(0) Obligation:

Clauses:

insert(X, void, tree(X, void, void)).
insert(X, tree(X, Left, Right), tree(X, Left, Right)).
insert(X, tree(Y, Left, Right), tree(Y, Left1, Right)) :- ','(less(X, Y), insert(X, Left, Left1)).
insert(X, tree(Y, Left, Right), tree(Y, Left, Right1)) :- ','(less(Y, X), insert(X, Right, Right1)).
less(0, s(X1)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

insert(g,a,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

less20(0, s(T55)).
less20(s(T60), s(T62)) :- less20(T60, T62).
less33(0, s(T95)).
less33(s(T102), s(T101)) :- less33(T102, T101).
p18(T40, T42, T43, T44) :- less20(T40, T42).
p18(T40, T42, T47, T48) :- ','(less20(T40, T42), insert1(s(T40), T47, T48)).
p43(0, s(T124), T125, T126) :- insert1(s(T124), T125, T126).
p43(s(T135), s(T134), T136, T137) :- p50(T135, T134, T136, T137).
p50(T135, T134, T136, T137) :- less33(T135, T134).
p50(T135, T134, T140, T141) :- ','(less33(T135, T134), insert1(s(T134), T140, T141)).
p68(0, s(T218), T219, T220) :- insert1(s(T218), T219, T220).
p68(s(T229), s(T228), T230, T231) :- p50(T229, T228, T230, T231).
insert1(T5, void, tree(T5, void, void)).
insert1(T9, tree(T9, T10, T11), tree(T9, T10, T11)).
insert1(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) :- insert1(0, T30, T31).
insert1(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) :- p18(T40, T42, T43, T44).
insert1(T77, tree(T82, T79, T83), tree(T82, T79, T84)) :- less33(T82, T77).
insert1(T77, tree(T82, T79, T87), tree(T82, T79, T88)) :- ','(less33(T82, T77), insert1(T77, T87, T88)).
insert1(T112, tree(T117, T114, T118), tree(T117, T114, T119)) :- p43(T117, T112, T118, T119).
insert1(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) :- insert1(0, T164, T165).
insert1(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) :- p18(T174, T176, T177, T178).
insert1(T191, tree(T196, T193, T197), tree(T196, T193, T198)) :- p43(T196, T191, T197, T198).
insert1(T206, tree(T211, T208, T212), tree(T211, T208, T213)) :- p68(T211, T206, T212, T213).
insert1(T237, tree(T237, T238, T239), tree(T237, T238, T239)).
insert1(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) :- insert1(0, T258, T259).
insert1(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) :- p18(T268, T270, T271, T272).
insert1(T285, tree(T290, T287, T291), tree(T290, T287, T292)) :- p68(T290, T285, T291, T292).
insert1(s(T312), tree(0, T302, T313), tree(0, T302, T314)) :- insert1(s(T312), T313, T314).
insert1(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) :- p50(T323, T322, T324, T325).
insert1(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) :- insert1(0, T346, T347).
insert1(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) :- p18(T356, T358, T359, T360).
insert1(T373, tree(T378, T375, T379), tree(T378, T375, T380)) :- p68(T378, T373, T379, T380).
insert1(s(T400), tree(0, T390, T401), tree(0, T390, T402)) :- insert1(s(T400), T401, T402).
insert1(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) :- p50(T411, T410, T412, T413).

Queries:

insert1(g,a,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:
insert1_in: (b,f,f)
p18_in: (b,f,f,f)
less20_in: (b,f)
less33_in: (f,b)
p43_in: (f,b,f,f)
p50_in: (f,b,f,f)
p68_in: (f,b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

insert1_in_gaa(T5, void, tree(T5, void, void)) → insert1_out_gaa(T5, void, tree(T5, void, void))
insert1_in_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11)) → insert1_out_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11))
insert1_in_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → U13_gaa(T29, T30, T20, T31, insert1_in_gaa(0, T30, T31))
insert1_in_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → U14_gaa(T40, T42, T43, T20, T44, p18_in_gaaa(T40, T42, T43, T44))
p18_in_gaaa(T40, T42, T43, T44) → U3_gaaa(T40, T42, T43, T44, less20_in_ga(T40, T42))
less20_in_ga(0, s(T55)) → less20_out_ga(0, s(T55))
less20_in_ga(s(T60), s(T62)) → U1_ga(T60, T62, less20_in_ga(T60, T62))
U1_ga(T60, T62, less20_out_ga(T60, T62)) → less20_out_ga(s(T60), s(T62))
U3_gaaa(T40, T42, T43, T44, less20_out_ga(T40, T42)) → p18_out_gaaa(T40, T42, T43, T44)
p18_in_gaaa(T40, T42, T47, T48) → U4_gaaa(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_gaaa(T40, T42, T47, T48, less20_out_ga(T40, T42)) → U5_gaaa(T40, T42, T47, T48, insert1_in_gaa(s(T40), T47, T48))
insert1_in_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → U15_gaa(T77, T82, T79, T83, T84, less33_in_ag(T82, T77))
less33_in_ag(0, s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T102), s(T101)) → U2_ag(T102, T101, less33_in_ag(T102, T101))
U2_ag(T102, T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U15_gaa(T77, T82, T79, T83, T84, less33_out_ag(T82, T77)) → insert1_out_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84))
insert1_in_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_gaa(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_gaa(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → U17_gaa(T77, T82, T79, T87, T88, insert1_in_gaa(T77, T87, T88))
insert1_in_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → U18_gaa(T112, T117, T114, T118, T119, p43_in_agaa(T117, T112, T118, T119))
p43_in_agaa(0, s(T124), T125, T126) → U6_agaa(T124, T125, T126, insert1_in_gaa(s(T124), T125, T126))
insert1_in_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) → U19_gaa(T163, T164, T154, T165, insert1_in_gaa(0, T164, T165))
insert1_in_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) → U20_gaa(T174, T176, T177, T154, T178, p18_in_gaaa(T174, T176, T177, T178))
U20_gaa(T174, T176, T177, T154, T178, p18_out_gaaa(T174, T176, T177, T178)) → insert1_out_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154))
insert1_in_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198)) → U21_gaa(T191, T196, T193, T197, T198, p43_in_agaa(T196, T191, T197, T198))
p43_in_agaa(s(T135), s(T134), T136, T137) → U7_agaa(T135, T134, T136, T137, p50_in_agaa(T135, T134, T136, T137))
p50_in_agaa(T135, T134, T136, T137) → U8_agaa(T135, T134, T136, T137, less33_in_ag(T135, T134))
U8_agaa(T135, T134, T136, T137, less33_out_ag(T135, T134)) → p50_out_agaa(T135, T134, T136, T137)
p50_in_agaa(T135, T134, T140, T141) → U9_agaa(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_agaa(T135, T134, T140, T141, less33_out_ag(T135, T134)) → U10_agaa(T135, T134, T140, T141, insert1_in_gaa(s(T134), T140, T141))
insert1_in_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → U22_gaa(T206, T211, T208, T212, T213, p68_in_agaa(T211, T206, T212, T213))
p68_in_agaa(0, s(T218), T219, T220) → U11_agaa(T218, T219, T220, insert1_in_gaa(s(T218), T219, T220))
insert1_in_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) → U23_gaa(T257, T258, T248, T259, insert1_in_gaa(0, T258, T259))
insert1_in_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) → U24_gaa(T268, T270, T271, T248, T272, p18_in_gaaa(T268, T270, T271, T272))
U24_gaa(T268, T270, T271, T248, T272, p18_out_gaaa(T268, T270, T271, T272)) → insert1_out_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248))
insert1_in_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292)) → U25_gaa(T285, T290, T287, T291, T292, p68_in_agaa(T290, T285, T291, T292))
p68_in_agaa(s(T229), s(T228), T230, T231) → U12_agaa(T229, T228, T230, T231, p50_in_agaa(T229, T228, T230, T231))
U12_agaa(T229, T228, T230, T231, p50_out_agaa(T229, T228, T230, T231)) → p68_out_agaa(s(T229), s(T228), T230, T231)
U25_gaa(T285, T290, T287, T291, T292, p68_out_agaa(T290, T285, T291, T292)) → insert1_out_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292))
insert1_in_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → U26_gaa(T312, T302, T313, T314, insert1_in_gaa(s(T312), T313, T314))
insert1_in_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → U27_gaa(T322, T323, T302, T324, T325, p50_in_agaa(T323, T322, T324, T325))
U27_gaa(T322, T323, T302, T324, T325, p50_out_agaa(T323, T322, T324, T325)) → insert1_out_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325))
insert1_in_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) → U28_gaa(T345, T346, T336, T347, insert1_in_gaa(0, T346, T347))
insert1_in_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) → U29_gaa(T356, T358, T359, T336, T360, p18_in_gaaa(T356, T358, T359, T360))
U29_gaa(T356, T358, T359, T336, T360, p18_out_gaaa(T356, T358, T359, T360)) → insert1_out_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336))
insert1_in_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380)) → U30_gaa(T373, T378, T375, T379, T380, p68_in_agaa(T378, T373, T379, T380))
U30_gaa(T373, T378, T375, T379, T380, p68_out_agaa(T378, T373, T379, T380)) → insert1_out_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380))
insert1_in_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402)) → U31_gaa(T400, T390, T401, T402, insert1_in_gaa(s(T400), T401, T402))
insert1_in_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) → U32_gaa(T410, T411, T390, T412, T413, p50_in_agaa(T411, T410, T412, T413))
U32_gaa(T410, T411, T390, T412, T413, p50_out_agaa(T411, T410, T412, T413)) → insert1_out_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413))
U31_gaa(T400, T390, T401, T402, insert1_out_gaa(s(T400), T401, T402)) → insert1_out_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402))
U28_gaa(T345, T346, T336, T347, insert1_out_gaa(0, T346, T347)) → insert1_out_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336))
U26_gaa(T312, T302, T313, T314, insert1_out_gaa(s(T312), T313, T314)) → insert1_out_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314))
U23_gaa(T257, T258, T248, T259, insert1_out_gaa(0, T258, T259)) → insert1_out_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248))
U11_agaa(T218, T219, T220, insert1_out_gaa(s(T218), T219, T220)) → p68_out_agaa(0, s(T218), T219, T220)
U22_gaa(T206, T211, T208, T212, T213, p68_out_agaa(T211, T206, T212, T213)) → insert1_out_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213))
U10_agaa(T135, T134, T140, T141, insert1_out_gaa(s(T134), T140, T141)) → p50_out_agaa(T135, T134, T140, T141)
U7_agaa(T135, T134, T136, T137, p50_out_agaa(T135, T134, T136, T137)) → p43_out_agaa(s(T135), s(T134), T136, T137)
U21_gaa(T191, T196, T193, T197, T198, p43_out_agaa(T196, T191, T197, T198)) → insert1_out_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198))
U19_gaa(T163, T164, T154, T165, insert1_out_gaa(0, T164, T165)) → insert1_out_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154))
U6_agaa(T124, T125, T126, insert1_out_gaa(s(T124), T125, T126)) → p43_out_agaa(0, s(T124), T125, T126)
U18_gaa(T112, T117, T114, T118, T119, p43_out_agaa(T117, T112, T118, T119)) → insert1_out_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119))
U17_gaa(T77, T82, T79, T87, T88, insert1_out_gaa(T77, T87, T88)) → insert1_out_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88))
U5_gaaa(T40, T42, T47, T48, insert1_out_gaa(s(T40), T47, T48)) → p18_out_gaaa(T40, T42, T47, T48)
U14_gaa(T40, T42, T43, T20, T44, p18_out_gaaa(T40, T42, T43, T44)) → insert1_out_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20))
U13_gaa(T29, T30, T20, T31, insert1_out_gaa(0, T30, T31)) → insert1_out_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20))

The argument filtering Pi contains the following mapping:
insert1_in_gaa(x1, x2, x3)  =  insert1_in_gaa(x1)
insert1_out_gaa(x1, x2, x3)  =  insert1_out_gaa
0  =  0
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x5)
s(x1)  =  s(x1)
U14_gaa(x1, x2, x3, x4, x5, x6)  =  U14_gaa(x6)
p18_in_gaaa(x1, x2, x3, x4)  =  p18_in_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x5)
less20_in_ga(x1, x2)  =  less20_in_ga(x1)
less20_out_ga(x1, x2)  =  less20_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
p18_out_gaaa(x1, x2, x3, x4)  =  p18_out_gaaa
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
U5_gaaa(x1, x2, x3, x4, x5)  =  U5_gaaa(x5)
U15_gaa(x1, x2, x3, x4, x5, x6)  =  U15_gaa(x6)
less33_in_ag(x1, x2)  =  less33_in_ag(x2)
less33_out_ag(x1, x2)  =  less33_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U16_gaa(x1, x2, x3, x4, x5, x6)  =  U16_gaa(x1, x6)
U17_gaa(x1, x2, x3, x4, x5, x6)  =  U17_gaa(x6)
U18_gaa(x1, x2, x3, x4, x5, x6)  =  U18_gaa(x6)
p43_in_agaa(x1, x2, x3, x4)  =  p43_in_agaa(x2)
U6_agaa(x1, x2, x3, x4)  =  U6_agaa(x4)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x5)
U20_gaa(x1, x2, x3, x4, x5, x6)  =  U20_gaa(x6)
U21_gaa(x1, x2, x3, x4, x5, x6)  =  U21_gaa(x6)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x5)
p50_in_agaa(x1, x2, x3, x4)  =  p50_in_agaa(x2)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x5)
p50_out_agaa(x1, x2, x3, x4)  =  p50_out_agaa(x1)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x1, x5)
U22_gaa(x1, x2, x3, x4, x5, x6)  =  U22_gaa(x6)
p68_in_agaa(x1, x2, x3, x4)  =  p68_in_agaa(x2)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x4)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x5)
U24_gaa(x1, x2, x3, x4, x5, x6)  =  U24_gaa(x6)
U25_gaa(x1, x2, x3, x4, x5, x6)  =  U25_gaa(x6)
U12_agaa(x1, x2, x3, x4, x5)  =  U12_agaa(x5)
p68_out_agaa(x1, x2, x3, x4)  =  p68_out_agaa(x1)
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x5)
U27_gaa(x1, x2, x3, x4, x5, x6)  =  U27_gaa(x6)
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U29_gaa(x1, x2, x3, x4, x5, x6)  =  U29_gaa(x6)
U30_gaa(x1, x2, x3, x4, x5, x6)  =  U30_gaa(x6)
U31_gaa(x1, x2, x3, x4, x5)  =  U31_gaa(x5)
U32_gaa(x1, x2, x3, x4, x5, x6)  =  U32_gaa(x6)
p43_out_agaa(x1, x2, x3, x4)  =  p43_out_agaa(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

insert1_in_gaa(T5, void, tree(T5, void, void)) → insert1_out_gaa(T5, void, tree(T5, void, void))
insert1_in_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11)) → insert1_out_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11))
insert1_in_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → U13_gaa(T29, T30, T20, T31, insert1_in_gaa(0, T30, T31))
insert1_in_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → U14_gaa(T40, T42, T43, T20, T44, p18_in_gaaa(T40, T42, T43, T44))
p18_in_gaaa(T40, T42, T43, T44) → U3_gaaa(T40, T42, T43, T44, less20_in_ga(T40, T42))
less20_in_ga(0, s(T55)) → less20_out_ga(0, s(T55))
less20_in_ga(s(T60), s(T62)) → U1_ga(T60, T62, less20_in_ga(T60, T62))
U1_ga(T60, T62, less20_out_ga(T60, T62)) → less20_out_ga(s(T60), s(T62))
U3_gaaa(T40, T42, T43, T44, less20_out_ga(T40, T42)) → p18_out_gaaa(T40, T42, T43, T44)
p18_in_gaaa(T40, T42, T47, T48) → U4_gaaa(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_gaaa(T40, T42, T47, T48, less20_out_ga(T40, T42)) → U5_gaaa(T40, T42, T47, T48, insert1_in_gaa(s(T40), T47, T48))
insert1_in_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → U15_gaa(T77, T82, T79, T83, T84, less33_in_ag(T82, T77))
less33_in_ag(0, s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T102), s(T101)) → U2_ag(T102, T101, less33_in_ag(T102, T101))
U2_ag(T102, T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U15_gaa(T77, T82, T79, T83, T84, less33_out_ag(T82, T77)) → insert1_out_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84))
insert1_in_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_gaa(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_gaa(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → U17_gaa(T77, T82, T79, T87, T88, insert1_in_gaa(T77, T87, T88))
insert1_in_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → U18_gaa(T112, T117, T114, T118, T119, p43_in_agaa(T117, T112, T118, T119))
p43_in_agaa(0, s(T124), T125, T126) → U6_agaa(T124, T125, T126, insert1_in_gaa(s(T124), T125, T126))
insert1_in_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) → U19_gaa(T163, T164, T154, T165, insert1_in_gaa(0, T164, T165))
insert1_in_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) → U20_gaa(T174, T176, T177, T154, T178, p18_in_gaaa(T174, T176, T177, T178))
U20_gaa(T174, T176, T177, T154, T178, p18_out_gaaa(T174, T176, T177, T178)) → insert1_out_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154))
insert1_in_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198)) → U21_gaa(T191, T196, T193, T197, T198, p43_in_agaa(T196, T191, T197, T198))
p43_in_agaa(s(T135), s(T134), T136, T137) → U7_agaa(T135, T134, T136, T137, p50_in_agaa(T135, T134, T136, T137))
p50_in_agaa(T135, T134, T136, T137) → U8_agaa(T135, T134, T136, T137, less33_in_ag(T135, T134))
U8_agaa(T135, T134, T136, T137, less33_out_ag(T135, T134)) → p50_out_agaa(T135, T134, T136, T137)
p50_in_agaa(T135, T134, T140, T141) → U9_agaa(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_agaa(T135, T134, T140, T141, less33_out_ag(T135, T134)) → U10_agaa(T135, T134, T140, T141, insert1_in_gaa(s(T134), T140, T141))
insert1_in_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → U22_gaa(T206, T211, T208, T212, T213, p68_in_agaa(T211, T206, T212, T213))
p68_in_agaa(0, s(T218), T219, T220) → U11_agaa(T218, T219, T220, insert1_in_gaa(s(T218), T219, T220))
insert1_in_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) → U23_gaa(T257, T258, T248, T259, insert1_in_gaa(0, T258, T259))
insert1_in_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) → U24_gaa(T268, T270, T271, T248, T272, p18_in_gaaa(T268, T270, T271, T272))
U24_gaa(T268, T270, T271, T248, T272, p18_out_gaaa(T268, T270, T271, T272)) → insert1_out_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248))
insert1_in_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292)) → U25_gaa(T285, T290, T287, T291, T292, p68_in_agaa(T290, T285, T291, T292))
p68_in_agaa(s(T229), s(T228), T230, T231) → U12_agaa(T229, T228, T230, T231, p50_in_agaa(T229, T228, T230, T231))
U12_agaa(T229, T228, T230, T231, p50_out_agaa(T229, T228, T230, T231)) → p68_out_agaa(s(T229), s(T228), T230, T231)
U25_gaa(T285, T290, T287, T291, T292, p68_out_agaa(T290, T285, T291, T292)) → insert1_out_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292))
insert1_in_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → U26_gaa(T312, T302, T313, T314, insert1_in_gaa(s(T312), T313, T314))
insert1_in_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → U27_gaa(T322, T323, T302, T324, T325, p50_in_agaa(T323, T322, T324, T325))
U27_gaa(T322, T323, T302, T324, T325, p50_out_agaa(T323, T322, T324, T325)) → insert1_out_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325))
insert1_in_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) → U28_gaa(T345, T346, T336, T347, insert1_in_gaa(0, T346, T347))
insert1_in_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) → U29_gaa(T356, T358, T359, T336, T360, p18_in_gaaa(T356, T358, T359, T360))
U29_gaa(T356, T358, T359, T336, T360, p18_out_gaaa(T356, T358, T359, T360)) → insert1_out_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336))
insert1_in_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380)) → U30_gaa(T373, T378, T375, T379, T380, p68_in_agaa(T378, T373, T379, T380))
U30_gaa(T373, T378, T375, T379, T380, p68_out_agaa(T378, T373, T379, T380)) → insert1_out_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380))
insert1_in_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402)) → U31_gaa(T400, T390, T401, T402, insert1_in_gaa(s(T400), T401, T402))
insert1_in_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) → U32_gaa(T410, T411, T390, T412, T413, p50_in_agaa(T411, T410, T412, T413))
U32_gaa(T410, T411, T390, T412, T413, p50_out_agaa(T411, T410, T412, T413)) → insert1_out_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413))
U31_gaa(T400, T390, T401, T402, insert1_out_gaa(s(T400), T401, T402)) → insert1_out_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402))
U28_gaa(T345, T346, T336, T347, insert1_out_gaa(0, T346, T347)) → insert1_out_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336))
U26_gaa(T312, T302, T313, T314, insert1_out_gaa(s(T312), T313, T314)) → insert1_out_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314))
U23_gaa(T257, T258, T248, T259, insert1_out_gaa(0, T258, T259)) → insert1_out_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248))
U11_agaa(T218, T219, T220, insert1_out_gaa(s(T218), T219, T220)) → p68_out_agaa(0, s(T218), T219, T220)
U22_gaa(T206, T211, T208, T212, T213, p68_out_agaa(T211, T206, T212, T213)) → insert1_out_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213))
U10_agaa(T135, T134, T140, T141, insert1_out_gaa(s(T134), T140, T141)) → p50_out_agaa(T135, T134, T140, T141)
U7_agaa(T135, T134, T136, T137, p50_out_agaa(T135, T134, T136, T137)) → p43_out_agaa(s(T135), s(T134), T136, T137)
U21_gaa(T191, T196, T193, T197, T198, p43_out_agaa(T196, T191, T197, T198)) → insert1_out_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198))
U19_gaa(T163, T164, T154, T165, insert1_out_gaa(0, T164, T165)) → insert1_out_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154))
U6_agaa(T124, T125, T126, insert1_out_gaa(s(T124), T125, T126)) → p43_out_agaa(0, s(T124), T125, T126)
U18_gaa(T112, T117, T114, T118, T119, p43_out_agaa(T117, T112, T118, T119)) → insert1_out_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119))
U17_gaa(T77, T82, T79, T87, T88, insert1_out_gaa(T77, T87, T88)) → insert1_out_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88))
U5_gaaa(T40, T42, T47, T48, insert1_out_gaa(s(T40), T47, T48)) → p18_out_gaaa(T40, T42, T47, T48)
U14_gaa(T40, T42, T43, T20, T44, p18_out_gaaa(T40, T42, T43, T44)) → insert1_out_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20))
U13_gaa(T29, T30, T20, T31, insert1_out_gaa(0, T30, T31)) → insert1_out_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20))

The argument filtering Pi contains the following mapping:
insert1_in_gaa(x1, x2, x3)  =  insert1_in_gaa(x1)
insert1_out_gaa(x1, x2, x3)  =  insert1_out_gaa
0  =  0
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x5)
s(x1)  =  s(x1)
U14_gaa(x1, x2, x3, x4, x5, x6)  =  U14_gaa(x6)
p18_in_gaaa(x1, x2, x3, x4)  =  p18_in_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x5)
less20_in_ga(x1, x2)  =  less20_in_ga(x1)
less20_out_ga(x1, x2)  =  less20_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
p18_out_gaaa(x1, x2, x3, x4)  =  p18_out_gaaa
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
U5_gaaa(x1, x2, x3, x4, x5)  =  U5_gaaa(x5)
U15_gaa(x1, x2, x3, x4, x5, x6)  =  U15_gaa(x6)
less33_in_ag(x1, x2)  =  less33_in_ag(x2)
less33_out_ag(x1, x2)  =  less33_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U16_gaa(x1, x2, x3, x4, x5, x6)  =  U16_gaa(x1, x6)
U17_gaa(x1, x2, x3, x4, x5, x6)  =  U17_gaa(x6)
U18_gaa(x1, x2, x3, x4, x5, x6)  =  U18_gaa(x6)
p43_in_agaa(x1, x2, x3, x4)  =  p43_in_agaa(x2)
U6_agaa(x1, x2, x3, x4)  =  U6_agaa(x4)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x5)
U20_gaa(x1, x2, x3, x4, x5, x6)  =  U20_gaa(x6)
U21_gaa(x1, x2, x3, x4, x5, x6)  =  U21_gaa(x6)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x5)
p50_in_agaa(x1, x2, x3, x4)  =  p50_in_agaa(x2)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x5)
p50_out_agaa(x1, x2, x3, x4)  =  p50_out_agaa(x1)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x1, x5)
U22_gaa(x1, x2, x3, x4, x5, x6)  =  U22_gaa(x6)
p68_in_agaa(x1, x2, x3, x4)  =  p68_in_agaa(x2)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x4)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x5)
U24_gaa(x1, x2, x3, x4, x5, x6)  =  U24_gaa(x6)
U25_gaa(x1, x2, x3, x4, x5, x6)  =  U25_gaa(x6)
U12_agaa(x1, x2, x3, x4, x5)  =  U12_agaa(x5)
p68_out_agaa(x1, x2, x3, x4)  =  p68_out_agaa(x1)
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x5)
U27_gaa(x1, x2, x3, x4, x5, x6)  =  U27_gaa(x6)
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U29_gaa(x1, x2, x3, x4, x5, x6)  =  U29_gaa(x6)
U30_gaa(x1, x2, x3, x4, x5, x6)  =  U30_gaa(x6)
U31_gaa(x1, x2, x3, x4, x5)  =  U31_gaa(x5)
U32_gaa(x1, x2, x3, x4, x5, x6)  =  U32_gaa(x6)
p43_out_agaa(x1, x2, x3, x4)  =  p43_out_agaa(x1)

(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:

INSERT1_IN_GAA(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → U13_GAA(T29, T30, T20, T31, insert1_in_gaa(0, T30, T31))
INSERT1_IN_GAA(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → INSERT1_IN_GAA(0, T30, T31)
INSERT1_IN_GAA(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → U14_GAA(T40, T42, T43, T20, T44, p18_in_gaaa(T40, T42, T43, T44))
INSERT1_IN_GAA(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → P18_IN_GAAA(T40, T42, T43, T44)
P18_IN_GAAA(T40, T42, T43, T44) → U3_GAAA(T40, T42, T43, T44, less20_in_ga(T40, T42))
P18_IN_GAAA(T40, T42, T43, T44) → LESS20_IN_GA(T40, T42)
LESS20_IN_GA(s(T60), s(T62)) → U1_GA(T60, T62, less20_in_ga(T60, T62))
LESS20_IN_GA(s(T60), s(T62)) → LESS20_IN_GA(T60, T62)
P18_IN_GAAA(T40, T42, T47, T48) → U4_GAAA(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_GAAA(T40, T42, T47, T48, less20_out_ga(T40, T42)) → U5_GAAA(T40, T42, T47, T48, insert1_in_gaa(s(T40), T47, T48))
U4_GAAA(T40, T42, T47, T48, less20_out_ga(T40, T42)) → INSERT1_IN_GAA(s(T40), T47, T48)
INSERT1_IN_GAA(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → U15_GAA(T77, T82, T79, T83, T84, less33_in_ag(T82, T77))
INSERT1_IN_GAA(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → LESS33_IN_AG(T82, T77)
LESS33_IN_AG(s(T102), s(T101)) → U2_AG(T102, T101, less33_in_ag(T102, T101))
LESS33_IN_AG(s(T102), s(T101)) → LESS33_IN_AG(T102, T101)
INSERT1_IN_GAA(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_GAA(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_GAA(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → U17_GAA(T77, T82, T79, T87, T88, insert1_in_gaa(T77, T87, T88))
U16_GAA(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → INSERT1_IN_GAA(T77, T87, T88)
INSERT1_IN_GAA(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → U18_GAA(T112, T117, T114, T118, T119, p43_in_agaa(T117, T112, T118, T119))
INSERT1_IN_GAA(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → P43_IN_AGAA(T117, T112, T118, T119)
P43_IN_AGAA(0, s(T124), T125, T126) → U6_AGAA(T124, T125, T126, insert1_in_gaa(s(T124), T125, T126))
P43_IN_AGAA(0, s(T124), T125, T126) → INSERT1_IN_GAA(s(T124), T125, T126)
INSERT1_IN_GAA(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) → U19_GAA(T163, T164, T154, T165, insert1_in_gaa(0, T164, T165))
INSERT1_IN_GAA(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) → U20_GAA(T174, T176, T177, T154, T178, p18_in_gaaa(T174, T176, T177, T178))
INSERT1_IN_GAA(T191, tree(T196, T193, T197), tree(T196, T193, T198)) → U21_GAA(T191, T196, T193, T197, T198, p43_in_agaa(T196, T191, T197, T198))
P43_IN_AGAA(s(T135), s(T134), T136, T137) → U7_AGAA(T135, T134, T136, T137, p50_in_agaa(T135, T134, T136, T137))
P43_IN_AGAA(s(T135), s(T134), T136, T137) → P50_IN_AGAA(T135, T134, T136, T137)
P50_IN_AGAA(T135, T134, T136, T137) → U8_AGAA(T135, T134, T136, T137, less33_in_ag(T135, T134))
P50_IN_AGAA(T135, T134, T136, T137) → LESS33_IN_AG(T135, T134)
P50_IN_AGAA(T135, T134, T140, T141) → U9_AGAA(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_AGAA(T135, T134, T140, T141, less33_out_ag(T135, T134)) → U10_AGAA(T135, T134, T140, T141, insert1_in_gaa(s(T134), T140, T141))
U9_AGAA(T135, T134, T140, T141, less33_out_ag(T135, T134)) → INSERT1_IN_GAA(s(T134), T140, T141)
INSERT1_IN_GAA(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → U22_GAA(T206, T211, T208, T212, T213, p68_in_agaa(T211, T206, T212, T213))
INSERT1_IN_GAA(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → P68_IN_AGAA(T211, T206, T212, T213)
P68_IN_AGAA(0, s(T218), T219, T220) → U11_AGAA(T218, T219, T220, insert1_in_gaa(s(T218), T219, T220))
P68_IN_AGAA(0, s(T218), T219, T220) → INSERT1_IN_GAA(s(T218), T219, T220)
INSERT1_IN_GAA(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) → U23_GAA(T257, T258, T248, T259, insert1_in_gaa(0, T258, T259))
INSERT1_IN_GAA(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) → U24_GAA(T268, T270, T271, T248, T272, p18_in_gaaa(T268, T270, T271, T272))
INSERT1_IN_GAA(T285, tree(T290, T287, T291), tree(T290, T287, T292)) → U25_GAA(T285, T290, T287, T291, T292, p68_in_agaa(T290, T285, T291, T292))
P68_IN_AGAA(s(T229), s(T228), T230, T231) → U12_AGAA(T229, T228, T230, T231, p50_in_agaa(T229, T228, T230, T231))
P68_IN_AGAA(s(T229), s(T228), T230, T231) → P50_IN_AGAA(T229, T228, T230, T231)
INSERT1_IN_GAA(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → U26_GAA(T312, T302, T313, T314, insert1_in_gaa(s(T312), T313, T314))
INSERT1_IN_GAA(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → INSERT1_IN_GAA(s(T312), T313, T314)
INSERT1_IN_GAA(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → U27_GAA(T322, T323, T302, T324, T325, p50_in_agaa(T323, T322, T324, T325))
INSERT1_IN_GAA(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → P50_IN_AGAA(T323, T322, T324, T325)
INSERT1_IN_GAA(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) → U28_GAA(T345, T346, T336, T347, insert1_in_gaa(0, T346, T347))
INSERT1_IN_GAA(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) → U29_GAA(T356, T358, T359, T336, T360, p18_in_gaaa(T356, T358, T359, T360))
INSERT1_IN_GAA(T373, tree(T378, T375, T379), tree(T378, T375, T380)) → U30_GAA(T373, T378, T375, T379, T380, p68_in_agaa(T378, T373, T379, T380))
INSERT1_IN_GAA(s(T400), tree(0, T390, T401), tree(0, T390, T402)) → U31_GAA(T400, T390, T401, T402, insert1_in_gaa(s(T400), T401, T402))
INSERT1_IN_GAA(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) → U32_GAA(T410, T411, T390, T412, T413, p50_in_agaa(T411, T410, T412, T413))

