(0) Obligation:

Clauses:

minsort([], []).
minsort(L, .(X, L1)) :- ','(min1(X, L), ','(remove(X, L, L2), minsort(L2, L1))).
min1(M, .(X, L)) :- min2(X, M, L).
min2(X, X, []).
min2(X, A, .(M, L)) :- ','(min(X, M, B), min2(B, A, L)).
min(X, Y, X) :- le(X, Y).
min(X, Y, Y) :- gt(X, Y).
remove(N, .(N, L), L).
remove(N, .(M, L), .(M, L1)) :- ','(notEq(N, M), remove(N, L, L1)).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).
notEq(s(X), s(Y)) :- notEq(X, Y).
notEq(s(X), 0).
notEq(0, s(X)).

Queries:

minsort(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

min215(T39, T39, []).
min215(T48, T52, .(T50, T51)) :- min25(T48, T50, X59).
min215(T48, T52, .(T50, T51)) :- ','(min25(T48, T50, T55), min215(T55, T52, T51)).
le30(s(T82), s(T83)) :- le30(T82, T83).
le30(0, s(T90)).
le30(0, 0).
gt44(s(T109), s(T110)) :- gt44(T109, T110).
gt44(s(T115), 0).
notEq62(s(T160), s(T161)) :- notEq62(T160, T161).
notEq62(s(T168), 0).
notEq62(0, s(T171)).
p61(T145, T146, T147, X170) :- notEq62(T145, T146).
p61(T184, T146, .(T184, T185), T185) :- notEq62(T184, T146).
p61(T192, T146, .(T193, T194), .(T193, X229)) :- ','(notEq62(T192, T146), p61(T192, T193, T194, X229)).
remove53(T135, T135, T136, T136).
remove53(T145, T146, T147, .(T146, X170)) :- p61(T145, T146, T147, X170).
min25(T70, T71, T70) :- le30(T70, T71).
min25(T97, T98, T98) :- gt44(T97, T98).
minsort1([], []).
minsort1(.(T25, T26), .(T27, T28)) :- min215(T25, T27, T26).
minsort1(.(T25, T26), .(T31, T32)) :- ','(min215(T25, T31, T26), remove53(T31, T25, T26, X20)).
minsort1(.(T25, T26), .(T31, T32)) :- ','(min215(T25, T31, T26), ','(remove53(T31, T25, T26, T122), minsort1(T122, T32))).

Queries:

minsort1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
minsort1_in: (b,f) (f,f)
min215_in: (b,f,b) (f,f,f) (b,f,f)
min25_in: (b,b,f) (f,f,f) (b,f,f)
le30_in: (b,b) (f,f) (b,f)
gt44_in: (b,b) (f,f) (b,f)
remove53_in: (f,b,b,f) (f,f,f,f)
p61_in: (f,b,b,f) (f,f,f,f)
notEq62_in: (f,b) (b,b) (f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → U14_GA(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → MIN215_IN_GAG(T25, T27, T26)
MIN215_IN_GAG(T48, T52, .(T50, T51)) → U1_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
MIN215_IN_GAG(T48, T52, .(T50, T51)) → MIN25_IN_GGA(T48, T50, X59)
MIN25_IN_GGA(T70, T71, T70) → U12_GGA(T70, T71, le30_in_gg(T70, T71))
MIN25_IN_GGA(T70, T71, T70) → LE30_IN_GG(T70, T71)
LE30_IN_GG(s(T82), s(T83)) → U4_GG(T82, T83, le30_in_gg(T82, T83))
LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)
MIN25_IN_GGA(T97, T98, T98) → U13_GGA(T97, T98, gt44_in_gg(T97, T98))
MIN25_IN_GGA(T97, T98, T98) → GT44_IN_GG(T97, T98)
GT44_IN_GG(s(T109), s(T110)) → U5_GG(T109, T110, gt44_in_gg(T109, T110))
GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)
MIN215_IN_GAG(T48, T52, .(T50, T51)) → U2_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_GAG(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)
MINSORT1_IN_GA(.(T25, T26), .(T31, T32)) → U15_GA(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_GA(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → REMOVE53_IN_AGGA(T31, T25, T26, X20)
REMOVE53_IN_AGGA(T145, T146, T147, .(T146, X170)) → U11_AGGA(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
REMOVE53_IN_AGGA(T145, T146, T147, .(T146, X170)) → P61_IN_AGGA(T145, T146, T147, X170)
P61_IN_AGGA(T145, T146, T147, X170) → U7_AGGA(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
P61_IN_AGGA(T145, T146, T147, X170) → NOTEQ62_IN_AG(T145, T146)
NOTEQ62_IN_AG(s(T160), s(T161)) → U6_AG(T160, T161, notEq62_in_ag(T160, T161))
NOTEQ62_IN_AG(s(T160), s(T161)) → NOTEQ62_IN_AG(T160, T161)
P61_IN_AGGA(T184, T146, .(T184, T185), T185) → U8_AGGA(T184, T146, T185, notEq62_in_gg(T184, T146))
P61_IN_AGGA(T184, T146, .(T184, T185), T185) → NOTEQ62_IN_GG(T184, T146)
NOTEQ62_IN_GG(s(T160), s(T161)) → U6_GG(T160, T161, notEq62_in_gg(T160, T161))
NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)
P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AGGA(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → NOTEQ62_IN_AG(T192, T146)
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_AGGA(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → P61_IN_AGGA(T192, T193, T194, X229)
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_GA(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_GA(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_GA(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U17_GA(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)
MINSORT1_IN_AA(.(T25, T26), .(T27, T28)) → U14_AA(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
MINSORT1_IN_AA(.(T25, T26), .(T27, T28)) → MIN215_IN_AAA(T25, T27, T26)
