(0) Obligation:

Clauses:

qsort([], []).
qsort(.(H, L), S) :- ','(split(L, H, A, B), ','(qsort(A, A1), ','(qsort(B, B1), append(A1, .(H, B1), S)))).
split([], Y, [], []).
split(.(X, Xs), Y, .(X, Ls), Bs) :- ','(le(X, Y), split(Xs, Y, Ls, Bs)).
split(.(X, Xs), Y, Ls, .(X, Bs)) :- ','(gt(X, Y), split(Xs, Y, Ls, Bs)).
append([], L, L).
append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).

Queries:

qsort(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

qsort14([]).
append23([], T33, T34, .(T33, T34)).
append23(.(T45, T46), T47, T48, .(T45, T50)) :- append23(T46, T47, T48, T50).
le36(s(T82), s(T83)) :- le36(T82, T83).
le36(0, s(T90)).
le36(0, 0).
split51([], T101, [], []).
split51(.(T114, T115), T116, .(T114, X167), X168) :- le36(T114, T116).
split51(.(T114, T115), T116, .(T114, X167), X168) :- ','(le36(T114, T116), split51(T115, T116, X167, X168)).
split51(.(T129, T130), T131, X199, .(T129, X200)) :- gt67(T129, T131).
split51(.(T129, T130), T131, X199, .(T129, X200)) :- ','(gt67(T129, T131), split51(T130, T131, X199, X200)).
gt67(s(T144), s(T145)) :- gt67(T144, T145).
gt67(s(T150), 0).
qsort79([], []).
qsort79(.(T163, T164), X265) :- split51(T164, T163, X261, X262).
qsort79(.(T163, T164), X265) :- ','(split51(T164, T163, T167, T168), qsort79(T167, X263)).
qsort79(.(T163, T164), X265) :- ','(split51(T164, T163, T167, T168), ','(qsort79(T167, T171), qsort79(T168, X264))).
qsort79(.(T163, T164), X265) :- ','(split51(T164, T163, T167, T168), ','(qsort79(T167, T171), ','(qsort79(T168, T172), append94(T171, T163, T172, X265)))).
append94([], T185, T186, .(T185, T186)).
append94(.(T195, T196), T197, T198, .(T195, X295)) :- append94(T196, T197, T198, X295).
p78(T94, X13, T155, T69, T9) :- qsort79(T94, X13).
p78(T94, T158, T155, T69, T9) :- ','(qsort79(T94, T158), append23(T155, T69, T158, T9)).
qsort1([], []).
qsort1(.(T14, []), T9) :- qsort14(X12).
qsort1(.(T14, []), T9) :- ','(qsort14(T17), qsort14(X13)).
qsort1(.(T14, []), T9) :- ','(qsort14(T17), ','(qsort14(T20), append23(T17, T14, T20, T9))).
qsort1(.(T69, .(T67, T68)), T9) :- le36(T67, T69).
qsort1(.(T69, .(T67, T68)), T9) :- ','(le36(T67, T69), split51(T68, T69, X93, X94)).
qsort1(.(T69, .(T67, T68)), T9) :- ','(le36(T67, T69), ','(split51(T68, T69, T93, T94), qsort1(.(T67, T93), X12))).
qsort1(.(T69, .(T67, T68)), T9) :- ','(le36(T67, T69), ','(split51(T68, T69, T93, T94), ','(qsort1(.(T67, T93), T155), p78(T94, X13, T155, T69, T9)))).
qsort1(.(T215, .(T213, T214)), T9) :- gt67(T213, T215).
qsort1(.(T215, .(T213, T214)), T9) :- ','(gt67(T213, T215), split51(T214, T215, X323, X324)).
qsort1(.(T215, .(T213, T214)), T9) :- ','(gt67(T213, T215), ','(split51(T214, T215, T222, T223), qsort79(T222, X12))).
qsort1(.(T215, .(T213, T214)), T9) :- ','(gt67(T213, T215), ','(split51(T214, T215, T222, T223), ','(qsort79(T222, T226), p78(.(T213, T223), X13, T226, T215, T9)))).

Queries:

qsort1(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:
qsort1_in: (b,f) (f,f)
append23_in: (b,b,b,f) (b,f,b,f) (f,b,f,f) (f,f,f,f)
le36_in: (b,b) (f,f) (f,b)
split51_in: (b,b,f,f) (f,f,f,f) (f,b,f,f)
gt67_in: (b,b) (f,f) (f,b)
qsort79_in: (f,f)
append94_in: (f,f,f,f)
p78_in: (f,f,f,b,f) (f,f,f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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

(4) Obligation:

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

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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)

(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:

QSORT1_IN_GA(.(T14, []), T9) → U19_GA(T14, T9, qsort14_in_a(X12))
QSORT1_IN_GA(.(T14, []), T9) → QSORT14_IN_A(X12)
QSORT1_IN_GA(.(T14, []), T9) → U20_GA(T14, T9, qsort14_in_a(T17))
U20_GA(T14, T9, qsort14_out_a(T17)) → U21_GA(T14, T9, qsort14_in_a(X13))
U20_GA(T14, T9, qsort14_out_a(T17)) → QSORT14_IN_A(X13)
U20_GA(T14, T9, qsort14_out_a(T17)) → U22_GA(T14, T9, T17, qsort14_in_a(T20))
U22_GA(T14, T9, T17, qsort14_out_a(T20)) → U23_GA(T14, T9, append23_in_ggga(T17, T14, T20, T9))
U22_GA(T14, T9, T17, qsort14_out_a(T20)) → APPEND23_IN_GGGA(T17, T14, T20, T9)
APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → U1_GGGA(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GGGA(T46, T47, T48, T50)
QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → U24_GA(T69, T67, T68, T9, le36_in_gg(T67, T69))
QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → LE36_IN_GG(T67, T69)
LE36_IN_GG(s(T82), s(T83)) → U2_GG(T82, T83, le36_in_gg(T82, T83))
LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_GA(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → SPLIT51_IN_GGAA(T68, T69, X93, X94)
SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_GGAA(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_GG(T114, T116)
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_GGAA(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_GGAA(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_GG(T129, T131)
GT67_IN_GG(s(T144), s(T145)) → U7_GG(T144, T145, gt67_in_gg(T144, T145))
GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_GGAA(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_GA(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_GA(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)
QSORT1_IN_AA(.(T14, []), T9) → U19_AA(T14, T9, qsort14_in_a(X12))
QSORT1_IN_AA(.(T14, []), T9) → QSORT14_IN_A(X12)
QSORT1_IN_AA(.(T14, []), T9) → U20_AA(T14, T9, qsort14_in_a(T17))
U20_AA(T14, T9, qsort14_out_a(T17)) → U21_AA(T14, T9, qsort14_in_a(X13))
U20_AA(T14, T9, qsort14_out_a(T17)) → QSORT14_IN_A(X13)
U20_AA(T14, T9, qsort14_out_a(T17)) → U22_AA(T14, T9, T17, qsort14_in_a(T20))
U22_AA(T14, T9, T17, qsort14_out_a(T20)) → U23_AA(T14, T9, append23_in_gaga(T17, T14, T20, T9))
U22_AA(T14, T9, T17, qsort14_out_a(T20)) → APPEND23_IN_GAGA(T17, T14, T20, T9)
APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → U1_GAGA(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)
QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, le36_in_aa(T67, T69))
QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → LE36_IN_AA(T67, T69)
LE36_IN_AA(s(T82), s(T83)) → U2_AA(T82, T83, le36_in_aa(T82, T83))
LE36_IN_AA(s(T82), s(T83)) → LE36_IN_AA(T82, T83)
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → SPLIT51_IN_AAAA(T68, T69, X93, X94)
SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AAAA(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_AA(T114, T116)
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_AAAA(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → SPLIT51_IN_AAAA(T115, T116, X167, X168)
SPLIT51_IN_AAAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AAAA(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
SPLIT51_IN_AAAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_AA(T129, T131)
GT67_IN_AA(s(T144), s(T145)) → U7_AA(T144, T145, gt67_in_aa(T144, T145))
GT67_IN_AA(s(T144), s(T145)) → GT67_IN_AA(T144, T145)
U5_AAAA(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_AAAA(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U5_AAAA(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AGAA(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_AG(T114, T116)
LE36_IN_AG(s(T82), s(T83)) → U2_AG(T82, T83, le36_in_ag(T82, T83))
LE36_IN_AG(s(T82), s(T83)) → LE36_IN_AG(T82, T83)
U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_AGAA(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T115, T116, X167, X168)
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AGAA(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_AG(T129, T131)
GT67_IN_AG(s(T144), s(T145)) → U7_AG(T144, T145, gt67_in_ag(T144, T145))
GT67_IN_AG(s(T144), s(T145)) → GT67_IN_AG(T144, T145)
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_AGAA(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_AA(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)
QSORT1_IN_AA(.(T215, .(T213, T214)), T9) → U30_AA(T215, T213, T214, T9, gt67_in_aa(T213, T215))
QSORT1_IN_AA(.(T215, .(T213, T214)), T9) → GT67_IN_AA(T213, T215)
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_AA(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → SPLIT51_IN_AGAA(T214, T215, X323, X324)
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_AA(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_AA(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → QSORT79_IN_AA(T222, X12)
QSORT79_IN_AA(.(T163, T164), X265) → U8_AA(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
QSORT79_IN_AA(.(T163, T164), X265) → SPLIT51_IN_AAAA(T164, T163, X261, X262)
QSORT79_IN_AA(.(T163, T164), X265) → U9_AA(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_AA(T163, T164, X265, qsort79_in_aa(T167, X263))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → QSORT79_IN_AA(T167, X263)
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_AA(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_AA(T163, T164, X265, qsort79_in_aa(T168, X264))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → QSORT79_IN_AA(T168, X264)
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_AA(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_AA(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_AA(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
U13_AA(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → APPEND94_IN_AAAA(T171, T163, T172, X265)
APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → U15_AAAA(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_AAAA(T196, T197, T198, X295)
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_AA(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_AA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_AA(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U34_AA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → P78_IN_AAAGA(.(T213, T223), X13, T226, T215, T9)
P78_IN_AAAGA(T94, X13, T155, T69, T9) → U16_AAAGA(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
P78_IN_AAAGA(T94, X13, T155, T69, T9) → QSORT79_IN_AA(T94, X13)
P78_IN_AAAGA(T94, T158, T155, T69, T9) → U17_AAAGA(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_AAAGA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_AAAGA(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
U17_AAAGA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → APPEND23_IN_AGAA(T155, T69, T158, T9)
APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → U1_AGAA(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_AA(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_AA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_AA(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
U28_AA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → P78_IN_AAAAA(T94, X13, T155, T69, T9)
P78_IN_AAAAA(T94, X13, T155, T69, T9) → U16_AAAAA(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
P78_IN_AAAAA(T94, X13, T155, T69, T9) → QSORT79_IN_AA(T94, X13)
P78_IN_AAAAA(T94, T158, T155, T69, T9) → U17_AAAAA(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_AAAAA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_AAAAA(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
U17_AAAAA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → APPEND23_IN_AAAA(T155, T69, T158, T9)
APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → U1_AAAA(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_GA(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_GA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_GA(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U28_GA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → P78_IN_AAAGA(T94, X13, T155, T69, T9)
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → U30_GA(T215, T213, T214, T9, gt67_in_gg(T213, T215))
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → GT67_IN_GG(T213, T215)
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_GA(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → SPLIT51_IN_GGAA(T214, T215, X323, X324)
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_GA(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_GA(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → QSORT79_IN_AA(T222, X12)
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_GA(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_GA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_GA(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U34_GA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → P78_IN_AAAGA(.(T213, T223), X13, T226, T215, T9)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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)
QSORT1_IN_GA(x1, x2)  =  QSORT1_IN_GA(x1)
U19_GA(x1, x2, x3)  =  U19_GA(x3)
QSORT14_IN_A(x1)  =  QSORT14_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)
LE36_IN_GG(x1, x2)  =  LE36_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x3)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x5)
SPLIT51_IN_GGAA(x1, x2, x3, x4)  =  SPLIT51_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)
GT67_IN_GG(x1, x2)  =  GT67_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)
QSORT1_IN_AA(x1, x2)  =  QSORT1_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)
LE36_IN_AA(x1, x2)  =  LE36_IN_AA
U2_AA(x1, x2, x3)  =  U2_AA(x3)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
SPLIT51_IN_AAAA(x1, x2, x3, x4)  =  SPLIT51_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)
GT67_IN_AA(x1, x2)  =  GT67_IN_AA
U7_AA(x1, x2, x3)  =  U7_AA(x3)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x6)
SPLIT51_IN_AGAA(x1, x2, x3, x4)  =  SPLIT51_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5, x6)  =  U3_AGAA(x3, x6)
LE36_IN_AG(x1, x2)  =  LE36_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x3)
U4_AGAA(x1, x2, x3, x4, x5, x6)  =  U4_AGAA(x6)
U5_AGAA(x1, x2, x3, x4, x5, x6)  =  U5_AGAA(x3, x6)
GT67_IN_AG(x1, x2)  =  GT67_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x3)
U6_AGAA(x1, x2, x3, x4, x5, x6)  =  U6_AGAA(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(x1, x5)
U33_AA(x1, x2, x3, x4, x5)  =  U33_AA(x5)
QSORT79_IN_AA(x1, x2)  =  QSORT79_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(x1, x6)
U35_AA(x1, x2, x3, x4, x5)  =  U35_AA(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)
U28_AA(x1, x2, x3, x4, x5, x6)  =  U28_AA(x6)
U29_AA(x1, x2, x3, x4, x5)  =  U29_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_GA(x1, x2, x3, x4, x5, x6)  =  U28_GA(x1, x6)
U29_GA(x1, x2, x3, x4, x5)  =  U29_GA(x5)
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

(6) Obligation:

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

QSORT1_IN_GA(.(T14, []), T9) → U19_GA(T14, T9, qsort14_in_a(X12))
QSORT1_IN_GA(.(T14, []), T9) → QSORT14_IN_A(X12)
QSORT1_IN_GA(.(T14, []), T9) → U20_GA(T14, T9, qsort14_in_a(T17))
U20_GA(T14, T9, qsort14_out_a(T17)) → U21_GA(T14, T9, qsort14_in_a(X13))
U20_GA(T14, T9, qsort14_out_a(T17)) → QSORT14_IN_A(X13)
U20_GA(T14, T9, qsort14_out_a(T17)) → U22_GA(T14, T9, T17, qsort14_in_a(T20))
U22_GA(T14, T9, T17, qsort14_out_a(T20)) → U23_GA(T14, T9, append23_in_ggga(T17, T14, T20, T9))
U22_GA(T14, T9, T17, qsort14_out_a(T20)) → APPEND23_IN_GGGA(T17, T14, T20, T9)
APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → U1_GGGA(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GGGA(T46, T47, T48, T50)
QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → U24_GA(T69, T67, T68, T9, le36_in_gg(T67, T69))
QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → LE36_IN_GG(T67, T69)
LE36_IN_GG(s(T82), s(T83)) → U2_GG(T82, T83, le36_in_gg(T82, T83))
LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_GA(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → SPLIT51_IN_GGAA(T68, T69, X93, X94)
SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_GGAA(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_GG(T114, T116)
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_GGAA(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_GGAA(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_GG(T129, T131)
GT67_IN_GG(s(T144), s(T145)) → U7_GG(T144, T145, gt67_in_gg(T144, T145))
GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_GGAA(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_GA(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_GA(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)
QSORT1_IN_AA(.(T14, []), T9) → U19_AA(T14, T9, qsort14_in_a(X12))
QSORT1_IN_AA(.(T14, []), T9) → QSORT14_IN_A(X12)
QSORT1_IN_AA(.(T14, []), T9) → U20_AA(T14, T9, qsort14_in_a(T17))
U20_AA(T14, T9, qsort14_out_a(T17)) → U21_AA(T14, T9, qsort14_in_a(X13))
U20_AA(T14, T9, qsort14_out_a(T17)) → QSORT14_IN_A(X13)
U20_AA(T14, T9, qsort14_out_a(T17)) → U22_AA(T14, T9, T17, qsort14_in_a(T20))
U22_AA(T14, T9, T17, qsort14_out_a(T20)) → U23_AA(T14, T9, append23_in_gaga(T17, T14, T20, T9))
U22_AA(T14, T9, T17, qsort14_out_a(T20)) → APPEND23_IN_GAGA(T17, T14, T20, T9)
APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → U1_GAGA(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)
QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, le36_in_aa(T67, T69))
QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → LE36_IN_AA(T67, T69)
LE36_IN_AA(s(T82), s(T83)) → U2_AA(T82, T83, le36_in_aa(T82, T83))
LE36_IN_AA(s(T82), s(T83)) → LE36_IN_AA(T82, T83)
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → SPLIT51_IN_AAAA(T68, T69, X93, X94)
SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AAAA(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_AA(T114, T116)
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_AAAA(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → SPLIT51_IN_AAAA(T115, T116, X167, X168)
SPLIT51_IN_AAAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AAAA(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
SPLIT51_IN_AAAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_AA(T129, T131)
GT67_IN_AA(s(T144), s(T145)) → U7_AA(T144, T145, gt67_in_aa(T144, T145))
GT67_IN_AA(s(T144), s(T145)) → GT67_IN_AA(T144, T145)
U5_AAAA(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_AAAA(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U5_AAAA(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AGAA(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_AG(T114, T116)
LE36_IN_AG(s(T82), s(T83)) → U2_AG(T82, T83, le36_in_ag(T82, T83))
LE36_IN_AG(s(T82), s(T83)) → LE36_IN_AG(T82, T83)
U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_AGAA(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T115, T116, X167, X168)
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AGAA(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_AG(T129, T131)
GT67_IN_AG(s(T144), s(T145)) → U7_AG(T144, T145, gt67_in_ag(T144, T145))
GT67_IN_AG(s(T144), s(T145)) → GT67_IN_AG(T144, T145)
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_AGAA(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_AA(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)
QSORT1_IN_AA(.(T215, .(T213, T214)), T9) → U30_AA(T215, T213, T214, T9, gt67_in_aa(T213, T215))
QSORT1_IN_AA(.(T215, .(T213, T214)), T9) → GT67_IN_AA(T213, T215)
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_AA(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → SPLIT51_IN_AGAA(T214, T215, X323, X324)
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_AA(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_AA(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → QSORT79_IN_AA(T222, X12)
QSORT79_IN_AA(.(T163, T164), X265) → U8_AA(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
QSORT79_IN_AA(.(T163, T164), X265) → SPLIT51_IN_AAAA(T164, T163, X261, X262)
QSORT79_IN_AA(.(T163, T164), X265) → U9_AA(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_AA(T163, T164, X265, qsort79_in_aa(T167, X263))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → QSORT79_IN_AA(T167, X263)
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_AA(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_AA(T163, T164, X265, qsort79_in_aa(T168, X264))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → QSORT79_IN_AA(T168, X264)
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_AA(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_AA(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_AA(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
U13_AA(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → APPEND94_IN_AAAA(T171, T163, T172, X265)
APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → U15_AAAA(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_AAAA(T196, T197, T198, X295)
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_AA(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_AA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_AA(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U34_AA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → P78_IN_AAAGA(.(T213, T223), X13, T226, T215, T9)
P78_IN_AAAGA(T94, X13, T155, T69, T9) → U16_AAAGA(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
P78_IN_AAAGA(T94, X13, T155, T69, T9) → QSORT79_IN_AA(T94, X13)
P78_IN_AAAGA(T94, T158, T155, T69, T9) → U17_AAAGA(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_AAAGA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_AAAGA(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
U17_AAAGA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → APPEND23_IN_AGAA(T155, T69, T158, T9)
APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → U1_AGAA(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_AA(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_AA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_AA(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
U28_AA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → P78_IN_AAAAA(T94, X13, T155, T69, T9)
P78_IN_AAAAA(T94, X13, T155, T69, T9) → U16_AAAAA(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
P78_IN_AAAAA(T94, X13, T155, T69, T9) → QSORT79_IN_AA(T94, X13)
P78_IN_AAAAA(T94, T158, T155, T69, T9) → U17_AAAAA(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_AAAAA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_AAAAA(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
U17_AAAAA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → APPEND23_IN_AAAA(T155, T69, T158, T9)
APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → U1_AAAA(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_GA(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_GA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_GA(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U28_GA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → P78_IN_AAAGA(T94, X13, T155, T69, T9)
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → U30_GA(T215, T213, T214, T9, gt67_in_gg(T213, T215))
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → GT67_IN_GG(T213, T215)
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_GA(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → SPLIT51_IN_GGAA(T214, T215, X323, X324)
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_GA(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_GA(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → QSORT79_IN_AA(T222, X12)
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_GA(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_GA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_GA(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U34_GA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → P78_IN_AAAGA(.(T213, T223), X13, T226, T215, T9)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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)
QSORT1_IN_GA(x1, x2)  =  QSORT1_IN_GA(x1)
U19_GA(x1, x2, x3)  =  U19_GA(x3)
QSORT14_IN_A(x1)  =  QSORT14_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)
LE36_IN_GG(x1, x2)  =  LE36_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x3)
U25_GA(x1, x2, x3, x4, x5)  =  U25_GA(x5)
SPLIT51_IN_GGAA(x1, x2, x3, x4)  =  SPLIT51_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)
GT67_IN_GG(x1, x2)  =  GT67_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)
QSORT1_IN_AA(x1, x2)  =  QSORT1_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)
LE36_IN_AA(x1, x2)  =  LE36_IN_AA
U2_AA(x1, x2, x3)  =  U2_AA(x3)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
SPLIT51_IN_AAAA(x1, x2, x3, x4)  =  SPLIT51_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)
GT67_IN_AA(x1, x2)  =  GT67_IN_AA
U7_AA(x1, x2, x3)  =  U7_AA(x3)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x6)
SPLIT51_IN_AGAA(x1, x2, x3, x4)  =  SPLIT51_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5, x6)  =  U3_AGAA(x3, x6)
LE36_IN_AG(x1, x2)  =  LE36_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x3)
U4_AGAA(x1, x2, x3, x4, x5, x6)  =  U4_AGAA(x6)
U5_AGAA(x1, x2, x3, x4, x5, x6)  =  U5_AGAA(x3, x6)
GT67_IN_AG(x1, x2)  =  GT67_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x3)
U6_AGAA(x1, x2, x3, x4, x5, x6)  =  U6_AGAA(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(x1, x5)
U33_AA(x1, x2, x3, x4, x5)  =  U33_AA(x5)
QSORT79_IN_AA(x1, x2)  =  QSORT79_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(x1, x6)
U35_AA(x1, x2, x3, x4, x5)  =  U35_AA(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)
U28_AA(x1, x2, x3, x4, x5, x6)  =  U28_AA(x6)
U29_AA(x1, x2, x3, x4, x5)  =  U29_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_GA(x1, x2, x3, x4, x5, x6)  =  U28_GA(x1, x6)
U29_GA(x1, x2, x3, x4, x5)  =  U29_GA(x5)
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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 16 SCCs with 95 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

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

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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

(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_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)

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

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_AAAAAPPEND23_IN_AAAA

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_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.



(15) NO

(16) Obligation:

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

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

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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

(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_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)

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

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

APPEND23_IN_AGAA(T47) → APPEND23_IN_AGAA(T47)

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_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).



(22) NO

(23) Obligation:

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

APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_AAAA(T196, T197, T198, X295)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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

(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(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_AAAA(T196, T197, T198, X295)

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

GT67_IN_AG(s(T144), s(T145)) → GT67_IN_AG(T144, T145)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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)
GT67_IN_AG(x1, x2)  =  GT67_IN_AG(x2)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

GT67_IN_AG(s(T144), s(T145)) → GT67_IN_AG(T144, T145)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT67_IN_AG(x1, x2)  =  GT67_IN_AG(x2)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

GT67_IN_AG(s(T145)) → GT67_IN_AG(T145)

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

(35) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • GT67_IN_AG(s(T145)) → GT67_IN_AG(T145)
    The graph contains the following edges 1 > 1

(36) YES

(37) Obligation:

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

LE36_IN_AG(s(T82), s(T83)) → LE36_IN_AG(T82, T83)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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)
LE36_IN_AG(x1, x2)  =  LE36_IN_AG(x2)

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:

LE36_IN_AG(s(T82), s(T83)) → LE36_IN_AG(T82, T83)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE36_IN_AG(x1, x2)  =  LE36_IN_AG(x2)

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:

LE36_IN_AG(s(T83)) → LE36_IN_AG(T83)

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

(42) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • LE36_IN_AG(s(T83)) → LE36_IN_AG(T83)
    The graph contains the following edges 1 > 1

(43) YES

(44) Obligation:

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

U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T115, T116, X167, X168)
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AGAA(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AGAA(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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)
SPLIT51_IN_AGAA(x1, x2, x3, x4)  =  SPLIT51_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5, x6)  =  U3_AGAA(x3, x6)
U5_AGAA(x1, x2, x3, x4, x5, x6)  =  U5_AGAA(x3, 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:

U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T115, T116, X167, X168)
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AGAA(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AGAA(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)

The TRS R consists of the following rules:

le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
SPLIT51_IN_AGAA(x1, x2, x3, x4)  =  SPLIT51_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5, x6)  =  U3_AGAA(x3, x6)
U5_AGAA(x1, x2, x3, x4, x5, x6)  =  U5_AGAA(x3, 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:

U3_AGAA(T116, le36_out_ag(T114)) → SPLIT51_IN_AGAA(T116)
SPLIT51_IN_AGAA(T116) → U3_AGAA(T116, le36_in_ag(T116))
SPLIT51_IN_AGAA(T131) → U5_AGAA(T131, gt67_in_ag(T131))
U5_AGAA(T131, gt67_out_ag) → SPLIT51_IN_AGAA(T131)

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0)
U7_ag(x0)

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

(49) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SPLIT51_IN_AGAA(T116) → U3_AGAA(T116, le36_in_ag(T116)) at position [1] we obtained the following new rules [LPAR04]:

SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0))

(50) Obligation:

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

U3_AGAA(T116, le36_out_ag(T114)) → SPLIT51_IN_AGAA(T116)
SPLIT51_IN_AGAA(T131) → U5_AGAA(T131, gt67_in_ag(T131))
U5_AGAA(T131, gt67_out_ag) → SPLIT51_IN_AGAA(T131)
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0))

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0)
U7_ag(x0)

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

(51) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SPLIT51_IN_AGAA(T131) → U5_AGAA(T131, gt67_in_ag(T131)) at position [1] we obtained the following new rules [LPAR04]:

SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(gt67_in_ag(x0)))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag)

(52) Obligation:

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

U3_AGAA(T116, le36_out_ag(T114)) → SPLIT51_IN_AGAA(T116)
U5_AGAA(T131, gt67_out_ag) → SPLIT51_IN_AGAA(T131)
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0))
SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(gt67_in_ag(x0)))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag)

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0)
U7_ag(x0)

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

(53) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_AGAA(T116, le36_out_ag(T114)) → SPLIT51_IN_AGAA(T116) we obtained the following new rules [LPAR04]:

U3_AGAA(s(z0), le36_out_ag(x1)) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(s(z0), le36_out_ag(0)) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(0, le36_out_ag(0)) → SPLIT51_IN_AGAA(0)

(54) Obligation:

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

U5_AGAA(T131, gt67_out_ag) → SPLIT51_IN_AGAA(T131)
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0))
SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(gt67_in_ag(x0)))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag)
U3_AGAA(s(z0), le36_out_ag(x1)) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(s(z0), le36_out_ag(0)) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(0, le36_out_ag(0)) → SPLIT51_IN_AGAA(0)

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0)
U7_ag(x0)

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

(55) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U5_AGAA(T131, gt67_out_ag) → SPLIT51_IN_AGAA(T131) we obtained the following new rules [LPAR04]:

U5_AGAA(s(z0), gt67_out_ag) → SPLIT51_IN_AGAA(s(z0))
U5_AGAA(0, gt67_out_ag) → SPLIT51_IN_AGAA(0)

(56) Obligation:

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

SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0))
SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(gt67_in_ag(x0)))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag)
U3_AGAA(s(z0), le36_out_ag(x1)) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(s(z0), le36_out_ag(0)) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(0, le36_out_ag(0)) → SPLIT51_IN_AGAA(0)
U5_AGAA(s(z0), gt67_out_ag) → SPLIT51_IN_AGAA(s(z0))
U5_AGAA(0, gt67_out_ag) → SPLIT51_IN_AGAA(0)

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0)
U7_ag(x0)

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

(57) DependencyGraphProof (EQUIVALENT transformation)

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

(58) Complex Obligation (AND)

(59) Obligation:

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

U3_AGAA(0, le36_out_ag(0)) → SPLIT51_IN_AGAA(0)
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag)
U5_AGAA(0, gt67_out_ag) → SPLIT51_IN_AGAA(0)

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0)
U7_ag(x0)

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

(60) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(61) Obligation:

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

U3_AGAA(0, le36_out_ag(0)) → SPLIT51_IN_AGAA(0)
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag)
U5_AGAA(0, gt67_out_ag) → SPLIT51_IN_AGAA(0)

R is empty.
The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0)
U7_ag(x0)

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

(62) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0)
U7_ag(x0)

(63) Obligation:

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

U3_AGAA(0, le36_out_ag(0)) → SPLIT51_IN_AGAA(0)
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag)
U5_AGAA(0, gt67_out_ag) → SPLIT51_IN_AGAA(0)

R is empty.
Q is empty.
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 = SPLIT51_IN_AGAA(0) evaluates to t =SPLIT51_IN_AGAA(0)

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




Rewriting sequence

SPLIT51_IN_AGAA(0)U3_AGAA(0, le36_out_ag(0))
with rule SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0)) at position [] and matcher [ ]

U3_AGAA(0, le36_out_ag(0))SPLIT51_IN_AGAA(0)
with rule U3_AGAA(0, le36_out_ag(0)) → SPLIT51_IN_AGAA(0)

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:

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

U3_AGAA(s(z0), le36_out_ag(x1)) → SPLIT51_IN_AGAA(s(z0))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(le36_in_ag(x0)))
U3_AGAA(s(z0), le36_out_ag(0)) → SPLIT51_IN_AGAA(s(z0))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0))
SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(gt67_in_ag(x0)))
U5_AGAA(s(z0), gt67_out_ag) → SPLIT51_IN_AGAA(s(z0))

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0)
U7_ag(x0)

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

(67) 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 = SPLIT51_IN_AGAA(s(x0)) evaluates to t =SPLIT51_IN_AGAA(s(x0))

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




Rewriting sequence

SPLIT51_IN_AGAA(s(x0))U3_AGAA(s(x0), le36_out_ag(0))
with rule SPLIT51_IN_AGAA(s(x0')) → U3_AGAA(s(x0'), le36_out_ag(0)) at position [] and matcher [x0' / x0]

U3_AGAA(s(x0), le36_out_ag(0))SPLIT51_IN_AGAA(s(x0))
with rule U3_AGAA(s(z0), le36_out_ag(x1)) → SPLIT51_IN_AGAA(s(z0))

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.



(68) NO

(69) Obligation:

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

GT67_IN_AA(s(T144), s(T145)) → GT67_IN_AA(T144, T145)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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)
GT67_IN_AA(x1, x2)  =  GT67_IN_AA

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

(70) UsableRulesProof (EQUIVALENT transformation)

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

(71) Obligation:

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

GT67_IN_AA(s(T144), s(T145)) → GT67_IN_AA(T144, T145)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT67_IN_AA(x1, x2)  =  GT67_IN_AA

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

(72) PiDPToQDPProof (SOUND transformation)

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

(73) Obligation:

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

GT67_IN_AAGT67_IN_AA

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

(74) 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 = GT67_IN_AA evaluates to t =GT67_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 GT67_IN_AA to GT67_IN_AA.



(75) NO

(76) Obligation:

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

LE36_IN_AA(s(T82), s(T83)) → LE36_IN_AA(T82, T83)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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)
LE36_IN_AA(x1, x2)  =  LE36_IN_AA

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

(77) UsableRulesProof (EQUIVALENT transformation)

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

(78) Obligation:

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

LE36_IN_AA(s(T82), s(T83)) → LE36_IN_AA(T82, T83)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE36_IN_AA(x1, x2)  =  LE36_IN_AA

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

(79) PiDPToQDPProof (SOUND transformation)

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

(80) Obligation:

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

LE36_IN_AALE36_IN_AA

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

(81) 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 = LE36_IN_AA evaluates to t =LE36_IN_AA

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 LE36_IN_AA to LE36_IN_AA.