The TRS R consists of the following rules:

insert1_in_gaa(T5, void, tree(T5, void, void)) → insert1_out_gaa(T5, void, tree(T5, void, void))
insert1_in_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11)) → insert1_out_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11))
insert1_in_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → U13_gaa(T29, T30, T20, T31, insert1_in_gaa(0, T30, T31))
insert1_in_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → U14_gaa(T40, T42, T43, T20, T44, p18_in_gaaa(T40, T42, T43, T44))
p18_in_gaaa(T40, T42, T43, T44) → U3_gaaa(T40, T42, T43, T44, less20_in_ga(T40, T42))
less20_in_ga(0, s(T55)) → less20_out_ga(0, s(T55))
less20_in_ga(s(T60), s(T62)) → U1_ga(T60, T62, less20_in_ga(T60, T62))
U1_ga(T60, T62, less20_out_ga(T60, T62)) → less20_out_ga(s(T60), s(T62))
U3_gaaa(T40, T42, T43, T44, less20_out_ga(T40, T42)) → p18_out_gaaa(T40, T42, T43, T44)
p18_in_gaaa(T40, T42, T47, T48) → U4_gaaa(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_gaaa(T40, T42, T47, T48, less20_out_ga(T40, T42)) → U5_gaaa(T40, T42, T47, T48, insert1_in_gaa(s(T40), T47, T48))
insert1_in_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → U15_gaa(T77, T82, T79, T83, T84, less33_in_ag(T82, T77))
less33_in_ag(0, s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T102), s(T101)) → U2_ag(T102, T101, less33_in_ag(T102, T101))
U2_ag(T102, T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U15_gaa(T77, T82, T79, T83, T84, less33_out_ag(T82, T77)) → insert1_out_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84))
insert1_in_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_gaa(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_gaa(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → U17_gaa(T77, T82, T79, T87, T88, insert1_in_gaa(T77, T87, T88))
insert1_in_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → U18_gaa(T112, T117, T114, T118, T119, p43_in_agaa(T117, T112, T118, T119))
p43_in_agaa(0, s(T124), T125, T126) → U6_agaa(T124, T125, T126, insert1_in_gaa(s(T124), T125, T126))
insert1_in_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) → U19_gaa(T163, T164, T154, T165, insert1_in_gaa(0, T164, T165))
insert1_in_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) → U20_gaa(T174, T176, T177, T154, T178, p18_in_gaaa(T174, T176, T177, T178))
U20_gaa(T174, T176, T177, T154, T178, p18_out_gaaa(T174, T176, T177, T178)) → insert1_out_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154))
insert1_in_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198)) → U21_gaa(T191, T196, T193, T197, T198, p43_in_agaa(T196, T191, T197, T198))
p43_in_agaa(s(T135), s(T134), T136, T137) → U7_agaa(T135, T134, T136, T137, p50_in_agaa(T135, T134, T136, T137))
p50_in_agaa(T135, T134, T136, T137) → U8_agaa(T135, T134, T136, T137, less33_in_ag(T135, T134))
U8_agaa(T135, T134, T136, T137, less33_out_ag(T135, T134)) → p50_out_agaa(T135, T134, T136, T137)
p50_in_agaa(T135, T134, T140, T141) → U9_agaa(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_agaa(T135, T134, T140, T141, less33_out_ag(T135, T134)) → U10_agaa(T135, T134, T140, T141, insert1_in_gaa(s(T134), T140, T141))
insert1_in_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → U22_gaa(T206, T211, T208, T212, T213, p68_in_agaa(T211, T206, T212, T213))
p68_in_agaa(0, s(T218), T219, T220) → U11_agaa(T218, T219, T220, insert1_in_gaa(s(T218), T219, T220))
insert1_in_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) → U23_gaa(T257, T258, T248, T259, insert1_in_gaa(0, T258, T259))
insert1_in_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) → U24_gaa(T268, T270, T271, T248, T272, p18_in_gaaa(T268, T270, T271, T272))
U24_gaa(T268, T270, T271, T248, T272, p18_out_gaaa(T268, T270, T271, T272)) → insert1_out_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248))
insert1_in_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292)) → U25_gaa(T285, T290, T287, T291, T292, p68_in_agaa(T290, T285, T291, T292))
p68_in_agaa(s(T229), s(T228), T230, T231) → U12_agaa(T229, T228, T230, T231, p50_in_agaa(T229, T228, T230, T231))
U12_agaa(T229, T228, T230, T231, p50_out_agaa(T229, T228, T230, T231)) → p68_out_agaa(s(T229), s(T228), T230, T231)
U25_gaa(T285, T290, T287, T291, T292, p68_out_agaa(T290, T285, T291, T292)) → insert1_out_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292))
insert1_in_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → U26_gaa(T312, T302, T313, T314, insert1_in_gaa(s(T312), T313, T314))
insert1_in_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → U27_gaa(T322, T323, T302, T324, T325, p50_in_agaa(T323, T322, T324, T325))
U27_gaa(T322, T323, T302, T324, T325, p50_out_agaa(T323, T322, T324, T325)) → insert1_out_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325))
insert1_in_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) → U28_gaa(T345, T346, T336, T347, insert1_in_gaa(0, T346, T347))
insert1_in_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) → U29_gaa(T356, T358, T359, T336, T360, p18_in_gaaa(T356, T358, T359, T360))
U29_gaa(T356, T358, T359, T336, T360, p18_out_gaaa(T356, T358, T359, T360)) → insert1_out_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336))
insert1_in_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380)) → U30_gaa(T373, T378, T375, T379, T380, p68_in_agaa(T378, T373, T379, T380))
U30_gaa(T373, T378, T375, T379, T380, p68_out_agaa(T378, T373, T379, T380)) → insert1_out_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380))
insert1_in_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402)) → U31_gaa(T400, T390, T401, T402, insert1_in_gaa(s(T400), T401, T402))
insert1_in_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) → U32_gaa(T410, T411, T390, T412, T413, p50_in_agaa(T411, T410, T412, T413))
U32_gaa(T410, T411, T390, T412, T413, p50_out_agaa(T411, T410, T412, T413)) → insert1_out_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413))
U31_gaa(T400, T390, T401, T402, insert1_out_gaa(s(T400), T401, T402)) → insert1_out_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402))
U28_gaa(T345, T346, T336, T347, insert1_out_gaa(0, T346, T347)) → insert1_out_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336))
U26_gaa(T312, T302, T313, T314, insert1_out_gaa(s(T312), T313, T314)) → insert1_out_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314))
U23_gaa(T257, T258, T248, T259, insert1_out_gaa(0, T258, T259)) → insert1_out_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248))
U11_agaa(T218, T219, T220, insert1_out_gaa(s(T218), T219, T220)) → p68_out_agaa(0, s(T218), T219, T220)
U22_gaa(T206, T211, T208, T212, T213, p68_out_agaa(T211, T206, T212, T213)) → insert1_out_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213))
U10_agaa(T135, T134, T140, T141, insert1_out_gaa(s(T134), T140, T141)) → p50_out_agaa(T135, T134, T140, T141)
U7_agaa(T135, T134, T136, T137, p50_out_agaa(T135, T134, T136, T137)) → p43_out_agaa(s(T135), s(T134), T136, T137)
U21_gaa(T191, T196, T193, T197, T198, p43_out_agaa(T196, T191, T197, T198)) → insert1_out_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198))
U19_gaa(T163, T164, T154, T165, insert1_out_gaa(0, T164, T165)) → insert1_out_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154))
U6_agaa(T124, T125, T126, insert1_out_gaa(s(T124), T125, T126)) → p43_out_agaa(0, s(T124), T125, T126)
U18_gaa(T112, T117, T114, T118, T119, p43_out_agaa(T117, T112, T118, T119)) → insert1_out_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119))
U17_gaa(T77, T82, T79, T87, T88, insert1_out_gaa(T77, T87, T88)) → insert1_out_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88))
U5_gaaa(T40, T42, T47, T48, insert1_out_gaa(s(T40), T47, T48)) → p18_out_gaaa(T40, T42, T47, T48)
U14_gaa(T40, T42, T43, T20, T44, p18_out_gaaa(T40, T42, T43, T44)) → insert1_out_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20))
U13_gaa(T29, T30, T20, T31, insert1_out_gaa(0, T30, T31)) → insert1_out_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20))

The argument filtering Pi contains the following mapping:
insert1_in_gaa(x1, x2, x3)  =  insert1_in_gaa(x1)
insert1_out_gaa(x1, x2, x3)  =  insert1_out_gaa
0  =  0
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x5)
s(x1)  =  s(x1)
U14_gaa(x1, x2, x3, x4, x5, x6)  =  U14_gaa(x6)
p18_in_gaaa(x1, x2, x3, x4)  =  p18_in_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x5)
less20_in_ga(x1, x2)  =  less20_in_ga(x1)
less20_out_ga(x1, x2)  =  less20_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
p18_out_gaaa(x1, x2, x3, x4)  =  p18_out_gaaa
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
U5_gaaa(x1, x2, x3, x4, x5)  =  U5_gaaa(x5)
U15_gaa(x1, x2, x3, x4, x5, x6)  =  U15_gaa(x6)
less33_in_ag(x1, x2)  =  less33_in_ag(x2)
less33_out_ag(x1, x2)  =  less33_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U16_gaa(x1, x2, x3, x4, x5, x6)  =  U16_gaa(x1, x6)
U17_gaa(x1, x2, x3, x4, x5, x6)  =  U17_gaa(x6)
U18_gaa(x1, x2, x3, x4, x5, x6)  =  U18_gaa(x6)
p43_in_agaa(x1, x2, x3, x4)  =  p43_in_agaa(x2)
U6_agaa(x1, x2, x3, x4)  =  U6_agaa(x4)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x5)
U20_gaa(x1, x2, x3, x4, x5, x6)  =  U20_gaa(x6)
U21_gaa(x1, x2, x3, x4, x5, x6)  =  U21_gaa(x6)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x5)
p50_in_agaa(x1, x2, x3, x4)  =  p50_in_agaa(x2)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x5)
p50_out_agaa(x1, x2, x3, x4)  =  p50_out_agaa(x1)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x1, x5)
U22_gaa(x1, x2, x3, x4, x5, x6)  =  U22_gaa(x6)
p68_in_agaa(x1, x2, x3, x4)  =  p68_in_agaa(x2)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x4)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x5)
U24_gaa(x1, x2, x3, x4, x5, x6)  =  U24_gaa(x6)
U25_gaa(x1, x2, x3, x4, x5, x6)  =  U25_gaa(x6)
U12_agaa(x1, x2, x3, x4, x5)  =  U12_agaa(x5)
p68_out_agaa(x1, x2, x3, x4)  =  p68_out_agaa(x1)
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x5)
U27_gaa(x1, x2, x3, x4, x5, x6)  =  U27_gaa(x6)
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U29_gaa(x1, x2, x3, x4, x5, x6)  =  U29_gaa(x6)
U30_gaa(x1, x2, x3, x4, x5, x6)  =  U30_gaa(x6)
U31_gaa(x1, x2, x3, x4, x5)  =  U31_gaa(x5)
U32_gaa(x1, x2, x3, x4, x5, x6)  =  U32_gaa(x6)
p43_out_agaa(x1, x2, x3, x4)  =  p43_out_agaa(x1)
INSERT1_IN_GAA(x1, x2, x3)  =  INSERT1_IN_GAA(x1)
U13_GAA(x1, x2, x3, x4, x5)  =  U13_GAA(x5)
U14_GAA(x1, x2, x3, x4, x5, x6)  =  U14_GAA(x6)
P18_IN_GAAA(x1, x2, x3, x4)  =  P18_IN_GAAA(x1)
U3_GAAA(x1, x2, x3, x4, x5)  =  U3_GAAA(x5)
LESS20_IN_GA(x1, x2)  =  LESS20_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U4_GAAA(x1, x2, x3, x4, x5)  =  U4_GAAA(x1, x5)
U5_GAAA(x1, x2, x3, x4, x5)  =  U5_GAAA(x5)
U15_GAA(x1, x2, x3, x4, x5, x6)  =  U15_GAA(x6)
LESS33_IN_AG(x1, x2)  =  LESS33_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x3)
U16_GAA(x1, x2, x3, x4, x5, x6)  =  U16_GAA(x1, x6)
U17_GAA(x1, x2, x3, x4, x5, x6)  =  U17_GAA(x6)
U18_GAA(x1, x2, x3, x4, x5, x6)  =  U18_GAA(x6)
P43_IN_AGAA(x1, x2, x3, x4)  =  P43_IN_AGAA(x2)
U6_AGAA(x1, x2, x3, x4)  =  U6_AGAA(x4)
U19_GAA(x1, x2, x3, x4, x5)  =  U19_GAA(x5)
U20_GAA(x1, x2, x3, x4, x5, x6)  =  U20_GAA(x6)
U21_GAA(x1, x2, x3, x4, x5, x6)  =  U21_GAA(x6)
U7_AGAA(x1, x2, x3, x4, x5)  =  U7_AGAA(x5)
P50_IN_AGAA(x1, x2, x3, x4)  =  P50_IN_AGAA(x2)
U8_AGAA(x1, x2, x3, x4, x5)  =  U8_AGAA(x5)
U9_AGAA(x1, x2, x3, x4, x5)  =  U9_AGAA(x2, x5)
U10_AGAA(x1, x2, x3, x4, x5)  =  U10_AGAA(x1, x5)
U22_GAA(x1, x2, x3, x4, x5, x6)  =  U22_GAA(x6)
P68_IN_AGAA(x1, x2, x3, x4)  =  P68_IN_AGAA(x2)
U11_AGAA(x1, x2, x3, x4)  =  U11_AGAA(x4)
U23_GAA(x1, x2, x3, x4, x5)  =  U23_GAA(x5)
U24_GAA(x1, x2, x3, x4, x5, x6)  =  U24_GAA(x6)
U25_GAA(x1, x2, x3, x4, x5, x6)  =  U25_GAA(x6)
U12_AGAA(x1, x2, x3, x4, x5)  =  U12_AGAA(x5)
U26_GAA(x1, x2, x3, x4, x5)  =  U26_GAA(x5)
U27_GAA(x1, x2, x3, x4, x5, x6)  =  U27_GAA(x6)
U28_GAA(x1, x2, x3, x4, x5)  =  U28_GAA(x5)
U29_GAA(x1, x2, x3, x4, x5, x6)  =  U29_GAA(x6)
U30_GAA(x1, x2, x3, x4, x5, x6)  =  U30_GAA(x6)
U31_GAA(x1, x2, x3, x4, x5)  =  U31_GAA(x5)
U32_GAA(x1, x2, x3, x4, x5, x6)  =  U32_GAA(x6)

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

(6) Obligation:

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

INSERT1_IN_GAA(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → U13_GAA(T29, T30, T20, T31, insert1_in_gaa(0, T30, T31))
INSERT1_IN_GAA(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → INSERT1_IN_GAA(0, T30, T31)
INSERT1_IN_GAA(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → U14_GAA(T40, T42, T43, T20, T44, p18_in_gaaa(T40, T42, T43, T44))
INSERT1_IN_GAA(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → P18_IN_GAAA(T40, T42, T43, T44)
P18_IN_GAAA(T40, T42, T43, T44) → U3_GAAA(T40, T42, T43, T44, less20_in_ga(T40, T42))
P18_IN_GAAA(T40, T42, T43, T44) → LESS20_IN_GA(T40, T42)
LESS20_IN_GA(s(T60), s(T62)) → U1_GA(T60, T62, less20_in_ga(T60, T62))
LESS20_IN_GA(s(T60), s(T62)) → LESS20_IN_GA(T60, T62)
P18_IN_GAAA(T40, T42, T47, T48) → U4_GAAA(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_GAAA(T40, T42, T47, T48, less20_out_ga(T40, T42)) → U5_GAAA(T40, T42, T47, T48, insert1_in_gaa(s(T40), T47, T48))
U4_GAAA(T40, T42, T47, T48, less20_out_ga(T40, T42)) → INSERT1_IN_GAA(s(T40), T47, T48)
INSERT1_IN_GAA(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → U15_GAA(T77, T82, T79, T83, T84, less33_in_ag(T82, T77))
INSERT1_IN_GAA(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → LESS33_IN_AG(T82, T77)
LESS33_IN_AG(s(T102), s(T101)) → U2_AG(T102, T101, less33_in_ag(T102, T101))
LESS33_IN_AG(s(T102), s(T101)) → LESS33_IN_AG(T102, T101)
INSERT1_IN_GAA(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_GAA(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_GAA(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → U17_GAA(T77, T82, T79, T87, T88, insert1_in_gaa(T77, T87, T88))
U16_GAA(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → INSERT1_IN_GAA(T77, T87, T88)
INSERT1_IN_GAA(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → U18_GAA(T112, T117, T114, T118, T119, p43_in_agaa(T117, T112, T118, T119))
INSERT1_IN_GAA(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → P43_IN_AGAA(T117, T112, T118, T119)
P43_IN_AGAA(0, s(T124), T125, T126) → U6_AGAA(T124, T125, T126, insert1_in_gaa(s(T124), T125, T126))
P43_IN_AGAA(0, s(T124), T125, T126) → INSERT1_IN_GAA(s(T124), T125, T126)
INSERT1_IN_GAA(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) → U19_GAA(T163, T164, T154, T165, insert1_in_gaa(0, T164, T165))
INSERT1_IN_GAA(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) → U20_GAA(T174, T176, T177, T154, T178, p18_in_gaaa(T174, T176, T177, T178))
INSERT1_IN_GAA(T191, tree(T196, T193, T197), tree(T196, T193, T198)) → U21_GAA(T191, T196, T193, T197, T198, p43_in_agaa(T196, T191, T197, T198))
P43_IN_AGAA(s(T135), s(T134), T136, T137) → U7_AGAA(T135, T134, T136, T137, p50_in_agaa(T135, T134, T136, T137))
P43_IN_AGAA(s(T135), s(T134), T136, T137) → P50_IN_AGAA(T135, T134, T136, T137)
P50_IN_AGAA(T135, T134, T136, T137) → U8_AGAA(T135, T134, T136, T137, less33_in_ag(T135, T134))
P50_IN_AGAA(T135, T134, T136, T137) → LESS33_IN_AG(T135, T134)
P50_IN_AGAA(T135, T134, T140, T141) → U9_AGAA(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_AGAA(T135, T134, T140, T141, less33_out_ag(T135, T134)) → U10_AGAA(T135, T134, T140, T141, insert1_in_gaa(s(T134), T140, T141))
U9_AGAA(T135, T134, T140, T141, less33_out_ag(T135, T134)) → INSERT1_IN_GAA(s(T134), T140, T141)
INSERT1_IN_GAA(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → U22_GAA(T206, T211, T208, T212, T213, p68_in_agaa(T211, T206, T212, T213))
INSERT1_IN_GAA(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → P68_IN_AGAA(T211, T206, T212, T213)
P68_IN_AGAA(0, s(T218), T219, T220) → U11_AGAA(T218, T219, T220, insert1_in_gaa(s(T218), T219, T220))
P68_IN_AGAA(0, s(T218), T219, T220) → INSERT1_IN_GAA(s(T218), T219, T220)
INSERT1_IN_GAA(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) → U23_GAA(T257, T258, T248, T259, insert1_in_gaa(0, T258, T259))
INSERT1_IN_GAA(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) → U24_GAA(T268, T270, T271, T248, T272, p18_in_gaaa(T268, T270, T271, T272))
INSERT1_IN_GAA(T285, tree(T290, T287, T291), tree(T290, T287, T292)) → U25_GAA(T285, T290, T287, T291, T292, p68_in_agaa(T290, T285, T291, T292))
P68_IN_AGAA(s(T229), s(T228), T230, T231) → U12_AGAA(T229, T228, T230, T231, p50_in_agaa(T229, T228, T230, T231))
P68_IN_AGAA(s(T229), s(T228), T230, T231) → P50_IN_AGAA(T229, T228, T230, T231)
INSERT1_IN_GAA(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → U26_GAA(T312, T302, T313, T314, insert1_in_gaa(s(T312), T313, T314))
INSERT1_IN_GAA(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → INSERT1_IN_GAA(s(T312), T313, T314)
INSERT1_IN_GAA(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → U27_GAA(T322, T323, T302, T324, T325, p50_in_agaa(T323, T322, T324, T325))
INSERT1_IN_GAA(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → P50_IN_AGAA(T323, T322, T324, T325)
INSERT1_IN_GAA(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) → U28_GAA(T345, T346, T336, T347, insert1_in_gaa(0, T346, T347))
INSERT1_IN_GAA(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) → U29_GAA(T356, T358, T359, T336, T360, p18_in_gaaa(T356, T358, T359, T360))
INSERT1_IN_GAA(T373, tree(T378, T375, T379), tree(T378, T375, T380)) → U30_GAA(T373, T378, T375, T379, T380, p68_in_agaa(T378, T373, T379, T380))
INSERT1_IN_GAA(s(T400), tree(0, T390, T401), tree(0, T390, T402)) → U31_GAA(T400, T390, T401, T402, insert1_in_gaa(s(T400), T401, T402))
INSERT1_IN_GAA(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) → U32_GAA(T410, T411, T390, T412, T413, p50_in_agaa(T411, T410, T412, T413))

The TRS R consists of the following rules:

insert1_in_gaa(T5, void, tree(T5, void, void)) → insert1_out_gaa(T5, void, tree(T5, void, void))
insert1_in_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11)) → insert1_out_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11))
insert1_in_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → U13_gaa(T29, T30, T20, T31, insert1_in_gaa(0, T30, T31))
insert1_in_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → U14_gaa(T40, T42, T43, T20, T44, p18_in_gaaa(T40, T42, T43, T44))
p18_in_gaaa(T40, T42, T43, T44) → U3_gaaa(T40, T42, T43, T44, less20_in_ga(T40, T42))
less20_in_ga(0, s(T55)) → less20_out_ga(0, s(T55))
less20_in_ga(s(T60), s(T62)) → U1_ga(T60, T62, less20_in_ga(T60, T62))
U1_ga(T60, T62, less20_out_ga(T60, T62)) → less20_out_ga(s(T60), s(T62))
U3_gaaa(T40, T42, T43, T44, less20_out_ga(T40, T42)) → p18_out_gaaa(T40, T42, T43, T44)
p18_in_gaaa(T40, T42, T47, T48) → U4_gaaa(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_gaaa(T40, T42, T47, T48, less20_out_ga(T40, T42)) → U5_gaaa(T40, T42, T47, T48, insert1_in_gaa(s(T40), T47, T48))
insert1_in_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → U15_gaa(T77, T82, T79, T83, T84, less33_in_ag(T82, T77))
less33_in_ag(0, s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T102), s(T101)) → U2_ag(T102, T101, less33_in_ag(T102, T101))
U2_ag(T102, T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U15_gaa(T77, T82, T79, T83, T84, less33_out_ag(T82, T77)) → insert1_out_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84))
insert1_in_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_gaa(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_gaa(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → U17_gaa(T77, T82, T79, T87, T88, insert1_in_gaa(T77, T87, T88))
insert1_in_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → U18_gaa(T112, T117, T114, T118, T119, p43_in_agaa(T117, T112, T118, T119))
p43_in_agaa(0, s(T124), T125, T126) → U6_agaa(T124, T125, T126, insert1_in_gaa(s(T124), T125, T126))
insert1_in_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) → U19_gaa(T163, T164, T154, T165, insert1_in_gaa(0, T164, T165))
insert1_in_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) → U20_gaa(T174, T176, T177, T154, T178, p18_in_gaaa(T174, T176, T177, T178))
U20_gaa(T174, T176, T177, T154, T178, p18_out_gaaa(T174, T176, T177, T178)) → insert1_out_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154))
insert1_in_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198)) → U21_gaa(T191, T196, T193, T197, T198, p43_in_agaa(T196, T191, T197, T198))
p43_in_agaa(s(T135), s(T134), T136, T137) → U7_agaa(T135, T134, T136, T137, p50_in_agaa(T135, T134, T136, T137))
p50_in_agaa(T135, T134, T136, T137) → U8_agaa(T135, T134, T136, T137, less33_in_ag(T135, T134))
U8_agaa(T135, T134, T136, T137, less33_out_ag(T135, T134)) → p50_out_agaa(T135, T134, T136, T137)
p50_in_agaa(T135, T134, T140, T141) → U9_agaa(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_agaa(T135, T134, T140, T141, less33_out_ag(T135, T134)) → U10_agaa(T135, T134, T140, T141, insert1_in_gaa(s(T134), T140, T141))
insert1_in_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → U22_gaa(T206, T211, T208, T212, T213, p68_in_agaa(T211, T206, T212, T213))
p68_in_agaa(0, s(T218), T219, T220) → U11_agaa(T218, T219, T220, insert1_in_gaa(s(T218), T219, T220))
insert1_in_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) → U23_gaa(T257, T258, T248, T259, insert1_in_gaa(0, T258, T259))
insert1_in_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) → U24_gaa(T268, T270, T271, T248, T272, p18_in_gaaa(T268, T270, T271, T272))
U24_gaa(T268, T270, T271, T248, T272, p18_out_gaaa(T268, T270, T271, T272)) → insert1_out_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248))
insert1_in_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292)) → U25_gaa(T285, T290, T287, T291, T292, p68_in_agaa(T290, T285, T291, T292))
p68_in_agaa(s(T229), s(T228), T230, T231) → U12_agaa(T229, T228, T230, T231, p50_in_agaa(T229, T228, T230, T231))
U12_agaa(T229, T228, T230, T231, p50_out_agaa(T229, T228, T230, T231)) → p68_out_agaa(s(T229), s(T228), T230, T231)
U25_gaa(T285, T290, T287, T291, T292, p68_out_agaa(T290, T285, T291, T292)) → insert1_out_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292))
insert1_in_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → U26_gaa(T312, T302, T313, T314, insert1_in_gaa(s(T312), T313, T314))
insert1_in_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → U27_gaa(T322, T323, T302, T324, T325, p50_in_agaa(T323, T322, T324, T325))
U27_gaa(T322, T323, T302, T324, T325, p50_out_agaa(T323, T322, T324, T325)) → insert1_out_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325))
insert1_in_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) → U28_gaa(T345, T346, T336, T347, insert1_in_gaa(0, T346, T347))
insert1_in_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) → U29_gaa(T356, T358, T359, T336, T360, p18_in_gaaa(T356, T358, T359, T360))
U29_gaa(T356, T358, T359, T336, T360, p18_out_gaaa(T356, T358, T359, T360)) → insert1_out_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336))
insert1_in_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380)) → U30_gaa(T373, T378, T375, T379, T380, p68_in_agaa(T378, T373, T379, T380))
U30_gaa(T373, T378, T375, T379, T380, p68_out_agaa(T378, T373, T379, T380)) → insert1_out_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380))
insert1_in_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402)) → U31_gaa(T400, T390, T401, T402, insert1_in_gaa(s(T400), T401, T402))
insert1_in_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) → U32_gaa(T410, T411, T390, T412, T413, p50_in_agaa(T411, T410, T412, T413))
U32_gaa(T410, T411, T390, T412, T413, p50_out_agaa(T411, T410, T412, T413)) → insert1_out_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413))
U31_gaa(T400, T390, T401, T402, insert1_out_gaa(s(T400), T401, T402)) → insert1_out_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402))
U28_gaa(T345, T346, T336, T347, insert1_out_gaa(0, T346, T347)) → insert1_out_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336))
U26_gaa(T312, T302, T313, T314, insert1_out_gaa(s(T312), T313, T314)) → insert1_out_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314))
U23_gaa(T257, T258, T248, T259, insert1_out_gaa(0, T258, T259)) → insert1_out_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248))
U11_agaa(T218, T219, T220, insert1_out_gaa(s(T218), T219, T220)) → p68_out_agaa(0, s(T218), T219, T220)
U22_gaa(T206, T211, T208, T212, T213, p68_out_agaa(T211, T206, T212, T213)) → insert1_out_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213))
U10_agaa(T135, T134, T140, T141, insert1_out_gaa(s(T134), T140, T141)) → p50_out_agaa(T135, T134, T140, T141)
U7_agaa(T135, T134, T136, T137, p50_out_agaa(T135, T134, T136, T137)) → p43_out_agaa(s(T135), s(T134), T136, T137)
U21_gaa(T191, T196, T193, T197, T198, p43_out_agaa(T196, T191, T197, T198)) → insert1_out_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198))
U19_gaa(T163, T164, T154, T165, insert1_out_gaa(0, T164, T165)) → insert1_out_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154))
U6_agaa(T124, T125, T126, insert1_out_gaa(s(T124), T125, T126)) → p43_out_agaa(0, s(T124), T125, T126)
U18_gaa(T112, T117, T114, T118, T119, p43_out_agaa(T117, T112, T118, T119)) → insert1_out_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119))
U17_gaa(T77, T82, T79, T87, T88, insert1_out_gaa(T77, T87, T88)) → insert1_out_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88))
U5_gaaa(T40, T42, T47, T48, insert1_out_gaa(s(T40), T47, T48)) → p18_out_gaaa(T40, T42, T47, T48)
U14_gaa(T40, T42, T43, T20, T44, p18_out_gaaa(T40, T42, T43, T44)) → insert1_out_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20))
U13_gaa(T29, T30, T20, T31, insert1_out_gaa(0, T30, T31)) → insert1_out_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20))

