cons(T45, T46), T47, T48, cons(T45, T50)) → U1_agaa(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
U1_agaa(T45, T46, T47, T48, T50, append23_out_agaa(T46, T47, T48, T50)) → append23_out_agaa(cons(T45, T46), T47, T48, cons(T45, T50))
U18_aaaga(T90, T157, T154, T69, T9, append23_out_agaa(T154, T69, T157, T9)) → p78_out_aaaga(T90, T157, T154, T69, T9)
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T90, X13, T154, T69, T9)) → qs1_out_ga(cons(T69, cons(T67, T68)), T9)
qs1_in_ga(cons(T214, cons(T212, T213)), T9) → U30_ga(T214, T212, T213, T9, geq62_in_gg(T212, T214))
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U31_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, X323, X324))
U31_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, X323, X324)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U30_ga(T214, T212, T213, T9, geq62_out_gg(T212, T214)) → U32_ga(T214, T212, T213, T9, split46_in_ggaa(T213, T214, T221, T222))
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U33_ga(T214, T212, T213, T9, qs79_in_aa(T221, X12))
U33_ga(T214, T212, T213, T9, qs79_out_aa(T221, X12)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)
U32_ga(T214, T212, T213, T9, split46_out_ggaa(T213, T214, T221, T222)) → U34_ga(T214, T212, T213, T9, T222, qs79_in_aa(T221, T225))
U34_ga(T214, T212, T213, T9, T222, qs79_out_aa(T221, T225)) → U35_ga(T214, T212, T213, T9, p78_in_aaaga(cons(T212, T222), X13, T225, T214, T9))
U35_ga(T214, T212, T213, T9, p78_out_aaaga(cons(T212, T222), X13, T225, T214, T9)) → qs1_out_ga(cons(T214, cons(T212, T213)), T9)