(82) NO

(83) Obligation:

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

SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AAAA(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → SPLIT51_IN_AAAA(T115, T116, X167, X168)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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)
SPLIT51_IN_AAAA(x1, x2, x3, x4)  =  SPLIT51_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)

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

(84) UsableRulesProof (EQUIVALENT transformation)

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

(85) Obligation:

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

SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AAAA(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → SPLIT51_IN_AAAA(T115, T116, X167, X168)

The TRS R consists of the following rules:

le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
SPLIT51_IN_AAAA(x1, x2, x3, x4)  =  SPLIT51_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)

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

(86) PiDPToQDPProof (SOUND transformation)

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

(87) Obligation:

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

SPLIT51_IN_AAAAU3_AAAA(le36_in_aa)
U3_AAAA(le36_out_aa(T114)) → SPLIT51_IN_AAAA

The TRS R consists of the following rules:

le36_in_aaU2_aa(le36_in_aa)
le36_in_aale36_out_aa(0)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))

The set Q consists of the following terms:

le36_in_aa
U2_aa(x0)

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

(88) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SPLIT51_IN_AAAAU3_AAAA(le36_in_aa) at position [0] we obtained the following new rules [LPAR04]:

SPLIT51_IN_AAAAU3_AAAA(U2_aa(le36_in_aa))
SPLIT51_IN_AAAAU3_AAAA(le36_out_aa(0))

(89) Obligation:

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

U3_AAAA(le36_out_aa(T114)) → SPLIT51_IN_AAAA
SPLIT51_IN_AAAAU3_AAAA(U2_aa(le36_in_aa))
SPLIT51_IN_AAAAU3_AAAA(le36_out_aa(0))

The TRS R consists of the following rules:

le36_in_aaU2_aa(le36_in_aa)
le36_in_aale36_out_aa(0)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))

The set Q consists of the following terms:

le36_in_aa
U2_aa(x0)

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

(90) 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 = SPLIT51_IN_AAAA evaluates to t =SPLIT51_IN_AAAA

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




Rewriting sequence

SPLIT51_IN_AAAAU3_AAAA(le36_out_aa(0))
with rule SPLIT51_IN_AAAAU3_AAAA(le36_out_aa(0)) at position [] and matcher [ ]

U3_AAAA(le36_out_aa(0))SPLIT51_IN_AAAA
with rule U3_AAAA(le36_out_aa(T114)) → SPLIT51_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.



(91) NO

(92) Obligation:

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

QSORT79_IN_AA(.(T163, T164), X265) → U9_AA(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → QSORT79_IN_AA(T167, X263)
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_AA(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → QSORT79_IN_AA(T168, X264)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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)
QSORT79_IN_AA(x1, x2)  =  QSORT79_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

(93) UsableRulesProof (EQUIVALENT transformation)

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

(94) Obligation:

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

QSORT79_IN_AA(.(T163, T164), X265) → U9_AA(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → QSORT79_IN_AA(T167, X263)
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_AA(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → QSORT79_IN_AA(T168, X264)

The TRS R consists of the following rules:

split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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)
QSORT79_IN_AA(x1, x2)  =  QSORT79_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

(95) PiDPToQDPProof (SOUND transformation)

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

(96) Obligation:

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

QSORT79_IN_AAU9_AA(split51_in_aaaa)
U9_AA(split51_out_aaaa) → QSORT79_IN_AA
U9_AA(split51_out_aaaa) → U11_AA(qsort79_in_aa)
U11_AA(qsort79_out_aa) → QSORT79_IN_AA

The TRS R consists of the following rules:

split51_in_aaaasplit51_out_aaaa
split51_in_aaaaU3_aaaa(le36_in_aa)
split51_in_aaaaU5_aaaa(gt67_in_aa)
qsort79_in_aaqsort79_out_aa
qsort79_in_aaU8_aa(split51_in_aaaa)
qsort79_in_aaU9_aa(split51_in_aaaa)
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U8_aa(split51_out_aaaa) → qsort79_out_aa
U9_aa(split51_out_aaaa) → U10_aa(qsort79_in_aa)
U9_aa(split51_out_aaaa) → U11_aa(qsort79_in_aa)
le36_in_aaU2_aa(le36_in_aa)
le36_in_aale36_out_aa(0)
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aaU7_aa(gt67_in_aa)
gt67_in_aagt67_out_aa(0)
U6_aaaa(split51_out_agaa) → split51_out_aaaa
U10_aa(qsort79_out_aa) → qsort79_out_aa
U11_aa(qsort79_out_aa) → U12_aa(qsort79_in_aa)
U11_aa(qsort79_out_aa) → U13_aa(qsort79_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U12_aa(qsort79_out_aa) → qsort79_out_aa
U13_aa(qsort79_out_aa) → U14_aa(append94_in_aaaa)
U3_agaa(T116, le36_out_ag(T114)) → split51_out_agaa
U3_agaa(T116, le36_out_ag(T114)) → U4_agaa(split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag) → split51_out_agaa
U5_agaa(T131, gt67_out_ag) → U6_agaa(split51_in_agaa(T131))
U14_aa(append94_out_aaaa) → qsort79_out_aa
le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
U4_agaa(split51_out_agaa) → split51_out_agaa
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U6_agaa(split51_out_agaa) → split51_out_agaa
append94_in_aaaaappend94_out_aaaa
append94_in_aaaaU15_aaaa(append94_in_aaaa)
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag
U15_aaaa(append94_out_aaaa) → append94_out_aaaa

The set Q consists of the following terms:

split51_in_aaaa
qsort79_in_aa
U3_aaaa(x0)
U5_aaaa(x0)
U8_aa(x0)
U9_aa(x0)
le36_in_aa
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U10_aa(x0)
U11_aa(x0)
U2_aa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U12_aa(x0)
U13_aa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
U14_aa(x0)
le36_in_ag(x0)
U4_agaa(x0)
gt67_in_ag(x0)
U6_agaa(x0)
append94_in_aaaa
U2_ag(x0)
U7_ag(x0)
U15_aaaa(x0)

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

(97) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QSORT79_IN_AAU9_AA(split51_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

QSORT79_IN_AAU9_AA(split51_out_aaaa)
QSORT79_IN_AAU9_AA(U3_aaaa(le36_in_aa))
QSORT79_IN_AAU9_AA(U5_aaaa(gt67_in_aa))

(98) Obligation:

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

U9_AA(split51_out_aaaa) → QSORT79_IN_AA
U9_AA(split51_out_aaaa) → U11_AA(qsort79_in_aa)
U11_AA(qsort79_out_aa) → QSORT79_IN_AA
QSORT79_IN_AAU9_AA(split51_out_aaaa)
QSORT79_IN_AAU9_AA(U3_aaaa(le36_in_aa))
QSORT79_IN_AAU9_AA(U5_aaaa(gt67_in_aa))

The TRS R consists of the following rules:

split51_in_aaaasplit51_out_aaaa
split51_in_aaaaU3_aaaa(le36_in_aa)
split51_in_aaaaU5_aaaa(gt67_in_aa)
qsort79_in_aaqsort79_out_aa
qsort79_in_aaU8_aa(split51_in_aaaa)
qsort79_in_aaU9_aa(split51_in_aaaa)
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U8_aa(split51_out_aaaa) → qsort79_out_aa
U9_aa(split51_out_aaaa) → U10_aa(qsort79_in_aa)
U9_aa(split51_out_aaaa) → U11_aa(qsort79_in_aa)
le36_in_aaU2_aa(le36_in_aa)
le36_in_aale36_out_aa(0)
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aaU7_aa(gt67_in_aa)
gt67_in_aagt67_out_aa(0)
U6_aaaa(split51_out_agaa) → split51_out_aaaa
U10_aa(qsort79_out_aa) → qsort79_out_aa
U11_aa(qsort79_out_aa) → U12_aa(qsort79_in_aa)
U11_aa(qsort79_out_aa) → U13_aa(qsort79_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U12_aa(qsort79_out_aa) → qsort79_out_aa
U13_aa(qsort79_out_aa) → U14_aa(append94_in_aaaa)
U3_agaa(T116, le36_out_ag(T114)) → split51_out_agaa
U3_agaa(T116, le36_out_ag(T114)) → U4_agaa(split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag) → split51_out_agaa
U5_agaa(T131, gt67_out_ag) → U6_agaa(split51_in_agaa(T131))
U14_aa(append94_out_aaaa) → qsort79_out_aa
le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
U4_agaa(split51_out_agaa) → split51_out_agaa
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U6_agaa(split51_out_agaa) → split51_out_agaa
append94_in_aaaaappend94_out_aaaa
append94_in_aaaaU15_aaaa(append94_in_aaaa)
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag
U15_aaaa(append94_out_aaaa) → append94_out_aaaa

The set Q consists of the following terms:

split51_in_aaaa
qsort79_in_aa
U3_aaaa(x0)
U5_aaaa(x0)
U8_aa(x0)
U9_aa(x0)
le36_in_aa
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U10_aa(x0)
U11_aa(x0)
U2_aa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U12_aa(x0)
U13_aa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
U14_aa(x0)
le36_in_ag(x0)
U4_agaa(x0)
gt67_in_ag(x0)
U6_agaa(x0)
append94_in_aaaa
U2_ag(x0)
U7_ag(x0)
U15_aaaa(x0)

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

(99) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U9_AA(split51_out_aaaa) → U11_AA(qsort79_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U9_AA(split51_out_aaaa) → U11_AA(qsort79_out_aa)
U9_AA(split51_out_aaaa) → U11_AA(U8_aa(split51_in_aaaa))
U9_AA(split51_out_aaaa) → U11_AA(U9_aa(split51_in_aaaa))

(100) Obligation:

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

U9_AA(split51_out_aaaa) → QSORT79_IN_AA
U11_AA(qsort79_out_aa) → QSORT79_IN_AA
QSORT79_IN_AAU9_AA(split51_out_aaaa)
QSORT79_IN_AAU9_AA(U3_aaaa(le36_in_aa))
QSORT79_IN_AAU9_AA(U5_aaaa(gt67_in_aa))
U9_AA(split51_out_aaaa) → U11_AA(qsort79_out_aa)
U9_AA(split51_out_aaaa) → U11_AA(U8_aa(split51_in_aaaa))
U9_AA(split51_out_aaaa) → U11_AA(U9_aa(split51_in_aaaa))

The TRS R consists of the following rules:

split51_in_aaaasplit51_out_aaaa
split51_in_aaaaU3_aaaa(le36_in_aa)
split51_in_aaaaU5_aaaa(gt67_in_aa)
qsort79_in_aaqsort79_out_aa
qsort79_in_aaU8_aa(split51_in_aaaa)
qsort79_in_aaU9_aa(split51_in_aaaa)
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U8_aa(split51_out_aaaa) → qsort79_out_aa
U9_aa(split51_out_aaaa) → U10_aa(qsort79_in_aa)
U9_aa(split51_out_aaaa) → U11_aa(qsort79_in_aa)
le36_in_aaU2_aa(le36_in_aa)
le36_in_aale36_out_aa(0)
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aaU7_aa(gt67_in_aa)
gt67_in_aagt67_out_aa(0)
U6_aaaa(split51_out_agaa) → split51_out_aaaa
U10_aa(qsort79_out_aa) → qsort79_out_aa
U11_aa(qsort79_out_aa) → U12_aa(qsort79_in_aa)
U11_aa(qsort79_out_aa) → U13_aa(qsort79_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U12_aa(qsort79_out_aa) → qsort79_out_aa
U13_aa(qsort79_out_aa) → U14_aa(append94_in_aaaa)
U3_agaa(T116, le36_out_ag(T114)) → split51_out_agaa
U3_agaa(T116, le36_out_ag(T114)) → U4_agaa(split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag) → split51_out_agaa
U5_agaa(T131, gt67_out_ag) → U6_agaa(split51_in_agaa(T131))
U14_aa(append94_out_aaaa) → qsort79_out_aa
le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
U4_agaa(split51_out_agaa) → split51_out_agaa
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U6_agaa(split51_out_agaa) → split51_out_agaa
append94_in_aaaaappend94_out_aaaa
append94_in_aaaaU15_aaaa(append94_in_aaaa)
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag
U15_aaaa(append94_out_aaaa) → append94_out_aaaa

The set Q consists of the following terms:

split51_in_aaaa
qsort79_in_aa
U3_aaaa(x0)
U5_aaaa(x0)
U8_aa(x0)
U9_aa(x0)
le36_in_aa
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U10_aa(x0)
U11_aa(x0)
U2_aa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U12_aa(x0)
U13_aa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
U14_aa(x0)
le36_in_ag(x0)
U4_agaa(x0)
gt67_in_ag(x0)
U6_agaa(x0)
append94_in_aaaa
U2_ag(x0)
U7_ag(x0)
U15_aaaa(x0)

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

(101) 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 = QSORT79_IN_AA evaluates to t =QSORT79_IN_AA

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




Rewriting sequence

QSORT79_IN_AAU9_AA(split51_out_aaaa)
with rule QSORT79_IN_AAU9_AA(split51_out_aaaa) at position [] and matcher [ ]

U9_AA(split51_out_aaaa)QSORT79_IN_AA
with rule U9_AA(split51_out_aaaa) → QSORT79_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.



(102) NO

(103) Obligation:

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

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

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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

(104) UsableRulesProof (EQUIVALENT transformation)

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

(105) Obligation:

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

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

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

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

(106) PiDPToQDPProof (SOUND transformation)

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

(107) Obligation:

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

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

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

(108) 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(.(T45, T46), T48) → APPEND23_IN_GAGA(T46, T48)
    The graph contains the following edges 1 > 1, 2 >= 2

(109) YES

(110) Obligation:

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

QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, le36_in_aa(T67, T69))
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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)
QSORT1_IN_AA(x1, x2)  =  QSORT1_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

(111) UsableRulesProof (EQUIVALENT transformation)

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

(112) Obligation:

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

QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, le36_in_aa(T67, T69))
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)

The TRS R consists of the following rules:

le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x6)
QSORT1_IN_AA(x1, x2)  =  QSORT1_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

(113) PiDPToQDPProof (SOUND transformation)

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

(114) Obligation:

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

QSORT1_IN_AAU24_AA(le36_in_aa)
U24_AA(le36_out_aa(T67)) → U26_AA(split51_in_aaaa)
U26_AA(split51_out_aaaa) → QSORT1_IN_AA

The TRS R consists of the following rules:

le36_in_aaU2_aa(le36_in_aa)
le36_in_aale36_out_aa(0)
split51_in_aaaasplit51_out_aaaa
split51_in_aaaaU3_aaaa(le36_in_aa)
split51_in_aaaaU5_aaaa(gt67_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aaU7_aa(gt67_in_aa)
gt67_in_aagt67_out_aa(0)
U6_aaaa(split51_out_agaa) → split51_out_aaaa
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U3_agaa(T116, le36_out_ag(T114)) → split51_out_agaa
U3_agaa(T116, le36_out_ag(T114)) → U4_agaa(split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag) → split51_out_agaa
U5_agaa(T131, gt67_out_ag) → U6_agaa(split51_in_agaa(T131))
le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
U4_agaa(split51_out_agaa) → split51_out_agaa
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U6_agaa(split51_out_agaa) → split51_out_agaa
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_aa
split51_in_aaaa
U2_aa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
le36_in_ag(x0)
U4_agaa(x0)
gt67_in_ag(x0)
U6_agaa(x0)
U2_ag(x0)
U7_ag(x0)

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

(115) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QSORT1_IN_AAU24_AA(le36_in_aa) at position [0] we obtained the following new rules [LPAR04]:

QSORT1_IN_AAU24_AA(U2_aa(le36_in_aa))
QSORT1_IN_AAU24_AA(le36_out_aa(0))

(116) Obligation:

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

U24_AA(le36_out_aa(T67)) → U26_AA(split51_in_aaaa)
U26_AA(split51_out_aaaa) → QSORT1_IN_AA
QSORT1_IN_AAU24_AA(U2_aa(le36_in_aa))
QSORT1_IN_AAU24_AA(le36_out_aa(0))

The TRS R consists of the following rules:

le36_in_aaU2_aa(le36_in_aa)
le36_in_aale36_out_aa(0)
split51_in_aaaasplit51_out_aaaa
split51_in_aaaaU3_aaaa(le36_in_aa)
split51_in_aaaaU5_aaaa(gt67_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aaU7_aa(gt67_in_aa)
gt67_in_aagt67_out_aa(0)
U6_aaaa(split51_out_agaa) → split51_out_aaaa
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U3_agaa(T116, le36_out_ag(T114)) → split51_out_agaa
U3_agaa(T116, le36_out_ag(T114)) → U4_agaa(split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag) → split51_out_agaa
U5_agaa(T131, gt67_out_ag) → U6_agaa(split51_in_agaa(T131))
le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
U4_agaa(split51_out_agaa) → split51_out_agaa
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U6_agaa(split51_out_agaa) → split51_out_agaa
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_aa
split51_in_aaaa
U2_aa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
le36_in_ag(x0)
U4_agaa(x0)
gt67_in_ag(x0)
U6_agaa(x0)
U2_ag(x0)
U7_ag(x0)