The argument filtering Pi contains the following mapping:
insert1_in_gaa(x1, x2, x3)  =  insert1_in_gaa(x1)
insert1_out_gaa(x1, x2, x3)  =  insert1_out_gaa
0  =  0
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x5)
s(x1)  =  s(x1)
U14_gaa(x1, x2, x3, x4, x5, x6)  =  U14_gaa(x6)
p18_in_gaaa(x1, x2, x3, x4)  =  p18_in_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x5)
less20_in_ga(x1, x2)  =  less20_in_ga(x1)
less20_out_ga(x1, x2)  =  less20_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
p18_out_gaaa(x1, x2, x3, x4)  =  p18_out_gaaa
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
U5_gaaa(x1, x2, x3, x4, x5)  =  U5_gaaa(x5)
U15_gaa(x1, x2, x3, x4, x5, x6)  =  U15_gaa(x6)
less33_in_ag(x1, x2)  =  less33_in_ag(x2)
less33_out_ag(x1, x2)  =  less33_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U16_gaa(x1, x2, x3, x4, x5, x6)  =  U16_gaa(x1, x6)
U17_gaa(x1, x2, x3, x4, x5, x6)  =  U17_gaa(x6)
U18_gaa(x1, x2, x3, x4, x5, x6)  =  U18_gaa(x6)
p43_in_agaa(x1, x2, x3, x4)  =  p43_in_agaa(x2)
U6_agaa(x1, x2, x3, x4)  =  U6_agaa(x4)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x5)
U20_gaa(x1, x2, x3, x4, x5, x6)  =  U20_gaa(x6)
U21_gaa(x1, x2, x3, x4, x5, x6)  =  U21_gaa(x6)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x5)
p50_in_agaa(x1, x2, x3, x4)  =  p50_in_agaa(x2)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x5)
p50_out_agaa(x1, x2, x3, x4)  =  p50_out_agaa(x1)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x1, x5)
U22_gaa(x1, x2, x3, x4, x5, x6)  =  U22_gaa(x6)
p68_in_agaa(x1, x2, x3, x4)  =  p68_in_agaa(x2)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x4)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x5)
U24_gaa(x1, x2, x3, x4, x5, x6)  =  U24_gaa(x6)
U25_gaa(x1, x2, x3, x4, x5, x6)  =  U25_gaa(x6)
U12_agaa(x1, x2, x3, x4, x5)  =  U12_agaa(x5)
p68_out_agaa(x1, x2, x3, x4)  =  p68_out_agaa(x1)
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x5)
U27_gaa(x1, x2, x3, x4, x5, x6)  =  U27_gaa(x6)
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U29_gaa(x1, x2, x3, x4, x5, x6)  =  U29_gaa(x6)
U30_gaa(x1, x2, x3, x4, x5, x6)  =  U30_gaa(x6)
U31_gaa(x1, x2, x3, x4, x5)  =  U31_gaa(x5)
U32_gaa(x1, x2, x3, x4, x5, x6)  =  U32_gaa(x6)
p43_out_agaa(x1, x2, x3, x4)  =  p43_out_agaa(x1)
INSERT1_IN_GAA(x1, x2, x3)  =  INSERT1_IN_GAA(x1)
U13_GAA(x1, x2, x3, x4, x5)  =  U13_GAA(x5)
U14_GAA(x1, x2, x3, x4, x5, x6)  =  U14_GAA(x6)
P18_IN_GAAA(x1, x2, x3, x4)  =  P18_IN_GAAA(x1)
U3_GAAA(x1, x2, x3, x4, x5)  =  U3_GAAA(x5)
LESS20_IN_GA(x1, x2)  =  LESS20_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U4_GAAA(x1, x2, x3, x4, x5)  =  U4_GAAA(x1, x5)
U5_GAAA(x1, x2, x3, x4, x5)  =  U5_GAAA(x5)
U15_GAA(x1, x2, x3, x4, x5, x6)  =  U15_GAA(x6)
LESS33_IN_AG(x1, x2)  =  LESS33_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x3)
U16_GAA(x1, x2, x3, x4, x5, x6)  =  U16_GAA(x1, x6)
U17_GAA(x1, x2, x3, x4, x5, x6)  =  U17_GAA(x6)
U18_GAA(x1, x2, x3, x4, x5, x6)  =  U18_GAA(x6)
P43_IN_AGAA(x1, x2, x3, x4)  =  P43_IN_AGAA(x2)
U6_AGAA(x1, x2, x3, x4)  =  U6_AGAA(x4)
U19_GAA(x1, x2, x3, x4, x5)  =  U19_GAA(x5)
U20_GAA(x1, x2, x3, x4, x5, x6)  =  U20_GAA(x6)
U21_GAA(x1, x2, x3, x4, x5, x6)  =  U21_GAA(x6)
U7_AGAA(x1, x2, x3, x4, x5)  =  U7_AGAA(x5)
P50_IN_AGAA(x1, x2, x3, x4)  =  P50_IN_AGAA(x2)
U8_AGAA(x1, x2, x3, x4, x5)  =  U8_AGAA(x5)
U9_AGAA(x1, x2, x3, x4, x5)  =  U9_AGAA(x2, x5)
U10_AGAA(x1, x2, x3, x4, x5)  =  U10_AGAA(x1, x5)
U22_GAA(x1, x2, x3, x4, x5, x6)  =  U22_GAA(x6)
P68_IN_AGAA(x1, x2, x3, x4)  =  P68_IN_AGAA(x2)
U11_AGAA(x1, x2, x3, x4)  =  U11_AGAA(x4)
U23_GAA(x1, x2, x3, x4, x5)  =  U23_GAA(x5)
U24_GAA(x1, x2, x3, x4, x5, x6)  =  U24_GAA(x6)
U25_GAA(x1, x2, x3, x4, x5, x6)  =  U25_GAA(x6)
U12_AGAA(x1, x2, x3, x4, x5)  =  U12_AGAA(x5)
U26_GAA(x1, x2, x3, x4, x5)  =  U26_GAA(x5)
U27_GAA(x1, x2, x3, x4, x5, x6)  =  U27_GAA(x6)
U28_GAA(x1, x2, x3, x4, x5)  =  U28_GAA(x5)
U29_GAA(x1, x2, x3, x4, x5, x6)  =  U29_GAA(x6)
U30_GAA(x1, x2, x3, x4, x5, x6)  =  U30_GAA(x6)
U31_GAA(x1, x2, x3, x4, x5)  =  U31_GAA(x5)
U32_GAA(x1, x2, x3, x4, x5, x6)  =  U32_GAA(x6)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 32 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESS33_IN_AG(s(T102), s(T101)) → LESS33_IN_AG(T102, T101)

The TRS R consists of the following rules:

insert1_in_gaa(T5, void, tree(T5, void, void)) → insert1_out_gaa(T5, void, tree(T5, void, void))
insert1_in_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11)) → insert1_out_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11))
insert1_in_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → U13_gaa(T29, T30, T20, T31, insert1_in_gaa(0, T30, T31))
insert1_in_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → U14_gaa(T40, T42, T43, T20, T44, p18_in_gaaa(T40, T42, T43, T44))
p18_in_gaaa(T40, T42, T43, T44) → U3_gaaa(T40, T42, T43, T44, less20_in_ga(T40, T42))
less20_in_ga(0, s(T55)) → less20_out_ga(0, s(T55))
less20_in_ga(s(T60), s(T62)) → U1_ga(T60, T62, less20_in_ga(T60, T62))
U1_ga(T60, T62, less20_out_ga(T60, T62)) → less20_out_ga(s(T60), s(T62))
U3_gaaa(T40, T42, T43, T44, less20_out_ga(T40, T42)) → p18_out_gaaa(T40, T42, T43, T44)
p18_in_gaaa(T40, T42, T47, T48) → U4_gaaa(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_gaaa(T40, T42, T47, T48, less20_out_ga(T40, T42)) → U5_gaaa(T40, T42, T47, T48, insert1_in_gaa(s(T40), T47, T48))
insert1_in_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → U15_gaa(T77, T82, T79, T83, T84, less33_in_ag(T82, T77))
less33_in_ag(0, s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T102), s(T101)) → U2_ag(T102, T101, less33_in_ag(T102, T101))
U2_ag(T102, T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U15_gaa(T77, T82, T79, T83, T84, less33_out_ag(T82, T77)) → insert1_out_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84))
insert1_in_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_gaa(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_gaa(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → U17_gaa(T77, T82, T79, T87, T88, insert1_in_gaa(T77, T87, T88))
insert1_in_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → U18_gaa(T112, T117, T114, T118, T119, p43_in_agaa(T117, T112, T118, T119))
p43_in_agaa(0, s(T124), T125, T126) → U6_agaa(T124, T125, T126, insert1_in_gaa(s(T124), T125, T126))
insert1_in_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) → U19_gaa(T163, T164, T154, T165, insert1_in_gaa(0, T164, T165))
insert1_in_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) → U20_gaa(T174, T176, T177, T154, T178, p18_in_gaaa(T174, T176, T177, T178))
U20_gaa(T174, T176, T177, T154, T178, p18_out_gaaa(T174, T176, T177, T178)) → insert1_out_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154))
insert1_in_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198)) → U21_gaa(T191, T196, T193, T197, T198, p43_in_agaa(T196, T191, T197, T198))
p43_in_agaa(s(T135), s(T134), T136, T137) → U7_agaa(T135, T134, T136, T137, p50_in_agaa(T135, T134, T136, T137))
p50_in_agaa(T135, T134, T136, T137) → U8_agaa(T135, T134, T136, T137, less33_in_ag(T135, T134))
U8_agaa(T135, T134, T136, T137, less33_out_ag(T135, T134)) → p50_out_agaa(T135, T134, T136, T137)
p50_in_agaa(T135, T134, T140, T141) → U9_agaa(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_agaa(T135, T134, T140, T141, less33_out_ag(T135, T134)) → U10_agaa(T135, T134, T140, T141, insert1_in_gaa(s(T134), T140, T141))
insert1_in_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → U22_gaa(T206, T211, T208, T212, T213, p68_in_agaa(T211, T206, T212, T213))
p68_in_agaa(0, s(T218), T219, T220) → U11_agaa(T218, T219, T220, insert1_in_gaa(s(T218), T219, T220))
insert1_in_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) → U23_gaa(T257, T258, T248, T259, insert1_in_gaa(0, T258, T259))
insert1_in_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) → U24_gaa(T268, T270, T271, T248, T272, p18_in_gaaa(T268, T270, T271, T272))
U24_gaa(T268, T270, T271, T248, T272, p18_out_gaaa(T268, T270, T271, T272)) → insert1_out_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248))
insert1_in_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292)) → U25_gaa(T285, T290, T287, T291, T292, p68_in_agaa(T290, T285, T291, T292))
p68_in_agaa(s(T229), s(T228), T230, T231) → U12_agaa(T229, T228, T230, T231, p50_in_agaa(T229, T228, T230, T231))
U12_agaa(T229, T228, T230, T231, p50_out_agaa(T229, T228, T230, T231)) → p68_out_agaa(s(T229), s(T228), T230, T231)
U25_gaa(T285, T290, T287, T291, T292, p68_out_agaa(T290, T285, T291, T292)) → insert1_out_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292))
insert1_in_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → U26_gaa(T312, T302, T313, T314, insert1_in_gaa(s(T312), T313, T314))
insert1_in_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → U27_gaa(T322, T323, T302, T324, T325, p50_in_agaa(T323, T322, T324, T325))
U27_gaa(T322, T323, T302, T324, T325, p50_out_agaa(T323, T322, T324, T325)) → insert1_out_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325))
insert1_in_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) → U28_gaa(T345, T346, T336, T347, insert1_in_gaa(0, T346, T347))
insert1_in_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) → U29_gaa(T356, T358, T359, T336, T360, p18_in_gaaa(T356, T358, T359, T360))
U29_gaa(T356, T358, T359, T336, T360, p18_out_gaaa(T356, T358, T359, T360)) → insert1_out_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336))
insert1_in_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380)) → U30_gaa(T373, T378, T375, T379, T380, p68_in_agaa(T378, T373, T379, T380))
U30_gaa(T373, T378, T375, T379, T380, p68_out_agaa(T378, T373, T379, T380)) → insert1_out_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380))
insert1_in_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402)) → U31_gaa(T400, T390, T401, T402, insert1_in_gaa(s(T400), T401, T402))
insert1_in_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) → U32_gaa(T410, T411, T390, T412, T413, p50_in_agaa(T411, T410, T412, T413))
U32_gaa(T410, T411, T390, T412, T413, p50_out_agaa(T411, T410, T412, T413)) → insert1_out_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413))
U31_gaa(T400, T390, T401, T402, insert1_out_gaa(s(T400), T401, T402)) → insert1_out_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402))
U28_gaa(T345, T346, T336, T347, insert1_out_gaa(0, T346, T347)) → insert1_out_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336))
U26_gaa(T312, T302, T313, T314, insert1_out_gaa(s(T312), T313, T314)) → insert1_out_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314))
U23_gaa(T257, T258, T248, T259, insert1_out_gaa(0, T258, T259)) → insert1_out_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248))
U11_agaa(T218, T219, T220, insert1_out_gaa(s(T218), T219, T220)) → p68_out_agaa(0, s(T218), T219, T220)
U22_gaa(T206, T211, T208, T212, T213, p68_out_agaa(T211, T206, T212, T213)) → insert1_out_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213))
U10_agaa(T135, T134, T140, T141, insert1_out_gaa(s(T134), T140, T141)) → p50_out_agaa(T135, T134, T140, T141)
U7_agaa(T135, T134, T136, T137, p50_out_agaa(T135, T134, T136, T137)) → p43_out_agaa(s(T135), s(T134), T136, T137)
U21_gaa(T191, T196, T193, T197, T198, p43_out_agaa(T196, T191, T197, T198)) → insert1_out_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198))
U19_gaa(T163, T164, T154, T165, insert1_out_gaa(0, T164, T165)) → insert1_out_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154))
U6_agaa(T124, T125, T126, insert1_out_gaa(s(T124), T125, T126)) → p43_out_agaa(0, s(T124), T125, T126)
U18_gaa(T112, T117, T114, T118, T119, p43_out_agaa(T117, T112, T118, T119)) → insert1_out_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119))
U17_gaa(T77, T82, T79, T87, T88, insert1_out_gaa(T77, T87, T88)) → insert1_out_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88))
U5_gaaa(T40, T42, T47, T48, insert1_out_gaa(s(T40), T47, T48)) → p18_out_gaaa(T40, T42, T47, T48)
U14_gaa(T40, T42, T43, T20, T44, p18_out_gaaa(T40, T42, T43, T44)) → insert1_out_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20))
U13_gaa(T29, T30, T20, T31, insert1_out_gaa(0, T30, T31)) → insert1_out_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20))

The argument filtering Pi contains the following mapping:
insert1_in_gaa(x1, x2, x3)  =  insert1_in_gaa(x1)
insert1_out_gaa(x1, x2, x3)  =  insert1_out_gaa
0  =  0
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x5)
s(x1)  =  s(x1)
U14_gaa(x1, x2, x3, x4, x5, x6)  =  U14_gaa(x6)
p18_in_gaaa(x1, x2, x3, x4)  =  p18_in_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x5)
less20_in_ga(x1, x2)  =  less20_in_ga(x1)
less20_out_ga(x1, x2)  =  less20_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
p18_out_gaaa(x1, x2, x3, x4)  =  p18_out_gaaa
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
U5_gaaa(x1, x2, x3, x4, x5)  =  U5_gaaa(x5)
U15_gaa(x1, x2, x3, x4, x5, x6)  =  U15_gaa(x6)
less33_in_ag(x1, x2)  =  less33_in_ag(x2)
less33_out_ag(x1, x2)  =  less33_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U16_gaa(x1, x2, x3, x4, x5, x6)  =  U16_gaa(x1, x6)
U17_gaa(x1, x2, x3, x4, x5, x6)  =  U17_gaa(x6)
U18_gaa(x1, x2, x3, x4, x5, x6)  =  U18_gaa(x6)
p43_in_agaa(x1, x2, x3, x4)  =  p43_in_agaa(x2)
U6_agaa(x1, x2, x3, x4)  =  U6_agaa(x4)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x5)
U20_gaa(x1, x2, x3, x4, x5, x6)  =  U20_gaa(x6)
U21_gaa(x1, x2, x3, x4, x5, x6)  =  U21_gaa(x6)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x5)
p50_in_agaa(x1, x2, x3, x4)  =  p50_in_agaa(x2)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x5)
p50_out_agaa(x1, x2, x3, x4)  =  p50_out_agaa(x1)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x1, x5)
U22_gaa(x1, x2, x3, x4, x5, x6)  =  U22_gaa(x6)
p68_in_agaa(x1, x2, x3, x4)  =  p68_in_agaa(x2)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x4)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x5)
U24_gaa(x1, x2, x3, x4, x5, x6)  =  U24_gaa(x6)
U25_gaa(x1, x2, x3, x4, x5, x6)  =  U25_gaa(x6)
U12_agaa(x1, x2, x3, x4, x5)  =  U12_agaa(x5)
p68_out_agaa(x1, x2, x3, x4)  =  p68_out_agaa(x1)
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x5)
U27_gaa(x1, x2, x3, x4, x5, x6)  =  U27_gaa(x6)
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U29_gaa(x1, x2, x3, x4, x5, x6)  =  U29_gaa(x6)
U30_gaa(x1, x2, x3, x4, x5, x6)  =  U30_gaa(x6)
U31_gaa(x1, x2, x3, x4, x5)  =  U31_gaa(x5)
U32_gaa(x1, x2, x3, x4, x5, x6)  =  U32_gaa(x6)
p43_out_agaa(x1, x2, x3, x4)  =  p43_out_agaa(x1)
LESS33_IN_AG(x1, x2)  =  LESS33_IN_AG(x2)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

LESS33_IN_AG(s(T102), s(T101)) → LESS33_IN_AG(T102, T101)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS33_IN_AG(x1, x2)  =  LESS33_IN_AG(x2)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

LESS33_IN_AG(s(T101)) → LESS33_IN_AG(T101)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • LESS33_IN_AG(s(T101)) → LESS33_IN_AG(T101)
    The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

LESS20_IN_GA(s(T60), s(T62)) → LESS20_IN_GA(T60, T62)

The TRS R consists of the following rules:

insert1_in_gaa(T5, void, tree(T5, void, void)) → insert1_out_gaa(T5, void, tree(T5, void, void))
insert1_in_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11)) → insert1_out_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11))
insert1_in_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → U13_gaa(T29, T30, T20, T31, insert1_in_gaa(0, T30, T31))
insert1_in_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → U14_gaa(T40, T42, T43, T20, T44, p18_in_gaaa(T40, T42, T43, T44))
p18_in_gaaa(T40, T42, T43, T44) → U3_gaaa(T40, T42, T43, T44, less20_in_ga(T40, T42))
less20_in_ga(0, s(T55)) → less20_out_ga(0, s(T55))
less20_in_ga(s(T60), s(T62)) → U1_ga(T60, T62, less20_in_ga(T60, T62))
U1_ga(T60, T62, less20_out_ga(T60, T62)) → less20_out_ga(s(T60), s(T62))
U3_gaaa(T40, T42, T43, T44, less20_out_ga(T40, T42)) → p18_out_gaaa(T40, T42, T43, T44)
p18_in_gaaa(T40, T42, T47, T48) → U4_gaaa(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_gaaa(T40, T42, T47, T48, less20_out_ga(T40, T42)) → U5_gaaa(T40, T42, T47, T48, insert1_in_gaa(s(T40), T47, T48))
insert1_in_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → U15_gaa(T77, T82, T79, T83, T84, less33_in_ag(T82, T77))
less33_in_ag(0, s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T102), s(T101)) → U2_ag(T102, T101, less33_in_ag(T102, T101))
U2_ag(T102, T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U15_gaa(T77, T82, T79, T83, T84, less33_out_ag(T82, T77)) → insert1_out_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84))
insert1_in_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_gaa(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_gaa(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → U17_gaa(T77, T82, T79, T87, T88, insert1_in_gaa(T77, T87, T88))
insert1_in_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → U18_gaa(T112, T117, T114, T118, T119, p43_in_agaa(T117, T112, T118, T119))
p43_in_agaa(0, s(T124), T125, T126) → U6_agaa(T124, T125, T126, insert1_in_gaa(s(T124), T125, T126))
insert1_in_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) → U19_gaa(T163, T164, T154, T165, insert1_in_gaa(0, T164, T165))
insert1_in_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) → U20_gaa(T174, T176, T177, T154, T178, p18_in_gaaa(T174, T176, T177, T178))
U20_gaa(T174, T176, T177, T154, T178, p18_out_gaaa(T174, T176, T177, T178)) → insert1_out_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154))
insert1_in_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198)) → U21_gaa(T191, T196, T193, T197, T198, p43_in_agaa(T196, T191, T197, T198))
p43_in_agaa(s(T135), s(T134), T136, T137) → U7_agaa(T135, T134, T136, T137, p50_in_agaa(T135, T134, T136, T137))
p50_in_agaa(T135, T134, T136, T137) → U8_agaa(T135, T134, T136, T137, less33_in_ag(T135, T134))
U8_agaa(T135, T134, T136, T137, less33_out_ag(T135, T134)) → p50_out_agaa(T135, T134, T136, T137)
p50_in_agaa(T135, T134, T140, T141) → U9_agaa(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_agaa(T135, T134, T140, T141, less33_out_ag(T135, T134)) → U10_agaa(T135, T134, T140, T141, insert1_in_gaa(s(T134), T140, T141))
insert1_in_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → U22_gaa(T206, T211, T208, T212, T213, p68_in_agaa(T211, T206, T212, T213))
p68_in_agaa(0, s(T218), T219, T220) → U11_agaa(T218, T219, T220, insert1_in_gaa(s(T218), T219, T220))
insert1_in_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) → U23_gaa(T257, T258, T248, T259, insert1_in_gaa(0, T258, T259))
insert1_in_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) → U24_gaa(T268, T270, T271, T248, T272, p18_in_gaaa(T268, T270, T271, T272))
U24_gaa(T268, T270, T271, T248, T272, p18_out_gaaa(T268, T270, T271, T272)) → insert1_out_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248))
insert1_in_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292)) → U25_gaa(T285, T290, T287, T291, T292, p68_in_agaa(T290, T285, T291, T292))
p68_in_agaa(s(T229), s(T228), T230, T231) → U12_agaa(T229, T228, T230, T231, p50_in_agaa(T229, T228, T230, T231))
U12_agaa(T229, T228, T230, T231, p50_out_agaa(T229, T228, T230, T231)) → p68_out_agaa(s(T229), s(T228), T230, T231)
U25_gaa(T285, T290, T287, T291, T292, p68_out_agaa(T290, T285, T291, T292)) → insert1_out_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292))
insert1_in_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → U26_gaa(T312, T302, T313, T314, insert1_in_gaa(s(T312), T313, T314))
insert1_in_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → U27_gaa(T322, T323, T302, T324, T325, p50_in_agaa(T323, T322, T324, T325))
U27_gaa(T322, T323, T302, T324, T325, p50_out_agaa(T323, T322, T324, T325)) → insert1_out_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325))
insert1_in_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) → U28_gaa(T345, T346, T336, T347, insert1_in_gaa(0, T346, T347))
insert1_in_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) → U29_gaa(T356, T358, T359, T336, T360, p18_in_gaaa(T356, T358, T359, T360))
U29_gaa(T356, T358, T359, T336, T360, p18_out_gaaa(T356, T358, T359, T360)) → insert1_out_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336))
insert1_in_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380)) → U30_gaa(T373, T378, T375, T379, T380, p68_in_agaa(T378, T373, T379, T380))
U30_gaa(T373, T378, T375, T379, T380, p68_out_agaa(T378, T373, T379, T380)) → insert1_out_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380))
insert1_in_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402)) → U31_gaa(T400, T390, T401, T402, insert1_in_gaa(s(T400), T401, T402))
insert1_in_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) → U32_gaa(T410, T411, T390, T412, T413, p50_in_agaa(T411, T410, T412, T413))
U32_gaa(T410, T411, T390, T412, T413, p50_out_agaa(T411, T410, T412, T413)) → insert1_out_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413))
U31_gaa(T400, T390, T401, T402, insert1_out_gaa(s(T400), T401, T402)) → insert1_out_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402))
U28_gaa(T345, T346, T336, T347, insert1_out_gaa(0, T346, T347)) → insert1_out_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336))
U26_gaa(T312, T302, T313, T314, insert1_out_gaa(s(T312), T313, T314)) → insert1_out_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314))
U23_gaa(T257, T258, T248, T259, insert1_out_gaa(0, T258, T259)) → insert1_out_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248))
U11_agaa(T218, T219, T220, insert1_out_gaa(s(T218), T219, T220)) → p68_out_agaa(0, s(T218), T219, T220)
U22_gaa(T206, T211, T208, T212, T213, p68_out_agaa(T211, T206, T212, T213)) → insert1_out_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213))
U10_agaa(T135, T134, T140, T141, insert1_out_gaa(s(T134), T140, T141)) → p50_out_agaa(T135, T134, T140, T141)
U7_agaa(T135, T134, T136, T137, p50_out_agaa(T135, T134, T136, T137)) → p43_out_agaa(s(T135), s(T134), T136, T137)
U21_gaa(T191, T196, T193, T197, T198, p43_out_agaa(T196, T191, T197, T198)) → insert1_out_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198))
U19_gaa(T163, T164, T154, T165, insert1_out_gaa(0, T164, T165)) → insert1_out_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154))
U6_agaa(T124, T125, T126, insert1_out_gaa(s(T124), T125, T126)) → p43_out_agaa(0, s(T124), T125, T126)
U18_gaa(T112, T117, T114, T118, T119, p43_out_agaa(T117, T112, T118, T119)) → insert1_out_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119))
U17_gaa(T77, T82, T79, T87, T88, insert1_out_gaa(T77, T87, T88)) → insert1_out_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88))
U5_gaaa(T40, T42, T47, T48, insert1_out_gaa(s(T40), T47, T48)) → p18_out_gaaa(T40, T42, T47, T48)
U14_gaa(T40, T42, T43, T20, T44, p18_out_gaaa(T40, T42, T43, T44)) → insert1_out_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20))
U13_gaa(T29, T30, T20, T31, insert1_out_gaa(0, T30, T31)) → insert1_out_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20))

The argument filtering Pi contains the following mapping:
insert1_in_gaa(x1, x2, x3)  =  insert1_in_gaa(x1)
insert1_out_gaa(x1, x2, x3)  =  insert1_out_gaa
0  =  0
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x5)
s(x1)  =  s(x1)
U14_gaa(x1, x2, x3, x4, x5, x6)  =  U14_gaa(x6)
p18_in_gaaa(x1, x2, x3, x4)  =  p18_in_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x5)
less20_in_ga(x1, x2)  =  less20_in_ga(x1)
less20_out_ga(x1, x2)  =  less20_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
p18_out_gaaa(x1, x2, x3, x4)  =  p18_out_gaaa
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
U5_gaaa(x1, x2, x3, x4, x5)  =  U5_gaaa(x5)
U15_gaa(x1, x2, x3, x4, x5, x6)  =  U15_gaa(x6)
less33_in_ag(x1, x2)  =  less33_in_ag(x2)
less33_out_ag(x1, x2)  =  less33_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U16_gaa(x1, x2, x3, x4, x5, x6)  =  U16_gaa(x1, x6)
U17_gaa(x1, x2, x3, x4, x5, x6)  =  U17_gaa(x6)
U18_gaa(x1, x2, x3, x4, x5, x6)  =  U18_gaa(x6)
p43_in_agaa(x1, x2, x3, x4)  =  p43_in_agaa(x2)
U6_agaa(x1, x2, x3, x4)  =  U6_agaa(x4)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x5)
U20_gaa(x1, x2, x3, x4, x5, x6)  =  U20_gaa(x6)
U21_gaa(x1, x2, x3, x4, x5, x6)  =  U21_gaa(x6)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x5)
p50_in_agaa(x1, x2, x3, x4)  =  p50_in_agaa(x2)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x5)
p50_out_agaa(x1, x2, x3, x4)  =  p50_out_agaa(x1)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x1, x5)
U22_gaa(x1, x2, x3, x4, x5, x6)  =  U22_gaa(x6)
p68_in_agaa(x1, x2, x3, x4)  =  p68_in_agaa(x2)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x4)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x5)
U24_gaa(x1, x2, x3, x4, x5, x6)  =  U24_gaa(x6)
U25_gaa(x1, x2, x3, x4, x5, x6)  =  U25_gaa(x6)
U12_agaa(x1, x2, x3, x4, x5)  =  U12_agaa(x5)
p68_out_agaa(x1, x2, x3, x4)  =  p68_out_agaa(x1)
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x5)
U27_gaa(x1, x2, x3, x4, x5, x6)  =  U27_gaa(x6)
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U29_gaa(x1, x2, x3, x4, x5, x6)  =  U29_gaa(x6)
U30_gaa(x1, x2, x3, x4, x5, x6)  =  U30_gaa(x6)
U31_gaa(x1, x2, x3, x4, x5)  =  U31_gaa(x5)
U32_gaa(x1, x2, x3, x4, x5, x6)  =  U32_gaa(x6)
p43_out_agaa(x1, x2, x3, x4)  =  p43_out_agaa(x1)
LESS20_IN_GA(x1, x2)  =  LESS20_IN_GA(x1)

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:

LESS20_IN_GA(s(T60), s(T62)) → LESS20_IN_GA(T60, T62)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS20_IN_GA(x1, x2)  =  LESS20_IN_GA(x1)

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:

LESS20_IN_GA(s(T60)) → LESS20_IN_GA(T60)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • LESS20_IN_GA(s(T60)) → LESS20_IN_GA(T60)
    The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

INSERT1_IN_GAA(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_GAA(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_GAA(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → INSERT1_IN_GAA(T77, T87, T88)
INSERT1_IN_GAA(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → INSERT1_IN_GAA(0, T30, T31)
INSERT1_IN_GAA(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → P43_IN_AGAA(T117, T112, T118, T119)
P43_IN_AGAA(0, s(T124), T125, T126) → INSERT1_IN_GAA(s(T124), T125, T126)
INSERT1_IN_GAA(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → P18_IN_GAAA(T40, T42, T43, T44)
P18_IN_GAAA(T40, T42, T47, T48) → U4_GAAA(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_GAAA(T40, T42, T47, T48, less20_out_ga(T40, T42)) → INSERT1_IN_GAA(s(T40), T47, T48)
INSERT1_IN_GAA(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → P68_IN_AGAA(T211, T206, T212, T213)
P68_IN_AGAA(0, s(T218), T219, T220) → INSERT1_IN_GAA(s(T218), T219, T220)
INSERT1_IN_GAA(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → INSERT1_IN_GAA(s(T312), T313, T314)
INSERT1_IN_GAA(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → P50_IN_AGAA(T323, T322, T324, T325)
P50_IN_AGAA(T135, T134, T140, T141) → U9_AGAA(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_AGAA(T135, T134, T140, T141, less33_out_ag(T135, T134)) → INSERT1_IN_GAA(s(T134), T140, T141)
P68_IN_AGAA(s(T229), s(T228), T230, T231) → P50_IN_AGAA(T229, T228, T230, T231)
P43_IN_AGAA(s(T135), s(T134), T136, T137) → P50_IN_AGAA(T135, T134, T136, T137)