The argument filtering Pi contains the following mapping:
qs1_in_ga(x1, x2)  =  qs1_in_ga(x1)
[]  =  []
qs1_out_ga(x1, x2)  =  qs1_out_ga
cons(x1, x2)  =  cons(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qs14_in_a(x1)  =  qs14_in_a
qs14_out_a(x1)  =  qs14_out_a(x1)
U20_ga(x1, x2, x3)  =  U20_ga(x1, x3)
U21_ga(x1, x2, x3)  =  U21_ga(x3)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x3, x4)
U23_ga(x1, x2, x3)  =  U23_ga(x3)
append23_in_ggga(x1, x2, x3, x4)  =  append23_in_ggga(x1, x2, x3)
append23_out_ggga(x1, x2, x3, x4)  =  append23_out_ggga(x4)
U1_ggga(x1, x2, x3, x4, x5, x6)  =  U1_ggga(x1, x6)
U24_ga(x1, x2, x3, x4, x5)  =  U24_ga(x1, x3, x5)
less36_in_gg(x1, x2)  =  less36_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less36_out_gg(x1, x2)  =  less36_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split46_in_ggaa(x1, x2, x3, x4)  =  split46_in_ggaa(x1, x2)
split46_out_ggaa(x1, x2, x3, x4)  =  split46_out_ggaa
U3_ggaa(x1, x2, x3, x4, x5, x6)  =  U3_ggaa(x2, x3, x6)
U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x2, x3, x6)
geq62_in_gg(x1, x2)  =  geq62_in_gg(x1, x2)
geq62_out_gg(x1, x2)  =  geq62_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x6)
U26_ga(x1, x2, x3, x4, x5)  =  U26_ga(x1, x5)
U27_ga(x1, x2, x3, x4, x5)  =  U27_ga(x5)
qs1_in_aa(x1, x2)  =  qs1_in_aa
qs1_out_aa(x1, x2)  =  qs1_out_aa
U19_aa(x1, x2, x3)  =  U19_aa(x3)
U20_aa(x1, x2, x3)  =  U20_aa(x3)
U21_aa(x1, x2, x3)  =  U21_aa(x3)
U22_aa(x1, x2, x3, x4)  =  U22_aa(x3, x4)
U23_aa(x1, x2, x3)  =  U23_aa(x3)
append23_in_gaga(x1, x2, x3, x4)  =  append23_in_gaga(x1, x3)
append23_out_gaga(x1, x2, x3, x4)  =  append23_out_gaga
U1_gaga(x1, x2, x3, x4, x5, x6)  =  U1_gaga(x6)
U24_aa(x1, x2, x3, x4, x5)  =  U24_aa(x5)
less36_in_aa(x1, x2)  =  less36_in_aa
less36_out_aa(x1, x2)  =  less36_out_aa(x1)
U2_aa(x1, x2, x3)  =  U2_aa(x3)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split46_in_aaaa(x1, x2, x3, x4)  =  split46_in_aaaa
split46_out_aaaa(x1, x2, x3, x4)  =  split46_out_aaaa
U3_aaaa(x1, x2, x3, x4, x5, x6)  =  U3_aaaa(x6)
U4_aaaa(x1, x2, x3, x4, x5, x6)  =  U4_aaaa(x6)
U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)
geq62_in_aa(x1, x2)  =  geq62_in_aa
geq62_out_aa(x1, x2)  =  geq62_out_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
U26_aa(x1, x2, x3, x4, x5)  =  U26_aa(x5)
U27_aa(x1, x2, x3, x4, x5)  =  U27_aa(x5)
U30_aa(x1, x2, x3, x4, x5)  =  U30_aa(x5)
U31_aa(x1, x2, x3, x4, x5)  =  U31_aa(x5)
U32_aa(x1, x2, x3, x4, x5)  =  U32_aa(x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qs79_in_aa(x1, x2)  =  qs79_in_aa
qs79_out_aa(x1, x2)  =  qs79_out_aa
U8_aa(x1, x2, x3, x4)  =  U8_aa(x4)
U9_aa(x1, x2, x3, x4)  =  U9_aa(x4)
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
U11_aa(x1, x2, x3, x4, x5)  =  U11_aa(x5)
U12_aa(x1, x2, x3, x4)  =  U12_aa(x4)
U13_aa(x1, x2, x3, x4, x5)  =  U13_aa(x5)
U14_aa(x1, x2, x3, x4)  =  U14_aa(x4)
append94_in_aaaa(x1, x2, x3, x4)  =  append94_in_aaaa
append94_out_aaaa(x1, x2, x3, x4)  =  append94_out_aaaa
U15_aaaa(x1, x2, x3, x4, x5, x6)  =  U15_aaaa(x6)
U34_aa(x1, x2, x3, x4, x5, x6)  =  U34_aa(x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(x5)
p78_in_aaaaa(x1, x2, x3, x4, x5)  =  p78_in_aaaaa
U16_aaaaa(x1, x2, x3, x4, x5, x6)  =  U16_aaaaa(x6)
p78_out_aaaaa(x1, x2, x3, x4, x5)  =  p78_out_aaaaa
U17_aaaaa(x1, x2, x3, x4, x5, x6)  =  U17_aaaaa(x6)
U18_aaaaa(x1, x2, x3, x4, x5, x6)  =  U18_aaaaa(x6)
append23_in_aaaa(x1, x2, x3, x4)  =  append23_in_aaaa
append23_out_aaaa(x1, x2, x3, x4)  =  append23_out_aaaa
U1_aaaa(x1, x2, x3, x4, x5, x6)  =  U1_aaaa(x6)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_aa(x5)
U28_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
p78_in_aaaga(x1, x2, x3, x4, x5)  =  p78_in_aaaga(x4)
U16_aaaga(x1, x2, x3, x4, x5, x6)  =  U16_aaaga(x6)
p78_out_aaaga(x1, x2, x3, x4, x5)  =  p78_out_aaaga
U17_aaaga(x1, x2, x3, x4, x5, x6)  =  U17_aaaga(x4, x6)
U18_aaaga(x1, x2, x3, x4, x5, x6)  =  U18_aaaga(x6)
append23_in_agaa(x1, x2, x3, x4)  =  append23_in_agaa(x2)
append23_out_agaa(x1, x2, x3, x4)  =  append23_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6)  =  U1_agaa(x6)
U30_ga(x1, x2, x3, x4, x5)  =  U30_ga(x1, x3, x5)
U31_ga(x1, x2, x3, x4, x5)  =  U31_ga(x5)
U32_ga(x1, x2, x3, x4, x5)  =  U32_ga(x1, x5)
U33_ga(x1, x2, x3, x4, x5)  =  U33_ga(x5)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x6)
U35_ga(x1, x2, x3, x4, x5)  =  U35_ga(x5)
APPEND23_IN_GGGA(x1, x2, x3, x4)  =  APPEND23_IN_GGGA(x1, x2, x3)

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

(215) UsableRulesProof (EQUIVALENT transformation)

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

(216) 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

(217) PiDPToQDPProof (SOUND transformation)

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

(218) 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.

(219) 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

(220) YES