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

(117) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_AA(le36_out_aa(T67)) → U26_AA(split51_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

U24_AA(le36_out_aa(y0)) → U26_AA(split51_out_aaaa)
U24_AA(le36_out_aa(y0)) → U26_AA(U3_aaaa(le36_in_aa))
U24_AA(le36_out_aa(y0)) → U26_AA(U5_aaaa(gt67_in_aa))

(118) Obligation:

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

U26_AA(split51_out_aaaa) → QSORT1_IN_AA
QSORT1_IN_AAU24_AA(U2_aa(le36_in_aa))
QSORT1_IN_AAU24_AA(le36_out_aa(0))
U24_AA(le36_out_aa(y0)) → U26_AA(split51_out_aaaa)
U24_AA(le36_out_aa(y0)) → U26_AA(U3_aaaa(le36_in_aa))
U24_AA(le36_out_aa(y0)) → U26_AA(U5_aaaa(gt67_in_aa))

The TRS R consists of the following rules:

le36_in_aaU2_aa(le36_in_aa)
le36_in_aale36_out_aa(0)
split51_in_aaaasplit51_out_aaaa
split51_in_aaaaU3_aaaa(le36_in_aa)
split51_in_aaaaU5_aaaa(gt67_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aaU7_aa(gt67_in_aa)
gt67_in_aagt67_out_aa(0)
U6_aaaa(split51_out_agaa) → split51_out_aaaa
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U3_agaa(T116, le36_out_ag(T114)) → split51_out_agaa
U3_agaa(T116, le36_out_ag(T114)) → U4_agaa(split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag) → split51_out_agaa
U5_agaa(T131, gt67_out_ag) → U6_agaa(split51_in_agaa(T131))
le36_in_ag(s(T83)) → U2_ag(le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0)
le36_in_ag(0) → le36_out_ag(0)
U4_agaa(split51_out_agaa) → split51_out_agaa
gt67_in_ag(s(T145)) → U7_ag(gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag
U6_agaa(split51_out_agaa) → split51_out_agaa
U2_ag(le36_out_ag(T82)) → le36_out_ag(s(T82))
U7_ag(gt67_out_ag) → gt67_out_ag

The set Q consists of the following terms:

le36_in_aa
split51_in_aaaa
U2_aa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
le36_in_ag(x0)
U4_agaa(x0)
gt67_in_ag(x0)
U6_agaa(x0)
U2_ag(x0)
U7_ag(x0)

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

(119) 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 = QSORT1_IN_AA evaluates to t =QSORT1_IN_AA

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




Rewriting sequence

QSORT1_IN_AAU24_AA(le36_out_aa(0))
with rule QSORT1_IN_AAU24_AA(le36_out_aa(0)) at position [] and matcher [ ]

U24_AA(le36_out_aa(0))U26_AA(split51_out_aaaa)
with rule U24_AA(le36_out_aa(y0)) → U26_AA(split51_out_aaaa) at position [] and matcher [y0 / 0]

U26_AA(split51_out_aaaa)QSORT1_IN_AA
with rule U26_AA(split51_out_aaaa) → QSORT1_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.



(120) NO

(121) Obligation:

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

GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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)
GT67_IN_GG(x1, x2)  =  GT67_IN_GG(x1, x2)

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

(122) UsableRulesProof (EQUIVALENT transformation)

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

(123) Obligation:

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

GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)

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

(124) PiDPToQDPProof (EQUIVALENT transformation)

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

(125) Obligation:

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

GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)

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

(126) 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:

  • GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)
    The graph contains the following edges 1 > 1, 2 > 2

(127) YES

(128) Obligation:

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

LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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)
LE36_IN_GG(x1, x2)  =  LE36_IN_GG(x1, x2)

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

(129) UsableRulesProof (EQUIVALENT transformation)

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

(130) Obligation:

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

LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)

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

(131) PiDPToQDPProof (EQUIVALENT transformation)

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

(132) Obligation:

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

LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)

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

(133) 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:

  • LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)
    The graph contains the following edges 1 > 1, 2 > 2

(134) YES

(135) Obligation:

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

SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_GGAA(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_GGAA(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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)
SPLIT51_IN_GGAA(x1, x2, x3, x4)  =  SPLIT51_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

(136) UsableRulesProof (EQUIVALENT transformation)

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

(137) Obligation:

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

SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_GGAA(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_GGAA(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)

The TRS R consists of the following rules:

le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
SPLIT51_IN_GGAA(x1, x2, x3, x4)  =  SPLIT51_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

(138) PiDPToQDPProof (SOUND transformation)

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

(139) Obligation:

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

SPLIT51_IN_GGAA(.(T114, T115), T116) → U3_GGAA(T115, T116, le36_in_gg(T114, T116))
U3_GGAA(T115, T116, le36_out_gg) → SPLIT51_IN_GGAA(T115, T116)
SPLIT51_IN_GGAA(.(T129, T130), T131) → U5_GGAA(T130, T131, gt67_in_gg(T129, T131))
U5_GGAA(T130, T131, gt67_out_gg) → SPLIT51_IN_GGAA(T130, T131)

The TRS R consists of the following rules:

le36_in_gg(s(T82), s(T83)) → U2_gg(le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg
le36_in_gg(0, 0) → le36_out_gg
gt67_in_gg(s(T144), s(T145)) → U7_gg(gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg
U2_gg(le36_out_gg) → le36_out_gg
U7_gg(gt67_out_gg) → gt67_out_gg

The set Q consists of the following terms:

le36_in_gg(x0, x1)
gt67_in_gg(x0, x1)
U2_gg(x0)
U7_gg(x0)

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

(140) 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(T115, T116, le36_out_gg) → SPLIT51_IN_GGAA(T115, T116)
    The graph contains the following edges 1 >= 1, 2 >= 2

  • U5_GGAA(T130, T131, gt67_out_gg) → SPLIT51_IN_GGAA(T130, T131)
    The graph contains the following edges 1 >= 1, 2 >= 2

  • SPLIT51_IN_GGAA(.(T114, T115), T116) → U3_GGAA(T115, T116, le36_in_gg(T114, T116))
    The graph contains the following edges 1 > 1, 2 >= 2

  • SPLIT51_IN_GGAA(.(T129, T130), T131) → U5_GGAA(T130, T131, gt67_in_gg(T129, T131))
    The graph contains the following edges 1 > 1, 2 >= 2

(141) YES

(142) Obligation:

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

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

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_ga(x1, x2, x3, x4, x5, x6)  =  U28_ga(x1, x6)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x5)
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

(143) UsableRulesProof (EQUIVALENT transformation)

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

(144) Obligation:

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

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

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

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

(145) PiDPToQDPProof (SOUND transformation)

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

(146) Obligation:

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

APPEND23_IN_GGGA(.(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.

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

(148) YES

(149) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
qsort1_in: (b,f) (f,f)
append23_in: (b,b,b,f) (b,f,b,f) (f,b,f,f) (f,f,f,f)
le36_in: (b,b) (f,f) (f,b)
split51_in: (b,b,f,f) (f,f,f,f) (f,b,f,f)
gt67_in: (b,b) (f,f) (f,b)
qsort79_in: (f,f)
append94_in: (f,f,f,f)
p78_in: (f,f,f,b,f) (f,f,f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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

(150) Obligation:

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

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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)

(151) 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:

QSORT1_IN_GA(.(T14, []), T9) → U19_GA(T14, T9, qsort14_in_a(X12))
QSORT1_IN_GA(.(T14, []), T9) → QSORT14_IN_A(X12)
QSORT1_IN_GA(.(T14, []), T9) → U20_GA(T14, T9, qsort14_in_a(T17))
U20_GA(T14, T9, qsort14_out_a(T17)) → U21_GA(T14, T9, qsort14_in_a(X13))
U20_GA(T14, T9, qsort14_out_a(T17)) → QSORT14_IN_A(X13)
U20_GA(T14, T9, qsort14_out_a(T17)) → U22_GA(T14, T9, T17, qsort14_in_a(T20))
U22_GA(T14, T9, T17, qsort14_out_a(T20)) → U23_GA(T14, T9, append23_in_ggga(T17, T14, T20, T9))
U22_GA(T14, T9, T17, qsort14_out_a(T20)) → APPEND23_IN_GGGA(T17, T14, T20, T9)
APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → U1_GGGA(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GGGA(T46, T47, T48, T50)
QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → U24_GA(T69, T67, T68, T9, le36_in_gg(T67, T69))
QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → LE36_IN_GG(T67, T69)
LE36_IN_GG(s(T82), s(T83)) → U2_GG(T82, T83, le36_in_gg(T82, T83))
LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_GA(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → SPLIT51_IN_GGAA(T68, T69, X93, X94)
SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_GGAA(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_GG(T114, T116)
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_GGAA(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_GGAA(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_GG(T129, T131)
GT67_IN_GG(s(T144), s(T145)) → U7_GG(T144, T145, gt67_in_gg(T144, T145))
GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_GGAA(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_GA(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_GA(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)
QSORT1_IN_AA(.(T14, []), T9) → U19_AA(T14, T9, qsort14_in_a(X12))
QSORT1_IN_AA(.(T14, []), T9) → QSORT14_IN_A(X12)
QSORT1_IN_AA(.(T14, []), T9) → U20_AA(T14, T9, qsort14_in_a(T17))
U20_AA(T14, T9, qsort14_out_a(T17)) → U21_AA(T14, T9, qsort14_in_a(X13))
U20_AA(T14, T9, qsort14_out_a(T17)) → QSORT14_IN_A(X13)
U20_AA(T14, T9, qsort14_out_a(T17)) → U22_AA(T14, T9, T17, qsort14_in_a(T20))
U22_AA(T14, T9, T17, qsort14_out_a(T20)) → U23_AA(T14, T9, append23_in_gaga(T17, T14, T20, T9))
U22_AA(T14, T9, T17, qsort14_out_a(T20)) → APPEND23_IN_GAGA(T17, T14, T20, T9)
APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → U1_GAGA(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)
QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, le36_in_aa(T67, T69))
QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → LE36_IN_AA(T67, T69)
LE36_IN_AA(s(T82), s(T83)) → U2_AA(T82, T83, le36_in_aa(T82, T83))
LE36_IN_AA(s(T82), s(T83)) → LE36_IN_AA(T82, T83)
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → SPLIT51_IN_AAAA(T68, T69, X93, X94)
SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AAAA(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_AA(T114, T116)
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_AAAA(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → SPLIT51_IN_AAAA(T115, T116, X167, X168)
SPLIT51_IN_AAAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AAAA(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
SPLIT51_IN_AAAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_AA(T129, T131)
GT67_IN_AA(s(T144), s(T145)) → U7_AA(T144, T145, gt67_in_aa(T144, T145))
GT67_IN_AA(s(T144), s(T145)) → GT67_IN_AA(T144, T145)
U5_AAAA(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_AAAA(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U5_AAAA(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AGAA(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_AG(T114, T116)
LE36_IN_AG(s(T82), s(T83)) → U2_AG(T82, T83, le36_in_ag(T82, T83))
LE36_IN_AG(s(T82), s(T83)) → LE36_IN_AG(T82, T83)
U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_AGAA(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T115, T116, X167, X168)
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AGAA(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_AG(T129, T131)
GT67_IN_AG(s(T144), s(T145)) → U7_AG(T144, T145, gt67_in_ag(T144, T145))
GT67_IN_AG(s(T144), s(T145)) → GT67_IN_AG(T144, T145)
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_AGAA(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_AA(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)
QSORT1_IN_AA(.(T215, .(T213, T214)), T9) → U30_AA(T215, T213, T214, T9, gt67_in_aa(T213, T215))
QSORT1_IN_AA(.(T215, .(T213, T214)), T9) → GT67_IN_AA(T213, T215)
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_AA(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → SPLIT51_IN_AGAA(T214, T215, X323, X324)
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_AA(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_AA(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → QSORT79_IN_AA(T222, X12)
QSORT79_IN_AA(.(T163, T164), X265) → U8_AA(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
QSORT79_IN_AA(.(T163, T164), X265) → SPLIT51_IN_AAAA(T164, T163, X261, X262)
QSORT79_IN_AA(.(T163, T164), X265) → U9_AA(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_AA(T163, T164, X265, qsort79_in_aa(T167, X263))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → QSORT79_IN_AA(T167, X263)
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_AA(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_AA(T163, T164, X265, qsort79_in_aa(T168, X264))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → QSORT79_IN_AA(T168, X264)
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_AA(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_AA(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_AA(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
U13_AA(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → APPEND94_IN_AAAA(T171, T163, T172, X265)
APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → U15_AAAA(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_AAAA(T196, T197, T198, X295)
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_AA(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_AA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_AA(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U34_AA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → P78_IN_AAAGA(.(T213, T223), X13, T226, T215, T9)
P78_IN_AAAGA(T94, X13, T155, T69, T9) → U16_AAAGA(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
P78_IN_AAAGA(T94, X13, T155, T69, T9) → QSORT79_IN_AA(T94, X13)
P78_IN_AAAGA(T94, T158, T155, T69, T9) → U17_AAAGA(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_AAAGA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_AAAGA(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
U17_AAAGA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → APPEND23_IN_AGAA(T155, T69, T158, T9)
APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → U1_AGAA(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_AA(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_AA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_AA(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
U28_AA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → P78_IN_AAAAA(T94, X13, T155, T69, T9)
P78_IN_AAAAA(T94, X13, T155, T69, T9) → U16_AAAAA(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
P78_IN_AAAAA(T94, X13, T155, T69, T9) → QSORT79_IN_AA(T94, X13)
P78_IN_AAAAA(T94, T158, T155, T69, T9) → U17_AAAAA(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_AAAAA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_AAAAA(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
U17_AAAAA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → APPEND23_IN_AAAA(T155, T69, T158, T9)
APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → U1_AAAA(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_GA(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_GA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_GA(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U28_GA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → P78_IN_AAAGA(T94, X13, T155, T69, T9)
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → U30_GA(T215, T213, T214, T9, gt67_in_gg(T213, T215))
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → GT67_IN_GG(T213, T215)
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_GA(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → SPLIT51_IN_GGAA(T214, T215, X323, X324)
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_GA(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_GA(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → QSORT79_IN_AA(T222, X12)
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_GA(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_GA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_GA(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U34_GA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → P78_IN_AAAGA(.(T213, T223), X13, T226, T215, T9)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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)
QSORT1_IN_GA(x1, x2)  =  QSORT1_IN_GA(x1)
U19_GA(x1, x2, x3)  =  U19_GA(x1, x3)
QSORT14_IN_A(x1)  =  QSORT14_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)
LE36_IN_GG(x1, x2)  =  LE36_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)
SPLIT51_IN_GGAA(x1, x2, x3, x4)  =  SPLIT51_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)
GT67_IN_GG(x1, x2)  =  GT67_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)
QSORT1_IN_AA(x1, x2)  =  QSORT1_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)
LE36_IN_AA(x1, x2)  =  LE36_IN_AA
U2_AA(x1, x2, x3)  =  U2_AA(x3)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
SPLIT51_IN_AAAA(x1, x2, x3, x4)  =  SPLIT51_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)
GT67_IN_AA(x1, x2)  =  GT67_IN_AA
U7_AA(x1, x2, x3)  =  U7_AA(x3)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x6)
SPLIT51_IN_AGAA(x1, x2, x3, x4)  =  SPLIT51_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5, x6)  =  U3_AGAA(x3, x6)
LE36_IN_AG(x1, x2)  =  LE36_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
U4_AGAA(x1, x2, x3, x4, x5, x6)  =  U4_AGAA(x3, x6)
U5_AGAA(x1, x2, x3, x4, x5, x6)  =  U5_AGAA(x3, x6)
GT67_IN_AG(x1, x2)  =  GT67_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x2, x3)
U6_AGAA(x1, x2, x3, x4, x5, x6)  =  U6_AGAA(x3, 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(x1, x5)
U33_AA(x1, x2, x3, x4, x5)  =  U33_AA(x5)
QSORT79_IN_AA(x1, x2)  =  QSORT79_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(x1, x6)
U35_AA(x1, x2, x3, x4, x5)  =  U35_AA(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)
U28_AA(x1, x2, x3, x4, x5, x6)  =  U28_AA(x6)
U29_AA(x1, x2, x3, x4, x5)  =  U29_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_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)
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

(152) Obligation:

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

QSORT1_IN_GA(.(T14, []), T9) → U19_GA(T14, T9, qsort14_in_a(X12))
QSORT1_IN_GA(.(T14, []), T9) → QSORT14_IN_A(X12)
QSORT1_IN_GA(.(T14, []), T9) → U20_GA(T14, T9, qsort14_in_a(T17))
U20_GA(T14, T9, qsort14_out_a(T17)) → U21_GA(T14, T9, qsort14_in_a(X13))
U20_GA(T14, T9, qsort14_out_a(T17)) → QSORT14_IN_A(X13)
U20_GA(T14, T9, qsort14_out_a(T17)) → U22_GA(T14, T9, T17, qsort14_in_a(T20))
U22_GA(T14, T9, T17, qsort14_out_a(T20)) → U23_GA(T14, T9, append23_in_ggga(T17, T14, T20, T9))
U22_GA(T14, T9, T17, qsort14_out_a(T20)) → APPEND23_IN_GGGA(T17, T14, T20, T9)
APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → U1_GGGA(T45, T46, T47, T48, T50, append23_in_ggga(T46, T47, T48, T50))
APPEND23_IN_GGGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GGGA(T46, T47, T48, T50)
QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → U24_GA(T69, T67, T68, T9, le36_in_gg(T67, T69))
QSORT1_IN_GA(.(T69, .(T67, T68)), T9) → LE36_IN_GG(T67, T69)
LE36_IN_GG(s(T82), s(T83)) → U2_GG(T82, T83, le36_in_gg(T82, T83))
LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_GA(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → SPLIT51_IN_GGAA(T68, T69, X93, X94)
SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_GGAA(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_GG(T114, T116)
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_GGAA(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_GGAA(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_GG(T129, T131)
GT67_IN_GG(s(T144), s(T145)) → U7_GG(T144, T145, gt67_in_gg(T144, T145))
GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_GGAA(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)
U24_GA(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_GA(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_GA(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)
QSORT1_IN_AA(.(T14, []), T9) → U19_AA(T14, T9, qsort14_in_a(X12))
QSORT1_IN_AA(.(T14, []), T9) → QSORT14_IN_A(X12)
QSORT1_IN_AA(.(T14, []), T9) → U20_AA(T14, T9, qsort14_in_a(T17))
U20_AA(T14, T9, qsort14_out_a(T17)) → U21_AA(T14, T9, qsort14_in_a(X13))
U20_AA(T14, T9, qsort14_out_a(T17)) → QSORT14_IN_A(X13)
U20_AA(T14, T9, qsort14_out_a(T17)) → U22_AA(T14, T9, T17, qsort14_in_a(T20))
U22_AA(T14, T9, T17, qsort14_out_a(T20)) → U23_AA(T14, T9, append23_in_gaga(T17, T14, T20, T9))
U22_AA(T14, T9, T17, qsort14_out_a(T20)) → APPEND23_IN_GAGA(T17, T14, T20, T9)
APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → U1_GAGA(T45, T46, T47, T48, T50, append23_in_gaga(T46, T47, T48, T50))
APPEND23_IN_GAGA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_GAGA(T46, T47, T48, T50)
QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, le36_in_aa(T67, T69))
QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → LE36_IN_AA(T67, T69)
LE36_IN_AA(s(T82), s(T83)) → U2_AA(T82, T83, le36_in_aa(T82, T83))
LE36_IN_AA(s(T82), s(T83)) → LE36_IN_AA(T82, T83)
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → SPLIT51_IN_AAAA(T68, T69, X93, X94)
SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AAAA(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_AA(T114, T116)
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_AAAA(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → SPLIT51_IN_AAAA(T115, T116, X167, X168)
SPLIT51_IN_AAAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AAAA(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
SPLIT51_IN_AAAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_AA(T129, T131)
GT67_IN_AA(s(T144), s(T145)) → U7_AA(T144, T145, gt67_in_aa(T144, T145))
GT67_IN_AA(s(T144), s(T145)) → GT67_IN_AA(T144, T145)
U5_AAAA(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_AAAA(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U5_AAAA(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AGAA(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → LE36_IN_AG(T114, T116)
LE36_IN_AG(s(T82), s(T83)) → U2_AG(T82, T83, le36_in_ag(T82, T83))
LE36_IN_AG(s(T82), s(T83)) → LE36_IN_AG(T82, T83)
U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_AGAA(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T115, T116, X167, X168)
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AGAA(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → GT67_IN_AG(T129, T131)
GT67_IN_AG(s(T144), s(T145)) → U7_AG(T144, T145, gt67_in_ag(T144, T145))
GT67_IN_AG(s(T144), s(T145)) → GT67_IN_AG(T144, T145)
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_AGAA(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_AA(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)
QSORT1_IN_AA(.(T215, .(T213, T214)), T9) → U30_AA(T215, T213, T214, T9, gt67_in_aa(T213, T215))
QSORT1_IN_AA(.(T215, .(T213, T214)), T9) → GT67_IN_AA(T213, T215)
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_AA(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → SPLIT51_IN_AGAA(T214, T215, X323, X324)
U30_AA(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_AA(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_AA(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → QSORT79_IN_AA(T222, X12)
QSORT79_IN_AA(.(T163, T164), X265) → U8_AA(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
QSORT79_IN_AA(.(T163, T164), X265) → SPLIT51_IN_AAAA(T164, T163, X261, X262)
QSORT79_IN_AA(.(T163, T164), X265) → U9_AA(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_AA(T163, T164, X265, qsort79_in_aa(T167, X263))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → QSORT79_IN_AA(T167, X263)
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_AA(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_AA(T163, T164, X265, qsort79_in_aa(T168, X264))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → QSORT79_IN_AA(T168, X264)
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_AA(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_AA(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_AA(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
U13_AA(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → APPEND94_IN_AAAA(T171, T163, T172, X265)
APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → U15_AAAA(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_AAAA(T196, T197, T198, X295)
U32_AA(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_AA(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_AA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_AA(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U34_AA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → P78_IN_AAAGA(.(T213, T223), X13, T226, T215, T9)
P78_IN_AAAGA(T94, X13, T155, T69, T9) → U16_AAAGA(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
P78_IN_AAAGA(T94, X13, T155, T69, T9) → QSORT79_IN_AA(T94, X13)
P78_IN_AAAGA(T94, T158, T155, T69, T9) → U17_AAAGA(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_AAAGA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_AAAGA(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
U17_AAAGA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → APPEND23_IN_AGAA(T155, T69, T158, T9)
APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → U1_AGAA(T45, T46, T47, T48, T50, append23_in_agaa(T46, T47, T48, T50))
APPEND23_IN_AGAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AGAA(T46, T47, T48, T50)
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_AA(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_AA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_AA(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
U28_AA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → P78_IN_AAAAA(T94, X13, T155, T69, T9)
P78_IN_AAAAA(T94, X13, T155, T69, T9) → U16_AAAAA(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
P78_IN_AAAAA(T94, X13, T155, T69, T9) → QSORT79_IN_AA(T94, X13)
P78_IN_AAAAA(T94, T158, T155, T69, T9) → U17_AAAAA(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_AAAAA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_AAAAA(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
U17_AAAAA(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → APPEND23_IN_AAAA(T155, T69, T158, T9)
APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → U1_AAAA(T45, T46, T47, T48, T50, append23_in_aaaa(T46, T47, T48, T50))
APPEND23_IN_AAAA(.(T45, T46), T47, T48, .(T45, T50)) → APPEND23_IN_AAAA(T46, T47, T48, T50)
U26_GA(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_GA(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_GA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_GA(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U28_GA(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → P78_IN_AAAGA(T94, X13, T155, T69, T9)
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → U30_GA(T215, T213, T214, T9, gt67_in_gg(T213, T215))
QSORT1_IN_GA(.(T215, .(T213, T214)), T9) → GT67_IN_GG(T213, T215)
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_GA(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → SPLIT51_IN_GGAA(T214, T215, X323, X324)
U30_GA(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_GA(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_GA(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → QSORT79_IN_AA(T222, X12)
U32_GA(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_GA(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_GA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_GA(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U34_GA(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → P78_IN_AAAGA(.(T213, T223), X13, T226, T215, T9)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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)
QSORT1_IN_GA(x1, x2)  =  QSORT1_IN_GA(x1)
U19_GA(x1, x2, x3)  =  U19_GA(x1, x3)
QSORT14_IN_A(x1)  =  QSORT14_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)
LE36_IN_GG(x1, x2)  =  LE36_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)
SPLIT51_IN_GGAA(x1, x2, x3, x4)  =  SPLIT51_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)
GT67_IN_GG(x1, x2)  =  GT67_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)
QSORT1_IN_AA(x1, x2)  =  QSORT1_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)
LE36_IN_AA(x1, x2)  =  LE36_IN_AA
U2_AA(x1, x2, x3)  =  U2_AA(x3)
U25_AA(x1, x2, x3, x4, x5)  =  U25_AA(x5)
SPLIT51_IN_AAAA(x1, x2, x3, x4)  =  SPLIT51_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)
GT67_IN_AA(x1, x2)  =  GT67_IN_AA
U7_AA(x1, x2, x3)  =  U7_AA(x3)
U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x6)
SPLIT51_IN_AGAA(x1, x2, x3, x4)  =  SPLIT51_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5, x6)  =  U3_AGAA(x3, x6)
LE36_IN_AG(x1, x2)  =  LE36_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
U4_AGAA(x1, x2, x3, x4, x5, x6)  =  U4_AGAA(x3, x6)
U5_AGAA(x1, x2, x3, x4, x5, x6)  =  U5_AGAA(x3, x6)
GT67_IN_AG(x1, x2)  =  GT67_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x2, x3)
U6_AGAA(x1, x2, x3, x4, x5, x6)  =  U6_AGAA(x3, 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(x1, x5)
U33_AA(x1, x2, x3, x4, x5)  =  U33_AA(x5)
QSORT79_IN_AA(x1, x2)  =  QSORT79_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(x1, x6)
U35_AA(x1, x2, x3, x4, x5)  =  U35_AA(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)
U28_AA(x1, x2, x3, x4, x5, x6)  =  U28_AA(x6)
U29_AA(x1, x2, x3, x4, x5)  =  U29_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_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)
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

(153) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 16 SCCs with 95 less nodes.

(154) Complex Obligation (AND)

(155) Obligation:

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

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

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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

(156) UsableRulesProof (EQUIVALENT transformation)

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

(157) Obligation:

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

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

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

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

(158) PiDPToQDPProof (SOUND transformation)

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

(159) 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.

(160) 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.



(161) NO

(162) Obligation:

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

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

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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

(163) UsableRulesProof (EQUIVALENT transformation)

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

(164) Obligation:

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

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

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

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

(165) PiDPToQDPProof (SOUND transformation)

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

(166) 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.

(167) 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).



(168) NO

(169) Obligation:

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

APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_AAAA(T196, T197, T198, X295)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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

(170) UsableRulesProof (EQUIVALENT transformation)

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

(171) Obligation:

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

APPEND94_IN_AAAA(.(T195, T196), T197, T198, .(T195, X295)) → APPEND94_IN_AAAA(T196, T197, T198, X295)

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

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

(172) PiDPToQDPProof (SOUND transformation)

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

(173) 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.

(174) 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.



(175) NO

(176) Obligation:

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

GT67_IN_AG(s(T144), s(T145)) → GT67_IN_AG(T144, T145)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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)
GT67_IN_AG(x1, x2)  =  GT67_IN_AG(x2)

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

(177) UsableRulesProof (EQUIVALENT transformation)

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

(178) Obligation:

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

GT67_IN_AG(s(T144), s(T145)) → GT67_IN_AG(T144, T145)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT67_IN_AG(x1, x2)  =  GT67_IN_AG(x2)

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

(179) PiDPToQDPProof (SOUND transformation)

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

(180) Obligation:

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

GT67_IN_AG(s(T145)) → GT67_IN_AG(T145)

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

(181) 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:

  • GT67_IN_AG(s(T145)) → GT67_IN_AG(T145)
    The graph contains the following edges 1 > 1

(182) YES

(183) Obligation:

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

LE36_IN_AG(s(T82), s(T83)) → LE36_IN_AG(T82, T83)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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)
LE36_IN_AG(x1, x2)  =  LE36_IN_AG(x2)

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

(184) UsableRulesProof (EQUIVALENT transformation)

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

(185) Obligation:

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

LE36_IN_AG(s(T82), s(T83)) → LE36_IN_AG(T82, T83)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE36_IN_AG(x1, x2)  =  LE36_IN_AG(x2)

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

(186) PiDPToQDPProof (SOUND transformation)

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

(187) Obligation:

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

LE36_IN_AG(s(T83)) → LE36_IN_AG(T83)

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

(188) 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:

  • LE36_IN_AG(s(T83)) → LE36_IN_AG(T83)
    The graph contains the following edges 1 > 1

(189) YES

(190) Obligation:

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

U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T115, T116, X167, X168)
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AGAA(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AGAA(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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)
SPLIT51_IN_AGAA(x1, x2, x3, x4)  =  SPLIT51_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5, x6)  =  U3_AGAA(x3, x6)
U5_AGAA(x1, x2, x3, x4, x5, x6)  =  U5_AGAA(x3, x6)

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

(191) UsableRulesProof (EQUIVALENT transformation)

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

(192) Obligation:

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

U3_AGAA(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T115, T116, X167, X168)
SPLIT51_IN_AGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AGAA(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
SPLIT51_IN_AGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_AGAA(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
U5_AGAA(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → SPLIT51_IN_AGAA(T130, T131, X199, X200)

The TRS R consists of the following rules:

le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
SPLIT51_IN_AGAA(x1, x2, x3, x4)  =  SPLIT51_IN_AGAA(x2)
U3_AGAA(x1, x2, x3, x4, x5, x6)  =  U3_AGAA(x3, x6)
U5_AGAA(x1, x2, x3, x4, x5, x6)  =  U5_AGAA(x3, x6)

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

(193) PiDPToQDPProof (SOUND transformation)

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

(194) Obligation:

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

U3_AGAA(T116, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T116)
SPLIT51_IN_AGAA(T116) → U3_AGAA(T116, le36_in_ag(T116))
SPLIT51_IN_AGAA(T131) → U5_AGAA(T131, gt67_in_ag(T131))
U5_AGAA(T131, gt67_out_ag(T131)) → SPLIT51_IN_AGAA(T131)

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(195) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SPLIT51_IN_AGAA(T116) → U3_AGAA(T116, le36_in_ag(T116)) at position [1] we obtained the following new rules [LPAR04]:

SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(x0, le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0, s(x0)))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0))

(196) Obligation:

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

U3_AGAA(T116, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T116)
SPLIT51_IN_AGAA(T131) → U5_AGAA(T131, gt67_in_ag(T131))
U5_AGAA(T131, gt67_out_ag(T131)) → SPLIT51_IN_AGAA(T131)
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(x0, le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0, s(x0)))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0))

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(197) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SPLIT51_IN_AGAA(T131) → U5_AGAA(T131, gt67_in_ag(T131)) at position [1] we obtained the following new rules [LPAR04]:

SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(x0, gt67_in_ag(x0)))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag(0))

(198) Obligation:

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

U3_AGAA(T116, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T116)
U5_AGAA(T131, gt67_out_ag(T131)) → SPLIT51_IN_AGAA(T131)
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(x0, le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0, s(x0)))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0))
SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(x0, gt67_in_ag(x0)))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag(0))

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(199) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_AGAA(T116, le36_out_ag(T114, T116)) → SPLIT51_IN_AGAA(T116) we obtained the following new rules [LPAR04]:

U3_AGAA(s(z0), le36_out_ag(x1, s(z0))) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(s(z0), le36_out_ag(0, s(z0))) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(0, le36_out_ag(0, 0)) → SPLIT51_IN_AGAA(0)

(200) Obligation:

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

U5_AGAA(T131, gt67_out_ag(T131)) → SPLIT51_IN_AGAA(T131)
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(x0, le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0, s(x0)))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0))
SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(x0, gt67_in_ag(x0)))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag(0))
U3_AGAA(s(z0), le36_out_ag(x1, s(z0))) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(s(z0), le36_out_ag(0, s(z0))) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(0, le36_out_ag(0, 0)) → SPLIT51_IN_AGAA(0)

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(201) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U5_AGAA(T131, gt67_out_ag(T131)) → SPLIT51_IN_AGAA(T131) we obtained the following new rules [LPAR04]:

U5_AGAA(s(z0), gt67_out_ag(s(z0))) → SPLIT51_IN_AGAA(s(z0))
U5_AGAA(0, gt67_out_ag(0)) → SPLIT51_IN_AGAA(0)

(202) Obligation:

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

SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(x0, le36_in_ag(x0)))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0, s(x0)))
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0))
SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(x0, gt67_in_ag(x0)))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag(0))
U3_AGAA(s(z0), le36_out_ag(x1, s(z0))) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(s(z0), le36_out_ag(0, s(z0))) → SPLIT51_IN_AGAA(s(z0))
U3_AGAA(0, le36_out_ag(0, 0)) → SPLIT51_IN_AGAA(0)
U5_AGAA(s(z0), gt67_out_ag(s(z0))) → SPLIT51_IN_AGAA(s(z0))
U5_AGAA(0, gt67_out_ag(0)) → SPLIT51_IN_AGAA(0)

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(203) DependencyGraphProof (EQUIVALENT transformation)

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

(204) Complex Obligation (AND)

(205) Obligation:

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

U3_AGAA(0, le36_out_ag(0, 0)) → SPLIT51_IN_AGAA(0)
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag(0))
U5_AGAA(0, gt67_out_ag(0)) → SPLIT51_IN_AGAA(0)

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(206) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(207) Obligation:

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

U3_AGAA(0, le36_out_ag(0, 0)) → SPLIT51_IN_AGAA(0)
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag(0))
U5_AGAA(0, gt67_out_ag(0)) → SPLIT51_IN_AGAA(0)

R is empty.
The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(208) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0, x1)
U7_ag(x0, x1)

(209) Obligation:

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

U3_AGAA(0, le36_out_ag(0, 0)) → SPLIT51_IN_AGAA(0)
SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0))
SPLIT51_IN_AGAA(0) → U5_AGAA(0, gt67_out_ag(0))
U5_AGAA(0, gt67_out_ag(0)) → SPLIT51_IN_AGAA(0)

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

(210) 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 = SPLIT51_IN_AGAA(0) evaluates to t =SPLIT51_IN_AGAA(0)

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




Rewriting sequence

SPLIT51_IN_AGAA(0)U3_AGAA(0, le36_out_ag(0, 0))
with rule SPLIT51_IN_AGAA(0) → U3_AGAA(0, le36_out_ag(0, 0)) at position [] and matcher [ ]

U3_AGAA(0, le36_out_ag(0, 0))SPLIT51_IN_AGAA(0)
with rule U3_AGAA(0, le36_out_ag(0, 0)) → SPLIT51_IN_AGAA(0)

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.