The TRS R consists of the following rules:

insert1_in_gaa(T5, void, tree(T5, void, void)) → insert1_out_gaa(T5, void, tree(T5, void, void))
insert1_in_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11)) → insert1_out_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11))
insert1_in_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → U13_gaa(T29, T30, T20, T31, insert1_in_gaa(0, T30, T31))
insert1_in_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → U14_gaa(T40, T42, T43, T20, T44, p18_in_gaaa(T40, T42, T43, T44))
p18_in_gaaa(T40, T42, T43, T44) → U3_gaaa(T40, T42, T43, T44, less20_in_ga(T40, T42))
less20_in_ga(0, s(T55)) → less20_out_ga(0, s(T55))
less20_in_ga(s(T60), s(T62)) → U1_ga(T60, T62, less20_in_ga(T60, T62))
U1_ga(T60, T62, less20_out_ga(T60, T62)) → less20_out_ga(s(T60), s(T62))
U3_gaaa(T40, T42, T43, T44, less20_out_ga(T40, T42)) → p18_out_gaaa(T40, T42, T43, T44)
p18_in_gaaa(T40, T42, T47, T48) → U4_gaaa(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_gaaa(T40, T42, T47, T48, less20_out_ga(T40, T42)) → U5_gaaa(T40, T42, T47, T48, insert1_in_gaa(s(T40), T47, T48))
insert1_in_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → U15_gaa(T77, T82, T79, T83, T84, less33_in_ag(T82, T77))
less33_in_ag(0, s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T102), s(T101)) → U2_ag(T102, T101, less33_in_ag(T102, T101))
U2_ag(T102, T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U15_gaa(T77, T82, T79, T83, T84, less33_out_ag(T82, T77)) → insert1_out_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84))
insert1_in_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_gaa(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_gaa(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → U17_gaa(T77, T82, T79, T87, T88, insert1_in_gaa(T77, T87, T88))
insert1_in_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → U18_gaa(T112, T117, T114, T118, T119, p43_in_agaa(T117, T112, T118, T119))
p43_in_agaa(0, s(T124), T125, T126) → U6_agaa(T124, T125, T126, insert1_in_gaa(s(T124), T125, T126))
insert1_in_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) → U19_gaa(T163, T164, T154, T165, insert1_in_gaa(0, T164, T165))
insert1_in_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) → U20_gaa(T174, T176, T177, T154, T178, p18_in_gaaa(T174, T176, T177, T178))
U20_gaa(T174, T176, T177, T154, T178, p18_out_gaaa(T174, T176, T177, T178)) → insert1_out_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154))
insert1_in_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198)) → U21_gaa(T191, T196, T193, T197, T198, p43_in_agaa(T196, T191, T197, T198))
p43_in_agaa(s(T135), s(T134), T136, T137) → U7_agaa(T135, T134, T136, T137, p50_in_agaa(T135, T134, T136, T137))
p50_in_agaa(T135, T134, T136, T137) → U8_agaa(T135, T134, T136, T137, less33_in_ag(T135, T134))
U8_agaa(T135, T134, T136, T137, less33_out_ag(T135, T134)) → p50_out_agaa(T135, T134, T136, T137)
p50_in_agaa(T135, T134, T140, T141) → U9_agaa(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_agaa(T135, T134, T140, T141, less33_out_ag(T135, T134)) → U10_agaa(T135, T134, T140, T141, insert1_in_gaa(s(T134), T140, T141))
insert1_in_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → U22_gaa(T206, T211, T208, T212, T213, p68_in_agaa(T211, T206, T212, T213))
p68_in_agaa(0, s(T218), T219, T220) → U11_agaa(T218, T219, T220, insert1_in_gaa(s(T218), T219, T220))
insert1_in_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) → U23_gaa(T257, T258, T248, T259, insert1_in_gaa(0, T258, T259))
insert1_in_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) → U24_gaa(T268, T270, T271, T248, T272, p18_in_gaaa(T268, T270, T271, T272))
U24_gaa(T268, T270, T271, T248, T272, p18_out_gaaa(T268, T270, T271, T272)) → insert1_out_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248))
insert1_in_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292)) → U25_gaa(T285, T290, T287, T291, T292, p68_in_agaa(T290, T285, T291, T292))
p68_in_agaa(s(T229), s(T228), T230, T231) → U12_agaa(T229, T228, T230, T231, p50_in_agaa(T229, T228, T230, T231))
U12_agaa(T229, T228, T230, T231, p50_out_agaa(T229, T228, T230, T231)) → p68_out_agaa(s(T229), s(T228), T230, T231)
U25_gaa(T285, T290, T287, T291, T292, p68_out_agaa(T290, T285, T291, T292)) → insert1_out_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292))
insert1_in_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → U26_gaa(T312, T302, T313, T314, insert1_in_gaa(s(T312), T313, T314))
insert1_in_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → U27_gaa(T322, T323, T302, T324, T325, p50_in_agaa(T323, T322, T324, T325))
U27_gaa(T322, T323, T302, T324, T325, p50_out_agaa(T323, T322, T324, T325)) → insert1_out_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325))
insert1_in_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) → U28_gaa(T345, T346, T336, T347, insert1_in_gaa(0, T346, T347))
insert1_in_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) → U29_gaa(T356, T358, T359, T336, T360, p18_in_gaaa(T356, T358, T359, T360))
U29_gaa(T356, T358, T359, T336, T360, p18_out_gaaa(T356, T358, T359, T360)) → insert1_out_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336))
insert1_in_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380)) → U30_gaa(T373, T378, T375, T379, T380, p68_in_agaa(T378, T373, T379, T380))
U30_gaa(T373, T378, T375, T379, T380, p68_out_agaa(T378, T373, T379, T380)) → insert1_out_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380))
insert1_in_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402)) → U31_gaa(T400, T390, T401, T402, insert1_in_gaa(s(T400), T401, T402))
insert1_in_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) → U32_gaa(T410, T411, T390, T412, T413, p50_in_agaa(T411, T410, T412, T413))
U32_gaa(T410, T411, T390, T412, T413, p50_out_agaa(T411, T410, T412, T413)) → insert1_out_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413))
U31_gaa(T400, T390, T401, T402, insert1_out_gaa(s(T400), T401, T402)) → insert1_out_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402))
U28_gaa(T345, T346, T336, T347, insert1_out_gaa(0, T346, T347)) → insert1_out_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336))
U26_gaa(T312, T302, T313, T314, insert1_out_gaa(s(T312), T313, T314)) → insert1_out_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314))
U23_gaa(T257, T258, T248, T259, insert1_out_gaa(0, T258, T259)) → insert1_out_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248))
U11_agaa(T218, T219, T220, insert1_out_gaa(s(T218), T219, T220)) → p68_out_agaa(0, s(T218), T219, T220)
U22_gaa(T206, T211, T208, T212, T213, p68_out_agaa(T211, T206, T212, T213)) → insert1_out_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213))
U10_agaa(T135, T134, T140, T141, insert1_out_gaa(s(T134), T140, T141)) → p50_out_agaa(T135, T134, T140, T141)
U7_agaa(T135, T134, T136, T137, p50_out_agaa(T135, T134, T136, T137)) → p43_out_agaa(s(T135), s(T134), T136, T137)
U21_gaa(T191, T196, T193, T197, T198, p43_out_agaa(T196, T191, T197, T198)) → insert1_out_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198))
U19_gaa(T163, T164, T154, T165, insert1_out_gaa(0, T164, T165)) → insert1_out_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154))
U6_agaa(T124, T125, T126, insert1_out_gaa(s(T124), T125, T126)) → p43_out_agaa(0, s(T124), T125, T126)
U18_gaa(T112, T117, T114, T118, T119, p43_out_agaa(T117, T112, T118, T119)) → insert1_out_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119))
U17_gaa(T77, T82, T79, T87, T88, insert1_out_gaa(T77, T87, T88)) → insert1_out_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88))
U5_gaaa(T40, T42, T47, T48, insert1_out_gaa(s(T40), T47, T48)) → p18_out_gaaa(T40, T42, T47, T48)
U14_gaa(T40, T42, T43, T20, T44, p18_out_gaaa(T40, T42, T43, T44)) → insert1_out_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20))
U13_gaa(T29, T30, T20, T31, insert1_out_gaa(0, T30, T31)) → insert1_out_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20))

The argument filtering Pi contains the following mapping:
insert1_in_gaa(x1, x2, x3)  =  insert1_in_gaa(x1)
insert1_out_gaa(x1, x2, x3)  =  insert1_out_gaa
0  =  0
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x5)
s(x1)  =  s(x1)
U14_gaa(x1, x2, x3, x4, x5, x6)  =  U14_gaa(x6)
p18_in_gaaa(x1, x2, x3, x4)  =  p18_in_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x5)
less20_in_ga(x1, x2)  =  less20_in_ga(x1)
less20_out_ga(x1, x2)  =  less20_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
p18_out_gaaa(x1, x2, x3, x4)  =  p18_out_gaaa
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
U5_gaaa(x1, x2, x3, x4, x5)  =  U5_gaaa(x5)
U15_gaa(x1, x2, x3, x4, x5, x6)  =  U15_gaa(x6)
less33_in_ag(x1, x2)  =  less33_in_ag(x2)
less33_out_ag(x1, x2)  =  less33_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
U16_gaa(x1, x2, x3, x4, x5, x6)  =  U16_gaa(x1, x6)
U17_gaa(x1, x2, x3, x4, x5, x6)  =  U17_gaa(x6)
U18_gaa(x1, x2, x3, x4, x5, x6)  =  U18_gaa(x6)
p43_in_agaa(x1, x2, x3, x4)  =  p43_in_agaa(x2)
U6_agaa(x1, x2, x3, x4)  =  U6_agaa(x4)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x5)
U20_gaa(x1, x2, x3, x4, x5, x6)  =  U20_gaa(x6)
U21_gaa(x1, x2, x3, x4, x5, x6)  =  U21_gaa(x6)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x5)
p50_in_agaa(x1, x2, x3, x4)  =  p50_in_agaa(x2)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x5)
p50_out_agaa(x1, x2, x3, x4)  =  p50_out_agaa(x1)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x1, x5)
U22_gaa(x1, x2, x3, x4, x5, x6)  =  U22_gaa(x6)
p68_in_agaa(x1, x2, x3, x4)  =  p68_in_agaa(x2)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x4)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x5)
U24_gaa(x1, x2, x3, x4, x5, x6)  =  U24_gaa(x6)
U25_gaa(x1, x2, x3, x4, x5, x6)  =  U25_gaa(x6)
U12_agaa(x1, x2, x3, x4, x5)  =  U12_agaa(x5)
p68_out_agaa(x1, x2, x3, x4)  =  p68_out_agaa(x1)
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x5)
U27_gaa(x1, x2, x3, x4, x5, x6)  =  U27_gaa(x6)
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U29_gaa(x1, x2, x3, x4, x5, x6)  =  U29_gaa(x6)
U30_gaa(x1, x2, x3, x4, x5, x6)  =  U30_gaa(x6)
U31_gaa(x1, x2, x3, x4, x5)  =  U31_gaa(x5)
U32_gaa(x1, x2, x3, x4, x5, x6)  =  U32_gaa(x6)
p43_out_agaa(x1, x2, x3, x4)  =  p43_out_agaa(x1)
INSERT1_IN_GAA(x1, x2, x3)  =  INSERT1_IN_GAA(x1)
P18_IN_GAAA(x1, x2, x3, x4)  =  P18_IN_GAAA(x1)
U4_GAAA(x1, x2, x3, x4, x5)  =  U4_GAAA(x1, x5)
U16_GAA(x1, x2, x3, x4, x5, x6)  =  U16_GAA(x1, x6)
P43_IN_AGAA(x1, x2, x3, x4)  =  P43_IN_AGAA(x2)
P50_IN_AGAA(x1, x2, x3, x4)  =  P50_IN_AGAA(x2)
U9_AGAA(x1, x2, x3, x4, x5)  =  U9_AGAA(x2, x5)
P68_IN_AGAA(x1, x2, x3, x4)  =  P68_IN_AGAA(x2)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

INSERT1_IN_GAA(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_GAA(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_GAA(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → INSERT1_IN_GAA(T77, T87, T88)
INSERT1_IN_GAA(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → INSERT1_IN_GAA(0, T30, T31)
INSERT1_IN_GAA(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → P43_IN_AGAA(T117, T112, T118, T119)
P43_IN_AGAA(0, s(T124), T125, T126) → INSERT1_IN_GAA(s(T124), T125, T126)
INSERT1_IN_GAA(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → P18_IN_GAAA(T40, T42, T43, T44)
P18_IN_GAAA(T40, T42, T47, T48) → U4_GAAA(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_GAAA(T40, T42, T47, T48, less20_out_ga(T40, T42)) → INSERT1_IN_GAA(s(T40), T47, T48)
INSERT1_IN_GAA(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → P68_IN_AGAA(T211, T206, T212, T213)
P68_IN_AGAA(0, s(T218), T219, T220) → INSERT1_IN_GAA(s(T218), T219, T220)
INSERT1_IN_GAA(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → INSERT1_IN_GAA(s(T312), T313, T314)
INSERT1_IN_GAA(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → P50_IN_AGAA(T323, T322, T324, T325)
P50_IN_AGAA(T135, T134, T140, T141) → U9_AGAA(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_AGAA(T135, T134, T140, T141, less33_out_ag(T135, T134)) → INSERT1_IN_GAA(s(T134), T140, T141)
P68_IN_AGAA(s(T229), s(T228), T230, T231) → P50_IN_AGAA(T229, T228, T230, T231)
P43_IN_AGAA(s(T135), s(T134), T136, T137) → P50_IN_AGAA(T135, T134, T136, T137)

The TRS R consists of the following rules:

less33_in_ag(0, s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T102), s(T101)) → U2_ag(T102, T101, less33_in_ag(T102, T101))
less20_in_ga(0, s(T55)) → less20_out_ga(0, s(T55))
less20_in_ga(s(T60), s(T62)) → U1_ga(T60, T62, less20_in_ga(T60, T62))
U2_ag(T102, T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U1_ga(T60, T62, less20_out_ga(T60, T62)) → less20_out_ga(s(T60), s(T62))

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
less20_in_ga(x1, x2)  =  less20_in_ga(x1)
less20_out_ga(x1, x2)  =  less20_out_ga
U1_ga(x1, x2, x3)  =  U1_ga(x3)
less33_in_ag(x1, x2)  =  less33_in_ag(x2)
less33_out_ag(x1, x2)  =  less33_out_ag(x1)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
INSERT1_IN_GAA(x1, x2, x3)  =  INSERT1_IN_GAA(x1)
P18_IN_GAAA(x1, x2, x3, x4)  =  P18_IN_GAAA(x1)
U4_GAAA(x1, x2, x3, x4, x5)  =  U4_GAAA(x1, x5)
U16_GAA(x1, x2, x3, x4, x5, x6)  =  U16_GAA(x1, x6)
P43_IN_AGAA(x1, x2, x3, x4)  =  P43_IN_AGAA(x2)
P50_IN_AGAA(x1, x2, x3, x4)  =  P50_IN_AGAA(x2)
U9_AGAA(x1, x2, x3, x4, x5)  =  U9_AGAA(x2, x5)
P68_IN_AGAA(x1, x2, x3, x4)  =  P68_IN_AGAA(x2)

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:

INSERT1_IN_GAA(T77) → U16_GAA(T77, less33_in_ag(T77))
U16_GAA(T77, less33_out_ag(T82)) → INSERT1_IN_GAA(T77)
INSERT1_IN_GAA(0) → INSERT1_IN_GAA(0)
INSERT1_IN_GAA(T112) → P43_IN_AGAA(T112)
P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
P18_IN_GAAA(T40) → U4_GAAA(T40, less20_in_ga(T40))
U4_GAAA(T40, less20_out_ga) → INSERT1_IN_GAA(s(T40))
INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(T134) → U9_AGAA(T134, less33_in_ag(T134))
U9_AGAA(T134, less33_out_ag(T135)) → INSERT1_IN_GAA(s(T134))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0)
less33_in_ag(s(T101)) → U2_ag(less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga
less20_in_ga(s(T60)) → U1_ga(less20_in_ga(T60))
U2_ag(less33_out_ag(T102)) → less33_out_ag(s(T102))
U1_ga(less20_out_ga) → less20_out_ga

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0)
U1_ga(x0)

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

(28) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule INSERT1_IN_GAA(T77) → U16_GAA(T77, less33_in_ag(T77)) at position [1] we obtained the following new rules [LPAR04]:

INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(less33_in_ag(x0)))

(29) Obligation:

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

U16_GAA(T77, less33_out_ag(T82)) → INSERT1_IN_GAA(T77)
INSERT1_IN_GAA(0) → INSERT1_IN_GAA(0)
INSERT1_IN_GAA(T112) → P43_IN_AGAA(T112)
P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
P18_IN_GAAA(T40) → U4_GAAA(T40, less20_in_ga(T40))
U4_GAAA(T40, less20_out_ga) → INSERT1_IN_GAA(s(T40))
INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(T134) → U9_AGAA(T134, less33_in_ag(T134))
U9_AGAA(T134, less33_out_ag(T135)) → INSERT1_IN_GAA(s(T134))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(less33_in_ag(x0)))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0)
less33_in_ag(s(T101)) → U2_ag(less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga
less20_in_ga(s(T60)) → U1_ga(less20_in_ga(T60))
U2_ag(less33_out_ag(T102)) → less33_out_ag(s(T102))
U1_ga(less20_out_ga) → less20_out_ga

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0)
U1_ga(x0)

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

(30) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P18_IN_GAAA(T40) → U4_GAAA(T40, less20_in_ga(T40)) at position [1] we obtained the following new rules [LPAR04]:

P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(less20_in_ga(x0)))

(31) Obligation:

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

U16_GAA(T77, less33_out_ag(T82)) → INSERT1_IN_GAA(T77)
INSERT1_IN_GAA(0) → INSERT1_IN_GAA(0)
INSERT1_IN_GAA(T112) → P43_IN_AGAA(T112)
P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
U4_GAAA(T40, less20_out_ga) → INSERT1_IN_GAA(s(T40))
INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(T134) → U9_AGAA(T134, less33_in_ag(T134))
U9_AGAA(T134, less33_out_ag(T135)) → INSERT1_IN_GAA(s(T134))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(less33_in_ag(x0)))
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(less20_in_ga(x0)))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0)
less33_in_ag(s(T101)) → U2_ag(less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga
less20_in_ga(s(T60)) → U1_ga(less20_in_ga(T60))
U2_ag(less33_out_ag(T102)) → less33_out_ag(s(T102))
U1_ga(less20_out_ga) → less20_out_ga

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0)
U1_ga(x0)

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

(32) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P50_IN_AGAA(T134) → U9_AGAA(T134, less33_in_ag(T134)) at position [1] we obtained the following new rules [LPAR04]:

P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(less33_in_ag(x0)))

(33) Obligation:

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

U16_GAA(T77, less33_out_ag(T82)) → INSERT1_IN_GAA(T77)
INSERT1_IN_GAA(0) → INSERT1_IN_GAA(0)
INSERT1_IN_GAA(T112) → P43_IN_AGAA(T112)
P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
U4_GAAA(T40, less20_out_ga) → INSERT1_IN_GAA(s(T40))
INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
U9_AGAA(T134, less33_out_ag(T135)) → INSERT1_IN_GAA(s(T134))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(less33_in_ag(x0)))
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(less20_in_ga(x0)))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(less33_in_ag(x0)))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0)
less33_in_ag(s(T101)) → U2_ag(less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga
less20_in_ga(s(T60)) → U1_ga(less20_in_ga(T60))
U2_ag(less33_out_ag(T102)) → less33_out_ag(s(T102))
U1_ga(less20_out_ga) → less20_out_ga

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0)
U1_ga(x0)

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

(34) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U16_GAA(T77, less33_out_ag(T82)) → INSERT1_IN_GAA(T77) we obtained the following new rules [LPAR04]:

U16_GAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(z0))
U16_GAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(z0))

(35) Obligation:

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

INSERT1_IN_GAA(0) → INSERT1_IN_GAA(0)
INSERT1_IN_GAA(T112) → P43_IN_AGAA(T112)
P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
U4_GAAA(T40, less20_out_ga) → INSERT1_IN_GAA(s(T40))
INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
U9_AGAA(T134, less33_out_ag(T135)) → INSERT1_IN_GAA(s(T134))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(less33_in_ag(x0)))
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(less20_in_ga(x0)))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(z0))
U16_GAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(z0))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0)
less33_in_ag(s(T101)) → U2_ag(less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga
less20_in_ga(s(T60)) → U1_ga(less20_in_ga(T60))
U2_ag(less33_out_ag(T102)) → less33_out_ag(s(T102))
U1_ga(less20_out_ga) → less20_out_ga

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0)
U1_ga(x0)

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

(36) DependencyGraphProof (EQUIVALENT transformation)

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

(37) Complex Obligation (AND)

(38) Obligation:

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

P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
INSERT1_IN_GAA(T112) → P43_IN_AGAA(T112)
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0))
U9_AGAA(T134, less33_out_ag(T135)) → INSERT1_IN_GAA(s(T134))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga)
U4_GAAA(T40, less20_out_ga) → INSERT1_IN_GAA(s(T40))
INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(less33_in_ag(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0))
U16_GAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(z0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(z0))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(less20_in_ga(x0)))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0)
less33_in_ag(s(T101)) → U2_ag(less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga
less20_in_ga(s(T60)) → U1_ga(less20_in_ga(T60))
U2_ag(less33_out_ag(T102)) → less33_out_ag(s(T102))
U1_ga(less20_out_ga) → less20_out_ga

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0)
U1_ga(x0)

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

(39) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule INSERT1_IN_GAA(T112) → P43_IN_AGAA(T112) we obtained the following new rules [LPAR04]:

INSERT1_IN_GAA(s(z0)) → P43_IN_AGAA(s(z0))

(40) Obligation:

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

P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0))
U9_AGAA(T134, less33_out_ag(T135)) → INSERT1_IN_GAA(s(T134))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga)
U4_GAAA(T40, less20_out_ga) → INSERT1_IN_GAA(s(T40))
INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(less33_in_ag(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0))
U16_GAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(z0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(z0))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(less20_in_ga(x0)))
INSERT1_IN_GAA(s(z0)) → P43_IN_AGAA(s(z0))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0)
less33_in_ag(s(T101)) → U2_ag(less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga
less20_in_ga(s(T60)) → U1_ga(less20_in_ga(T60))
U2_ag(less33_out_ag(T102)) → less33_out_ag(s(T102))
U1_ga(less20_out_ga) → less20_out_ga

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0)
U1_ga(x0)

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

(41) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U9_AGAA(T134, less33_out_ag(T135)) → INSERT1_IN_GAA(s(T134)) we obtained the following new rules [LPAR04]:

U9_AGAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(s(z0)))
U9_AGAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(s(z0)))

(42) Obligation:

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

P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga)
U4_GAAA(T40, less20_out_ga) → INSERT1_IN_GAA(s(T40))
INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(less33_in_ag(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0))
U16_GAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(z0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(z0))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(less20_in_ga(x0)))
INSERT1_IN_GAA(s(z0)) → P43_IN_AGAA(s(z0))
U9_AGAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(s(z0)))
U9_AGAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(s(z0)))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0)
less33_in_ag(s(T101)) → U2_ag(less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga
less20_in_ga(s(T60)) → U1_ga(less20_in_ga(T60))
U2_ag(less33_out_ag(T102)) → less33_out_ag(s(T102))
U1_ga(less20_out_ga) → less20_out_ga

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0)
U1_ga(x0)

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

(43) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U4_GAAA(T40, less20_out_ga) → INSERT1_IN_GAA(s(T40)) we obtained the following new rules [LPAR04]:

U4_GAAA(0, less20_out_ga) → INSERT1_IN_GAA(s(0))
U4_GAAA(s(z0), less20_out_ga) → INSERT1_IN_GAA(s(s(z0)))

(44) Obligation:

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

P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga)
INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(less33_in_ag(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0))
U16_GAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(z0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(z0))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(less20_in_ga(x0)))
INSERT1_IN_GAA(s(z0)) → P43_IN_AGAA(s(z0))
U9_AGAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(s(z0)))
U9_AGAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(s(z0)))
U4_GAAA(0, less20_out_ga) → INSERT1_IN_GAA(s(0))
U4_GAAA(s(z0), less20_out_ga) → INSERT1_IN_GAA(s(s(z0)))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0)
less33_in_ag(s(T101)) → U2_ag(less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga
less20_in_ga(s(T60)) → U1_ga(less20_in_ga(T60))
U2_ag(less33_out_ag(T102)) → less33_out_ag(s(T102))
U1_ga(less20_out_ga) → less20_out_ga

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0)
U1_ga(x0)

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

(45) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206) we obtained the following new rules [LPAR04]:

INSERT1_IN_GAA(s(z0)) → P68_IN_AGAA(s(z0))
INSERT1_IN_GAA(s(s(z0))) → P68_IN_AGAA(s(s(z0)))
INSERT1_IN_GAA(s(0)) → P68_IN_AGAA(s(0))

(46) Obligation:

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

P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(less33_in_ag(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0))
U16_GAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(z0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(z0))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(less20_in_ga(x0)))
INSERT1_IN_GAA(s(z0)) → P43_IN_AGAA(s(z0))
U9_AGAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(s(z0)))
U9_AGAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(s(z0)))
U4_GAAA(0, less20_out_ga) → INSERT1_IN_GAA(s(0))
U4_GAAA(s(z0), less20_out_ga) → INSERT1_IN_GAA(s(s(z0)))
INSERT1_IN_GAA(s(z0)) → P68_IN_AGAA(s(z0))
INSERT1_IN_GAA(s(s(z0))) → P68_IN_AGAA(s(s(z0)))
INSERT1_IN_GAA(s(0)) → P68_IN_AGAA(s(0))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0)
less33_in_ag(s(T101)) → U2_ag(less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga
less20_in_ga(s(T60)) → U1_ga(less20_in_ga(T60))
U2_ag(less33_out_ag(T102)) → less33_out_ag(s(T102))
U1_ga(less20_out_ga) → less20_out_ga

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0)
U1_ga(x0)

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

(47) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134) we obtained the following new rules [LPAR04]:

P43_IN_AGAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))

(48) Obligation:

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

P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(less33_in_ag(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0))
U16_GAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(z0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(z0))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(less20_in_ga(x0)))
INSERT1_IN_GAA(s(z0)) → P43_IN_AGAA(s(z0))
U9_AGAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(s(z0)))
U9_AGAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(s(z0)))
U4_GAAA(0, less20_out_ga) → INSERT1_IN_GAA(s(0))
U4_GAAA(s(z0), less20_out_ga) → INSERT1_IN_GAA(s(s(z0)))
INSERT1_IN_GAA(s(z0)) → P68_IN_AGAA(s(z0))
INSERT1_IN_GAA(s(s(z0))) → P68_IN_AGAA(s(s(z0)))
INSERT1_IN_GAA(s(0)) → P68_IN_AGAA(s(0))
P43_IN_AGAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0)
less33_in_ag(s(T101)) → U2_ag(less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga
less20_in_ga(s(T60)) → U1_ga(less20_in_ga(T60))
U2_ag(less33_out_ag(T102)) → less33_out_ag(s(T102))
U1_ga(less20_out_ga) → less20_out_ga

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0)
U1_ga(x0)

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

(49) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40) we obtained the following new rules [LPAR04]:

INSERT1_IN_GAA(s(0)) → P18_IN_GAAA(0)
INSERT1_IN_GAA(s(s(y_0))) → P18_IN_GAAA(s(y_0))

(50) Obligation:

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

P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0))
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(less33_in_ag(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0))
U16_GAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(z0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(z0))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(less20_in_ga(x0)))
INSERT1_IN_GAA(s(z0)) → P43_IN_AGAA(s(z0))
U9_AGAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(s(z0)))
U9_AGAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(s(z0)))
U4_GAAA(0, less20_out_ga) → INSERT1_IN_GAA(s(0))
U4_GAAA(s(z0), less20_out_ga) → INSERT1_IN_GAA(s(s(z0)))
INSERT1_IN_GAA(s(z0)) → P68_IN_AGAA(s(z0))
INSERT1_IN_GAA(s(s(z0))) → P68_IN_AGAA(s(s(z0)))
INSERT1_IN_GAA(s(0)) → P68_IN_AGAA(s(0))
P43_IN_AGAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))
INSERT1_IN_GAA(s(0)) → P18_IN_GAAA(0)
INSERT1_IN_GAA(s(s(y_0))) → P18_IN_GAAA(s(y_0))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0)
less33_in_ag(s(T101)) → U2_ag(less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga
less20_in_ga(s(T60)) → U1_ga(less20_in_ga(T60))
U2_ag(less33_out_ag(T102)) → less33_out_ag(s(T102))
U1_ga(less20_out_ga) → less20_out_ga

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0)
U1_ga(x0)

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

(51) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322) we obtained the following new rules [LPAR04]:

INSERT1_IN_GAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))

(52) Obligation:

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

P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0))
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(less33_in_ag(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0))
U16_GAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(z0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(z0))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(less20_in_ga(x0)))
INSERT1_IN_GAA(s(z0)) → P43_IN_AGAA(s(z0))
U9_AGAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(s(z0)))
U9_AGAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(s(z0)))
U4_GAAA(0, less20_out_ga) → INSERT1_IN_GAA(s(0))
U4_GAAA(s(z0), less20_out_ga) → INSERT1_IN_GAA(s(s(z0)))
INSERT1_IN_GAA(s(z0)) → P68_IN_AGAA(s(z0))
INSERT1_IN_GAA(s(s(z0))) → P68_IN_AGAA(s(s(z0)))
INSERT1_IN_GAA(s(0)) → P68_IN_AGAA(s(0))
P43_IN_AGAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))
INSERT1_IN_GAA(s(0)) → P18_IN_GAAA(0)
INSERT1_IN_GAA(s(s(y_0))) → P18_IN_GAAA(s(y_0))
INSERT1_IN_GAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0)
less33_in_ag(s(T101)) → U2_ag(less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga
less20_in_ga(s(T60)) → U1_ga(less20_in_ga(T60))
U2_ag(less33_out_ag(T102)) → less33_out_ag(s(T102))
U1_ga(less20_out_ga) → less20_out_ga

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0)
U1_ga(x0)

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

(53) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228) we obtained the following new rules [LPAR04]:

P68_IN_AGAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))

(54) Obligation:

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

P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0))
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(less33_in_ag(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0))
U16_GAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(z0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(z0))
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(less20_in_ga(x0)))
INSERT1_IN_GAA(s(z0)) → P43_IN_AGAA(s(z0))
U9_AGAA(s(z0), less33_out_ag(0)) → INSERT1_IN_GAA(s(s(z0)))
U9_AGAA(s(z0), less33_out_ag(x1)) → INSERT1_IN_GAA(s(s(z0)))
U4_GAAA(0, less20_out_ga) → INSERT1_IN_GAA(s(0))
U4_GAAA(s(z0), less20_out_ga) → INSERT1_IN_GAA(s(s(z0)))
INSERT1_IN_GAA(s(z0)) → P68_IN_AGAA(s(z0))
INSERT1_IN_GAA(s(s(z0))) → P68_IN_AGAA(s(s(z0)))
INSERT1_IN_GAA(s(0)) → P68_IN_AGAA(s(0))
P43_IN_AGAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))
INSERT1_IN_GAA(s(0)) → P18_IN_GAAA(0)
INSERT1_IN_GAA(s(s(y_0))) → P18_IN_GAAA(s(y_0))
INSERT1_IN_GAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))
P68_IN_AGAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0)
less33_in_ag(s(T101)) → U2_ag(less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga
less20_in_ga(s(T60)) → U1_ga(less20_in_ga(T60))
U2_ag(less33_out_ag(T102)) → less33_out_ag(s(T102))
U1_ga(less20_out_ga) → less20_out_ga

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0)
U1_ga(x0)

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

(55) NonTerminationProof (EQUIVALENT transformation)

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

s = INSERT1_IN_GAA(s(T312)) evaluates to t =INSERT1_IN_GAA(s(T312))

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 INSERT1_IN_GAA(s(T312)) to INSERT1_IN_GAA(s(T312)).



(56) NO

(57) Obligation:

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

INSERT1_IN_GAA(0) → INSERT1_IN_GAA(0)

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0)
less33_in_ag(s(T101)) → U2_ag(less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga
less20_in_ga(s(T60)) → U1_ga(less20_in_ga(T60))
U2_ag(less33_out_ag(T102)) → less33_out_ag(s(T102))
U1_ga(less20_out_ga) → less20_out_ga

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0)
U1_ga(x0)

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

(58) 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.

(59) Obligation:

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

INSERT1_IN_GAA(0) → INSERT1_IN_GAA(0)

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

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0)
U1_ga(x0)

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

(60) 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].

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0)
U1_ga(x0)

(61) Obligation:

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

INSERT1_IN_GAA(0) → INSERT1_IN_GAA(0)

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

(62) NonTerminationProof (EQUIVALENT transformation)

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

s = INSERT1_IN_GAA(0) evaluates to t =INSERT1_IN_GAA(0)

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 INSERT1_IN_GAA(0) to INSERT1_IN_GAA(0).