MIN215_IN_AAA(T48, T52, .(T50, T51)) → U1_AAA(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
MIN215_IN_AAA(T48, T52, .(T50, T51)) → MIN25_IN_AAA(T48, T50, X59)
MIN25_IN_AAA(T70, T71, T70) → U12_AAA(T70, T71, le30_in_aa(T70, T71))
MIN25_IN_AAA(T70, T71, T70) → LE30_IN_AA(T70, T71)
LE30_IN_AA(s(T82), s(T83)) → U4_AA(T82, T83, le30_in_aa(T82, T83))
LE30_IN_AA(s(T82), s(T83)) → LE30_IN_AA(T82, T83)
MIN25_IN_AAA(T97, T98, T98) → U13_AAA(T97, T98, gt44_in_aa(T97, T98))
MIN25_IN_AAA(T97, T98, T98) → GT44_IN_AA(T97, T98)
GT44_IN_AA(s(T109), s(T110)) → U5_AA(T109, T110, gt44_in_aa(T109, T110))
GT44_IN_AA(s(T109), s(T110)) → GT44_IN_AA(T109, T110)
MIN215_IN_AAA(T48, T52, .(T50, T51)) → U2_AAA(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_AAA(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_AAA(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U2_AAA(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)
MIN215_IN_GAA(T48, T52, .(T50, T51)) → U1_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
MIN215_IN_GAA(T48, T52, .(T50, T51)) → MIN25_IN_GAA(T48, T50, X59)
MIN25_IN_GAA(T70, T71, T70) → U12_GAA(T70, T71, le30_in_ga(T70, T71))
MIN25_IN_GAA(T70, T71, T70) → LE30_IN_GA(T70, T71)
LE30_IN_GA(s(T82), s(T83)) → U4_GA(T82, T83, le30_in_ga(T82, T83))
LE30_IN_GA(s(T82), s(T83)) → LE30_IN_GA(T82, T83)
MIN25_IN_GAA(T97, T98, T98) → U13_GAA(T97, T98, gt44_in_ga(T97, T98))
MIN25_IN_GAA(T97, T98, T98) → GT44_IN_GA(T97, T98)
GT44_IN_GA(s(T109), s(T110)) → U5_GA(T109, T110, gt44_in_ga(T109, T110))
GT44_IN_GA(s(T109), s(T110)) → GT44_IN_GA(T109, T110)
MIN215_IN_GAA(T48, T52, .(T50, T51)) → U2_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_GAA(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)
MINSORT1_IN_AA(.(T25, T26), .(T31, T32)) → U15_AA(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → REMOVE53_IN_AAAA(T31, T25, T26, X20)
REMOVE53_IN_AAAA(T145, T146, T147, .(T146, X170)) → U11_AAAA(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
REMOVE53_IN_AAAA(T145, T146, T147, .(T146, X170)) → P61_IN_AAAA(T145, T146, T147, X170)
P61_IN_AAAA(T145, T146, T147, X170) → U7_AAAA(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
P61_IN_AAAA(T145, T146, T147, X170) → NOTEQ62_IN_AA(T145, T146)
NOTEQ62_IN_AA(s(T160), s(T161)) → U6_AA(T160, T161, notEq62_in_aa(T160, T161))
NOTEQ62_IN_AA(s(T160), s(T161)) → NOTEQ62_IN_AA(T160, T161)
P61_IN_AAAA(T184, T146, .(T184, T185), T185) → U8_AAAA(T184, T146, T185, notEq62_in_aa(T184, T146))
P61_IN_AAAA(T184, T146, .(T184, T185), T185) → NOTEQ62_IN_AA(T184, T146)
P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AAAA(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → NOTEQ62_IN_AA(T192, T146)
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_AAAA(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → P61_IN_AAAA(T192, T193, T194, X229)
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_AA(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MINSORT1_IN_GA(x1, x2)  =  MINSORT1_IN_GA(x1)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x1, x2, x5)