(211) NO

(212) Obligation:

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

U3_AGAA(s(z0), le36_out_ag(x1, s(z0))) → SPLIT51_IN_AGAA(s(z0))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), U2_ag(x0, le36_in_ag(x0)))
U3_AGAA(s(z0), le36_out_ag(0, s(z0))) → SPLIT51_IN_AGAA(s(z0))
SPLIT51_IN_AGAA(s(x0)) → U3_AGAA(s(x0), le36_out_ag(0, s(x0)))
SPLIT51_IN_AGAA(s(x0)) → U5_AGAA(s(x0), U7_ag(x0, gt67_in_ag(x0)))
U5_AGAA(s(z0), gt67_out_ag(s(z0))) → SPLIT51_IN_AGAA(s(z0))

The TRS R consists of the following rules:

le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_ag(x0)
gt67_in_ag(x0)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(213) 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 = SPLIT51_IN_AGAA(s(x0)) evaluates to t =SPLIT51_IN_AGAA(s(x0))

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




Rewriting sequence

SPLIT51_IN_AGAA(s(x0))U3_AGAA(s(x0), le36_out_ag(0, s(x0)))
with rule SPLIT51_IN_AGAA(s(x0')) → U3_AGAA(s(x0'), le36_out_ag(0, s(x0'))) at position [] and matcher [x0' / x0]

U3_AGAA(s(x0), le36_out_ag(0, s(x0)))SPLIT51_IN_AGAA(s(x0))
with rule U3_AGAA(s(z0), le36_out_ag(x1, s(z0))) → SPLIT51_IN_AGAA(s(z0))

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.



(214) NO

(215) Obligation:

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

GT67_IN_AA(s(T144), s(T145)) → GT67_IN_AA(T144, T145)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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)
GT67_IN_AA(x1, x2)  =  GT67_IN_AA

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

(216) UsableRulesProof (EQUIVALENT transformation)

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

(217) Obligation:

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

GT67_IN_AA(s(T144), s(T145)) → GT67_IN_AA(T144, T145)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT67_IN_AA(x1, x2)  =  GT67_IN_AA

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

(218) PiDPToQDPProof (SOUND transformation)

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

(219) Obligation:

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

GT67_IN_AAGT67_IN_AA

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

(220) 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 = GT67_IN_AA evaluates to t =GT67_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 GT67_IN_AA to GT67_IN_AA.



(221) NO

(222) Obligation:

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

LE36_IN_AA(s(T82), s(T83)) → LE36_IN_AA(T82, T83)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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)
LE36_IN_AA(x1, x2)  =  LE36_IN_AA

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

(223) UsableRulesProof (EQUIVALENT transformation)

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

(224) Obligation:

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

LE36_IN_AA(s(T82), s(T83)) → LE36_IN_AA(T82, T83)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE36_IN_AA(x1, x2)  =  LE36_IN_AA

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

(225) PiDPToQDPProof (SOUND transformation)

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

(226) Obligation:

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

LE36_IN_AALE36_IN_AA

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

(227) 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 = LE36_IN_AA evaluates to t =LE36_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 LE36_IN_AA to LE36_IN_AA.



(228) NO

(229) Obligation:

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

SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AAAA(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → SPLIT51_IN_AAAA(T115, T116, X167, X168)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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)
SPLIT51_IN_AAAA(x1, x2, x3, x4)  =  SPLIT51_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)

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

(230) UsableRulesProof (EQUIVALENT transformation)

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

(231) Obligation:

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

SPLIT51_IN_AAAA(.(T114, T115), T116, .(T114, X167), X168) → U3_AAAA(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_AAAA(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → SPLIT51_IN_AAAA(T115, T116, X167, X168)

The TRS R consists of the following rules:

le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
SPLIT51_IN_AAAA(x1, x2, x3, x4)  =  SPLIT51_IN_AAAA
U3_AAAA(x1, x2, x3, x4, x5, x6)  =  U3_AAAA(x6)

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

(232) PiDPToQDPProof (SOUND transformation)

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

(233) Obligation:

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

SPLIT51_IN_AAAAU3_AAAA(le36_in_aa)
U3_AAAA(le36_out_aa(T114)) → SPLIT51_IN_AAAA

The TRS R consists of the following rules:

le36_in_aaU2_aa(le36_in_aa)
le36_in_aale36_out_aa(0)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))

The set Q consists of the following terms:

le36_in_aa
U2_aa(x0)

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

(234) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SPLIT51_IN_AAAAU3_AAAA(le36_in_aa) at position [0] we obtained the following new rules [LPAR04]:

SPLIT51_IN_AAAAU3_AAAA(U2_aa(le36_in_aa))
SPLIT51_IN_AAAAU3_AAAA(le36_out_aa(0))

(235) Obligation:

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

U3_AAAA(le36_out_aa(T114)) → SPLIT51_IN_AAAA
SPLIT51_IN_AAAAU3_AAAA(U2_aa(le36_in_aa))
SPLIT51_IN_AAAAU3_AAAA(le36_out_aa(0))

The TRS R consists of the following rules:

le36_in_aaU2_aa(le36_in_aa)
le36_in_aale36_out_aa(0)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))

The set Q consists of the following terms:

le36_in_aa
U2_aa(x0)

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

(236) 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 = SPLIT51_IN_AAAA evaluates to t =SPLIT51_IN_AAAA

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




Rewriting sequence

SPLIT51_IN_AAAAU3_AAAA(le36_out_aa(0))
with rule SPLIT51_IN_AAAAU3_AAAA(le36_out_aa(0)) at position [] and matcher [ ]

U3_AAAA(le36_out_aa(0))SPLIT51_IN_AAAA
with rule U3_AAAA(le36_out_aa(T114)) → SPLIT51_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.



(237) NO

(238) Obligation:

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

QSORT79_IN_AA(.(T163, T164), X265) → U9_AA(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → QSORT79_IN_AA(T167, X263)
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_AA(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → QSORT79_IN_AA(T168, X264)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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)
QSORT79_IN_AA(x1, x2)  =  QSORT79_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

(239) UsableRulesProof (EQUIVALENT transformation)

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

(240) Obligation:

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

QSORT79_IN_AA(.(T163, T164), X265) → U9_AA(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → QSORT79_IN_AA(T167, X263)
U9_AA(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_AA(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_AA(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → QSORT79_IN_AA(T168, X264)

The TRS R consists of the following rules:

split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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)
QSORT79_IN_AA(x1, x2)  =  QSORT79_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

(241) PiDPToQDPProof (SOUND transformation)

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

(242) Obligation:

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

QSORT79_IN_AAU9_AA(split51_in_aaaa)
U9_AA(split51_out_aaaa) → QSORT79_IN_AA
U9_AA(split51_out_aaaa) → U11_AA(qsort79_in_aa)
U11_AA(qsort79_out_aa) → QSORT79_IN_AA

The TRS R consists of the following rules:

split51_in_aaaasplit51_out_aaaa
split51_in_aaaaU3_aaaa(le36_in_aa)
split51_in_aaaaU5_aaaa(gt67_in_aa)
qsort79_in_aaqsort79_out_aa
qsort79_in_aaU8_aa(split51_in_aaaa)
qsort79_in_aaU9_aa(split51_in_aaaa)
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U8_aa(split51_out_aaaa) → qsort79_out_aa
U9_aa(split51_out_aaaa) → U10_aa(qsort79_in_aa)
U9_aa(split51_out_aaaa) → U11_aa(qsort79_in_aa)
le36_in_aaU2_aa(le36_in_aa)
le36_in_aale36_out_aa(0)
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aaU7_aa(gt67_in_aa)
gt67_in_aagt67_out_aa(0)
U6_aaaa(split51_out_agaa(T131)) → split51_out_aaaa
U10_aa(qsort79_out_aa) → qsort79_out_aa
U11_aa(qsort79_out_aa) → U12_aa(qsort79_in_aa)
U11_aa(qsort79_out_aa) → U13_aa(qsort79_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa(T101)
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U12_aa(qsort79_out_aa) → qsort79_out_aa
U13_aa(qsort79_out_aa) → U14_aa(append94_in_aaaa)
U3_agaa(T116, le36_out_ag(T114, T116)) → split51_out_agaa(T116)
U3_agaa(T116, le36_out_ag(T114, T116)) → U4_agaa(T116, split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag(T131)) → split51_out_agaa(T131)
U5_agaa(T131, gt67_out_ag(T131)) → U6_agaa(T131, split51_in_agaa(T131))
U14_aa(append94_out_aaaa) → qsort79_out_aa
le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
U4_agaa(T116, split51_out_agaa(T116)) → split51_out_agaa(T116)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U6_agaa(T131, split51_out_agaa(T131)) → split51_out_agaa(T131)
append94_in_aaaaappend94_out_aaaa
append94_in_aaaaU15_aaaa(append94_in_aaaa)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))
U15_aaaa(append94_out_aaaa) → append94_out_aaaa

The set Q consists of the following terms:

split51_in_aaaa
qsort79_in_aa
U3_aaaa(x0)
U5_aaaa(x0)
U8_aa(x0)
U9_aa(x0)
le36_in_aa
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U10_aa(x0)
U11_aa(x0)
U2_aa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U12_aa(x0)
U13_aa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
U14_aa(x0)
le36_in_ag(x0)
U4_agaa(x0, x1)
gt67_in_ag(x0)
U6_agaa(x0, x1)
append94_in_aaaa
U2_ag(x0, x1)
U7_ag(x0, x1)
U15_aaaa(x0)

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

(243) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QSORT79_IN_AAU9_AA(split51_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

QSORT79_IN_AAU9_AA(split51_out_aaaa)
QSORT79_IN_AAU9_AA(U3_aaaa(le36_in_aa))
QSORT79_IN_AAU9_AA(U5_aaaa(gt67_in_aa))

(244) Obligation:

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

U9_AA(split51_out_aaaa) → QSORT79_IN_AA
U9_AA(split51_out_aaaa) → U11_AA(qsort79_in_aa)
U11_AA(qsort79_out_aa) → QSORT79_IN_AA
QSORT79_IN_AAU9_AA(split51_out_aaaa)
QSORT79_IN_AAU9_AA(U3_aaaa(le36_in_aa))
QSORT79_IN_AAU9_AA(U5_aaaa(gt67_in_aa))

The TRS R consists of the following rules:

split51_in_aaaasplit51_out_aaaa
split51_in_aaaaU3_aaaa(le36_in_aa)
split51_in_aaaaU5_aaaa(gt67_in_aa)
qsort79_in_aaqsort79_out_aa
qsort79_in_aaU8_aa(split51_in_aaaa)
qsort79_in_aaU9_aa(split51_in_aaaa)
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U8_aa(split51_out_aaaa) → qsort79_out_aa
U9_aa(split51_out_aaaa) → U10_aa(qsort79_in_aa)
U9_aa(split51_out_aaaa) → U11_aa(qsort79_in_aa)
le36_in_aaU2_aa(le36_in_aa)
le36_in_aale36_out_aa(0)
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aaU7_aa(gt67_in_aa)
gt67_in_aagt67_out_aa(0)
U6_aaaa(split51_out_agaa(T131)) → split51_out_aaaa
U10_aa(qsort79_out_aa) → qsort79_out_aa
U11_aa(qsort79_out_aa) → U12_aa(qsort79_in_aa)
U11_aa(qsort79_out_aa) → U13_aa(qsort79_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa(T101)
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U12_aa(qsort79_out_aa) → qsort79_out_aa
U13_aa(qsort79_out_aa) → U14_aa(append94_in_aaaa)
U3_agaa(T116, le36_out_ag(T114, T116)) → split51_out_agaa(T116)
U3_agaa(T116, le36_out_ag(T114, T116)) → U4_agaa(T116, split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag(T131)) → split51_out_agaa(T131)
U5_agaa(T131, gt67_out_ag(T131)) → U6_agaa(T131, split51_in_agaa(T131))
U14_aa(append94_out_aaaa) → qsort79_out_aa
le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
U4_agaa(T116, split51_out_agaa(T116)) → split51_out_agaa(T116)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U6_agaa(T131, split51_out_agaa(T131)) → split51_out_agaa(T131)
append94_in_aaaaappend94_out_aaaa
append94_in_aaaaU15_aaaa(append94_in_aaaa)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))
U15_aaaa(append94_out_aaaa) → append94_out_aaaa

The set Q consists of the following terms:

split51_in_aaaa
qsort79_in_aa
U3_aaaa(x0)
U5_aaaa(x0)
U8_aa(x0)
U9_aa(x0)
le36_in_aa
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U10_aa(x0)
U11_aa(x0)
U2_aa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U12_aa(x0)
U13_aa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
U14_aa(x0)
le36_in_ag(x0)
U4_agaa(x0, x1)
gt67_in_ag(x0)
U6_agaa(x0, x1)
append94_in_aaaa
U2_ag(x0, x1)
U7_ag(x0, x1)
U15_aaaa(x0)

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

(245) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U9_AA(split51_out_aaaa) → U11_AA(qsort79_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U9_AA(split51_out_aaaa) → U11_AA(qsort79_out_aa)
U9_AA(split51_out_aaaa) → U11_AA(U8_aa(split51_in_aaaa))
U9_AA(split51_out_aaaa) → U11_AA(U9_aa(split51_in_aaaa))

(246) Obligation:

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

U9_AA(split51_out_aaaa) → QSORT79_IN_AA
U11_AA(qsort79_out_aa) → QSORT79_IN_AA
QSORT79_IN_AAU9_AA(split51_out_aaaa)
QSORT79_IN_AAU9_AA(U3_aaaa(le36_in_aa))
QSORT79_IN_AAU9_AA(U5_aaaa(gt67_in_aa))
U9_AA(split51_out_aaaa) → U11_AA(qsort79_out_aa)
U9_AA(split51_out_aaaa) → U11_AA(U8_aa(split51_in_aaaa))
U9_AA(split51_out_aaaa) → U11_AA(U9_aa(split51_in_aaaa))

The TRS R consists of the following rules:

split51_in_aaaasplit51_out_aaaa
split51_in_aaaaU3_aaaa(le36_in_aa)
split51_in_aaaaU5_aaaa(gt67_in_aa)
qsort79_in_aaqsort79_out_aa
qsort79_in_aaU8_aa(split51_in_aaaa)
qsort79_in_aaU9_aa(split51_in_aaaa)
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U8_aa(split51_out_aaaa) → qsort79_out_aa
U9_aa(split51_out_aaaa) → U10_aa(qsort79_in_aa)
U9_aa(split51_out_aaaa) → U11_aa(qsort79_in_aa)
le36_in_aaU2_aa(le36_in_aa)
le36_in_aale36_out_aa(0)
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aaU7_aa(gt67_in_aa)
gt67_in_aagt67_out_aa(0)
U6_aaaa(split51_out_agaa(T131)) → split51_out_aaaa
U10_aa(qsort79_out_aa) → qsort79_out_aa
U11_aa(qsort79_out_aa) → U12_aa(qsort79_in_aa)
U11_aa(qsort79_out_aa) → U13_aa(qsort79_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa(T101)
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U12_aa(qsort79_out_aa) → qsort79_out_aa
U13_aa(qsort79_out_aa) → U14_aa(append94_in_aaaa)
U3_agaa(T116, le36_out_ag(T114, T116)) → split51_out_agaa(T116)
U3_agaa(T116, le36_out_ag(T114, T116)) → U4_agaa(T116, split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag(T131)) → split51_out_agaa(T131)
U5_agaa(T131, gt67_out_ag(T131)) → U6_agaa(T131, split51_in_agaa(T131))
U14_aa(append94_out_aaaa) → qsort79_out_aa
le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
U4_agaa(T116, split51_out_agaa(T116)) → split51_out_agaa(T116)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U6_agaa(T131, split51_out_agaa(T131)) → split51_out_agaa(T131)
append94_in_aaaaappend94_out_aaaa
append94_in_aaaaU15_aaaa(append94_in_aaaa)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))
U15_aaaa(append94_out_aaaa) → append94_out_aaaa

The set Q consists of the following terms:

split51_in_aaaa
qsort79_in_aa
U3_aaaa(x0)
U5_aaaa(x0)
U8_aa(x0)
U9_aa(x0)
le36_in_aa
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U10_aa(x0)
U11_aa(x0)
U2_aa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U12_aa(x0)
U13_aa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
U14_aa(x0)
le36_in_ag(x0)
U4_agaa(x0, x1)
gt67_in_ag(x0)
U6_agaa(x0, x1)
append94_in_aaaa
U2_ag(x0, x1)
U7_ag(x0, x1)
U15_aaaa(x0)

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

(247) 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 = QSORT79_IN_AA evaluates to t =QSORT79_IN_AA

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




Rewriting sequence

QSORT79_IN_AAU9_AA(split51_out_aaaa)
with rule QSORT79_IN_AAU9_AA(split51_out_aaaa) at position [] and matcher [ ]

U9_AA(split51_out_aaaa)QSORT79_IN_AA
with rule U9_AA(split51_out_aaaa) → QSORT79_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.