(63) NO

(64) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
insert1_in: (b,f,f)
p18_in: (b,f,f,f)
less20_in: (b,f)
less33_in: (f,b)
p43_in: (f,b,f,f)
p50_in: (f,b,f,f)
p68_in: (f,b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

insert1_in_gaa(T5, void, tree(T5, void, void)) → insert1_out_gaa(T5, void, tree(T5, void, void))
insert1_in_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11)) → insert1_out_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11))
insert1_in_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → U13_gaa(T29, T30, T20, T31, insert1_in_gaa(0, T30, T31))
insert1_in_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → U14_gaa(T40, T42, T43, T20, T44, p18_in_gaaa(T40, T42, T43, T44))
p18_in_gaaa(T40, T42, T43, T44) → U3_gaaa(T40, T42, T43, T44, less20_in_ga(T40, T42))
less20_in_ga(0, s(T55)) → less20_out_ga(0, s(T55))
less20_in_ga(s(T60), s(T62)) → U1_ga(T60, T62, less20_in_ga(T60, T62))
U1_ga(T60, T62, less20_out_ga(T60, T62)) → less20_out_ga(s(T60), s(T62))
U3_gaaa(T40, T42, T43, T44, less20_out_ga(T40, T42)) → p18_out_gaaa(T40, T42, T43, T44)
p18_in_gaaa(T40, T42, T47, T48) → U4_gaaa(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_gaaa(T40, T42, T47, T48, less20_out_ga(T40, T42)) → U5_gaaa(T40, T42, T47, T48, insert1_in_gaa(s(T40), T47, T48))
insert1_in_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → U15_gaa(T77, T82, T79, T83, T84, less33_in_ag(T82, T77))
less33_in_ag(0, s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T102), s(T101)) → U2_ag(T102, T101, less33_in_ag(T102, T101))
U2_ag(T102, T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U15_gaa(T77, T82, T79, T83, T84, less33_out_ag(T82, T77)) → insert1_out_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84))
insert1_in_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_gaa(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_gaa(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → U17_gaa(T77, T82, T79, T87, T88, insert1_in_gaa(T77, T87, T88))
insert1_in_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → U18_gaa(T112, T117, T114, T118, T119, p43_in_agaa(T117, T112, T118, T119))
p43_in_agaa(0, s(T124), T125, T126) → U6_agaa(T124, T125, T126, insert1_in_gaa(s(T124), T125, T126))
insert1_in_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) → U19_gaa(T163, T164, T154, T165, insert1_in_gaa(0, T164, T165))
insert1_in_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) → U20_gaa(T174, T176, T177, T154, T178, p18_in_gaaa(T174, T176, T177, T178))
U20_gaa(T174, T176, T177, T154, T178, p18_out_gaaa(T174, T176, T177, T178)) → insert1_out_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154))
insert1_in_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198)) → U21_gaa(T191, T196, T193, T197, T198, p43_in_agaa(T196, T191, T197, T198))
p43_in_agaa(s(T135), s(T134), T136, T137) → U7_agaa(T135, T134, T136, T137, p50_in_agaa(T135, T134, T136, T137))
p50_in_agaa(T135, T134, T136, T137) → U8_agaa(T135, T134, T136, T137, less33_in_ag(T135, T134))
U8_agaa(T135, T134, T136, T137, less33_out_ag(T135, T134)) → p50_out_agaa(T135, T134, T136, T137)
p50_in_agaa(T135, T134, T140, T141) → U9_agaa(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_agaa(T135, T134, T140, T141, less33_out_ag(T135, T134)) → U10_agaa(T135, T134, T140, T141, insert1_in_gaa(s(T134), T140, T141))
insert1_in_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → U22_gaa(T206, T211, T208, T212, T213, p68_in_agaa(T211, T206, T212, T213))
p68_in_agaa(0, s(T218), T219, T220) → U11_agaa(T218, T219, T220, insert1_in_gaa(s(T218), T219, T220))
insert1_in_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) → U23_gaa(T257, T258, T248, T259, insert1_in_gaa(0, T258, T259))
insert1_in_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) → U24_gaa(T268, T270, T271, T248, T272, p18_in_gaaa(T268, T270, T271, T272))
U24_gaa(T268, T270, T271, T248, T272, p18_out_gaaa(T268, T270, T271, T272)) → insert1_out_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248))
insert1_in_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292)) → U25_gaa(T285, T290, T287, T291, T292, p68_in_agaa(T290, T285, T291, T292))
p68_in_agaa(s(T229), s(T228), T230, T231) → U12_agaa(T229, T228, T230, T231, p50_in_agaa(T229, T228, T230, T231))
U12_agaa(T229, T228, T230, T231, p50_out_agaa(T229, T228, T230, T231)) → p68_out_agaa(s(T229), s(T228), T230, T231)
U25_gaa(T285, T290, T287, T291, T292, p68_out_agaa(T290, T285, T291, T292)) → insert1_out_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292))
insert1_in_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → U26_gaa(T312, T302, T313, T314, insert1_in_gaa(s(T312), T313, T314))
insert1_in_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → U27_gaa(T322, T323, T302, T324, T325, p50_in_agaa(T323, T322, T324, T325))
U27_gaa(T322, T323, T302, T324, T325, p50_out_agaa(T323, T322, T324, T325)) → insert1_out_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325))
insert1_in_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) → U28_gaa(T345, T346, T336, T347, insert1_in_gaa(0, T346, T347))
insert1_in_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) → U29_gaa(T356, T358, T359, T336, T360, p18_in_gaaa(T356, T358, T359, T360))
U29_gaa(T356, T358, T359, T336, T360, p18_out_gaaa(T356, T358, T359, T360)) → insert1_out_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336))
insert1_in_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380)) → U30_gaa(T373, T378, T375, T379, T380, p68_in_agaa(T378, T373, T379, T380))
U30_gaa(T373, T378, T375, T379, T380, p68_out_agaa(T378, T373, T379, T380)) → insert1_out_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380))
insert1_in_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402)) → U31_gaa(T400, T390, T401, T402, insert1_in_gaa(s(T400), T401, T402))
insert1_in_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) → U32_gaa(T410, T411, T390, T412, T413, p50_in_agaa(T411, T410, T412, T413))
U32_gaa(T410, T411, T390, T412, T413, p50_out_agaa(T411, T410, T412, T413)) → insert1_out_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413))
U31_gaa(T400, T390, T401, T402, insert1_out_gaa(s(T400), T401, T402)) → insert1_out_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402))
U28_gaa(T345, T346, T336, T347, insert1_out_gaa(0, T346, T347)) → insert1_out_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336))
U26_gaa(T312, T302, T313, T314, insert1_out_gaa(s(T312), T313, T314)) → insert1_out_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314))
U23_gaa(T257, T258, T248, T259, insert1_out_gaa(0, T258, T259)) → insert1_out_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248))
U11_agaa(T218, T219, T220, insert1_out_gaa(s(T218), T219, T220)) → p68_out_agaa(0, s(T218), T219, T220)
U22_gaa(T206, T211, T208, T212, T213, p68_out_agaa(T211, T206, T212, T213)) → insert1_out_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213))
U10_agaa(T135, T134, T140, T141, insert1_out_gaa(s(T134), T140, T141)) → p50_out_agaa(T135, T134, T140, T141)
U7_agaa(T135, T134, T136, T137, p50_out_agaa(T135, T134, T136, T137)) → p43_out_agaa(s(T135), s(T134), T136, T137)
U21_gaa(T191, T196, T193, T197, T198, p43_out_agaa(T196, T191, T197, T198)) → insert1_out_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198))
U19_gaa(T163, T164, T154, T165, insert1_out_gaa(0, T164, T165)) → insert1_out_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154))
U6_agaa(T124, T125, T126, insert1_out_gaa(s(T124), T125, T126)) → p43_out_agaa(0, s(T124), T125, T126)
U18_gaa(T112, T117, T114, T118, T119, p43_out_agaa(T117, T112, T118, T119)) → insert1_out_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119))
U17_gaa(T77, T82, T79, T87, T88, insert1_out_gaa(T77, T87, T88)) → insert1_out_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88))
U5_gaaa(T40, T42, T47, T48, insert1_out_gaa(s(T40), T47, T48)) → p18_out_gaaa(T40, T42, T47, T48)
U14_gaa(T40, T42, T43, T20, T44, p18_out_gaaa(T40, T42, T43, T44)) → insert1_out_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20))
U13_gaa(T29, T30, T20, T31, insert1_out_gaa(0, T30, T31)) → insert1_out_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20))

The argument filtering Pi contains the following mapping:
insert1_in_gaa(x1, x2, x3)  =  insert1_in_gaa(x1)
insert1_out_gaa(x1, x2, x3)  =  insert1_out_gaa(x1)
0  =  0
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x5)
s(x1)  =  s(x1)
U14_gaa(x1, x2, x3, x4, x5, x6)  =  U14_gaa(x1, x6)
p18_in_gaaa(x1, x2, x3, x4)  =  p18_in_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x1, x5)
less20_in_ga(x1, x2)  =  less20_in_ga(x1)
less20_out_ga(x1, x2)  =  less20_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
p18_out_gaaa(x1, x2, x3, x4)  =  p18_out_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
U5_gaaa(x1, x2, x3, x4, x5)  =  U5_gaaa(x1, x5)
U15_gaa(x1, x2, x3, x4, x5, x6)  =  U15_gaa(x1, x6)
less33_in_ag(x1, x2)  =  less33_in_ag(x2)
less33_out_ag(x1, x2)  =  less33_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U16_gaa(x1, x2, x3, x4, x5, x6)  =  U16_gaa(x1, x6)
U17_gaa(x1, x2, x3, x4, x5, x6)  =  U17_gaa(x1, x6)
U18_gaa(x1, x2, x3, x4, x5, x6)  =  U18_gaa(x1, x6)
p43_in_agaa(x1, x2, x3, x4)  =  p43_in_agaa(x2)
U6_agaa(x1, x2, x3, x4)  =  U6_agaa(x1, x4)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x5)
U20_gaa(x1, x2, x3, x4, x5, x6)  =  U20_gaa(x1, x6)
U21_gaa(x1, x2, x3, x4, x5, x6)  =  U21_gaa(x1, x6)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x2, x5)
p50_in_agaa(x1, x2, x3, x4)  =  p50_in_agaa(x2)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x2, x5)
p50_out_agaa(x1, x2, x3, x4)  =  p50_out_agaa(x1, x2)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x1, x2, x5)
U22_gaa(x1, x2, x3, x4, x5, x6)  =  U22_gaa(x1, x6)
p68_in_agaa(x1, x2, x3, x4)  =  p68_in_agaa(x2)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x1, x4)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x5)
U24_gaa(x1, x2, x3, x4, x5, x6)  =  U24_gaa(x1, x6)
U25_gaa(x1, x2, x3, x4, x5, x6)  =  U25_gaa(x1, x6)
U12_agaa(x1, x2, x3, x4, x5)  =  U12_agaa(x2, x5)
p68_out_agaa(x1, x2, x3, x4)  =  p68_out_agaa(x1, x2)
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x1, x5)
U27_gaa(x1, x2, x3, x4, x5, x6)  =  U27_gaa(x1, x6)
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U29_gaa(x1, x2, x3, x4, x5, x6)  =  U29_gaa(x1, x6)
U30_gaa(x1, x2, x3, x4, x5, x6)  =  U30_gaa(x1, x6)
U31_gaa(x1, x2, x3, x4, x5)  =  U31_gaa(x1, x5)
U32_gaa(x1, x2, x3, x4, x5, x6)  =  U32_gaa(x1, x6)
p43_out_agaa(x1, x2, x3, x4)  =  p43_out_agaa(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(65) Obligation:

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

insert1_in_gaa(T5, void, tree(T5, void, void)) → insert1_out_gaa(T5, void, tree(T5, void, void))
insert1_in_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11)) → insert1_out_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11))
insert1_in_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → U13_gaa(T29, T30, T20, T31, insert1_in_gaa(0, T30, T31))
insert1_in_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → U14_gaa(T40, T42, T43, T20, T44, p18_in_gaaa(T40, T42, T43, T44))
p18_in_gaaa(T40, T42, T43, T44) → U3_gaaa(T40, T42, T43, T44, less20_in_ga(T40, T42))
less20_in_ga(0, s(T55)) → less20_out_ga(0, s(T55))
less20_in_ga(s(T60), s(T62)) → U1_ga(T60, T62, less20_in_ga(T60, T62))
U1_ga(T60, T62, less20_out_ga(T60, T62)) → less20_out_ga(s(T60), s(T62))
U3_gaaa(T40, T42, T43, T44, less20_out_ga(T40, T42)) → p18_out_gaaa(T40, T42, T43, T44)
p18_in_gaaa(T40, T42, T47, T48) → U4_gaaa(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_gaaa(T40, T42, T47, T48, less20_out_ga(T40, T42)) → U5_gaaa(T40, T42, T47, T48, insert1_in_gaa(s(T40), T47, T48))
insert1_in_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → U15_gaa(T77, T82, T79, T83, T84, less33_in_ag(T82, T77))
less33_in_ag(0, s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T102), s(T101)) → U2_ag(T102, T101, less33_in_ag(T102, T101))
U2_ag(T102, T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U15_gaa(T77, T82, T79, T83, T84, less33_out_ag(T82, T77)) → insert1_out_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84))
insert1_in_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_gaa(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_gaa(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → U17_gaa(T77, T82, T79, T87, T88, insert1_in_gaa(T77, T87, T88))
insert1_in_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → U18_gaa(T112, T117, T114, T118, T119, p43_in_agaa(T117, T112, T118, T119))
p43_in_agaa(0, s(T124), T125, T126) → U6_agaa(T124, T125, T126, insert1_in_gaa(s(T124), T125, T126))
insert1_in_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) → U19_gaa(T163, T164, T154, T165, insert1_in_gaa(0, T164, T165))
insert1_in_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) → U20_gaa(T174, T176, T177, T154, T178, p18_in_gaaa(T174, T176, T177, T178))
U20_gaa(T174, T176, T177, T154, T178, p18_out_gaaa(T174, T176, T177, T178)) → insert1_out_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154))
insert1_in_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198)) → U21_gaa(T191, T196, T193, T197, T198, p43_in_agaa(T196, T191, T197, T198))
p43_in_agaa(s(T135), s(T134), T136, T137) → U7_agaa(T135, T134, T136, T137, p50_in_agaa(T135, T134, T136, T137))
p50_in_agaa(T135, T134, T136, T137) → U8_agaa(T135, T134, T136, T137, less33_in_ag(T135, T134))
U8_agaa(T135, T134, T136, T137, less33_out_ag(T135, T134)) → p50_out_agaa(T135, T134, T136, T137)
p50_in_agaa(T135, T134, T140, T141) → U9_agaa(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_agaa(T135, T134, T140, T141, less33_out_ag(T135, T134)) → U10_agaa(T135, T134, T140, T141, insert1_in_gaa(s(T134), T140, T141))
insert1_in_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → U22_gaa(T206, T211, T208, T212, T213, p68_in_agaa(T211, T206, T212, T213))
p68_in_agaa(0, s(T218), T219, T220) → U11_agaa(T218, T219, T220, insert1_in_gaa(s(T218), T219, T220))
insert1_in_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) → U23_gaa(T257, T258, T248, T259, insert1_in_gaa(0, T258, T259))
insert1_in_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) → U24_gaa(T268, T270, T271, T248, T272, p18_in_gaaa(T268, T270, T271, T272))
U24_gaa(T268, T270, T271, T248, T272, p18_out_gaaa(T268, T270, T271, T272)) → insert1_out_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248))
insert1_in_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292)) → U25_gaa(T285, T290, T287, T291, T292, p68_in_agaa(T290, T285, T291, T292))
p68_in_agaa(s(T229), s(T228), T230, T231) → U12_agaa(T229, T228, T230, T231, p50_in_agaa(T229, T228, T230, T231))
U12_agaa(T229, T228, T230, T231, p50_out_agaa(T229, T228, T230, T231)) → p68_out_agaa(s(T229), s(T228), T230, T231)
U25_gaa(T285, T290, T287, T291, T292, p68_out_agaa(T290, T285, T291, T292)) → insert1_out_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292))
insert1_in_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → U26_gaa(T312, T302, T313, T314, insert1_in_gaa(s(T312), T313, T314))
insert1_in_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → U27_gaa(T322, T323, T302, T324, T325, p50_in_agaa(T323, T322, T324, T325))
U27_gaa(T322, T323, T302, T324, T325, p50_out_agaa(T323, T322, T324, T325)) → insert1_out_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325))
insert1_in_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) → U28_gaa(T345, T346, T336, T347, insert1_in_gaa(0, T346, T347))
insert1_in_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) → U29_gaa(T356, T358, T359, T336, T360, p18_in_gaaa(T356, T358, T359, T360))
U29_gaa(T356, T358, T359, T336, T360, p18_out_gaaa(T356, T358, T359, T360)) → insert1_out_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336))
insert1_in_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380)) → U30_gaa(T373, T378, T375, T379, T380, p68_in_agaa(T378, T373, T379, T380))
U30_gaa(T373, T378, T375, T379, T380, p68_out_agaa(T378, T373, T379, T380)) → insert1_out_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380))
insert1_in_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402)) → U31_gaa(T400, T390, T401, T402, insert1_in_gaa(s(T400), T401, T402))
insert1_in_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) → U32_gaa(T410, T411, T390, T412, T413, p50_in_agaa(T411, T410, T412, T413))
U32_gaa(T410, T411, T390, T412, T413, p50_out_agaa(T411, T410, T412, T413)) → insert1_out_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413))
U31_gaa(T400, T390, T401, T402, insert1_out_gaa(s(T400), T401, T402)) → insert1_out_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402))
U28_gaa(T345, T346, T336, T347, insert1_out_gaa(0, T346, T347)) → insert1_out_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336))
U26_gaa(T312, T302, T313, T314, insert1_out_gaa(s(T312), T313, T314)) → insert1_out_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314))
U23_gaa(T257, T258, T248, T259, insert1_out_gaa(0, T258, T259)) → insert1_out_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248))
U11_agaa(T218, T219, T220, insert1_out_gaa(s(T218), T219, T220)) → p68_out_agaa(0, s(T218), T219, T220)
U22_gaa(T206, T211, T208, T212, T213, p68_out_agaa(T211, T206, T212, T213)) → insert1_out_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213))
U10_agaa(T135, T134, T140, T141, insert1_out_gaa(s(T134), T140, T141)) → p50_out_agaa(T135, T134, T140, T141)
U7_agaa(T135, T134, T136, T137, p50_out_agaa(T135, T134, T136, T137)) → p43_out_agaa(s(T135), s(T134), T136, T137)
U21_gaa(T191, T196, T193, T197, T198, p43_out_agaa(T196, T191, T197, T198)) → insert1_out_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198))
U19_gaa(T163, T164, T154, T165, insert1_out_gaa(0, T164, T165)) → insert1_out_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154))
U6_agaa(T124, T125, T126, insert1_out_gaa(s(T124), T125, T126)) → p43_out_agaa(0, s(T124), T125, T126)
U18_gaa(T112, T117, T114, T118, T119, p43_out_agaa(T117, T112, T118, T119)) → insert1_out_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119))
U17_gaa(T77, T82, T79, T87, T88, insert1_out_gaa(T77, T87, T88)) → insert1_out_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88))
U5_gaaa(T40, T42, T47, T48, insert1_out_gaa(s(T40), T47, T48)) → p18_out_gaaa(T40, T42, T47, T48)
U14_gaa(T40, T42, T43, T20, T44, p18_out_gaaa(T40, T42, T43, T44)) → insert1_out_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20))
U13_gaa(T29, T30, T20, T31, insert1_out_gaa(0, T30, T31)) → insert1_out_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20))

The argument filtering Pi contains the following mapping:
insert1_in_gaa(x1, x2, x3)  =  insert1_in_gaa(x1)
insert1_out_gaa(x1, x2, x3)  =  insert1_out_gaa(x1)
0  =  0
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x5)
s(x1)  =  s(x1)
U14_gaa(x1, x2, x3, x4, x5, x6)  =  U14_gaa(x1, x6)
p18_in_gaaa(x1, x2, x3, x4)  =  p18_in_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x1, x5)
less20_in_ga(x1, x2)  =  less20_in_ga(x1)
less20_out_ga(x1, x2)  =  less20_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
p18_out_gaaa(x1, x2, x3, x4)  =  p18_out_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
U5_gaaa(x1, x2, x3, x4, x5)  =  U5_gaaa(x1, x5)
U15_gaa(x1, x2, x3, x4, x5, x6)  =  U15_gaa(x1, x6)
less33_in_ag(x1, x2)  =  less33_in_ag(x2)
less33_out_ag(x1, x2)  =  less33_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U16_gaa(x1, x2, x3, x4, x5, x6)  =  U16_gaa(x1, x6)
U17_gaa(x1, x2, x3, x4, x5, x6)  =  U17_gaa(x1, x6)
U18_gaa(x1, x2, x3, x4, x5, x6)  =  U18_gaa(x1, x6)
p43_in_agaa(x1, x2, x3, x4)  =  p43_in_agaa(x2)
U6_agaa(x1, x2, x3, x4)  =  U6_agaa(x1, x4)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x5)
U20_gaa(x1, x2, x3, x4, x5, x6)  =  U20_gaa(x1, x6)
U21_gaa(x1, x2, x3, x4, x5, x6)  =  U21_gaa(x1, x6)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x2, x5)
p50_in_agaa(x1, x2, x3, x4)  =  p50_in_agaa(x2)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x2, x5)
p50_out_agaa(x1, x2, x3, x4)  =  p50_out_agaa(x1, x2)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x1, x2, x5)
U22_gaa(x1, x2, x3, x4, x5, x6)  =  U22_gaa(x1, x6)
p68_in_agaa(x1, x2, x3, x4)  =  p68_in_agaa(x2)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x1, x4)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x5)
U24_gaa(x1, x2, x3, x4, x5, x6)  =  U24_gaa(x1, x6)
U25_gaa(x1, x2, x3, x4, x5, x6)  =  U25_gaa(x1, x6)
U12_agaa(x1, x2, x3, x4, x5)  =  U12_agaa(x2, x5)
p68_out_agaa(x1, x2, x3, x4)  =  p68_out_agaa(x1, x2)
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x1, x5)
U27_gaa(x1, x2, x3, x4, x5, x6)  =  U27_gaa(x1, x6)
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U29_gaa(x1, x2, x3, x4, x5, x6)  =  U29_gaa(x1, x6)
U30_gaa(x1, x2, x3, x4, x5, x6)  =  U30_gaa(x1, x6)
U31_gaa(x1, x2, x3, x4, x5)  =  U31_gaa(x1, x5)
U32_gaa(x1, x2, x3, x4, x5, x6)  =  U32_gaa(x1, x6)
p43_out_agaa(x1, x2, x3, x4)  =  p43_out_agaa(x1, x2)

(66) 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:

INSERT1_IN_GAA(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → U13_GAA(T29, T30, T20, T31, insert1_in_gaa(0, T30, T31))
INSERT1_IN_GAA(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → INSERT1_IN_GAA(0, T30, T31)
INSERT1_IN_GAA(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → U14_GAA(T40, T42, T43, T20, T44, p18_in_gaaa(T40, T42, T43, T44))
INSERT1_IN_GAA(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → P18_IN_GAAA(T40, T42, T43, T44)
P18_IN_GAAA(T40, T42, T43, T44) → U3_GAAA(T40, T42, T43, T44, less20_in_ga(T40, T42))
P18_IN_GAAA(T40, T42, T43, T44) → LESS20_IN_GA(T40, T42)
LESS20_IN_GA(s(T60), s(T62)) → U1_GA(T60, T62, less20_in_ga(T60, T62))
LESS20_IN_GA(s(T60), s(T62)) → LESS20_IN_GA(T60, T62)
P18_IN_GAAA(T40, T42, T47, T48) → U4_GAAA(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_GAAA(T40, T42, T47, T48, less20_out_ga(T40, T42)) → U5_GAAA(T40, T42, T47, T48, insert1_in_gaa(s(T40), T47, T48))
U4_GAAA(T40, T42, T47, T48, less20_out_ga(T40, T42)) → INSERT1_IN_GAA(s(T40), T47, T48)
INSERT1_IN_GAA(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → U15_GAA(T77, T82, T79, T83, T84, less33_in_ag(T82, T77))
INSERT1_IN_GAA(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → LESS33_IN_AG(T82, T77)
LESS33_IN_AG(s(T102), s(T101)) → U2_AG(T102, T101, less33_in_ag(T102, T101))
LESS33_IN_AG(s(T102), s(T101)) → LESS33_IN_AG(T102, T101)
INSERT1_IN_GAA(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_GAA(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_GAA(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → U17_GAA(T77, T82, T79, T87, T88, insert1_in_gaa(T77, T87, T88))
U16_GAA(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → INSERT1_IN_GAA(T77, T87, T88)
INSERT1_IN_GAA(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → U18_GAA(T112, T117, T114, T118, T119, p43_in_agaa(T117, T112, T118, T119))
INSERT1_IN_GAA(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → P43_IN_AGAA(T117, T112, T118, T119)
P43_IN_AGAA(0, s(T124), T125, T126) → U6_AGAA(T124, T125, T126, insert1_in_gaa(s(T124), T125, T126))
P43_IN_AGAA(0, s(T124), T125, T126) → INSERT1_IN_GAA(s(T124), T125, T126)
INSERT1_IN_GAA(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) → U19_GAA(T163, T164, T154, T165, insert1_in_gaa(0, T164, T165))
INSERT1_IN_GAA(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) → U20_GAA(T174, T176, T177, T154, T178, p18_in_gaaa(T174, T176, T177, T178))
INSERT1_IN_GAA(T191, tree(T196, T193, T197), tree(T196, T193, T198)) → U21_GAA(T191, T196, T193, T197, T198, p43_in_agaa(T196, T191, T197, T198))
P43_IN_AGAA(s(T135), s(T134), T136, T137) → U7_AGAA(T135, T134, T136, T137, p50_in_agaa(T135, T134, T136, T137))
P43_IN_AGAA(s(T135), s(T134), T136, T137) → P50_IN_AGAA(T135, T134, T136, T137)
P50_IN_AGAA(T135, T134, T136, T137) → U8_AGAA(T135, T134, T136, T137, less33_in_ag(T135, T134))
P50_IN_AGAA(T135, T134, T136, T137) → LESS33_IN_AG(T135, T134)
P50_IN_AGAA(T135, T134, T140, T141) → U9_AGAA(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_AGAA(T135, T134, T140, T141, less33_out_ag(T135, T134)) → U10_AGAA(T135, T134, T140, T141, insert1_in_gaa(s(T134), T140, T141))
U9_AGAA(T135, T134, T140, T141, less33_out_ag(T135, T134)) → INSERT1_IN_GAA(s(T134), T140, T141)
INSERT1_IN_GAA(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → U22_GAA(T206, T211, T208, T212, T213, p68_in_agaa(T211, T206, T212, T213))
INSERT1_IN_GAA(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → P68_IN_AGAA(T211, T206, T212, T213)
P68_IN_AGAA(0, s(T218), T219, T220) → U11_AGAA(T218, T219, T220, insert1_in_gaa(s(T218), T219, T220))
P68_IN_AGAA(0, s(T218), T219, T220) → INSERT1_IN_GAA(s(T218), T219, T220)
INSERT1_IN_GAA(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) → U23_GAA(T257, T258, T248, T259, insert1_in_gaa(0, T258, T259))
INSERT1_IN_GAA(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) → U24_GAA(T268, T270, T271, T248, T272, p18_in_gaaa(T268, T270, T271, T272))
INSERT1_IN_GAA(T285, tree(T290, T287, T291), tree(T290, T287, T292)) → U25_GAA(T285, T290, T287, T291, T292, p68_in_agaa(T290, T285, T291, T292))
P68_IN_AGAA(s(T229), s(T228), T230, T231) → U12_AGAA(T229, T228, T230, T231, p50_in_agaa(T229, T228, T230, T231))
P68_IN_AGAA(s(T229), s(T228), T230, T231) → P50_IN_AGAA(T229, T228, T230, T231)
INSERT1_IN_GAA(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → U26_GAA(T312, T302, T313, T314, insert1_in_gaa(s(T312), T313, T314))
INSERT1_IN_GAA(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → INSERT1_IN_GAA(s(T312), T313, T314)
INSERT1_IN_GAA(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → U27_GAA(T322, T323, T302, T324, T325, p50_in_agaa(T323, T322, T324, T325))
INSERT1_IN_GAA(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → P50_IN_AGAA(T323, T322, T324, T325)
INSERT1_IN_GAA(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) → U28_GAA(T345, T346, T336, T347, insert1_in_gaa(0, T346, T347))
INSERT1_IN_GAA(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) → U29_GAA(T356, T358, T359, T336, T360, p18_in_gaaa(T356, T358, T359, T360))
INSERT1_IN_GAA(T373, tree(T378, T375, T379), tree(T378, T375, T380)) → U30_GAA(T373, T378, T375, T379, T380, p68_in_agaa(T378, T373, T379, T380))
INSERT1_IN_GAA(s(T400), tree(0, T390, T401), tree(0, T390, T402)) → U31_GAA(T400, T390, T401, T402, insert1_in_gaa(s(T400), T401, T402))
INSERT1_IN_GAA(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) → U32_GAA(T410, T411, T390, T412, T413, p50_in_agaa(T411, T410, T412, T413))

The TRS R consists of the following rules:

insert1_in_gaa(T5, void, tree(T5, void, void)) → insert1_out_gaa(T5, void, tree(T5, void, void))
insert1_in_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11)) → insert1_out_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11))
insert1_in_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → U13_gaa(T29, T30, T20, T31, insert1_in_gaa(0, T30, T31))
insert1_in_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → U14_gaa(T40, T42, T43, T20, T44, p18_in_gaaa(T40, T42, T43, T44))
p18_in_gaaa(T40, T42, T43, T44) → U3_gaaa(T40, T42, T43, T44, less20_in_ga(T40, T42))
less20_in_ga(0, s(T55)) → less20_out_ga(0, s(T55))
less20_in_ga(s(T60), s(T62)) → U1_ga(T60, T62, less20_in_ga(T60, T62))
U1_ga(T60, T62, less20_out_ga(T60, T62)) → less20_out_ga(s(T60), s(T62))
U3_gaaa(T40, T42, T43, T44, less20_out_ga(T40, T42)) → p18_out_gaaa(T40, T42, T43, T44)
p18_in_gaaa(T40, T42, T47, T48) → U4_gaaa(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_gaaa(T40, T42, T47, T48, less20_out_ga(T40, T42)) → U5_gaaa(T40, T42, T47, T48, insert1_in_gaa(s(T40), T47, T48))
insert1_in_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → U15_gaa(T77, T82, T79, T83, T84, less33_in_ag(T82, T77))
less33_in_ag(0, s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T102), s(T101)) → U2_ag(T102, T101, less33_in_ag(T102, T101))
U2_ag(T102, T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U15_gaa(T77, T82, T79, T83, T84, less33_out_ag(T82, T77)) → insert1_out_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84))
insert1_in_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_gaa(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_gaa(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → U17_gaa(T77, T82, T79, T87, T88, insert1_in_gaa(T77, T87, T88))
insert1_in_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → U18_gaa(T112, T117, T114, T118, T119, p43_in_agaa(T117, T112, T118, T119))
p43_in_agaa(0, s(T124), T125, T126) → U6_agaa(T124, T125, T126, insert1_in_gaa(s(T124), T125, T126))
insert1_in_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) → U19_gaa(T163, T164, T154, T165, insert1_in_gaa(0, T164, T165))
insert1_in_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) → U20_gaa(T174, T176, T177, T154, T178, p18_in_gaaa(T174, T176, T177, T178))
U20_gaa(T174, T176, T177, T154, T178, p18_out_gaaa(T174, T176, T177, T178)) → insert1_out_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154))
insert1_in_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198)) → U21_gaa(T191, T196, T193, T197, T198, p43_in_agaa(T196, T191, T197, T198))
p43_in_agaa(s(T135), s(T134), T136, T137) → U7_agaa(T135, T134, T136, T137, p50_in_agaa(T135, T134, T136, T137))
p50_in_agaa(T135, T134, T136, T137) → U8_agaa(T135, T134, T136, T137, less33_in_ag(T135, T134))
U8_agaa(T135, T134, T136, T137, less33_out_ag(T135, T134)) → p50_out_agaa(T135, T134, T136, T137)
p50_in_agaa(T135, T134, T140, T141) → U9_agaa(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_agaa(T135, T134, T140, T141, less33_out_ag(T135, T134)) → U10_agaa(T135, T134, T140, T141, insert1_in_gaa(s(T134), T140, T141))
insert1_in_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → U22_gaa(T206, T211, T208, T212, T213, p68_in_agaa(T211, T206, T212, T213))
p68_in_agaa(0, s(T218), T219, T220) → U11_agaa(T218, T219, T220, insert1_in_gaa(s(T218), T219, T220))
insert1_in_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) → U23_gaa(T257, T258, T248, T259, insert1_in_gaa(0, T258, T259))
insert1_in_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) → U24_gaa(T268, T270, T271, T248, T272, p18_in_gaaa(T268, T270, T271, T272))
U24_gaa(T268, T270, T271, T248, T272, p18_out_gaaa(T268, T270, T271, T272)) → insert1_out_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248))
insert1_in_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292)) → U25_gaa(T285, T290, T287, T291, T292, p68_in_agaa(T290, T285, T291, T292))
p68_in_agaa(s(T229), s(T228), T230, T231) → U12_agaa(T229, T228, T230, T231, p50_in_agaa(T229, T228, T230, T231))
U12_agaa(T229, T228, T230, T231, p50_out_agaa(T229, T228, T230, T231)) → p68_out_agaa(s(T229), s(T228), T230, T231)
U25_gaa(T285, T290, T287, T291, T292, p68_out_agaa(T290, T285, T291, T292)) → insert1_out_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292))
insert1_in_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → U26_gaa(T312, T302, T313, T314, insert1_in_gaa(s(T312), T313, T314))
insert1_in_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → U27_gaa(T322, T323, T302, T324, T325, p50_in_agaa(T323, T322, T324, T325))
U27_gaa(T322, T323, T302, T324, T325, p50_out_agaa(T323, T322, T324, T325)) → insert1_out_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325))
insert1_in_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) → U28_gaa(T345, T346, T336, T347, insert1_in_gaa(0, T346, T347))
insert1_in_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) → U29_gaa(T356, T358, T359, T336, T360, p18_in_gaaa(T356, T358, T359, T360))
U29_gaa(T356, T358, T359, T336, T360, p18_out_gaaa(T356, T358, T359, T360)) → insert1_out_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336))
insert1_in_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380)) → U30_gaa(T373, T378, T375, T379, T380, p68_in_agaa(T378, T373, T379, T380))
U30_gaa(T373, T378, T375, T379, T380, p68_out_agaa(T378, T373, T379, T380)) → insert1_out_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380))
insert1_in_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402)) → U31_gaa(T400, T390, T401, T402, insert1_in_gaa(s(T400), T401, T402))
insert1_in_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) → U32_gaa(T410, T411, T390, T412, T413, p50_in_agaa(T411, T410, T412, T413))
U32_gaa(T410, T411, T390, T412, T413, p50_out_agaa(T411, T410, T412, T413)) → insert1_out_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413))
U31_gaa(T400, T390, T401, T402, insert1_out_gaa(s(T400), T401, T402)) → insert1_out_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402))
U28_gaa(T345, T346, T336, T347, insert1_out_gaa(0, T346, T347)) → insert1_out_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336))
U26_gaa(T312, T302, T313, T314, insert1_out_gaa(s(T312), T313, T314)) → insert1_out_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314))
U23_gaa(T257, T258, T248, T259, insert1_out_gaa(0, T258, T259)) → insert1_out_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248))
U11_agaa(T218, T219, T220, insert1_out_gaa(s(T218), T219, T220)) → p68_out_agaa(0, s(T218), T219, T220)
U22_gaa(T206, T211, T208, T212, T213, p68_out_agaa(T211, T206, T212, T213)) → insert1_out_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213))
U10_agaa(T135, T134, T140, T141, insert1_out_gaa(s(T134), T140, T141)) → p50_out_agaa(T135, T134, T140, T141)
U7_agaa(T135, T134, T136, T137, p50_out_agaa(T135, T134, T136, T137)) → p43_out_agaa(s(T135), s(T134), T136, T137)
U21_gaa(T191, T196, T193, T197, T198, p43_out_agaa(T196, T191, T197, T198)) → insert1_out_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198))
U19_gaa(T163, T164, T154, T165, insert1_out_gaa(0, T164, T165)) → insert1_out_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154))
U6_agaa(T124, T125, T126, insert1_out_gaa(s(T124), T125, T126)) → p43_out_agaa(0, s(T124), T125, T126)
U18_gaa(T112, T117, T114, T118, T119, p43_out_agaa(T117, T112, T118, T119)) → insert1_out_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119))
U17_gaa(T77, T82, T79, T87, T88, insert1_out_gaa(T77, T87, T88)) → insert1_out_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88))
U5_gaaa(T40, T42, T47, T48, insert1_out_gaa(s(T40), T47, T48)) → p18_out_gaaa(T40, T42, T47, T48)
U14_gaa(T40, T42, T43, T20, T44, p18_out_gaaa(T40, T42, T43, T44)) → insert1_out_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20))
U13_gaa(T29, T30, T20, T31, insert1_out_gaa(0, T30, T31)) → insert1_out_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20))