MIN215_IN_GAG(x1, x2, x3)  =  MIN215_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x1, x3, x4, x5)
MIN25_IN_GGA(x1, x2, x3)  =  MIN25_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3)  =  U12_GGA(x1, x2, x3)
LE30_IN_GG(x1, x2)  =  LE30_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)
U13_GGA(x1, x2, x3)  =  U13_GGA(x1, x2, x3)
GT44_IN_GG(x1, x2)  =  GT44_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x1, x3, x4, x5)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x1, x3, x4, x5)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x2, x5)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x1, x2, x5)
REMOVE53_IN_AGGA(x1, x2, x3, x4)  =  REMOVE53_IN_AGGA(x2, x3)
U11_AGGA(x1, x2, x3, x4, x5)  =  U11_AGGA(x2, x3, x5)
P61_IN_AGGA(x1, x2, x3, x4)  =  P61_IN_AGGA(x2, x3)
U7_AGGA(x1, x2, x3, x4, x5)  =  U7_AGGA(x2, x3, x5)
NOTEQ62_IN_AG(x1, x2)  =  NOTEQ62_IN_AG(x2)
U6_AG(x1, x2, x3)  =  U6_AG(x2, x3)
U8_AGGA(x1, x2, x3, x4)  =  U8_AGGA(x1, x2, x3, x4)
NOTEQ62_IN_GG(x1, x2)  =  NOTEQ62_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x1, x2, x3)
U9_AGGA(x1, x2, x3, x4, x5, x6)  =  U9_AGGA(x2, x3, x4, x6)
U10_AGGA(x1, x2, x3, x4, x5, x6)  =  U10_AGGA(x2, x3, x4, x6)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x2, x5)
U18_GA(x1, x2, x3, x4, x5)  =  U18_GA(x1, x2, x5)
MINSORT1_IN_AA(x1, x2)  =  MINSORT1_IN_AA
U14_AA(x1, x2, x3, x4, x5)  =  U14_AA(x5)
MIN215_IN_AAA(x1, x2, x3)  =  MIN215_IN_AAA
U1_AAA(x1, x2, x3, x4, x5)  =  U1_AAA(x5)
MIN25_IN_AAA(x1, x2, x3)  =  MIN25_IN_AAA
U12_AAA(x1, x2, x3)  =  U12_AAA(x3)
LE30_IN_AA(x1, x2)  =  LE30_IN_AA
U4_AA(x1, x2, x3)  =  U4_AA(x3)
U13_AAA(x1, x2, x3)  =  U13_AAA(x3)
GT44_IN_AA(x1, x2)  =  GT44_IN_AA
U5_AA(x1, x2, x3)  =  U5_AA(x3)
U2_AAA(x1, x2, x3, x4, x5)  =  U2_AAA(x5)
U3_AAA(x1, x2, x3, x4, x5)  =  U3_AAA(x5)
MIN215_IN_GAA(x1, x2, x3)  =  MIN215_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x1, x5)
MIN25_IN_GAA(x1, x2, x3)  =  MIN25_IN_GAA(x1)
U12_GAA(x1, x2, x3)  =  U12_GAA(x1, x3)
LE30_IN_GA(x1, x2)  =  LE30_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U13_GAA(x1, x2, x3)  =  U13_GAA(x1, x3)
GT44_IN_GA(x1, x2)  =  GT44_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x1, x5)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x1, x5)
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U16_AA(x1, x2, x3, x4, x5)  =  U16_AA(x5)
REMOVE53_IN_AAAA(x1, x2, x3, x4)  =  REMOVE53_IN_AAAA
U11_AAAA(x1, x2, x3, x4, x5)  =  U11_AAAA(x5)
P61_IN_AAAA(x1, x2, x3, x4)  =  P61_IN_AAAA
U7_AAAA(x1, x2, x3, x4, x5)  =  U7_AAAA(x5)
NOTEQ62_IN_AA(x1, x2)  =  NOTEQ62_IN_AA
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U8_AAAA(x1, x2, x3, x4)  =  U8_AAAA(x4)
U9_AAAA(x1, x2, x3, x4, x5, x6)  =  U9_AAAA(x6)
U10_AAAA(x1, x2, x3, x4, x5, x6)  =  U10_AAAA(x6)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
U18_AA(x1, x2, x3, x4, x5)  =  U18_AA(x5)