(248) NO

(249) Obligation:

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

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

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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

(250) UsableRulesProof (EQUIVALENT transformation)

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

(251) Obligation:

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

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

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

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

(252) PiDPToQDPProof (SOUND transformation)

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

(253) Obligation:

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

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

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

(254) 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(.(T45, T46), T48) → APPEND23_IN_GAGA(T46, T48)
    The graph contains the following edges 1 > 1, 2 >= 2

(255) YES

(256) Obligation:

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

QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, le36_in_aa(T67, T69))
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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)
QSORT1_IN_AA(x1, x2)  =  QSORT1_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

(257) UsableRulesProof (EQUIVALENT transformation)

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

(258) Obligation:

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

QSORT1_IN_AA(.(T69, .(T67, T68)), T9) → U24_AA(T69, T67, T68, T9, le36_in_aa(T67, T69))
U24_AA(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_AA(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_AA(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → QSORT1_IN_AA(.(T67, T93), X12)

The TRS R consists of the following rules:

le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, x6)
QSORT1_IN_AA(x1, x2)  =  QSORT1_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

(259) PiDPToQDPProof (SOUND transformation)

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

(260) Obligation:

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

QSORT1_IN_AAU24_AA(le36_in_aa)
U24_AA(le36_out_aa(T67)) → U26_AA(split51_in_aaaa)
U26_AA(split51_out_aaaa) → QSORT1_IN_AA

The TRS R consists of the following rules:

le36_in_aaU2_aa(le36_in_aa)
le36_in_aale36_out_aa(0)
split51_in_aaaasplit51_out_aaaa
split51_in_aaaaU3_aaaa(le36_in_aa)
split51_in_aaaaU5_aaaa(gt67_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aaU7_aa(gt67_in_aa)
gt67_in_aagt67_out_aa(0)
U6_aaaa(split51_out_agaa(T131)) → split51_out_aaaa
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa(T101)
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U3_agaa(T116, le36_out_ag(T114, T116)) → split51_out_agaa(T116)
U3_agaa(T116, le36_out_ag(T114, T116)) → U4_agaa(T116, split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag(T131)) → split51_out_agaa(T131)
U5_agaa(T131, gt67_out_ag(T131)) → U6_agaa(T131, split51_in_agaa(T131))
le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
U4_agaa(T116, split51_out_agaa(T116)) → split51_out_agaa(T116)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U6_agaa(T131, split51_out_agaa(T131)) → split51_out_agaa(T131)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_aa
split51_in_aaaa
U2_aa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
le36_in_ag(x0)
U4_agaa(x0, x1)
gt67_in_ag(x0)
U6_agaa(x0, x1)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(261) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule QSORT1_IN_AAU24_AA(le36_in_aa) at position [0] we obtained the following new rules [LPAR04]:

QSORT1_IN_AAU24_AA(U2_aa(le36_in_aa))
QSORT1_IN_AAU24_AA(le36_out_aa(0))

(262) Obligation:

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

U24_AA(le36_out_aa(T67)) → U26_AA(split51_in_aaaa)
U26_AA(split51_out_aaaa) → QSORT1_IN_AA
QSORT1_IN_AAU24_AA(U2_aa(le36_in_aa))
QSORT1_IN_AAU24_AA(le36_out_aa(0))

The TRS R consists of the following rules:

le36_in_aaU2_aa(le36_in_aa)
le36_in_aale36_out_aa(0)
split51_in_aaaasplit51_out_aaaa
split51_in_aaaaU3_aaaa(le36_in_aa)
split51_in_aaaaU5_aaaa(gt67_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aaU7_aa(gt67_in_aa)
gt67_in_aagt67_out_aa(0)
U6_aaaa(split51_out_agaa(T131)) → split51_out_aaaa
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa(T101)
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U3_agaa(T116, le36_out_ag(T114, T116)) → split51_out_agaa(T116)
U3_agaa(T116, le36_out_ag(T114, T116)) → U4_agaa(T116, split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag(T131)) → split51_out_agaa(T131)
U5_agaa(T131, gt67_out_ag(T131)) → U6_agaa(T131, split51_in_agaa(T131))
le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
U4_agaa(T116, split51_out_agaa(T116)) → split51_out_agaa(T116)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U6_agaa(T131, split51_out_agaa(T131)) → split51_out_agaa(T131)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_aa
split51_in_aaaa
U2_aa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
le36_in_ag(x0)
U4_agaa(x0, x1)
gt67_in_ag(x0)
U6_agaa(x0, x1)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(263) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U24_AA(le36_out_aa(T67)) → U26_AA(split51_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

U24_AA(le36_out_aa(y0)) → U26_AA(split51_out_aaaa)
U24_AA(le36_out_aa(y0)) → U26_AA(U3_aaaa(le36_in_aa))
U24_AA(le36_out_aa(y0)) → U26_AA(U5_aaaa(gt67_in_aa))

(264) Obligation:

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

U26_AA(split51_out_aaaa) → QSORT1_IN_AA
QSORT1_IN_AAU24_AA(U2_aa(le36_in_aa))
QSORT1_IN_AAU24_AA(le36_out_aa(0))
U24_AA(le36_out_aa(y0)) → U26_AA(split51_out_aaaa)
U24_AA(le36_out_aa(y0)) → U26_AA(U3_aaaa(le36_in_aa))
U24_AA(le36_out_aa(y0)) → U26_AA(U5_aaaa(gt67_in_aa))

The TRS R consists of the following rules:

le36_in_aaU2_aa(le36_in_aa)
le36_in_aale36_out_aa(0)
split51_in_aaaasplit51_out_aaaa
split51_in_aaaaU3_aaaa(le36_in_aa)
split51_in_aaaaU5_aaaa(gt67_in_aa)
U2_aa(le36_out_aa(T82)) → le36_out_aa(s(T82))
U3_aaaa(le36_out_aa(T114)) → split51_out_aaaa
U3_aaaa(le36_out_aa(T114)) → U4_aaaa(split51_in_aaaa)
U5_aaaa(gt67_out_aa(T131)) → split51_out_aaaa
U5_aaaa(gt67_out_aa(T131)) → U6_aaaa(split51_in_agaa(T131))
U4_aaaa(split51_out_aaaa) → split51_out_aaaa
gt67_in_aaU7_aa(gt67_in_aa)
gt67_in_aagt67_out_aa(0)
U6_aaaa(split51_out_agaa(T131)) → split51_out_aaaa
U7_aa(gt67_out_aa(T145)) → gt67_out_aa(s(T145))
split51_in_agaa(T101) → split51_out_agaa(T101)
split51_in_agaa(T116) → U3_agaa(T116, le36_in_ag(T116))
split51_in_agaa(T131) → U5_agaa(T131, gt67_in_ag(T131))
U3_agaa(T116, le36_out_ag(T114, T116)) → split51_out_agaa(T116)
U3_agaa(T116, le36_out_ag(T114, T116)) → U4_agaa(T116, split51_in_agaa(T116))
U5_agaa(T131, gt67_out_ag(T131)) → split51_out_agaa(T131)
U5_agaa(T131, gt67_out_ag(T131)) → U6_agaa(T131, split51_in_agaa(T131))
le36_in_ag(s(T83)) → U2_ag(T83, le36_in_ag(T83))
le36_in_ag(s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0) → le36_out_ag(0, 0)
U4_agaa(T116, split51_out_agaa(T116)) → split51_out_agaa(T116)
gt67_in_ag(s(T145)) → U7_ag(T145, gt67_in_ag(T145))
gt67_in_ag(0) → gt67_out_ag(0)
U6_agaa(T131, split51_out_agaa(T131)) → split51_out_agaa(T131)
U2_ag(T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U7_ag(T145, gt67_out_ag(T145)) → gt67_out_ag(s(T145))

The set Q consists of the following terms:

le36_in_aa
split51_in_aaaa
U2_aa(x0)
U3_aaaa(x0)
U5_aaaa(x0)
U4_aaaa(x0)
gt67_in_aa
U6_aaaa(x0)
U7_aa(x0)
split51_in_agaa(x0)
U3_agaa(x0, x1)
U5_agaa(x0, x1)
le36_in_ag(x0)
U4_agaa(x0, x1)
gt67_in_ag(x0)
U6_agaa(x0, x1)
U2_ag(x0, x1)
U7_ag(x0, x1)

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

(265) 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 = QSORT1_IN_AA evaluates to t =QSORT1_IN_AA

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




Rewriting sequence

QSORT1_IN_AAU24_AA(le36_out_aa(0))
with rule QSORT1_IN_AAU24_AA(le36_out_aa(0)) at position [] and matcher [ ]

U24_AA(le36_out_aa(0))U26_AA(split51_out_aaaa)
with rule U24_AA(le36_out_aa(y0)) → U26_AA(split51_out_aaaa) at position [] and matcher [y0 / 0]

U26_AA(split51_out_aaaa)QSORT1_IN_AA
with rule U26_AA(split51_out_aaaa) → QSORT1_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.



(266) NO

(267) Obligation:

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

GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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)
GT67_IN_GG(x1, x2)  =  GT67_IN_GG(x1, x2)

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

(268) UsableRulesProof (EQUIVALENT transformation)

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

(269) Obligation:

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

GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)

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

(270) PiDPToQDPProof (EQUIVALENT transformation)

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

(271) Obligation:

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

GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)

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

(272) 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:

  • GT67_IN_GG(s(T144), s(T145)) → GT67_IN_GG(T144, T145)
    The graph contains the following edges 1 > 1, 2 > 2

(273) YES

(274) Obligation:

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

LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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)
LE36_IN_GG(x1, x2)  =  LE36_IN_GG(x1, x2)

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

(275) UsableRulesProof (EQUIVALENT transformation)

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

(276) Obligation:

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

LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)

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

(277) PiDPToQDPProof (EQUIVALENT transformation)

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

(278) Obligation:

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

LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)

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

(279) 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:

  • LE36_IN_GG(s(T82), s(T83)) → LE36_IN_GG(T82, T83)
    The graph contains the following edges 1 > 1, 2 > 2

(280) YES

(281) Obligation:

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

SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_GGAA(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_GGAA(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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)
SPLIT51_IN_GGAA(x1, x2, x3, x4)  =  SPLIT51_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

(282) UsableRulesProof (EQUIVALENT transformation)

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

(283) Obligation:

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

SPLIT51_IN_GGAA(.(T114, T115), T116, .(T114, X167), X168) → U3_GGAA(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_GGAA(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116, X167, X168)
SPLIT51_IN_GGAA(.(T129, T130), T131, X199, .(T129, X200)) → U5_GGAA(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
U5_GGAA(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131, X199, X200)

The TRS R consists of the following rules:

le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
SPLIT51_IN_GGAA(x1, x2, x3, x4)  =  SPLIT51_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

(284) PiDPToQDPProof (SOUND transformation)

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

(285) Obligation:

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

SPLIT51_IN_GGAA(.(T114, T115), T116) → U3_GGAA(T114, T115, T116, le36_in_gg(T114, T116))
U3_GGAA(T114, T115, T116, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116)
SPLIT51_IN_GGAA(.(T129, T130), T131) → U5_GGAA(T129, T130, T131, gt67_in_gg(T129, T131))
U5_GGAA(T129, T130, T131, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131)

The TRS R consists of the following rules:

le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))

The set Q consists of the following terms:

le36_in_gg(x0, x1)
gt67_in_gg(x0, x1)
U2_gg(x0, x1, x2)
U7_gg(x0, x1, x2)

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

(286) 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(T114, T115, T116, le36_out_gg(T114, T116)) → SPLIT51_IN_GGAA(T115, T116)
    The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 2

  • U5_GGAA(T129, T130, T131, gt67_out_gg(T129, T131)) → SPLIT51_IN_GGAA(T130, T131)
    The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 2

  • SPLIT51_IN_GGAA(.(T114, T115), T116) → U3_GGAA(T114, T115, T116, le36_in_gg(T114, T116))
    The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3

  • SPLIT51_IN_GGAA(.(T129, T130), T131) → U5_GGAA(T129, T130, T131, gt67_in_gg(T129, T131))
    The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3

(287) YES

(288) Obligation:

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

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

The TRS R consists of the following rules:

qsort1_in_ga([], []) → qsort1_out_ga([], [])
qsort1_in_ga(.(T14, []), T9) → U19_ga(T14, T9, qsort14_in_a(X12))
qsort14_in_a([]) → qsort14_out_a([])
U19_ga(T14, T9, qsort14_out_a(X12)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T14, []), T9) → U20_ga(T14, T9, qsort14_in_a(T17))
U20_ga(T14, T9, qsort14_out_a(T17)) → U21_ga(T14, T9, qsort14_in_a(X13))
U21_ga(T14, T9, qsort14_out_a(X13)) → qsort1_out_ga(.(T14, []), T9)
U20_ga(T14, T9, qsort14_out_a(T17)) → U22_ga(T14, T9, T17, qsort14_in_a(T20))
U22_ga(T14, T9, T17, qsort14_out_a(T20)) → U23_ga(T14, T9, append23_in_ggga(T17, T14, T20, T9))
append23_in_ggga([], T33, T34, .(T33, T34)) → append23_out_ggga([], T33, T34, .(T33, T34))
append23_in_ggga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_ga(T14, T9, append23_out_ggga(T17, T14, T20, T9)) → qsort1_out_ga(.(T14, []), T9)
qsort1_in_ga(.(T69, .(T67, T68)), T9) → U24_ga(T69, T67, T68, T9, le36_in_gg(T67, T69))
le36_in_gg(s(T82), s(T83)) → U2_gg(T82, T83, le36_in_gg(T82, T83))
le36_in_gg(0, s(T90)) → le36_out_gg(0, s(T90))
le36_in_gg(0, 0) → le36_out_gg(0, 0)
U2_gg(T82, T83, le36_out_gg(T82, T83)) → le36_out_gg(s(T82), s(T83))
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U25_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, X93, X94))
split51_in_ggaa([], T101, [], []) → split51_out_ggaa([], T101, [], [])
split51_in_ggaa(.(T114, T115), T116, .(T114, X167), X168) → U3_ggaa(T114, T115, T116, X167, X168, le36_in_gg(T114, T116))
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U3_ggaa(T114, T115, T116, X167, X168, le36_out_gg(T114, T116)) → U4_ggaa(T114, T115, T116, X167, X168, split51_in_ggaa(T115, T116, X167, X168))
split51_in_ggaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_ggaa(T129, T130, T131, X199, X200, gt67_in_gg(T129, T131))
gt67_in_gg(s(T144), s(T145)) → U7_gg(T144, T145, gt67_in_gg(T144, T145))
gt67_in_gg(s(T150), 0) → gt67_out_gg(s(T150), 0)
U7_gg(T144, T145, gt67_out_gg(T144, T145)) → gt67_out_gg(s(T144), s(T145))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U5_ggaa(T129, T130, T131, X199, X200, gt67_out_gg(T129, T131)) → U6_ggaa(T129, T130, T131, X199, X200, split51_in_ggaa(T130, T131, X199, X200))
U6_ggaa(T129, T130, T131, X199, X200, split51_out_ggaa(T130, T131, X199, X200)) → split51_out_ggaa(.(T129, T130), T131, X199, .(T129, X200))
U4_ggaa(T114, T115, T116, X167, X168, split51_out_ggaa(T115, T116, X167, X168)) → split51_out_ggaa(.(T114, T115), T116, .(T114, X167), X168)
U25_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, X93, X94)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U24_ga(T69, T67, T68, T9, le36_out_gg(T67, T69)) → U26_ga(T69, T67, T68, T9, split51_in_ggaa(T68, T69, T93, T94))
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U27_ga(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa([], []) → qsort1_out_aa([], [])
qsort1_in_aa(.(T14, []), T9) → U19_aa(T14, T9, qsort14_in_a(X12))
U19_aa(T14, T9, qsort14_out_a(X12)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T14, []), T9) → U20_aa(T14, T9, qsort14_in_a(T17))
U20_aa(T14, T9, qsort14_out_a(T17)) → U21_aa(T14, T9, qsort14_in_a(X13))
U21_aa(T14, T9, qsort14_out_a(X13)) → qsort1_out_aa(.(T14, []), T9)
U20_aa(T14, T9, qsort14_out_a(T17)) → U22_aa(T14, T9, T17, qsort14_in_a(T20))
U22_aa(T14, T9, T17, qsort14_out_a(T20)) → U23_aa(T14, T9, append23_in_gaga(T17, T14, T20, T9))
append23_in_gaga([], T33, T34, .(T33, T34)) → append23_out_gaga([], T33, T34, .(T33, T34))
append23_in_gaga(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U23_aa(T14, T9, append23_out_gaga(T17, T14, T20, T9)) → qsort1_out_aa(.(T14, []), T9)
qsort1_in_aa(.(T69, .(T67, T68)), T9) → U24_aa(T69, T67, T68, T9, le36_in_aa(T67, T69))
le36_in_aa(s(T82), s(T83)) → U2_aa(T82, T83, le36_in_aa(T82, T83))
le36_in_aa(0, s(T90)) → le36_out_aa(0, s(T90))
le36_in_aa(0, 0) → le36_out_aa(0, 0)
U2_aa(T82, T83, le36_out_aa(T82, T83)) → le36_out_aa(s(T82), s(T83))
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U25_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, X93, X94))
split51_in_aaaa([], T101, [], []) → split51_out_aaaa([], T101, [], [])
split51_in_aaaa(.(T114, T115), T116, .(T114, X167), X168) → U3_aaaa(T114, T115, T116, X167, X168, le36_in_aa(T114, T116))
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U3_aaaa(T114, T115, T116, X167, X168, le36_out_aa(T114, T116)) → U4_aaaa(T114, T115, T116, X167, X168, split51_in_aaaa(T115, T116, X167, X168))
split51_in_aaaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_aaaa(T129, T130, T131, X199, X200, gt67_in_aa(T129, T131))
gt67_in_aa(s(T144), s(T145)) → U7_aa(T144, T145, gt67_in_aa(T144, T145))
gt67_in_aa(s(T150), 0) → gt67_out_aa(s(T150), 0)
U7_aa(T144, T145, gt67_out_aa(T144, T145)) → gt67_out_aa(s(T144), s(T145))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U5_aaaa(T129, T130, T131, X199, X200, gt67_out_aa(T129, T131)) → U6_aaaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
split51_in_agaa([], T101, [], []) → split51_out_agaa([], T101, [], [])
split51_in_agaa(.(T114, T115), T116, .(T114, X167), X168) → U3_agaa(T114, T115, T116, X167, X168, le36_in_ag(T114, T116))
le36_in_ag(s(T82), s(T83)) → U2_ag(T82, T83, le36_in_ag(T82, T83))
le36_in_ag(0, s(T90)) → le36_out_ag(0, s(T90))
le36_in_ag(0, 0) → le36_out_ag(0, 0)
U2_ag(T82, T83, le36_out_ag(T82, T83)) → le36_out_ag(s(T82), s(T83))
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U3_agaa(T114, T115, T116, X167, X168, le36_out_ag(T114, T116)) → U4_agaa(T114, T115, T116, X167, X168, split51_in_agaa(T115, T116, X167, X168))
split51_in_agaa(.(T129, T130), T131, X199, .(T129, X200)) → U5_agaa(T129, T130, T131, X199, X200, gt67_in_ag(T129, T131))
gt67_in_ag(s(T144), s(T145)) → U7_ag(T144, T145, gt67_in_ag(T144, T145))
gt67_in_ag(s(T150), 0) → gt67_out_ag(s(T150), 0)
U7_ag(T144, T145, gt67_out_ag(T144, T145)) → gt67_out_ag(s(T144), s(T145))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U5_agaa(T129, T130, T131, X199, X200, gt67_out_ag(T129, T131)) → U6_agaa(T129, T130, T131, X199, X200, split51_in_agaa(T130, T131, X199, X200))
U6_agaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_agaa(.(T129, T130), T131, X199, .(T129, X200))
U4_agaa(T114, T115, T116, X167, X168, split51_out_agaa(T115, T116, X167, X168)) → split51_out_agaa(.(T114, T115), T116, .(T114, X167), X168)
U6_aaaa(T129, T130, T131, X199, X200, split51_out_agaa(T130, T131, X199, X200)) → split51_out_aaaa(.(T129, T130), T131, X199, .(T129, X200))
U4_aaaa(T114, T115, T116, X167, X168, split51_out_aaaa(T115, T116, X167, X168)) → split51_out_aaaa(.(T114, T115), T116, .(T114, X167), X168)
U25_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, X93, X94)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U24_aa(T69, T67, T68, T9, le36_out_aa(T67, T69)) → U26_aa(T69, T67, T68, T9, split51_in_aaaa(T68, T69, T93, T94))
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U27_aa(T69, T67, T68, T9, qsort1_in_aa(.(T67, T93), X12))
qsort1_in_aa(.(T215, .(T213, T214)), T9) → U30_aa(T215, T213, T214, T9, gt67_in_aa(T213, T215))
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U31_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, X323, X324))
U31_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, X323, X324)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U30_aa(T215, T213, T214, T9, gt67_out_aa(T213, T215)) → U32_aa(T215, T213, T214, T9, split51_in_agaa(T214, T215, T222, T223))
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U33_aa(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
qsort79_in_aa([], []) → qsort79_out_aa([], [])
qsort79_in_aa(.(T163, T164), X265) → U8_aa(T163, T164, X265, split51_in_aaaa(T164, T163, X261, X262))
U8_aa(T163, T164, X265, split51_out_aaaa(T164, T163, X261, X262)) → qsort79_out_aa(.(T163, T164), X265)
qsort79_in_aa(.(T163, T164), X265) → U9_aa(T163, T164, X265, split51_in_aaaa(T164, T163, T167, T168))
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U10_aa(T163, T164, X265, qsort79_in_aa(T167, X263))
U10_aa(T163, T164, X265, qsort79_out_aa(T167, X263)) → qsort79_out_aa(.(T163, T164), X265)
U9_aa(T163, T164, X265, split51_out_aaaa(T164, T163, T167, T168)) → U11_aa(T163, T164, X265, T168, qsort79_in_aa(T167, T171))
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U12_aa(T163, T164, X265, qsort79_in_aa(T168, X264))
U12_aa(T163, T164, X265, qsort79_out_aa(T168, X264)) → qsort79_out_aa(.(T163, T164), X265)
U11_aa(T163, T164, X265, T168, qsort79_out_aa(T167, T171)) → U13_aa(T163, T164, X265, T171, qsort79_in_aa(T168, T172))
U13_aa(T163, T164, X265, T171, qsort79_out_aa(T168, T172)) → U14_aa(T163, T164, X265, append94_in_aaaa(T171, T163, T172, X265))
append94_in_aaaa([], T185, T186, .(T185, T186)) → append94_out_aaaa([], T185, T186, .(T185, T186))
append94_in_aaaa(.(T195, T196), T197, T198, .(T195, X295)) → U15_aaaa(T195, T196, T197, T198, X295, append94_in_aaaa(T196, T197, T198, X295))
U15_aaaa(T195, T196, T197, T198, X295, append94_out_aaaa(T196, T197, T198, X295)) → append94_out_aaaa(.(T195, T196), T197, T198, .(T195, X295))
U14_aa(T163, T164, X265, append94_out_aaaa(T171, T163, T172, X265)) → qsort79_out_aa(.(T163, T164), X265)
U33_aa(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U32_aa(T215, T213, T214, T9, split51_out_agaa(T214, T215, T222, T223)) → U34_aa(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_aa(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_aa(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
p78_in_aaaga(T94, X13, T155, T69, T9) → U16_aaaga(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaga(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaga(T94, X13, T155, T69, T9)
p78_in_aaaga(T94, T158, T155, T69, T9) → U17_aaaga(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaga(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaga(T94, T158, T155, T69, T9, append23_in_agaa(T155, T69, T158, T9))
append23_in_agaa([], T33, T34, .(T33, T34)) → append23_out_agaa([], T33, T34, .(T33, T34))
append23_in_agaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaga(T94, T158, T155, T69, T9, append23_out_agaa(T155, T69, T158, T9)) → p78_out_aaaga(T94, T158, T155, T69, T9)
U35_aa(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_aa(.(T215, .(T213, T214)), T9)
U27_aa(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U26_aa(T69, T67, T68, T9, split51_out_aaaa(T68, T69, T93, T94)) → U28_aa(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_aa(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_aa(T69, T67, T68, T9, p78_in_aaaaa(T94, X13, T155, T69, T9))
p78_in_aaaaa(T94, X13, T155, T69, T9) → U16_aaaaa(T94, X13, T155, T69, T9, qsort79_in_aa(T94, X13))
U16_aaaaa(T94, X13, T155, T69, T9, qsort79_out_aa(T94, X13)) → p78_out_aaaaa(T94, X13, T155, T69, T9)
p78_in_aaaaa(T94, T158, T155, T69, T9) → U17_aaaaa(T94, T158, T155, T69, T9, qsort79_in_aa(T94, T158))
U17_aaaaa(T94, T158, T155, T69, T9, qsort79_out_aa(T94, T158)) → U18_aaaaa(T94, T158, T155, T69, T9, append23_in_aaaa(T155, T69, T158, T9))
append23_in_aaaa([], T33, T34, .(T33, T34)) → append23_out_aaaa([], T33, T34, .(T33, T34))
append23_in_aaaa(.(T45, T46), T47, T48, .(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(.(T45, T46), T47, T48, .(T45, T50))
U18_aaaaa(T94, T158, T155, T69, T9, append23_out_aaaa(T155, T69, T158, T9)) → p78_out_aaaaa(T94, T158, T155, T69, T9)
U29_aa(T69, T67, T68, T9, p78_out_aaaaa(T94, X13, T155, T69, T9)) → qsort1_out_aa(.(T69, .(T67, T68)), T9)
U27_ga(T69, T67, T68, T9, qsort1_out_aa(.(T67, T93), X12)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
U26_ga(T69, T67, T68, T9, split51_out_ggaa(T68, T69, T93, T94)) → U28_ga(T69, T67, T68, T9, T94, qsort1_in_aa(.(T67, T93), T155))
U28_ga(T69, T67, T68, T9, T94, qsort1_out_aa(.(T67, T93), T155)) → U29_ga(T69, T67, T68, T9, p78_in_aaaga(T94, X13, T155, T69, T9))
U29_ga(T69, T67, T68, T9, p78_out_aaaga(T94, X13, T155, T69, T9)) → qsort1_out_ga(.(T69, .(T67, T68)), T9)
qsort1_in_ga(.(T215, .(T213, T214)), T9) → U30_ga(T215, T213, T214, T9, gt67_in_gg(T213, T215))
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U31_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, X323, X324))
U31_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, X323, X324)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U30_ga(T215, T213, T214, T9, gt67_out_gg(T213, T215)) → U32_ga(T215, T213, T214, T9, split51_in_ggaa(T214, T215, T222, T223))
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U33_ga(T215, T213, T214, T9, qsort79_in_aa(T222, X12))
U33_ga(T215, T213, T214, T9, qsort79_out_aa(T222, X12)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)
U32_ga(T215, T213, T214, T9, split51_out_ggaa(T214, T215, T222, T223)) → U34_ga(T215, T213, T214, T9, T223, qsort79_in_aa(T222, T226))
U34_ga(T215, T213, T214, T9, T223, qsort79_out_aa(T222, T226)) → U35_ga(T215, T213, T214, T9, p78_in_aaaga(.(T213, T223), X13, T226, T215, T9))
U35_ga(T215, T213, T214, T9, p78_out_aaaga(.(T213, T223), X13, T226, T215, T9)) → qsort1_out_ga(.(T215, .(T213, T214)), T9)

The argument filtering Pi contains the following mapping:
qsort1_in_ga(x1, x2)  =  qsort1_in_ga(x1)
[]  =  []
qsort1_out_ga(x1, x2)  =  qsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
qsort14_in_a(x1)  =  qsort14_in_a
qsort14_out_a(x1)  =  qsort14_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)
le36_in_gg(x1, x2)  =  le36_in_gg(x1, x2)
s(x1)  =  s(x1)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
0  =  0
le36_out_gg(x1, x2)  =  le36_out_gg(x1, x2)
U25_ga(x1, x2, x3, x4, x5)  =  U25_ga(x1, x2, x3, x5)
split51_in_ggaa(x1, x2, x3, x4)  =  split51_in_ggaa(x1, x2)
split51_out_ggaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_gg(x1, x2)  =  gt67_in_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
gt67_out_gg(x1, x2)  =  gt67_out_gg(x1, x2)
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)
qsort1_in_aa(x1, x2)  =  qsort1_in_aa
qsort1_out_aa(x1, x2)  =  qsort1_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)
le36_in_aa(x1, x2)  =  le36_in_aa
U2_aa(x1, x2, x3)  =  U2_aa(x3)
le36_out_aa(x1, x2)  =  le36_out_aa(x1)
U25_aa(x1, x2, x3, x4, x5)  =  U25_aa(x5)
split51_in_aaaa(x1, x2, x3, x4)  =  split51_in_aaaa
split51_out_aaaa(x1, x2, x3, x4)  =  split51_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)
gt67_in_aa(x1, x2)  =  gt67_in_aa
U7_aa(x1, x2, x3)  =  U7_aa(x3)
gt67_out_aa(x1, x2)  =  gt67_out_aa(x2)
U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x6)
split51_in_agaa(x1, x2, x3, x4)  =  split51_in_agaa(x2)
split51_out_agaa(x1, x2, x3, x4)  =  split51_out_agaa(x2)
U3_agaa(x1, x2, x3, x4, x5, x6)  =  U3_agaa(x3, x6)
le36_in_ag(x1, x2)  =  le36_in_ag(x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
le36_out_ag(x1, x2)  =  le36_out_ag(x1, x2)
U4_agaa(x1, x2, x3, x4, x5, x6)  =  U4_agaa(x3, x6)
U5_agaa(x1, x2, x3, x4, x5, x6)  =  U5_agaa(x3, x6)
gt67_in_ag(x1, x2)  =  gt67_in_ag(x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
gt67_out_ag(x1, x2)  =  gt67_out_ag(x2)
U6_agaa(x1, x2, x3, x4, x5, x6)  =  U6_agaa(x3, 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(x1, x5)
U33_aa(x1, x2, x3, x4, x5)  =  U33_aa(x5)
qsort79_in_aa(x1, x2)  =  qsort79_in_aa
qsort79_out_aa(x1, x2)  =  qsort79_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(x1, x6)
U35_aa(x1, x2, x3, x4, x5)  =  U35_aa(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)
U28_aa(x1, x2, x3, x4, x5, x6)  =  U28_aa(x6)
U29_aa(x1, x2, x3, x4, x5)  =  U29_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_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)
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

(289) UsableRulesProof (EQUIVALENT transformation)

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

(290) Obligation:

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

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

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

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

(291) PiDPToQDPProof (SOUND transformation)

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

(292) Obligation:

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

APPEND23_IN_GGGA(.(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.

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

(294) YES