The argument filtering Pi contains the following mapping:
insert1_in_gaa(x1, x2, x3)  =  insert1_in_gaa(x1)
insert1_out_gaa(x1, x2, x3)  =  insert1_out_gaa(x1)
0  =  0
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x5)
s(x1)  =  s(x1)
U14_gaa(x1, x2, x3, x4, x5, x6)  =  U14_gaa(x1, x6)
p18_in_gaaa(x1, x2, x3, x4)  =  p18_in_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x1, x5)
less20_in_ga(x1, x2)  =  less20_in_ga(x1)
less20_out_ga(x1, x2)  =  less20_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
p18_out_gaaa(x1, x2, x3, x4)  =  p18_out_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
U5_gaaa(x1, x2, x3, x4, x5)  =  U5_gaaa(x1, x5)
U15_gaa(x1, x2, x3, x4, x5, x6)  =  U15_gaa(x1, x6)
less33_in_ag(x1, x2)  =  less33_in_ag(x2)
less33_out_ag(x1, x2)  =  less33_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U16_gaa(x1, x2, x3, x4, x5, x6)  =  U16_gaa(x1, x6)
U17_gaa(x1, x2, x3, x4, x5, x6)  =  U17_gaa(x1, x6)
U18_gaa(x1, x2, x3, x4, x5, x6)  =  U18_gaa(x1, x6)
p43_in_agaa(x1, x2, x3, x4)  =  p43_in_agaa(x2)
U6_agaa(x1, x2, x3, x4)  =  U6_agaa(x1, x4)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x5)
U20_gaa(x1, x2, x3, x4, x5, x6)  =  U20_gaa(x1, x6)
U21_gaa(x1, x2, x3, x4, x5, x6)  =  U21_gaa(x1, x6)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x2, x5)
p50_in_agaa(x1, x2, x3, x4)  =  p50_in_agaa(x2)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x2, x5)
p50_out_agaa(x1, x2, x3, x4)  =  p50_out_agaa(x1, x2)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x1, x2, x5)
U22_gaa(x1, x2, x3, x4, x5, x6)  =  U22_gaa(x1, x6)
p68_in_agaa(x1, x2, x3, x4)  =  p68_in_agaa(x2)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x1, x4)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x5)
U24_gaa(x1, x2, x3, x4, x5, x6)  =  U24_gaa(x1, x6)
U25_gaa(x1, x2, x3, x4, x5, x6)  =  U25_gaa(x1, x6)
U12_agaa(x1, x2, x3, x4, x5)  =  U12_agaa(x2, x5)
p68_out_agaa(x1, x2, x3, x4)  =  p68_out_agaa(x1, x2)
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x1, x5)
U27_gaa(x1, x2, x3, x4, x5, x6)  =  U27_gaa(x1, x6)
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U29_gaa(x1, x2, x3, x4, x5, x6)  =  U29_gaa(x1, x6)
U30_gaa(x1, x2, x3, x4, x5, x6)  =  U30_gaa(x1, x6)
U31_gaa(x1, x2, x3, x4, x5)  =  U31_gaa(x1, x5)
U32_gaa(x1, x2, x3, x4, x5, x6)  =  U32_gaa(x1, x6)
p43_out_agaa(x1, x2, x3, x4)  =  p43_out_agaa(x1, x2)
INSERT1_IN_GAA(x1, x2, x3)  =  INSERT1_IN_GAA(x1)
U13_GAA(x1, x2, x3, x4, x5)  =  U13_GAA(x5)
U14_GAA(x1, x2, x3, x4, x5, x6)  =  U14_GAA(x1, x6)
P18_IN_GAAA(x1, x2, x3, x4)  =  P18_IN_GAAA(x1)
U3_GAAA(x1, x2, x3, x4, x5)  =  U3_GAAA(x1, x5)
LESS20_IN_GA(x1, x2)  =  LESS20_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U4_GAAA(x1, x2, x3, x4, x5)  =  U4_GAAA(x1, x5)
U5_GAAA(x1, x2, x3, x4, x5)  =  U5_GAAA(x1, x5)
U15_GAA(x1, x2, x3, x4, x5, x6)  =  U15_GAA(x1, x6)
LESS33_IN_AG(x1, x2)  =  LESS33_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
U16_GAA(x1, x2, x3, x4, x5, x6)  =  U16_GAA(x1, x6)
U17_GAA(x1, x2, x3, x4, x5, x6)  =  U17_GAA(x1, x6)
U18_GAA(x1, x2, x3, x4, x5, x6)  =  U18_GAA(x1, x6)
P43_IN_AGAA(x1, x2, x3, x4)  =  P43_IN_AGAA(x2)
U6_AGAA(x1, x2, x3, x4)  =  U6_AGAA(x1, x4)
U19_GAA(x1, x2, x3, x4, x5)  =  U19_GAA(x5)
U20_GAA(x1, x2, x3, x4, x5, x6)  =  U20_GAA(x1, x6)
U21_GAA(x1, x2, x3, x4, x5, x6)  =  U21_GAA(x1, x6)
U7_AGAA(x1, x2, x3, x4, x5)  =  U7_AGAA(x2, x5)
P50_IN_AGAA(x1, x2, x3, x4)  =  P50_IN_AGAA(x2)
U8_AGAA(x1, x2, x3, x4, x5)  =  U8_AGAA(x2, x5)
U9_AGAA(x1, x2, x3, x4, x5)  =  U9_AGAA(x2, x5)
U10_AGAA(x1, x2, x3, x4, x5)  =  U10_AGAA(x1, x2, x5)
U22_GAA(x1, x2, x3, x4, x5, x6)  =  U22_GAA(x1, x6)
P68_IN_AGAA(x1, x2, x3, x4)  =  P68_IN_AGAA(x2)
U11_AGAA(x1, x2, x3, x4)  =  U11_AGAA(x1, x4)
U23_GAA(x1, x2, x3, x4, x5)  =  U23_GAA(x5)
U24_GAA(x1, x2, x3, x4, x5, x6)  =  U24_GAA(x1, x6)
U25_GAA(x1, x2, x3, x4, x5, x6)  =  U25_GAA(x1, x6)
U12_AGAA(x1, x2, x3, x4, x5)  =  U12_AGAA(x2, x5)
U26_GAA(x1, x2, x3, x4, x5)  =  U26_GAA(x1, x5)
U27_GAA(x1, x2, x3, x4, x5, x6)  =  U27_GAA(x1, x6)
U28_GAA(x1, x2, x3, x4, x5)  =  U28_GAA(x5)
U29_GAA(x1, x2, x3, x4, x5, x6)  =  U29_GAA(x1, x6)
U30_GAA(x1, x2, x3, x4, x5, x6)  =  U30_GAA(x1, x6)
U31_GAA(x1, x2, x3, x4, x5)  =  U31_GAA(x1, x5)
U32_GAA(x1, x2, x3, x4, x5, x6)  =  U32_GAA(x1, x6)

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

(67) Obligation:

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

INSERT1_IN_GAA(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → U13_GAA(T29, T30, T20, T31, insert1_in_gaa(0, T30, T31))
INSERT1_IN_GAA(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → INSERT1_IN_GAA(0, T30, T31)
INSERT1_IN_GAA(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → U14_GAA(T40, T42, T43, T20, T44, p18_in_gaaa(T40, T42, T43, T44))
INSERT1_IN_GAA(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → P18_IN_GAAA(T40, T42, T43, T44)
P18_IN_GAAA(T40, T42, T43, T44) → U3_GAAA(T40, T42, T43, T44, less20_in_ga(T40, T42))
P18_IN_GAAA(T40, T42, T43, T44) → LESS20_IN_GA(T40, T42)
LESS20_IN_GA(s(T60), s(T62)) → U1_GA(T60, T62, less20_in_ga(T60, T62))
LESS20_IN_GA(s(T60), s(T62)) → LESS20_IN_GA(T60, T62)
P18_IN_GAAA(T40, T42, T47, T48) → U4_GAAA(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_GAAA(T40, T42, T47, T48, less20_out_ga(T40, T42)) → U5_GAAA(T40, T42, T47, T48, insert1_in_gaa(s(T40), T47, T48))
U4_GAAA(T40, T42, T47, T48, less20_out_ga(T40, T42)) → INSERT1_IN_GAA(s(T40), T47, T48)
INSERT1_IN_GAA(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → U15_GAA(T77, T82, T79, T83, T84, less33_in_ag(T82, T77))
INSERT1_IN_GAA(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → LESS33_IN_AG(T82, T77)
LESS33_IN_AG(s(T102), s(T101)) → U2_AG(T102, T101, less33_in_ag(T102, T101))
LESS33_IN_AG(s(T102), s(T101)) → LESS33_IN_AG(T102, T101)
INSERT1_IN_GAA(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_GAA(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_GAA(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → U17_GAA(T77, T82, T79, T87, T88, insert1_in_gaa(T77, T87, T88))
U16_GAA(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → INSERT1_IN_GAA(T77, T87, T88)
INSERT1_IN_GAA(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → U18_GAA(T112, T117, T114, T118, T119, p43_in_agaa(T117, T112, T118, T119))
INSERT1_IN_GAA(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → P43_IN_AGAA(T117, T112, T118, T119)
P43_IN_AGAA(0, s(T124), T125, T126) → U6_AGAA(T124, T125, T126, insert1_in_gaa(s(T124), T125, T126))
P43_IN_AGAA(0, s(T124), T125, T126) → INSERT1_IN_GAA(s(T124), T125, T126)
INSERT1_IN_GAA(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) → U19_GAA(T163, T164, T154, T165, insert1_in_gaa(0, T164, T165))
INSERT1_IN_GAA(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) → U20_GAA(T174, T176, T177, T154, T178, p18_in_gaaa(T174, T176, T177, T178))
INSERT1_IN_GAA(T191, tree(T196, T193, T197), tree(T196, T193, T198)) → U21_GAA(T191, T196, T193, T197, T198, p43_in_agaa(T196, T191, T197, T198))
P43_IN_AGAA(s(T135), s(T134), T136, T137) → U7_AGAA(T135, T134, T136, T137, p50_in_agaa(T135, T134, T136, T137))
P43_IN_AGAA(s(T135), s(T134), T136, T137) → P50_IN_AGAA(T135, T134, T136, T137)
P50_IN_AGAA(T135, T134, T136, T137) → U8_AGAA(T135, T134, T136, T137, less33_in_ag(T135, T134))
P50_IN_AGAA(T135, T134, T136, T137) → LESS33_IN_AG(T135, T134)
P50_IN_AGAA(T135, T134, T140, T141) → U9_AGAA(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_AGAA(T135, T134, T140, T141, less33_out_ag(T135, T134)) → U10_AGAA(T135, T134, T140, T141, insert1_in_gaa(s(T134), T140, T141))
U9_AGAA(T135, T134, T140, T141, less33_out_ag(T135, T134)) → INSERT1_IN_GAA(s(T134), T140, T141)
INSERT1_IN_GAA(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → U22_GAA(T206, T211, T208, T212, T213, p68_in_agaa(T211, T206, T212, T213))
INSERT1_IN_GAA(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → P68_IN_AGAA(T211, T206, T212, T213)
P68_IN_AGAA(0, s(T218), T219, T220) → U11_AGAA(T218, T219, T220, insert1_in_gaa(s(T218), T219, T220))
P68_IN_AGAA(0, s(T218), T219, T220) → INSERT1_IN_GAA(s(T218), T219, T220)
INSERT1_IN_GAA(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) → U23_GAA(T257, T258, T248, T259, insert1_in_gaa(0, T258, T259))
INSERT1_IN_GAA(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) → U24_GAA(T268, T270, T271, T248, T272, p18_in_gaaa(T268, T270, T271, T272))
INSERT1_IN_GAA(T285, tree(T290, T287, T291), tree(T290, T287, T292)) → U25_GAA(T285, T290, T287, T291, T292, p68_in_agaa(T290, T285, T291, T292))
P68_IN_AGAA(s(T229), s(T228), T230, T231) → U12_AGAA(T229, T228, T230, T231, p50_in_agaa(T229, T228, T230, T231))
P68_IN_AGAA(s(T229), s(T228), T230, T231) → P50_IN_AGAA(T229, T228, T230, T231)
INSERT1_IN_GAA(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → U26_GAA(T312, T302, T313, T314, insert1_in_gaa(s(T312), T313, T314))
INSERT1_IN_GAA(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → INSERT1_IN_GAA(s(T312), T313, T314)
INSERT1_IN_GAA(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → U27_GAA(T322, T323, T302, T324, T325, p50_in_agaa(T323, T322, T324, T325))
INSERT1_IN_GAA(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → P50_IN_AGAA(T323, T322, T324, T325)
INSERT1_IN_GAA(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) → U28_GAA(T345, T346, T336, T347, insert1_in_gaa(0, T346, T347))
INSERT1_IN_GAA(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) → U29_GAA(T356, T358, T359, T336, T360, p18_in_gaaa(T356, T358, T359, T360))
INSERT1_IN_GAA(T373, tree(T378, T375, T379), tree(T378, T375, T380)) → U30_GAA(T373, T378, T375, T379, T380, p68_in_agaa(T378, T373, T379, T380))
INSERT1_IN_GAA(s(T400), tree(0, T390, T401), tree(0, T390, T402)) → U31_GAA(T400, T390, T401, T402, insert1_in_gaa(s(T400), T401, T402))
INSERT1_IN_GAA(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) → U32_GAA(T410, T411, T390, T412, T413, p50_in_agaa(T411, T410, T412, T413))

The TRS R consists of the following rules:

insert1_in_gaa(T5, void, tree(T5, void, void)) → insert1_out_gaa(T5, void, tree(T5, void, void))
insert1_in_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11)) → insert1_out_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11))
insert1_in_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → U13_gaa(T29, T30, T20, T31, insert1_in_gaa(0, T30, T31))
insert1_in_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → U14_gaa(T40, T42, T43, T20, T44, p18_in_gaaa(T40, T42, T43, T44))
p18_in_gaaa(T40, T42, T43, T44) → U3_gaaa(T40, T42, T43, T44, less20_in_ga(T40, T42))
less20_in_ga(0, s(T55)) → less20_out_ga(0, s(T55))
less20_in_ga(s(T60), s(T62)) → U1_ga(T60, T62, less20_in_ga(T60, T62))
U1_ga(T60, T62, less20_out_ga(T60, T62)) → less20_out_ga(s(T60), s(T62))
U3_gaaa(T40, T42, T43, T44, less20_out_ga(T40, T42)) → p18_out_gaaa(T40, T42, T43, T44)
p18_in_gaaa(T40, T42, T47, T48) → U4_gaaa(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_gaaa(T40, T42, T47, T48, less20_out_ga(T40, T42)) → U5_gaaa(T40, T42, T47, T48, insert1_in_gaa(s(T40), T47, T48))
insert1_in_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → U15_gaa(T77, T82, T79, T83, T84, less33_in_ag(T82, T77))
less33_in_ag(0, s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T102), s(T101)) → U2_ag(T102, T101, less33_in_ag(T102, T101))
U2_ag(T102, T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U15_gaa(T77, T82, T79, T83, T84, less33_out_ag(T82, T77)) → insert1_out_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84))
insert1_in_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_gaa(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_gaa(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → U17_gaa(T77, T82, T79, T87, T88, insert1_in_gaa(T77, T87, T88))
insert1_in_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → U18_gaa(T112, T117, T114, T118, T119, p43_in_agaa(T117, T112, T118, T119))
p43_in_agaa(0, s(T124), T125, T126) → U6_agaa(T124, T125, T126, insert1_in_gaa(s(T124), T125, T126))
insert1_in_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) → U19_gaa(T163, T164, T154, T165, insert1_in_gaa(0, T164, T165))
insert1_in_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) → U20_gaa(T174, T176, T177, T154, T178, p18_in_gaaa(T174, T176, T177, T178))
U20_gaa(T174, T176, T177, T154, T178, p18_out_gaaa(T174, T176, T177, T178)) → insert1_out_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154))
insert1_in_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198)) → U21_gaa(T191, T196, T193, T197, T198, p43_in_agaa(T196, T191, T197, T198))
p43_in_agaa(s(T135), s(T134), T136, T137) → U7_agaa(T135, T134, T136, T137, p50_in_agaa(T135, T134, T136, T137))
p50_in_agaa(T135, T134, T136, T137) → U8_agaa(T135, T134, T136, T137, less33_in_ag(T135, T134))
U8_agaa(T135, T134, T136, T137, less33_out_ag(T135, T134)) → p50_out_agaa(T135, T134, T136, T137)
p50_in_agaa(T135, T134, T140, T141) → U9_agaa(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_agaa(T135, T134, T140, T141, less33_out_ag(T135, T134)) → U10_agaa(T135, T134, T140, T141, insert1_in_gaa(s(T134), T140, T141))
insert1_in_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → U22_gaa(T206, T211, T208, T212, T213, p68_in_agaa(T211, T206, T212, T213))
p68_in_agaa(0, s(T218), T219, T220) → U11_agaa(T218, T219, T220, insert1_in_gaa(s(T218), T219, T220))
insert1_in_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) → U23_gaa(T257, T258, T248, T259, insert1_in_gaa(0, T258, T259))
insert1_in_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) → U24_gaa(T268, T270, T271, T248, T272, p18_in_gaaa(T268, T270, T271, T272))
U24_gaa(T268, T270, T271, T248, T272, p18_out_gaaa(T268, T270, T271, T272)) → insert1_out_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248))
insert1_in_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292)) → U25_gaa(T285, T290, T287, T291, T292, p68_in_agaa(T290, T285, T291, T292))
p68_in_agaa(s(T229), s(T228), T230, T231) → U12_agaa(T229, T228, T230, T231, p50_in_agaa(T229, T228, T230, T231))
U12_agaa(T229, T228, T230, T231, p50_out_agaa(T229, T228, T230, T231)) → p68_out_agaa(s(T229), s(T228), T230, T231)
U25_gaa(T285, T290, T287, T291, T292, p68_out_agaa(T290, T285, T291, T292)) → insert1_out_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292))
insert1_in_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → U26_gaa(T312, T302, T313, T314, insert1_in_gaa(s(T312), T313, T314))
insert1_in_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → U27_gaa(T322, T323, T302, T324, T325, p50_in_agaa(T323, T322, T324, T325))
U27_gaa(T322, T323, T302, T324, T325, p50_out_agaa(T323, T322, T324, T325)) → insert1_out_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325))
insert1_in_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) → U28_gaa(T345, T346, T336, T347, insert1_in_gaa(0, T346, T347))
insert1_in_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) → U29_gaa(T356, T358, T359, T336, T360, p18_in_gaaa(T356, T358, T359, T360))
U29_gaa(T356, T358, T359, T336, T360, p18_out_gaaa(T356, T358, T359, T360)) → insert1_out_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336))
insert1_in_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380)) → U30_gaa(T373, T378, T375, T379, T380, p68_in_agaa(T378, T373, T379, T380))
U30_gaa(T373, T378, T375, T379, T380, p68_out_agaa(T378, T373, T379, T380)) → insert1_out_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380))
insert1_in_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402)) → U31_gaa(T400, T390, T401, T402, insert1_in_gaa(s(T400), T401, T402))
insert1_in_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) → U32_gaa(T410, T411, T390, T412, T413, p50_in_agaa(T411, T410, T412, T413))
U32_gaa(T410, T411, T390, T412, T413, p50_out_agaa(T411, T410, T412, T413)) → insert1_out_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413))
U31_gaa(T400, T390, T401, T402, insert1_out_gaa(s(T400), T401, T402)) → insert1_out_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402))
U28_gaa(T345, T346, T336, T347, insert1_out_gaa(0, T346, T347)) → insert1_out_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336))
U26_gaa(T312, T302, T313, T314, insert1_out_gaa(s(T312), T313, T314)) → insert1_out_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314))
U23_gaa(T257, T258, T248, T259, insert1_out_gaa(0, T258, T259)) → insert1_out_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248))
U11_agaa(T218, T219, T220, insert1_out_gaa(s(T218), T219, T220)) → p68_out_agaa(0, s(T218), T219, T220)
U22_gaa(T206, T211, T208, T212, T213, p68_out_agaa(T211, T206, T212, T213)) → insert1_out_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213))
U10_agaa(T135, T134, T140, T141, insert1_out_gaa(s(T134), T140, T141)) → p50_out_agaa(T135, T134, T140, T141)
U7_agaa(T135, T134, T136, T137, p50_out_agaa(T135, T134, T136, T137)) → p43_out_agaa(s(T135), s(T134), T136, T137)
U21_gaa(T191, T196, T193, T197, T198, p43_out_agaa(T196, T191, T197, T198)) → insert1_out_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198))
U19_gaa(T163, T164, T154, T165, insert1_out_gaa(0, T164, T165)) → insert1_out_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154))
U6_agaa(T124, T125, T126, insert1_out_gaa(s(T124), T125, T126)) → p43_out_agaa(0, s(T124), T125, T126)
U18_gaa(T112, T117, T114, T118, T119, p43_out_agaa(T117, T112, T118, T119)) → insert1_out_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119))
U17_gaa(T77, T82, T79, T87, T88, insert1_out_gaa(T77, T87, T88)) → insert1_out_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88))
U5_gaaa(T40, T42, T47, T48, insert1_out_gaa(s(T40), T47, T48)) → p18_out_gaaa(T40, T42, T47, T48)
U14_gaa(T40, T42, T43, T20, T44, p18_out_gaaa(T40, T42, T43, T44)) → insert1_out_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20))
U13_gaa(T29, T30, T20, T31, insert1_out_gaa(0, T30, T31)) → insert1_out_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20))

The argument filtering Pi contains the following mapping:
insert1_in_gaa(x1, x2, x3)  =  insert1_in_gaa(x1)
insert1_out_gaa(x1, x2, x3)  =  insert1_out_gaa(x1)
0  =  0
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x5)
s(x1)  =  s(x1)
U14_gaa(x1, x2, x3, x4, x5, x6)  =  U14_gaa(x1, x6)
p18_in_gaaa(x1, x2, x3, x4)  =  p18_in_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x1, x5)
less20_in_ga(x1, x2)  =  less20_in_ga(x1)
less20_out_ga(x1, x2)  =  less20_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
p18_out_gaaa(x1, x2, x3, x4)  =  p18_out_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
U5_gaaa(x1, x2, x3, x4, x5)  =  U5_gaaa(x1, x5)
U15_gaa(x1, x2, x3, x4, x5, x6)  =  U15_gaa(x1, x6)
less33_in_ag(x1, x2)  =  less33_in_ag(x2)
less33_out_ag(x1, x2)  =  less33_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U16_gaa(x1, x2, x3, x4, x5, x6)  =  U16_gaa(x1, x6)
U17_gaa(x1, x2, x3, x4, x5, x6)  =  U17_gaa(x1, x6)
U18_gaa(x1, x2, x3, x4, x5, x6)  =  U18_gaa(x1, x6)
p43_in_agaa(x1, x2, x3, x4)  =  p43_in_agaa(x2)
U6_agaa(x1, x2, x3, x4)  =  U6_agaa(x1, x4)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x5)
U20_gaa(x1, x2, x3, x4, x5, x6)  =  U20_gaa(x1, x6)
U21_gaa(x1, x2, x3, x4, x5, x6)  =  U21_gaa(x1, x6)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x2, x5)
p50_in_agaa(x1, x2, x3, x4)  =  p50_in_agaa(x2)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x2, x5)
p50_out_agaa(x1, x2, x3, x4)  =  p50_out_agaa(x1, x2)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x1, x2, x5)
U22_gaa(x1, x2, x3, x4, x5, x6)  =  U22_gaa(x1, x6)
p68_in_agaa(x1, x2, x3, x4)  =  p68_in_agaa(x2)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x1, x4)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x5)
U24_gaa(x1, x2, x3, x4, x5, x6)  =  U24_gaa(x1, x6)
U25_gaa(x1, x2, x3, x4, x5, x6)  =  U25_gaa(x1, x6)
U12_agaa(x1, x2, x3, x4, x5)  =  U12_agaa(x2, x5)
p68_out_agaa(x1, x2, x3, x4)  =  p68_out_agaa(x1, x2)
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x1, x5)
U27_gaa(x1, x2, x3, x4, x5, x6)  =  U27_gaa(x1, x6)
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U29_gaa(x1, x2, x3, x4, x5, x6)  =  U29_gaa(x1, x6)
U30_gaa(x1, x2, x3, x4, x5, x6)  =  U30_gaa(x1, x6)
U31_gaa(x1, x2, x3, x4, x5)  =  U31_gaa(x1, x5)
U32_gaa(x1, x2, x3, x4, x5, x6)  =  U32_gaa(x1, x6)
p43_out_agaa(x1, x2, x3, x4)  =  p43_out_agaa(x1, x2)
INSERT1_IN_GAA(x1, x2, x3)  =  INSERT1_IN_GAA(x1)
U13_GAA(x1, x2, x3, x4, x5)  =  U13_GAA(x5)
U14_GAA(x1, x2, x3, x4, x5, x6)  =  U14_GAA(x1, x6)
P18_IN_GAAA(x1, x2, x3, x4)  =  P18_IN_GAAA(x1)
U3_GAAA(x1, x2, x3, x4, x5)  =  U3_GAAA(x1, x5)
LESS20_IN_GA(x1, x2)  =  LESS20_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U4_GAAA(x1, x2, x3, x4, x5)  =  U4_GAAA(x1, x5)
U5_GAAA(x1, x2, x3, x4, x5)  =  U5_GAAA(x1, x5)
U15_GAA(x1, x2, x3, x4, x5, x6)  =  U15_GAA(x1, x6)
LESS33_IN_AG(x1, x2)  =  LESS33_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
U16_GAA(x1, x2, x3, x4, x5, x6)  =  U16_GAA(x1, x6)
U17_GAA(x1, x2, x3, x4, x5, x6)  =  U17_GAA(x1, x6)
U18_GAA(x1, x2, x3, x4, x5, x6)  =  U18_GAA(x1, x6)
P43_IN_AGAA(x1, x2, x3, x4)  =  P43_IN_AGAA(x2)
U6_AGAA(x1, x2, x3, x4)  =  U6_AGAA(x1, x4)
U19_GAA(x1, x2, x3, x4, x5)  =  U19_GAA(x5)
U20_GAA(x1, x2, x3, x4, x5, x6)  =  U20_GAA(x1, x6)
U21_GAA(x1, x2, x3, x4, x5, x6)  =  U21_GAA(x1, x6)
U7_AGAA(x1, x2, x3, x4, x5)  =  U7_AGAA(x2, x5)
P50_IN_AGAA(x1, x2, x3, x4)  =  P50_IN_AGAA(x2)
U8_AGAA(x1, x2, x3, x4, x5)  =  U8_AGAA(x2, x5)
U9_AGAA(x1, x2, x3, x4, x5)  =  U9_AGAA(x2, x5)
U10_AGAA(x1, x2, x3, x4, x5)  =  U10_AGAA(x1, x2, x5)
U22_GAA(x1, x2, x3, x4, x5, x6)  =  U22_GAA(x1, x6)
P68_IN_AGAA(x1, x2, x3, x4)  =  P68_IN_AGAA(x2)
U11_AGAA(x1, x2, x3, x4)  =  U11_AGAA(x1, x4)
U23_GAA(x1, x2, x3, x4, x5)  =  U23_GAA(x5)
U24_GAA(x1, x2, x3, x4, x5, x6)  =  U24_GAA(x1, x6)
U25_GAA(x1, x2, x3, x4, x5, x6)  =  U25_GAA(x1, x6)
U12_AGAA(x1, x2, x3, x4, x5)  =  U12_AGAA(x2, x5)
U26_GAA(x1, x2, x3, x4, x5)  =  U26_GAA(x1, x5)
U27_GAA(x1, x2, x3, x4, x5, x6)  =  U27_GAA(x1, x6)
U28_GAA(x1, x2, x3, x4, x5)  =  U28_GAA(x5)
U29_GAA(x1, x2, x3, x4, x5, x6)  =  U29_GAA(x1, x6)
U30_GAA(x1, x2, x3, x4, x5, x6)  =  U30_GAA(x1, x6)
U31_GAA(x1, x2, x3, x4, x5)  =  U31_GAA(x1, x5)
U32_GAA(x1, x2, x3, x4, x5, x6)  =  U32_GAA(x1, x6)

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

(68) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 32 less nodes.