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

(6) Obligation:

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

MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → U14_GA(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → MIN215_IN_GAG(T25, T27, T26)
MIN215_IN_GAG(T48, T52, .(T50, T51)) → U1_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
MIN215_IN_GAG(T48, T52, .(T50, T51)) → MIN25_IN_GGA(T48, T50, X59)
MIN25_IN_GGA(T70, T71, T70) → U12_GGA(T70, T71, le30_in_gg(T70, T71))
MIN25_IN_GGA(T70, T71, T70) → LE30_IN_GG(T70, T71)
LE30_IN_GG(s(T82), s(T83)) → U4_GG(T82, T83, le30_in_gg(T82, T83))
LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)
MIN25_IN_GGA(T97, T98, T98) → U13_GGA(T97, T98, gt44_in_gg(T97, T98))
MIN25_IN_GGA(T97, T98, T98) → GT44_IN_GG(T97, T98)
GT44_IN_GG(s(T109), s(T110)) → U5_GG(T109, T110, gt44_in_gg(T109, T110))
GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)
MIN215_IN_GAG(T48, T52, .(T50, T51)) → U2_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_GAG(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)
MINSORT1_IN_GA(.(T25, T26), .(T31, T32)) → U15_GA(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_GA(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → REMOVE53_IN_AGGA(T31, T25, T26, X20)
REMOVE53_IN_AGGA(T145, T146, T147, .(T146, X170)) → U11_AGGA(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
REMOVE53_IN_AGGA(T145, T146, T147, .(T146, X170)) → P61_IN_AGGA(T145, T146, T147, X170)
P61_IN_AGGA(T145, T146, T147, X170) → U7_AGGA(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
P61_IN_AGGA(T145, T146, T147, X170) → NOTEQ62_IN_AG(T145, T146)
NOTEQ62_IN_AG(s(T160), s(T161)) → U6_AG(T160, T161, notEq62_in_ag(T160, T161))
NOTEQ62_IN_AG(s(T160), s(T161)) → NOTEQ62_IN_AG(T160, T161)
P61_IN_AGGA(T184, T146, .(T184, T185), T185) → U8_AGGA(T184, T146, T185, notEq62_in_gg(T184, T146))
P61_IN_AGGA(T184, T146, .(T184, T185), T185) → NOTEQ62_IN_GG(T184, T146)
NOTEQ62_IN_GG(s(T160), s(T161)) → U6_GG(T160, T161, notEq62_in_gg(T160, T161))
NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)
P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AGGA(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → NOTEQ62_IN_AG(T192, T146)
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_AGGA(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → P61_IN_AGGA(T192, T193, T194, X229)
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_GA(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_GA(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_GA(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U17_GA(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)
MINSORT1_IN_AA(.(T25, T26), .(T27, T28)) → U14_AA(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
MINSORT1_IN_AA(.(T25, T26), .(T27, T28)) → MIN215_IN_AAA(T25, T27, T26)
MIN215_IN_AAA(T48, T52, .(T50, T51)) → U1_AAA(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
MIN215_IN_AAA(T48, T52, .(T50, T51)) → MIN25_IN_AAA(T48, T50, X59)
MIN25_IN_AAA(T70, T71, T70) → U12_AAA(T70, T71, le30_in_aa(T70, T71))
MIN25_IN_AAA(T70, T71, T70) → LE30_IN_AA(T70, T71)
LE30_IN_AA(s(T82), s(T83)) → U4_AA(T82, T83, le30_in_aa(T82, T83))
LE30_IN_AA(s(T82), s(T83)) → LE30_IN_AA(T82, T83)
MIN25_IN_AAA(T97, T98, T98) → U13_AAA(T97, T98, gt44_in_aa(T97, T98))
MIN25_IN_AAA(T97, T98, T98) → GT44_IN_AA(T97, T98)
GT44_IN_AA(s(T109), s(T110)) → U5_AA(T109, T110, gt44_in_aa(T109, T110))
GT44_IN_AA(s(T109), s(T110)) → GT44_IN_AA(T109, T110)
MIN215_IN_AAA(T48, T52, .(T50, T51)) → U2_AAA(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_AAA(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_AAA(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U2_AAA(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)
MIN215_IN_GAA(T48, T52, .(T50, T51)) → U1_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
MIN215_IN_GAA(T48, T52, .(T50, T51)) → MIN25_IN_GAA(T48, T50, X59)
MIN25_IN_GAA(T70, T71, T70) → U12_GAA(T70, T71, le30_in_ga(T70, T71))
MIN25_IN_GAA(T70, T71, T70) → LE30_IN_GA(T70, T71)
LE30_IN_GA(s(T82), s(T83)) → U4_GA(T82, T83, le30_in_ga(T82, T83))
LE30_IN_GA(s(T82), s(T83)) → LE30_IN_GA(T82, T83)
MIN25_IN_GAA(T97, T98, T98) → U13_GAA(T97, T98, gt44_in_ga(T97, T98))
MIN25_IN_GAA(T97, T98, T98) → GT44_IN_GA(T97, T98)
GT44_IN_GA(s(T109), s(T110)) → U5_GA(T109, T110, gt44_in_ga(T109, T110))
GT44_IN_GA(s(T109), s(T110)) → GT44_IN_GA(T109, T110)