(69) Complex Obligation (AND)

(70) Obligation:

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

LESS33_IN_AG(s(T102), s(T101)) → LESS33_IN_AG(T102, T101)

The TRS R consists of the following rules:

insert1_in_gaa(T5, void, tree(T5, void, void)) → insert1_out_gaa(T5, void, tree(T5, void, void))
insert1_in_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11)) → insert1_out_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11))
insert1_in_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → U13_gaa(T29, T30, T20, T31, insert1_in_gaa(0, T30, T31))
insert1_in_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → U14_gaa(T40, T42, T43, T20, T44, p18_in_gaaa(T40, T42, T43, T44))
p18_in_gaaa(T40, T42, T43, T44) → U3_gaaa(T40, T42, T43, T44, less20_in_ga(T40, T42))
less20_in_ga(0, s(T55)) → less20_out_ga(0, s(T55))
less20_in_ga(s(T60), s(T62)) → U1_ga(T60, T62, less20_in_ga(T60, T62))
U1_ga(T60, T62, less20_out_ga(T60, T62)) → less20_out_ga(s(T60), s(T62))
U3_gaaa(T40, T42, T43, T44, less20_out_ga(T40, T42)) → p18_out_gaaa(T40, T42, T43, T44)
p18_in_gaaa(T40, T42, T47, T48) → U4_gaaa(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_gaaa(T40, T42, T47, T48, less20_out_ga(T40, T42)) → U5_gaaa(T40, T42, T47, T48, insert1_in_gaa(s(T40), T47, T48))
insert1_in_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → U15_gaa(T77, T82, T79, T83, T84, less33_in_ag(T82, T77))
less33_in_ag(0, s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T102), s(T101)) → U2_ag(T102, T101, less33_in_ag(T102, T101))
U2_ag(T102, T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U15_gaa(T77, T82, T79, T83, T84, less33_out_ag(T82, T77)) → insert1_out_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84))
insert1_in_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_gaa(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_gaa(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → U17_gaa(T77, T82, T79, T87, T88, insert1_in_gaa(T77, T87, T88))
insert1_in_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → U18_gaa(T112, T117, T114, T118, T119, p43_in_agaa(T117, T112, T118, T119))
p43_in_agaa(0, s(T124), T125, T126) → U6_agaa(T124, T125, T126, insert1_in_gaa(s(T124), T125, T126))
insert1_in_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) → U19_gaa(T163, T164, T154, T165, insert1_in_gaa(0, T164, T165))
insert1_in_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) → U20_gaa(T174, T176, T177, T154, T178, p18_in_gaaa(T174, T176, T177, T178))
U20_gaa(T174, T176, T177, T154, T178, p18_out_gaaa(T174, T176, T177, T178)) → insert1_out_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154))
insert1_in_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198)) → U21_gaa(T191, T196, T193, T197, T198, p43_in_agaa(T196, T191, T197, T198))
p43_in_agaa(s(T135), s(T134), T136, T137) → U7_agaa(T135, T134, T136, T137, p50_in_agaa(T135, T134, T136, T137))
p50_in_agaa(T135, T134, T136, T137) → U8_agaa(T135, T134, T136, T137, less33_in_ag(T135, T134))
U8_agaa(T135, T134, T136, T137, less33_out_ag(T135, T134)) → p50_out_agaa(T135, T134, T136, T137)
p50_in_agaa(T135, T134, T140, T141) → U9_agaa(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_agaa(T135, T134, T140, T141, less33_out_ag(T135, T134)) → U10_agaa(T135, T134, T140, T141, insert1_in_gaa(s(T134), T140, T141))
insert1_in_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → U22_gaa(T206, T211, T208, T212, T213, p68_in_agaa(T211, T206, T212, T213))
p68_in_agaa(0, s(T218), T219, T220) → U11_agaa(T218, T219, T220, insert1_in_gaa(s(T218), T219, T220))
insert1_in_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) → U23_gaa(T257, T258, T248, T259, insert1_in_gaa(0, T258, T259))
insert1_in_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) → U24_gaa(T268, T270, T271, T248, T272, p18_in_gaaa(T268, T270, T271, T272))
U24_gaa(T268, T270, T271, T248, T272, p18_out_gaaa(T268, T270, T271, T272)) → insert1_out_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248))
insert1_in_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292)) → U25_gaa(T285, T290, T287, T291, T292, p68_in_agaa(T290, T285, T291, T292))
p68_in_agaa(s(T229), s(T228), T230, T231) → U12_agaa(T229, T228, T230, T231, p50_in_agaa(T229, T228, T230, T231))
U12_agaa(T229, T228, T230, T231, p50_out_agaa(T229, T228, T230, T231)) → p68_out_agaa(s(T229), s(T228), T230, T231)
U25_gaa(T285, T290, T287, T291, T292, p68_out_agaa(T290, T285, T291, T292)) → insert1_out_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292))
insert1_in_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → U26_gaa(T312, T302, T313, T314, insert1_in_gaa(s(T312), T313, T314))
insert1_in_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → U27_gaa(T322, T323, T302, T324, T325, p50_in_agaa(T323, T322, T324, T325))
U27_gaa(T322, T323, T302, T324, T325, p50_out_agaa(T323, T322, T324, T325)) → insert1_out_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325))
insert1_in_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) → U28_gaa(T345, T346, T336, T347, insert1_in_gaa(0, T346, T347))
insert1_in_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) → U29_gaa(T356, T358, T359, T336, T360, p18_in_gaaa(T356, T358, T359, T360))
U29_gaa(T356, T358, T359, T336, T360, p18_out_gaaa(T356, T358, T359, T360)) → insert1_out_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336))
insert1_in_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380)) → U30_gaa(T373, T378, T375, T379, T380, p68_in_agaa(T378, T373, T379, T380))
U30_gaa(T373, T378, T375, T379, T380, p68_out_agaa(T378, T373, T379, T380)) → insert1_out_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380))
insert1_in_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402)) → U31_gaa(T400, T390, T401, T402, insert1_in_gaa(s(T400), T401, T402))
insert1_in_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) → U32_gaa(T410, T411, T390, T412, T413, p50_in_agaa(T411, T410, T412, T413))
U32_gaa(T410, T411, T390, T412, T413, p50_out_agaa(T411, T410, T412, T413)) → insert1_out_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413))
U31_gaa(T400, T390, T401, T402, insert1_out_gaa(s(T400), T401, T402)) → insert1_out_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402))
U28_gaa(T345, T346, T336, T347, insert1_out_gaa(0, T346, T347)) → insert1_out_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336))
U26_gaa(T312, T302, T313, T314, insert1_out_gaa(s(T312), T313, T314)) → insert1_out_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314))
U23_gaa(T257, T258, T248, T259, insert1_out_gaa(0, T258, T259)) → insert1_out_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248))
U11_agaa(T218, T219, T220, insert1_out_gaa(s(T218), T219, T220)) → p68_out_agaa(0, s(T218), T219, T220)
U22_gaa(T206, T211, T208, T212, T213, p68_out_agaa(T211, T206, T212, T213)) → insert1_out_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213))
U10_agaa(T135, T134, T140, T141, insert1_out_gaa(s(T134), T140, T141)) → p50_out_agaa(T135, T134, T140, T141)
U7_agaa(T135, T134, T136, T137, p50_out_agaa(T135, T134, T136, T137)) → p43_out_agaa(s(T135), s(T134), T136, T137)
U21_gaa(T191, T196, T193, T197, T198, p43_out_agaa(T196, T191, T197, T198)) → insert1_out_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198))
U19_gaa(T163, T164, T154, T165, insert1_out_gaa(0, T164, T165)) → insert1_out_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154))
U6_agaa(T124, T125, T126, insert1_out_gaa(s(T124), T125, T126)) → p43_out_agaa(0, s(T124), T125, T126)
U18_gaa(T112, T117, T114, T118, T119, p43_out_agaa(T117, T112, T118, T119)) → insert1_out_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119))
U17_gaa(T77, T82, T79, T87, T88, insert1_out_gaa(T77, T87, T88)) → insert1_out_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88))
U5_gaaa(T40, T42, T47, T48, insert1_out_gaa(s(T40), T47, T48)) → p18_out_gaaa(T40, T42, T47, T48)
U14_gaa(T40, T42, T43, T20, T44, p18_out_gaaa(T40, T42, T43, T44)) → insert1_out_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20))
U13_gaa(T29, T30, T20, T31, insert1_out_gaa(0, T30, T31)) → insert1_out_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20))

The argument filtering Pi contains the following mapping:
insert1_in_gaa(x1, x2, x3)  =  insert1_in_gaa(x1)
insert1_out_gaa(x1, x2, x3)  =  insert1_out_gaa(x1)
0  =  0
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x5)
s(x1)  =  s(x1)
U14_gaa(x1, x2, x3, x4, x5, x6)  =  U14_gaa(x1, x6)
p18_in_gaaa(x1, x2, x3, x4)  =  p18_in_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x1, x5)
less20_in_ga(x1, x2)  =  less20_in_ga(x1)
less20_out_ga(x1, x2)  =  less20_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
p18_out_gaaa(x1, x2, x3, x4)  =  p18_out_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
U5_gaaa(x1, x2, x3, x4, x5)  =  U5_gaaa(x1, x5)
U15_gaa(x1, x2, x3, x4, x5, x6)  =  U15_gaa(x1, x6)
less33_in_ag(x1, x2)  =  less33_in_ag(x2)
less33_out_ag(x1, x2)  =  less33_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U16_gaa(x1, x2, x3, x4, x5, x6)  =  U16_gaa(x1, x6)
U17_gaa(x1, x2, x3, x4, x5, x6)  =  U17_gaa(x1, x6)
U18_gaa(x1, x2, x3, x4, x5, x6)  =  U18_gaa(x1, x6)
p43_in_agaa(x1, x2, x3, x4)  =  p43_in_agaa(x2)
U6_agaa(x1, x2, x3, x4)  =  U6_agaa(x1, x4)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x5)
U20_gaa(x1, x2, x3, x4, x5, x6)  =  U20_gaa(x1, x6)
U21_gaa(x1, x2, x3, x4, x5, x6)  =  U21_gaa(x1, x6)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x2, x5)
p50_in_agaa(x1, x2, x3, x4)  =  p50_in_agaa(x2)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x2, x5)
p50_out_agaa(x1, x2, x3, x4)  =  p50_out_agaa(x1, x2)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x1, x2, x5)
U22_gaa(x1, x2, x3, x4, x5, x6)  =  U22_gaa(x1, x6)
p68_in_agaa(x1, x2, x3, x4)  =  p68_in_agaa(x2)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x1, x4)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x5)
U24_gaa(x1, x2, x3, x4, x5, x6)  =  U24_gaa(x1, x6)
U25_gaa(x1, x2, x3, x4, x5, x6)  =  U25_gaa(x1, x6)
U12_agaa(x1, x2, x3, x4, x5)  =  U12_agaa(x2, x5)
p68_out_agaa(x1, x2, x3, x4)  =  p68_out_agaa(x1, x2)
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x1, x5)
U27_gaa(x1, x2, x3, x4, x5, x6)  =  U27_gaa(x1, x6)
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U29_gaa(x1, x2, x3, x4, x5, x6)  =  U29_gaa(x1, x6)
U30_gaa(x1, x2, x3, x4, x5, x6)  =  U30_gaa(x1, x6)
U31_gaa(x1, x2, x3, x4, x5)  =  U31_gaa(x1, x5)
U32_gaa(x1, x2, x3, x4, x5, x6)  =  U32_gaa(x1, x6)
p43_out_agaa(x1, x2, x3, x4)  =  p43_out_agaa(x1, x2)
LESS33_IN_AG(x1, x2)  =  LESS33_IN_AG(x2)

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

(71) UsableRulesProof (EQUIVALENT transformation)

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

(72) Obligation:

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

LESS33_IN_AG(s(T102), s(T101)) → LESS33_IN_AG(T102, T101)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS33_IN_AG(x1, x2)  =  LESS33_IN_AG(x2)

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

(73) PiDPToQDPProof (SOUND transformation)

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

(74) Obligation:

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

LESS33_IN_AG(s(T101)) → LESS33_IN_AG(T101)

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

(75) 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:

  • LESS33_IN_AG(s(T101)) → LESS33_IN_AG(T101)
    The graph contains the following edges 1 > 1

(76) YES

(77) Obligation:

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

LESS20_IN_GA(s(T60), s(T62)) → LESS20_IN_GA(T60, T62)

The TRS R consists of the following rules:

insert1_in_gaa(T5, void, tree(T5, void, void)) → insert1_out_gaa(T5, void, tree(T5, void, void))
insert1_in_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11)) → insert1_out_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11))
insert1_in_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → U13_gaa(T29, T30, T20, T31, insert1_in_gaa(0, T30, T31))
insert1_in_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → U14_gaa(T40, T42, T43, T20, T44, p18_in_gaaa(T40, T42, T43, T44))
p18_in_gaaa(T40, T42, T43, T44) → U3_gaaa(T40, T42, T43, T44, less20_in_ga(T40, T42))
less20_in_ga(0, s(T55)) → less20_out_ga(0, s(T55))
less20_in_ga(s(T60), s(T62)) → U1_ga(T60, T62, less20_in_ga(T60, T62))
U1_ga(T60, T62, less20_out_ga(T60, T62)) → less20_out_ga(s(T60), s(T62))
U3_gaaa(T40, T42, T43, T44, less20_out_ga(T40, T42)) → p18_out_gaaa(T40, T42, T43, T44)
p18_in_gaaa(T40, T42, T47, T48) → U4_gaaa(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_gaaa(T40, T42, T47, T48, less20_out_ga(T40, T42)) → U5_gaaa(T40, T42, T47, T48, insert1_in_gaa(s(T40), T47, T48))
insert1_in_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → U15_gaa(T77, T82, T79, T83, T84, less33_in_ag(T82, T77))
less33_in_ag(0, s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T102), s(T101)) → U2_ag(T102, T101, less33_in_ag(T102, T101))
U2_ag(T102, T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U15_gaa(T77, T82, T79, T83, T84, less33_out_ag(T82, T77)) → insert1_out_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84))
insert1_in_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_gaa(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_gaa(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → U17_gaa(T77, T82, T79, T87, T88, insert1_in_gaa(T77, T87, T88))
insert1_in_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → U18_gaa(T112, T117, T114, T118, T119, p43_in_agaa(T117, T112, T118, T119))
p43_in_agaa(0, s(T124), T125, T126) → U6_agaa(T124, T125, T126, insert1_in_gaa(s(T124), T125, T126))
insert1_in_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) → U19_gaa(T163, T164, T154, T165, insert1_in_gaa(0, T164, T165))
insert1_in_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) → U20_gaa(T174, T176, T177, T154, T178, p18_in_gaaa(T174, T176, T177, T178))
U20_gaa(T174, T176, T177, T154, T178, p18_out_gaaa(T174, T176, T177, T178)) → insert1_out_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154))
insert1_in_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198)) → U21_gaa(T191, T196, T193, T197, T198, p43_in_agaa(T196, T191, T197, T198))
p43_in_agaa(s(T135), s(T134), T136, T137) → U7_agaa(T135, T134, T136, T137, p50_in_agaa(T135, T134, T136, T137))
p50_in_agaa(T135, T134, T136, T137) → U8_agaa(T135, T134, T136, T137, less33_in_ag(T135, T134))
U8_agaa(T135, T134, T136, T137, less33_out_ag(T135, T134)) → p50_out_agaa(T135, T134, T136, T137)
p50_in_agaa(T135, T134, T140, T141) → U9_agaa(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_agaa(T135, T134, T140, T141, less33_out_ag(T135, T134)) → U10_agaa(T135, T134, T140, T141, insert1_in_gaa(s(T134), T140, T141))
insert1_in_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → U22_gaa(T206, T211, T208, T212, T213, p68_in_agaa(T211, T206, T212, T213))
p68_in_agaa(0, s(T218), T219, T220) → U11_agaa(T218, T219, T220, insert1_in_gaa(s(T218), T219, T220))
insert1_in_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) → U23_gaa(T257, T258, T248, T259, insert1_in_gaa(0, T258, T259))
insert1_in_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) → U24_gaa(T268, T270, T271, T248, T272, p18_in_gaaa(T268, T270, T271, T272))
U24_gaa(T268, T270, T271, T248, T272, p18_out_gaaa(T268, T270, T271, T272)) → insert1_out_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248))
insert1_in_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292)) → U25_gaa(T285, T290, T287, T291, T292, p68_in_agaa(T290, T285, T291, T292))
p68_in_agaa(s(T229), s(T228), T230, T231) → U12_agaa(T229, T228, T230, T231, p50_in_agaa(T229, T228, T230, T231))
U12_agaa(T229, T228, T230, T231, p50_out_agaa(T229, T228, T230, T231)) → p68_out_agaa(s(T229), s(T228), T230, T231)
U25_gaa(T285, T290, T287, T291, T292, p68_out_agaa(T290, T285, T291, T292)) → insert1_out_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292))
insert1_in_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → U26_gaa(T312, T302, T313, T314, insert1_in_gaa(s(T312), T313, T314))
insert1_in_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → U27_gaa(T322, T323, T302, T324, T325, p50_in_agaa(T323, T322, T324, T325))
U27_gaa(T322, T323, T302, T324, T325, p50_out_agaa(T323, T322, T324, T325)) → insert1_out_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325))
insert1_in_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) → U28_gaa(T345, T346, T336, T347, insert1_in_gaa(0, T346, T347))
insert1_in_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) → U29_gaa(T356, T358, T359, T336, T360, p18_in_gaaa(T356, T358, T359, T360))
U29_gaa(T356, T358, T359, T336, T360, p18_out_gaaa(T356, T358, T359, T360)) → insert1_out_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336))
insert1_in_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380)) → U30_gaa(T373, T378, T375, T379, T380, p68_in_agaa(T378, T373, T379, T380))
U30_gaa(T373, T378, T375, T379, T380, p68_out_agaa(T378, T373, T379, T380)) → insert1_out_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380))
insert1_in_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402)) → U31_gaa(T400, T390, T401, T402, insert1_in_gaa(s(T400), T401, T402))
insert1_in_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) → U32_gaa(T410, T411, T390, T412, T413, p50_in_agaa(T411, T410, T412, T413))
U32_gaa(T410, T411, T390, T412, T413, p50_out_agaa(T411, T410, T412, T413)) → insert1_out_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413))
U31_gaa(T400, T390, T401, T402, insert1_out_gaa(s(T400), T401, T402)) → insert1_out_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402))
U28_gaa(T345, T346, T336, T347, insert1_out_gaa(0, T346, T347)) → insert1_out_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336))
U26_gaa(T312, T302, T313, T314, insert1_out_gaa(s(T312), T313, T314)) → insert1_out_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314))
U23_gaa(T257, T258, T248, T259, insert1_out_gaa(0, T258, T259)) → insert1_out_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248))
U11_agaa(T218, T219, T220, insert1_out_gaa(s(T218), T219, T220)) → p68_out_agaa(0, s(T218), T219, T220)
U22_gaa(T206, T211, T208, T212, T213, p68_out_agaa(T211, T206, T212, T213)) → insert1_out_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213))
U10_agaa(T135, T134, T140, T141, insert1_out_gaa(s(T134), T140, T141)) → p50_out_agaa(T135, T134, T140, T141)
U7_agaa(T135, T134, T136, T137, p50_out_agaa(T135, T134, T136, T137)) → p43_out_agaa(s(T135), s(T134), T136, T137)
U21_gaa(T191, T196, T193, T197, T198, p43_out_agaa(T196, T191, T197, T198)) → insert1_out_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198))
U19_gaa(T163, T164, T154, T165, insert1_out_gaa(0, T164, T165)) → insert1_out_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154))
U6_agaa(T124, T125, T126, insert1_out_gaa(s(T124), T125, T126)) → p43_out_agaa(0, s(T124), T125, T126)
U18_gaa(T112, T117, T114, T118, T119, p43_out_agaa(T117, T112, T118, T119)) → insert1_out_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119))
U17_gaa(T77, T82, T79, T87, T88, insert1_out_gaa(T77, T87, T88)) → insert1_out_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88))
U5_gaaa(T40, T42, T47, T48, insert1_out_gaa(s(T40), T47, T48)) → p18_out_gaaa(T40, T42, T47, T48)
U14_gaa(T40, T42, T43, T20, T44, p18_out_gaaa(T40, T42, T43, T44)) → insert1_out_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20))
U13_gaa(T29, T30, T20, T31, insert1_out_gaa(0, T30, T31)) → insert1_out_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20))

The argument filtering Pi contains the following mapping:
insert1_in_gaa(x1, x2, x3)  =  insert1_in_gaa(x1)
insert1_out_gaa(x1, x2, x3)  =  insert1_out_gaa(x1)
0  =  0
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x5)
s(x1)  =  s(x1)
U14_gaa(x1, x2, x3, x4, x5, x6)  =  U14_gaa(x1, x6)
p18_in_gaaa(x1, x2, x3, x4)  =  p18_in_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x1, x5)
less20_in_ga(x1, x2)  =  less20_in_ga(x1)
less20_out_ga(x1, x2)  =  less20_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
p18_out_gaaa(x1, x2, x3, x4)  =  p18_out_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
U5_gaaa(x1, x2, x3, x4, x5)  =  U5_gaaa(x1, x5)
U15_gaa(x1, x2, x3, x4, x5, x6)  =  U15_gaa(x1, x6)
less33_in_ag(x1, x2)  =  less33_in_ag(x2)
less33_out_ag(x1, x2)  =  less33_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U16_gaa(x1, x2, x3, x4, x5, x6)  =  U16_gaa(x1, x6)
U17_gaa(x1, x2, x3, x4, x5, x6)  =  U17_gaa(x1, x6)
U18_gaa(x1, x2, x3, x4, x5, x6)  =  U18_gaa(x1, x6)
p43_in_agaa(x1, x2, x3, x4)  =  p43_in_agaa(x2)
U6_agaa(x1, x2, x3, x4)  =  U6_agaa(x1, x4)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x5)
U20_gaa(x1, x2, x3, x4, x5, x6)  =  U20_gaa(x1, x6)
U21_gaa(x1, x2, x3, x4, x5, x6)  =  U21_gaa(x1, x6)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x2, x5)
p50_in_agaa(x1, x2, x3, x4)  =  p50_in_agaa(x2)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x2, x5)
p50_out_agaa(x1, x2, x3, x4)  =  p50_out_agaa(x1, x2)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x1, x2, x5)
U22_gaa(x1, x2, x3, x4, x5, x6)  =  U22_gaa(x1, x6)
p68_in_agaa(x1, x2, x3, x4)  =  p68_in_agaa(x2)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x1, x4)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x5)
U24_gaa(x1, x2, x3, x4, x5, x6)  =  U24_gaa(x1, x6)
U25_gaa(x1, x2, x3, x4, x5, x6)  =  U25_gaa(x1, x6)
U12_agaa(x1, x2, x3, x4, x5)  =  U12_agaa(x2, x5)
p68_out_agaa(x1, x2, x3, x4)  =  p68_out_agaa(x1, x2)
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x1, x5)
U27_gaa(x1, x2, x3, x4, x5, x6)  =  U27_gaa(x1, x6)
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U29_gaa(x1, x2, x3, x4, x5, x6)  =  U29_gaa(x1, x6)
U30_gaa(x1, x2, x3, x4, x5, x6)  =  U30_gaa(x1, x6)
U31_gaa(x1, x2, x3, x4, x5)  =  U31_gaa(x1, x5)
U32_gaa(x1, x2, x3, x4, x5, x6)  =  U32_gaa(x1, x6)
p43_out_agaa(x1, x2, x3, x4)  =  p43_out_agaa(x1, x2)
LESS20_IN_GA(x1, x2)  =  LESS20_IN_GA(x1)

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

(78) UsableRulesProof (EQUIVALENT transformation)

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

(79) Obligation:

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

LESS20_IN_GA(s(T60), s(T62)) → LESS20_IN_GA(T60, T62)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS20_IN_GA(x1, x2)  =  LESS20_IN_GA(x1)

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

(80) PiDPToQDPProof (SOUND transformation)

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

(81) Obligation:

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

LESS20_IN_GA(s(T60)) → LESS20_IN_GA(T60)

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

(82) 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:

  • LESS20_IN_GA(s(T60)) → LESS20_IN_GA(T60)
    The graph contains the following edges 1 > 1

(83) YES

(84) Obligation:

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

INSERT1_IN_GAA(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_GAA(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_GAA(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → INSERT1_IN_GAA(T77, T87, T88)
INSERT1_IN_GAA(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → INSERT1_IN_GAA(0, T30, T31)
INSERT1_IN_GAA(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → P43_IN_AGAA(T117, T112, T118, T119)
P43_IN_AGAA(0, s(T124), T125, T126) → INSERT1_IN_GAA(s(T124), T125, T126)
INSERT1_IN_GAA(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → P18_IN_GAAA(T40, T42, T43, T44)
P18_IN_GAAA(T40, T42, T47, T48) → U4_GAAA(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_GAAA(T40, T42, T47, T48, less20_out_ga(T40, T42)) → INSERT1_IN_GAA(s(T40), T47, T48)
INSERT1_IN_GAA(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → P68_IN_AGAA(T211, T206, T212, T213)
P68_IN_AGAA(0, s(T218), T219, T220) → INSERT1_IN_GAA(s(T218), T219, T220)
INSERT1_IN_GAA(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → INSERT1_IN_GAA(s(T312), T313, T314)
INSERT1_IN_GAA(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → P50_IN_AGAA(T323, T322, T324, T325)
P50_IN_AGAA(T135, T134, T140, T141) → U9_AGAA(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_AGAA(T135, T134, T140, T141, less33_out_ag(T135, T134)) → INSERT1_IN_GAA(s(T134), T140, T141)
P68_IN_AGAA(s(T229), s(T228), T230, T231) → P50_IN_AGAA(T229, T228, T230, T231)
P43_IN_AGAA(s(T135), s(T134), T136, T137) → P50_IN_AGAA(T135, T134, T136, T137)

The TRS R consists of the following rules:

insert1_in_gaa(T5, void, tree(T5, void, void)) → insert1_out_gaa(T5, void, tree(T5, void, void))
insert1_in_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11)) → insert1_out_gaa(T9, tree(T9, T10, T11), tree(T9, T10, T11))
insert1_in_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → U13_gaa(T29, T30, T20, T31, insert1_in_gaa(0, T30, T31))
insert1_in_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → U14_gaa(T40, T42, T43, T20, T44, p18_in_gaaa(T40, T42, T43, T44))
p18_in_gaaa(T40, T42, T43, T44) → U3_gaaa(T40, T42, T43, T44, less20_in_ga(T40, T42))
less20_in_ga(0, s(T55)) → less20_out_ga(0, s(T55))
less20_in_ga(s(T60), s(T62)) → U1_ga(T60, T62, less20_in_ga(T60, T62))
U1_ga(T60, T62, less20_out_ga(T60, T62)) → less20_out_ga(s(T60), s(T62))
U3_gaaa(T40, T42, T43, T44, less20_out_ga(T40, T42)) → p18_out_gaaa(T40, T42, T43, T44)
p18_in_gaaa(T40, T42, T47, T48) → U4_gaaa(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_gaaa(T40, T42, T47, T48, less20_out_ga(T40, T42)) → U5_gaaa(T40, T42, T47, T48, insert1_in_gaa(s(T40), T47, T48))
insert1_in_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84)) → U15_gaa(T77, T82, T79, T83, T84, less33_in_ag(T82, T77))
less33_in_ag(0, s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T102), s(T101)) → U2_ag(T102, T101, less33_in_ag(T102, T101))
U2_ag(T102, T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U15_gaa(T77, T82, T79, T83, T84, less33_out_ag(T82, T77)) → insert1_out_gaa(T77, tree(T82, T79, T83), tree(T82, T79, T84))
insert1_in_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_gaa(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_gaa(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → U17_gaa(T77, T82, T79, T87, T88, insert1_in_gaa(T77, T87, T88))
insert1_in_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → U18_gaa(T112, T117, T114, T118, T119, p43_in_agaa(T117, T112, T118, T119))
p43_in_agaa(0, s(T124), T125, T126) → U6_agaa(T124, T125, T126, insert1_in_gaa(s(T124), T125, T126))
insert1_in_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154)) → U19_gaa(T163, T164, T154, T165, insert1_in_gaa(0, T164, T165))
insert1_in_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154)) → U20_gaa(T174, T176, T177, T154, T178, p18_in_gaaa(T174, T176, T177, T178))
U20_gaa(T174, T176, T177, T154, T178, p18_out_gaaa(T174, T176, T177, T178)) → insert1_out_gaa(s(T174), tree(s(T176), T177, T154), tree(s(T176), T178, T154))
insert1_in_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198)) → U21_gaa(T191, T196, T193, T197, T198, p43_in_agaa(T196, T191, T197, T198))
p43_in_agaa(s(T135), s(T134), T136, T137) → U7_agaa(T135, T134, T136, T137, p50_in_agaa(T135, T134, T136, T137))
p50_in_agaa(T135, T134, T136, T137) → U8_agaa(T135, T134, T136, T137, less33_in_ag(T135, T134))
U8_agaa(T135, T134, T136, T137, less33_out_ag(T135, T134)) → p50_out_agaa(T135, T134, T136, T137)
p50_in_agaa(T135, T134, T140, T141) → U9_agaa(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_agaa(T135, T134, T140, T141, less33_out_ag(T135, T134)) → U10_agaa(T135, T134, T140, T141, insert1_in_gaa(s(T134), T140, T141))
insert1_in_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → U22_gaa(T206, T211, T208, T212, T213, p68_in_agaa(T211, T206, T212, T213))
p68_in_agaa(0, s(T218), T219, T220) → U11_agaa(T218, T219, T220, insert1_in_gaa(s(T218), T219, T220))
insert1_in_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248)) → U23_gaa(T257, T258, T248, T259, insert1_in_gaa(0, T258, T259))
insert1_in_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248)) → U24_gaa(T268, T270, T271, T248, T272, p18_in_gaaa(T268, T270, T271, T272))
U24_gaa(T268, T270, T271, T248, T272, p18_out_gaaa(T268, T270, T271, T272)) → insert1_out_gaa(s(T268), tree(s(T270), T271, T248), tree(s(T270), T272, T248))
insert1_in_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292)) → U25_gaa(T285, T290, T287, T291, T292, p68_in_agaa(T290, T285, T291, T292))
p68_in_agaa(s(T229), s(T228), T230, T231) → U12_agaa(T229, T228, T230, T231, p50_in_agaa(T229, T228, T230, T231))
U12_agaa(T229, T228, T230, T231, p50_out_agaa(T229, T228, T230, T231)) → p68_out_agaa(s(T229), s(T228), T230, T231)
U25_gaa(T285, T290, T287, T291, T292, p68_out_agaa(T290, T285, T291, T292)) → insert1_out_gaa(T285, tree(T290, T287, T291), tree(T290, T287, T292))
insert1_in_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → U26_gaa(T312, T302, T313, T314, insert1_in_gaa(s(T312), T313, T314))
insert1_in_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → U27_gaa(T322, T323, T302, T324, T325, p50_in_agaa(T323, T322, T324, T325))
U27_gaa(T322, T323, T302, T324, T325, p50_out_agaa(T323, T322, T324, T325)) → insert1_out_gaa(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325))
insert1_in_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336)) → U28_gaa(T345, T346, T336, T347, insert1_in_gaa(0, T346, T347))
insert1_in_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336)) → U29_gaa(T356, T358, T359, T336, T360, p18_in_gaaa(T356, T358, T359, T360))
U29_gaa(T356, T358, T359, T336, T360, p18_out_gaaa(T356, T358, T359, T360)) → insert1_out_gaa(s(T356), tree(s(T358), T359, T336), tree(s(T358), T360, T336))
insert1_in_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380)) → U30_gaa(T373, T378, T375, T379, T380, p68_in_agaa(T378, T373, T379, T380))
U30_gaa(T373, T378, T375, T379, T380, p68_out_agaa(T378, T373, T379, T380)) → insert1_out_gaa(T373, tree(T378, T375, T379), tree(T378, T375, T380))
insert1_in_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402)) → U31_gaa(T400, T390, T401, T402, insert1_in_gaa(s(T400), T401, T402))
insert1_in_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413)) → U32_gaa(T410, T411, T390, T412, T413, p50_in_agaa(T411, T410, T412, T413))
U32_gaa(T410, T411, T390, T412, T413, p50_out_agaa(T411, T410, T412, T413)) → insert1_out_gaa(s(T410), tree(s(T411), T390, T412), tree(s(T411), T390, T413))
U31_gaa(T400, T390, T401, T402, insert1_out_gaa(s(T400), T401, T402)) → insert1_out_gaa(s(T400), tree(0, T390, T401), tree(0, T390, T402))
U28_gaa(T345, T346, T336, T347, insert1_out_gaa(0, T346, T347)) → insert1_out_gaa(0, tree(s(T345), T346, T336), tree(s(T345), T347, T336))
U26_gaa(T312, T302, T313, T314, insert1_out_gaa(s(T312), T313, T314)) → insert1_out_gaa(s(T312), tree(0, T302, T313), tree(0, T302, T314))
U23_gaa(T257, T258, T248, T259, insert1_out_gaa(0, T258, T259)) → insert1_out_gaa(0, tree(s(T257), T258, T248), tree(s(T257), T259, T248))
U11_agaa(T218, T219, T220, insert1_out_gaa(s(T218), T219, T220)) → p68_out_agaa(0, s(T218), T219, T220)
U22_gaa(T206, T211, T208, T212, T213, p68_out_agaa(T211, T206, T212, T213)) → insert1_out_gaa(T206, tree(T211, T208, T212), tree(T211, T208, T213))
U10_agaa(T135, T134, T140, T141, insert1_out_gaa(s(T134), T140, T141)) → p50_out_agaa(T135, T134, T140, T141)
U7_agaa(T135, T134, T136, T137, p50_out_agaa(T135, T134, T136, T137)) → p43_out_agaa(s(T135), s(T134), T136, T137)
U21_gaa(T191, T196, T193, T197, T198, p43_out_agaa(T196, T191, T197, T198)) → insert1_out_gaa(T191, tree(T196, T193, T197), tree(T196, T193, T198))
U19_gaa(T163, T164, T154, T165, insert1_out_gaa(0, T164, T165)) → insert1_out_gaa(0, tree(s(T163), T164, T154), tree(s(T163), T165, T154))
U6_agaa(T124, T125, T126, insert1_out_gaa(s(T124), T125, T126)) → p43_out_agaa(0, s(T124), T125, T126)
U18_gaa(T112, T117, T114, T118, T119, p43_out_agaa(T117, T112, T118, T119)) → insert1_out_gaa(T112, tree(T117, T114, T118), tree(T117, T114, T119))
U17_gaa(T77, T82, T79, T87, T88, insert1_out_gaa(T77, T87, T88)) → insert1_out_gaa(T77, tree(T82, T79, T87), tree(T82, T79, T88))
U5_gaaa(T40, T42, T47, T48, insert1_out_gaa(s(T40), T47, T48)) → p18_out_gaaa(T40, T42, T47, T48)
U14_gaa(T40, T42, T43, T20, T44, p18_out_gaaa(T40, T42, T43, T44)) → insert1_out_gaa(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20))
U13_gaa(T29, T30, T20, T31, insert1_out_gaa(0, T30, T31)) → insert1_out_gaa(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20))

The argument filtering Pi contains the following mapping:
insert1_in_gaa(x1, x2, x3)  =  insert1_in_gaa(x1)
insert1_out_gaa(x1, x2, x3)  =  insert1_out_gaa(x1)
0  =  0
U13_gaa(x1, x2, x3, x4, x5)  =  U13_gaa(x5)
s(x1)  =  s(x1)
U14_gaa(x1, x2, x3, x4, x5, x6)  =  U14_gaa(x1, x6)
p18_in_gaaa(x1, x2, x3, x4)  =  p18_in_gaaa(x1)
U3_gaaa(x1, x2, x3, x4, x5)  =  U3_gaaa(x1, x5)
less20_in_ga(x1, x2)  =  less20_in_ga(x1)
less20_out_ga(x1, x2)  =  less20_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
p18_out_gaaa(x1, x2, x3, x4)  =  p18_out_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5)  =  U4_gaaa(x1, x5)
U5_gaaa(x1, x2, x3, x4, x5)  =  U5_gaaa(x1, x5)
U15_gaa(x1, x2, x3, x4, x5, x6)  =  U15_gaa(x1, x6)
less33_in_ag(x1, x2)  =  less33_in_ag(x2)
less33_out_ag(x1, x2)  =  less33_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
U16_gaa(x1, x2, x3, x4, x5, x6)  =  U16_gaa(x1, x6)
U17_gaa(x1, x2, x3, x4, x5, x6)  =  U17_gaa(x1, x6)
U18_gaa(x1, x2, x3, x4, x5, x6)  =  U18_gaa(x1, x6)
p43_in_agaa(x1, x2, x3, x4)  =  p43_in_agaa(x2)
U6_agaa(x1, x2, x3, x4)  =  U6_agaa(x1, x4)
U19_gaa(x1, x2, x3, x4, x5)  =  U19_gaa(x5)
U20_gaa(x1, x2, x3, x4, x5, x6)  =  U20_gaa(x1, x6)
U21_gaa(x1, x2, x3, x4, x5, x6)  =  U21_gaa(x1, x6)
U7_agaa(x1, x2, x3, x4, x5)  =  U7_agaa(x2, x5)
p50_in_agaa(x1, x2, x3, x4)  =  p50_in_agaa(x2)
U8_agaa(x1, x2, x3, x4, x5)  =  U8_agaa(x2, x5)
p50_out_agaa(x1, x2, x3, x4)  =  p50_out_agaa(x1, x2)
U9_agaa(x1, x2, x3, x4, x5)  =  U9_agaa(x2, x5)
U10_agaa(x1, x2, x3, x4, x5)  =  U10_agaa(x1, x2, x5)
U22_gaa(x1, x2, x3, x4, x5, x6)  =  U22_gaa(x1, x6)
p68_in_agaa(x1, x2, x3, x4)  =  p68_in_agaa(x2)
U11_agaa(x1, x2, x3, x4)  =  U11_agaa(x1, x4)
U23_gaa(x1, x2, x3, x4, x5)  =  U23_gaa(x5)
U24_gaa(x1, x2, x3, x4, x5, x6)  =  U24_gaa(x1, x6)
U25_gaa(x1, x2, x3, x4, x5, x6)  =  U25_gaa(x1, x6)
U12_agaa(x1, x2, x3, x4, x5)  =  U12_agaa(x2, x5)
p68_out_agaa(x1, x2, x3, x4)  =  p68_out_agaa(x1, x2)
U26_gaa(x1, x2, x3, x4, x5)  =  U26_gaa(x1, x5)
U27_gaa(x1, x2, x3, x4, x5, x6)  =  U27_gaa(x1, x6)
U28_gaa(x1, x2, x3, x4, x5)  =  U28_gaa(x5)
U29_gaa(x1, x2, x3, x4, x5, x6)  =  U29_gaa(x1, x6)
U30_gaa(x1, x2, x3, x4, x5, x6)  =  U30_gaa(x1, x6)
U31_gaa(x1, x2, x3, x4, x5)  =  U31_gaa(x1, x5)
U32_gaa(x1, x2, x3, x4, x5, x6)  =  U32_gaa(x1, x6)
p43_out_agaa(x1, x2, x3, x4)  =  p43_out_agaa(x1, x2)
INSERT1_IN_GAA(x1, x2, x3)  =  INSERT1_IN_GAA(x1)
P18_IN_GAAA(x1, x2, x3, x4)  =  P18_IN_GAAA(x1)
U4_GAAA(x1, x2, x3, x4, x5)  =  U4_GAAA(x1, x5)
U16_GAA(x1, x2, x3, x4, x5, x6)  =  U16_GAA(x1, x6)
P43_IN_AGAA(x1, x2, x3, x4)  =  P43_IN_AGAA(x2)
P50_IN_AGAA(x1, x2, x3, x4)  =  P50_IN_AGAA(x2)
U9_AGAA(x1, x2, x3, x4, x5)  =  U9_AGAA(x2, x5)
P68_IN_AGAA(x1, x2, x3, x4)  =  P68_IN_AGAA(x2)

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

(85) UsableRulesProof (EQUIVALENT transformation)

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

(86) Obligation:

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

INSERT1_IN_GAA(T77, tree(T82, T79, T87), tree(T82, T79, T88)) → U16_GAA(T77, T82, T79, T87, T88, less33_in_ag(T82, T77))
U16_GAA(T77, T82, T79, T87, T88, less33_out_ag(T82, T77)) → INSERT1_IN_GAA(T77, T87, T88)
INSERT1_IN_GAA(0, tree(s(T29), T30, T20), tree(s(T29), T31, T20)) → INSERT1_IN_GAA(0, T30, T31)
INSERT1_IN_GAA(T112, tree(T117, T114, T118), tree(T117, T114, T119)) → P43_IN_AGAA(T117, T112, T118, T119)
P43_IN_AGAA(0, s(T124), T125, T126) → INSERT1_IN_GAA(s(T124), T125, T126)
INSERT1_IN_GAA(s(T40), tree(s(T42), T43, T20), tree(s(T42), T44, T20)) → P18_IN_GAAA(T40, T42, T43, T44)
P18_IN_GAAA(T40, T42, T47, T48) → U4_GAAA(T40, T42, T47, T48, less20_in_ga(T40, T42))
U4_GAAA(T40, T42, T47, T48, less20_out_ga(T40, T42)) → INSERT1_IN_GAA(s(T40), T47, T48)
INSERT1_IN_GAA(T206, tree(T211, T208, T212), tree(T211, T208, T213)) → P68_IN_AGAA(T211, T206, T212, T213)
P68_IN_AGAA(0, s(T218), T219, T220) → INSERT1_IN_GAA(s(T218), T219, T220)
INSERT1_IN_GAA(s(T312), tree(0, T302, T313), tree(0, T302, T314)) → INSERT1_IN_GAA(s(T312), T313, T314)
INSERT1_IN_GAA(s(T322), tree(s(T323), T302, T324), tree(s(T323), T302, T325)) → P50_IN_AGAA(T323, T322, T324, T325)
P50_IN_AGAA(T135, T134, T140, T141) → U9_AGAA(T135, T134, T140, T141, less33_in_ag(T135, T134))
U9_AGAA(T135, T134, T140, T141, less33_out_ag(T135, T134)) → INSERT1_IN_GAA(s(T134), T140, T141)
P68_IN_AGAA(s(T229), s(T228), T230, T231) → P50_IN_AGAA(T229, T228, T230, T231)
P43_IN_AGAA(s(T135), s(T134), T136, T137) → P50_IN_AGAA(T135, T134, T136, T137)

The TRS R consists of the following rules:

less33_in_ag(0, s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T102), s(T101)) → U2_ag(T102, T101, less33_in_ag(T102, T101))
less20_in_ga(0, s(T55)) → less20_out_ga(0, s(T55))
less20_in_ga(s(T60), s(T62)) → U1_ga(T60, T62, less20_in_ga(T60, T62))
U2_ag(T102, T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U1_ga(T60, T62, less20_out_ga(T60, T62)) → less20_out_ga(s(T60), s(T62))

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
less20_in_ga(x1, x2)  =  less20_in_ga(x1)
less20_out_ga(x1, x2)  =  less20_out_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
less33_in_ag(x1, x2)  =  less33_in_ag(x2)
less33_out_ag(x1, x2)  =  less33_out_ag(x1, x2)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
INSERT1_IN_GAA(x1, x2, x3)  =  INSERT1_IN_GAA(x1)
P18_IN_GAAA(x1, x2, x3, x4)  =  P18_IN_GAAA(x1)
U4_GAAA(x1, x2, x3, x4, x5)  =  U4_GAAA(x1, x5)
U16_GAA(x1, x2, x3, x4, x5, x6)  =  U16_GAA(x1, x6)
P43_IN_AGAA(x1, x2, x3, x4)  =  P43_IN_AGAA(x2)
P50_IN_AGAA(x1, x2, x3, x4)  =  P50_IN_AGAA(x2)
U9_AGAA(x1, x2, x3, x4, x5)  =  U9_AGAA(x2, x5)
P68_IN_AGAA(x1, x2, x3, x4)  =  P68_IN_AGAA(x2)

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

(87) PiDPToQDPProof (SOUND transformation)

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

(88) Obligation:

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

INSERT1_IN_GAA(T77) → U16_GAA(T77, less33_in_ag(T77))
U16_GAA(T77, less33_out_ag(T82, T77)) → INSERT1_IN_GAA(T77)
INSERT1_IN_GAA(0) → INSERT1_IN_GAA(0)
INSERT1_IN_GAA(T112) → P43_IN_AGAA(T112)
P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
P18_IN_GAAA(T40) → U4_GAAA(T40, less20_in_ga(T40))
U4_GAAA(T40, less20_out_ga(T40)) → INSERT1_IN_GAA(s(T40))
INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(T134) → U9_AGAA(T134, less33_in_ag(T134))
U9_AGAA(T134, less33_out_ag(T135, T134)) → INSERT1_IN_GAA(s(T134))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T101)) → U2_ag(T101, less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga(0)
less20_in_ga(s(T60)) → U1_ga(T60, less20_in_ga(T60))
U2_ag(T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U1_ga(T60, less20_out_ga(T60)) → less20_out_ga(s(T60))

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0, x1)
U1_ga(x0, x1)

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

(89) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule INSERT1_IN_GAA(T77) → U16_GAA(T77, less33_in_ag(T77)) at position [1] we obtained the following new rules [LPAR04]:

INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0, s(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(x0, less33_in_ag(x0)))

(90) Obligation:

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

U16_GAA(T77, less33_out_ag(T82, T77)) → INSERT1_IN_GAA(T77)
INSERT1_IN_GAA(0) → INSERT1_IN_GAA(0)
INSERT1_IN_GAA(T112) → P43_IN_AGAA(T112)
P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
P18_IN_GAAA(T40) → U4_GAAA(T40, less20_in_ga(T40))
U4_GAAA(T40, less20_out_ga(T40)) → INSERT1_IN_GAA(s(T40))
INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(T134) → U9_AGAA(T134, less33_in_ag(T134))
U9_AGAA(T134, less33_out_ag(T135, T134)) → INSERT1_IN_GAA(s(T134))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0, s(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(x0, less33_in_ag(x0)))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T101)) → U2_ag(T101, less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga(0)
less20_in_ga(s(T60)) → U1_ga(T60, less20_in_ga(T60))
U2_ag(T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U1_ga(T60, less20_out_ga(T60)) → less20_out_ga(s(T60))

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0, x1)
U1_ga(x0, x1)

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

(91) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P18_IN_GAAA(T40) → U4_GAAA(T40, less20_in_ga(T40)) at position [1] we obtained the following new rules [LPAR04]:

P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga(0))
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(x0, less20_in_ga(x0)))

(92) Obligation:

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

U16_GAA(T77, less33_out_ag(T82, T77)) → INSERT1_IN_GAA(T77)
INSERT1_IN_GAA(0) → INSERT1_IN_GAA(0)
INSERT1_IN_GAA(T112) → P43_IN_AGAA(T112)
P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
U4_GAAA(T40, less20_out_ga(T40)) → INSERT1_IN_GAA(s(T40))
INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(T134) → U9_AGAA(T134, less33_in_ag(T134))
U9_AGAA(T134, less33_out_ag(T135, T134)) → INSERT1_IN_GAA(s(T134))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0, s(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga(0))
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(x0, less20_in_ga(x0)))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T101)) → U2_ag(T101, less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga(0)
less20_in_ga(s(T60)) → U1_ga(T60, less20_in_ga(T60))
U2_ag(T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U1_ga(T60, less20_out_ga(T60)) → less20_out_ga(s(T60))

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0, x1)
U1_ga(x0, x1)

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

(93) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P50_IN_AGAA(T134) → U9_AGAA(T134, less33_in_ag(T134)) at position [1] we obtained the following new rules [LPAR04]:

P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0, s(x0)))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(x0, less33_in_ag(x0)))

(94) Obligation:

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

U16_GAA(T77, less33_out_ag(T82, T77)) → INSERT1_IN_GAA(T77)
INSERT1_IN_GAA(0) → INSERT1_IN_GAA(0)
INSERT1_IN_GAA(T112) → P43_IN_AGAA(T112)
P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
U4_GAAA(T40, less20_out_ga(T40)) → INSERT1_IN_GAA(s(T40))
INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
U9_AGAA(T134, less33_out_ag(T135, T134)) → INSERT1_IN_GAA(s(T134))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0, s(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga(0))
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(x0, less20_in_ga(x0)))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0, s(x0)))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(x0, less33_in_ag(x0)))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T101)) → U2_ag(T101, less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga(0)
less20_in_ga(s(T60)) → U1_ga(T60, less20_in_ga(T60))
U2_ag(T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U1_ga(T60, less20_out_ga(T60)) → less20_out_ga(s(T60))

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0, x1)
U1_ga(x0, x1)

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

(95) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U16_GAA(T77, less33_out_ag(T82, T77)) → INSERT1_IN_GAA(T77) we obtained the following new rules [LPAR04]:

U16_GAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(z0))
U16_GAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(z0))

(96) Obligation:

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

INSERT1_IN_GAA(0) → INSERT1_IN_GAA(0)
INSERT1_IN_GAA(T112) → P43_IN_AGAA(T112)
P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
U4_GAAA(T40, less20_out_ga(T40)) → INSERT1_IN_GAA(s(T40))
INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
U9_AGAA(T134, less33_out_ag(T135, T134)) → INSERT1_IN_GAA(s(T134))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0, s(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga(0))
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(x0, less20_in_ga(x0)))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0, s(x0)))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(z0))
U16_GAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(z0))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T101)) → U2_ag(T101, less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga(0)
less20_in_ga(s(T60)) → U1_ga(T60, less20_in_ga(T60))
U2_ag(T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U1_ga(T60, less20_out_ga(T60)) → less20_out_ga(s(T60))

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0, x1)
U1_ga(x0, x1)

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

(97) DependencyGraphProof (EQUIVALENT transformation)

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

(98) Complex Obligation (AND)

(99) Obligation:

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

P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
INSERT1_IN_GAA(T112) → P43_IN_AGAA(T112)
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0, s(x0)))
U9_AGAA(T134, less33_out_ag(T135, T134)) → INSERT1_IN_GAA(s(T134))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga(0))
U4_GAAA(T40, less20_out_ga(T40)) → INSERT1_IN_GAA(s(T40))
INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0, s(x0)))
U16_GAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(z0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(z0))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(x0, less20_in_ga(x0)))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T101)) → U2_ag(T101, less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga(0)
less20_in_ga(s(T60)) → U1_ga(T60, less20_in_ga(T60))
U2_ag(T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U1_ga(T60, less20_out_ga(T60)) → less20_out_ga(s(T60))

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0, x1)
U1_ga(x0, x1)

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

(100) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule INSERT1_IN_GAA(T112) → P43_IN_AGAA(T112) we obtained the following new rules [LPAR04]:

INSERT1_IN_GAA(s(z0)) → P43_IN_AGAA(s(z0))

(101) Obligation:

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

P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0, s(x0)))
U9_AGAA(T134, less33_out_ag(T135, T134)) → INSERT1_IN_GAA(s(T134))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga(0))
U4_GAAA(T40, less20_out_ga(T40)) → INSERT1_IN_GAA(s(T40))
INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0, s(x0)))
U16_GAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(z0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(z0))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(x0, less20_in_ga(x0)))
INSERT1_IN_GAA(s(z0)) → P43_IN_AGAA(s(z0))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T101)) → U2_ag(T101, less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga(0)
less20_in_ga(s(T60)) → U1_ga(T60, less20_in_ga(T60))
U2_ag(T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U1_ga(T60, less20_out_ga(T60)) → less20_out_ga(s(T60))

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0, x1)
U1_ga(x0, x1)

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

(102) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U9_AGAA(T134, less33_out_ag(T135, T134)) → INSERT1_IN_GAA(s(T134)) we obtained the following new rules [LPAR04]:

U9_AGAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(s(z0)))
U9_AGAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(s(z0)))

(103) Obligation:

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

P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0, s(x0)))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga(0))
U4_GAAA(T40, less20_out_ga(T40)) → INSERT1_IN_GAA(s(T40))
INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0, s(x0)))
U16_GAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(z0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(z0))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(x0, less20_in_ga(x0)))
INSERT1_IN_GAA(s(z0)) → P43_IN_AGAA(s(z0))
U9_AGAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(s(z0)))
U9_AGAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(s(z0)))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T101)) → U2_ag(T101, less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga(0)
less20_in_ga(s(T60)) → U1_ga(T60, less20_in_ga(T60))
U2_ag(T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U1_ga(T60, less20_out_ga(T60)) → less20_out_ga(s(T60))

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0, x1)
U1_ga(x0, x1)

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

(104) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U4_GAAA(T40, less20_out_ga(T40)) → INSERT1_IN_GAA(s(T40)) we obtained the following new rules [LPAR04]:

U4_GAAA(0, less20_out_ga(0)) → INSERT1_IN_GAA(s(0))
U4_GAAA(s(z0), less20_out_ga(s(z0))) → INSERT1_IN_GAA(s(s(z0)))

(105) Obligation:

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

P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0, s(x0)))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga(0))
INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206)
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0, s(x0)))
U16_GAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(z0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(z0))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(x0, less20_in_ga(x0)))
INSERT1_IN_GAA(s(z0)) → P43_IN_AGAA(s(z0))
U9_AGAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(s(z0)))
U9_AGAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(s(z0)))
U4_GAAA(0, less20_out_ga(0)) → INSERT1_IN_GAA(s(0))
U4_GAAA(s(z0), less20_out_ga(s(z0))) → INSERT1_IN_GAA(s(s(z0)))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T101)) → U2_ag(T101, less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga(0)
less20_in_ga(s(T60)) → U1_ga(T60, less20_in_ga(T60))
U2_ag(T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U1_ga(T60, less20_out_ga(T60)) → less20_out_ga(s(T60))

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0, x1)
U1_ga(x0, x1)

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

(106) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule INSERT1_IN_GAA(T206) → P68_IN_AGAA(T206) we obtained the following new rules [LPAR04]:

INSERT1_IN_GAA(s(z0)) → P68_IN_AGAA(s(z0))
INSERT1_IN_GAA(s(s(z0))) → P68_IN_AGAA(s(s(z0)))
INSERT1_IN_GAA(s(0)) → P68_IN_AGAA(s(0))

(107) Obligation:

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

P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0, s(x0)))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga(0))
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0, s(x0)))
U16_GAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(z0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(z0))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(x0, less20_in_ga(x0)))
INSERT1_IN_GAA(s(z0)) → P43_IN_AGAA(s(z0))
U9_AGAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(s(z0)))
U9_AGAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(s(z0)))
U4_GAAA(0, less20_out_ga(0)) → INSERT1_IN_GAA(s(0))
U4_GAAA(s(z0), less20_out_ga(s(z0))) → INSERT1_IN_GAA(s(s(z0)))
INSERT1_IN_GAA(s(z0)) → P68_IN_AGAA(s(z0))
INSERT1_IN_GAA(s(s(z0))) → P68_IN_AGAA(s(s(z0)))
INSERT1_IN_GAA(s(0)) → P68_IN_AGAA(s(0))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T101)) → U2_ag(T101, less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga(0)
less20_in_ga(s(T60)) → U1_ga(T60, less20_in_ga(T60))
U2_ag(T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U1_ga(T60, less20_out_ga(T60)) → less20_out_ga(s(T60))

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0, x1)
U1_ga(x0, x1)

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

(108) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule P43_IN_AGAA(s(T134)) → P50_IN_AGAA(T134) we obtained the following new rules [LPAR04]:

P43_IN_AGAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))

(109) Obligation:

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

P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0, s(x0)))
INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40)
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga(0))
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0, s(x0)))
U16_GAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(z0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(z0))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(x0, less20_in_ga(x0)))
INSERT1_IN_GAA(s(z0)) → P43_IN_AGAA(s(z0))
U9_AGAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(s(z0)))
U9_AGAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(s(z0)))
U4_GAAA(0, less20_out_ga(0)) → INSERT1_IN_GAA(s(0))
U4_GAAA(s(z0), less20_out_ga(s(z0))) → INSERT1_IN_GAA(s(s(z0)))
INSERT1_IN_GAA(s(z0)) → P68_IN_AGAA(s(z0))
INSERT1_IN_GAA(s(s(z0))) → P68_IN_AGAA(s(s(z0)))
INSERT1_IN_GAA(s(0)) → P68_IN_AGAA(s(0))
P43_IN_AGAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T101)) → U2_ag(T101, less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga(0)
less20_in_ga(s(T60)) → U1_ga(T60, less20_in_ga(T60))
U2_ag(T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U1_ga(T60, less20_out_ga(T60)) → less20_out_ga(s(T60))

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0, x1)
U1_ga(x0, x1)

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

(110) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule INSERT1_IN_GAA(s(T40)) → P18_IN_GAAA(T40) we obtained the following new rules [LPAR04]:

INSERT1_IN_GAA(s(0)) → P18_IN_GAAA(0)
INSERT1_IN_GAA(s(s(y_0))) → P18_IN_GAAA(s(y_0))

(111) Obligation:

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

P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0, s(x0)))
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga(0))
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322)
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0, s(x0)))
U16_GAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(z0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(z0))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(x0, less20_in_ga(x0)))
INSERT1_IN_GAA(s(z0)) → P43_IN_AGAA(s(z0))
U9_AGAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(s(z0)))
U9_AGAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(s(z0)))
U4_GAAA(0, less20_out_ga(0)) → INSERT1_IN_GAA(s(0))
U4_GAAA(s(z0), less20_out_ga(s(z0))) → INSERT1_IN_GAA(s(s(z0)))
INSERT1_IN_GAA(s(z0)) → P68_IN_AGAA(s(z0))
INSERT1_IN_GAA(s(s(z0))) → P68_IN_AGAA(s(s(z0)))
INSERT1_IN_GAA(s(0)) → P68_IN_AGAA(s(0))
P43_IN_AGAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))
INSERT1_IN_GAA(s(0)) → P18_IN_GAAA(0)
INSERT1_IN_GAA(s(s(y_0))) → P18_IN_GAAA(s(y_0))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T101)) → U2_ag(T101, less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga(0)
less20_in_ga(s(T60)) → U1_ga(T60, less20_in_ga(T60))
U2_ag(T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U1_ga(T60, less20_out_ga(T60)) → less20_out_ga(s(T60))

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0, x1)
U1_ga(x0, x1)

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

(112) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule INSERT1_IN_GAA(s(T322)) → P50_IN_AGAA(T322) we obtained the following new rules [LPAR04]:

INSERT1_IN_GAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))

(113) Obligation:

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

P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0, s(x0)))
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga(0))
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0, s(x0)))
U16_GAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(z0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(z0))
P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228)
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(x0, less20_in_ga(x0)))
INSERT1_IN_GAA(s(z0)) → P43_IN_AGAA(s(z0))
U9_AGAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(s(z0)))
U9_AGAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(s(z0)))
U4_GAAA(0, less20_out_ga(0)) → INSERT1_IN_GAA(s(0))
U4_GAAA(s(z0), less20_out_ga(s(z0))) → INSERT1_IN_GAA(s(s(z0)))
INSERT1_IN_GAA(s(z0)) → P68_IN_AGAA(s(z0))
INSERT1_IN_GAA(s(s(z0))) → P68_IN_AGAA(s(s(z0)))
INSERT1_IN_GAA(s(0)) → P68_IN_AGAA(s(0))
P43_IN_AGAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))
INSERT1_IN_GAA(s(0)) → P18_IN_GAAA(0)
INSERT1_IN_GAA(s(s(y_0))) → P18_IN_GAAA(s(y_0))
INSERT1_IN_GAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T101)) → U2_ag(T101, less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga(0)
less20_in_ga(s(T60)) → U1_ga(T60, less20_in_ga(T60))
U2_ag(T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U1_ga(T60, less20_out_ga(T60)) → less20_out_ga(s(T60))

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0, x1)
U1_ga(x0, x1)

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

(114) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule P68_IN_AGAA(s(T228)) → P50_IN_AGAA(T228) we obtained the following new rules [LPAR04]:

P68_IN_AGAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))

(115) Obligation:

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

P43_IN_AGAA(s(T124)) → INSERT1_IN_GAA(s(T124))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), less33_out_ag(0, s(x0)))
P18_IN_GAAA(0) → U4_GAAA(0, less20_out_ga(0))
P68_IN_AGAA(s(T218)) → INSERT1_IN_GAA(s(T218))
INSERT1_IN_GAA(s(T312)) → INSERT1_IN_GAA(s(T312))
P50_IN_AGAA(s(x0)) → U9_AGAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), less33_out_ag(0, s(x0)))
U16_GAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(z0))
INSERT1_IN_GAA(s(x0)) → U16_GAA(s(x0), U2_ag(x0, less33_in_ag(x0)))
U16_GAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(z0))
P18_IN_GAAA(s(x0)) → U4_GAAA(s(x0), U1_ga(x0, less20_in_ga(x0)))
INSERT1_IN_GAA(s(z0)) → P43_IN_AGAA(s(z0))
U9_AGAA(s(z0), less33_out_ag(0, s(z0))) → INSERT1_IN_GAA(s(s(z0)))
U9_AGAA(s(z0), less33_out_ag(x1, s(z0))) → INSERT1_IN_GAA(s(s(z0)))
U4_GAAA(0, less20_out_ga(0)) → INSERT1_IN_GAA(s(0))
U4_GAAA(s(z0), less20_out_ga(s(z0))) → INSERT1_IN_GAA(s(s(z0)))
INSERT1_IN_GAA(s(z0)) → P68_IN_AGAA(s(z0))
INSERT1_IN_GAA(s(s(z0))) → P68_IN_AGAA(s(s(z0)))
INSERT1_IN_GAA(s(0)) → P68_IN_AGAA(s(0))
P43_IN_AGAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))
INSERT1_IN_GAA(s(0)) → P18_IN_GAAA(0)
INSERT1_IN_GAA(s(s(y_0))) → P18_IN_GAAA(s(y_0))
INSERT1_IN_GAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))
P68_IN_AGAA(s(s(y_0))) → P50_IN_AGAA(s(y_0))

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T101)) → U2_ag(T101, less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga(0)
less20_in_ga(s(T60)) → U1_ga(T60, less20_in_ga(T60))
U2_ag(T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U1_ga(T60, less20_out_ga(T60)) → less20_out_ga(s(T60))

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0, x1)
U1_ga(x0, x1)

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

(116) 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 = INSERT1_IN_GAA(s(T312)) evaluates to t =INSERT1_IN_GAA(s(T312))

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 INSERT1_IN_GAA(s(T312)) to INSERT1_IN_GAA(s(T312)).



(117) NO

(118) Obligation:

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

INSERT1_IN_GAA(0) → INSERT1_IN_GAA(0)

The TRS R consists of the following rules:

less33_in_ag(s(T95)) → less33_out_ag(0, s(T95))
less33_in_ag(s(T101)) → U2_ag(T101, less33_in_ag(T101))
less20_in_ga(0) → less20_out_ga(0)
less20_in_ga(s(T60)) → U1_ga(T60, less20_in_ga(T60))
U2_ag(T101, less33_out_ag(T102, T101)) → less33_out_ag(s(T102), s(T101))
U1_ga(T60, less20_out_ga(T60)) → less20_out_ga(s(T60))

The set Q consists of the following terms:

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0, x1)
U1_ga(x0, x1)

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

(119) 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.

(120) Obligation:

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

INSERT1_IN_GAA(0) → INSERT1_IN_GAA(0)

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

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0, x1)
U1_ga(x0, x1)

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

(121) 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].

less33_in_ag(x0)
less20_in_ga(x0)
U2_ag(x0, x1)
U1_ga(x0, x1)

(122) Obligation:

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

INSERT1_IN_GAA(0) → INSERT1_IN_GAA(0)

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

(123) NonTerminationProof (EQUIVALENT transformation)

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

s = INSERT1_IN_GAA(0) evaluates to t =INSERT1_IN_GAA(0)

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 INSERT1_IN_GAA(0) to INSERT1_IN_GAA(0).



(124) NO