MIN215_IN_GAA(T48, T52, .(T50, T51)) → U2_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_GAA(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)
MINSORT1_IN_AA(.(T25, T26), .(T31, T32)) → U15_AA(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → REMOVE53_IN_AAAA(T31, T25, T26, X20)
REMOVE53_IN_AAAA(T145, T146, T147, .(T146, X170)) → U11_AAAA(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
REMOVE53_IN_AAAA(T145, T146, T147, .(T146, X170)) → P61_IN_AAAA(T145, T146, T147, X170)
P61_IN_AAAA(T145, T146, T147, X170) → U7_AAAA(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
P61_IN_AAAA(T145, T146, T147, X170) → NOTEQ62_IN_AA(T145, T146)
NOTEQ62_IN_AA(s(T160), s(T161)) → U6_AA(T160, T161, notEq62_in_aa(T160, T161))
NOTEQ62_IN_AA(s(T160), s(T161)) → NOTEQ62_IN_AA(T160, T161)
P61_IN_AAAA(T184, T146, .(T184, T185), T185) → U8_AAAA(T184, T146, T185, notEq62_in_aa(T184, T146))
P61_IN_AAAA(T184, T146, .(T184, T185), T185) → NOTEQ62_IN_AA(T184, T146)
P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AAAA(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → NOTEQ62_IN_AA(T192, T146)
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_AAAA(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → P61_IN_AAAA(T192, T193, T194, X229)
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_AA(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MINSORT1_IN_GA(x1, x2)  =  MINSORT1_IN_GA(x1)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x1, x2, x5)
MIN215_IN_GAG(x1, x2, x3)  =  MIN215_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x1, x3, x4, x5)
MIN25_IN_GGA(x1, x2, x3)  =  MIN25_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3)  =  U12_GGA(x1, x2, x3)
LE30_IN_GG(x1, x2)  =  LE30_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)
U13_GGA(x1, x2, x3)  =  U13_GGA(x1, x2, x3)
GT44_IN_GG(x1, x2)  =  GT44_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x1, x3, x4, x5)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x1, x3, x4, x5)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x2, x5)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x1, x2, x5)
REMOVE53_IN_AGGA(x1, x2, x3, x4)  =  REMOVE53_IN_AGGA(x2, x3)
U11_AGGA(x1, x2, x3, x4, x5)  =  U11_AGGA(x2, x3, x5)
P61_IN_AGGA(x1, x2, x3, x4)  =  P61_IN_AGGA(x2, x3)
U7_AGGA(x1, x2, x3, x4, x5)  =  U7_AGGA(x2, x3, x5)
NOTEQ62_IN_AG(x1, x2)  =  NOTEQ62_IN_AG(x2)
U6_AG(x1, x2, x3)  =  U6_AG(x2, x3)
U8_AGGA(x1, x2, x3, x4)  =  U8_AGGA(x1, x2, x3, x4)
NOTEQ62_IN_GG(x1, x2)  =  NOTEQ62_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x1, x2, x3)
U9_AGGA(x1, x2, x3, x4, x5, x6)  =  U9_AGGA(x2, x3, x4, x6)
U10_AGGA(x1, x2, x3, x4, x5, x6)  =  U10_AGGA(x2, x3, x4, x6)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x1, x2, x5)
U18_GA(x1, x2, x3, x4, x5)  =  U18_GA(x1, x2, x5)
MINSORT1_IN_AA(x1, x2)  =  MINSORT1_IN_AA
U14_AA(x1, x2, x3, x4, x5)  =  U14_AA(x5)
MIN215_IN_AAA(x1, x2, x3)  =  MIN215_IN_AAA
U1_AAA(x1, x2, x3, x4, x5)  =  U1_AAA(x5)
MIN25_IN_AAA(x1, x2, x3)  =  MIN25_IN_AAA
U12_AAA(x1, x2, x3)  =  U12_AAA(x3)
LE30_IN_AA(x1, x2)  =  LE30_IN_AA
U4_AA(x1, x2, x3)  =  U4_AA(x3)
U13_AAA(x1, x2, x3)  =  U13_AAA(x3)
GT44_IN_AA(x1, x2)  =  GT44_IN_AA
U5_AA(x1, x2, x3)  =  U5_AA(x3)
U2_AAA(x1, x2, x3, x4, x5)  =  U2_AAA(x5)
U3_AAA(x1, x2, x3, x4, x5)  =  U3_AAA(x5)
MIN215_IN_GAA(x1, x2, x3)  =  MIN215_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x1, x5)
MIN25_IN_GAA(x1, x2, x3)  =  MIN25_IN_GAA(x1)
U12_GAA(x1, x2, x3)  =  U12_GAA(x1, x3)
LE30_IN_GA(x1, x2)  =  LE30_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U13_GAA(x1, x2, x3)  =  U13_GAA(x1, x3)
GT44_IN_GA(x1, x2)  =  GT44_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x1, x5)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x1, x5)
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U16_AA(x1, x2, x3, x4, x5)  =  U16_AA(x5)
REMOVE53_IN_AAAA(x1, x2, x3, x4)  =  REMOVE53_IN_AAAA
U11_AAAA(x1, x2, x3, x4, x5)  =  U11_AAAA(x5)
P61_IN_AAAA(x1, x2, x3, x4)  =  P61_IN_AAAA
U7_AAAA(x1, x2, x3, x4, x5)  =  U7_AAAA(x5)
NOTEQ62_IN_AA(x1, x2)  =  NOTEQ62_IN_AA
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U8_AAAA(x1, x2, x3, x4)  =  U8_AAAA(x4)
U9_AAAA(x1, x2, x3, x4, x5, x6)  =  U9_AAAA(x6)
U10_AAAA(x1, x2, x3, x4, x5, x6)  =  U10_AAAA(x6)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
U18_AA(x1, x2, x3, x4, x5)  =  U18_AA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 14 SCCs with 61 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

NOTEQ62_IN_AA(s(T160), s(T161)) → NOTEQ62_IN_AA(T160, T161)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
NOTEQ62_IN_AA(x1, x2)  =  NOTEQ62_IN_AA

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:

NOTEQ62_IN_AA(s(T160), s(T161)) → NOTEQ62_IN_AA(T160, T161)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
NOTEQ62_IN_AA(x1, x2)  =  NOTEQ62_IN_AA

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:

NOTEQ62_IN_AANOTEQ62_IN_AA

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

(14) NonTerminationProof (EQUIVALENT transformation)

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

s = NOTEQ62_IN_AA evaluates to t =NOTEQ62_IN_AA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from NOTEQ62_IN_AA to NOTEQ62_IN_AA.



(15) NO

(16) Obligation:

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

P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AAAA(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → P61_IN_AAAA(T192, T193, T194, X229)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
P61_IN_AAAA(x1, x2, x3, x4)  =  P61_IN_AAAA
U9_AAAA(x1, x2, x3, x4, x5, x6)  =  U9_AAAA(x6)

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:

P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AAAA(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → P61_IN_AAAA(T192, T193, T194, X229)

The TRS R consists of the following rules:

notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
P61_IN_AAAA(x1, x2, x3, x4)  =  P61_IN_AAAA
U9_AAAA(x1, x2, x3, x4, x5, x6)  =  U9_AAAA(x6)

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:

P61_IN_AAAAU9_AAAA(notEq62_in_aa)
U9_AAAA(notEq62_out_aa) → P61_IN_AAAA

The TRS R consists of the following rules:

notEq62_in_aaU6_aa(notEq62_in_aa)
notEq62_in_aanotEq62_out_aa
U6_aa(notEq62_out_aa) → notEq62_out_aa

The set Q consists of the following terms:

notEq62_in_aa
U6_aa(x0)

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

(21) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P61_IN_AAAAU9_AAAA(notEq62_in_aa) at position [0] we obtained the following new rules [LPAR04]:

P61_IN_AAAAU9_AAAA(U6_aa(notEq62_in_aa))
P61_IN_AAAAU9_AAAA(notEq62_out_aa)

(22) Obligation:

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

U9_AAAA(notEq62_out_aa) → P61_IN_AAAA
P61_IN_AAAAU9_AAAA(U6_aa(notEq62_in_aa))
P61_IN_AAAAU9_AAAA(notEq62_out_aa)

The TRS R consists of the following rules:

notEq62_in_aaU6_aa(notEq62_in_aa)
notEq62_in_aanotEq62_out_aa
U6_aa(notEq62_out_aa) → notEq62_out_aa

The set Q consists of the following terms:

notEq62_in_aa
U6_aa(x0)

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

(23) NonTerminationProof (EQUIVALENT transformation)

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

s = P61_IN_AAAA evaluates to t =P61_IN_AAAA

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




Rewriting sequence

P61_IN_AAAAU9_AAAA(notEq62_out_aa)
with rule P61_IN_AAAAU9_AAAA(notEq62_out_aa) at position [] and matcher [ ]

U9_AAAA(notEq62_out_aa)P61_IN_AAAA
with rule U9_AAAA(notEq62_out_aa) → P61_IN_AAAA

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


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



(24) NO

(25) Obligation:

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

GT44_IN_GA(s(T109), s(T110)) → GT44_IN_GA(T109, T110)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
GT44_IN_GA(x1, x2)  =  GT44_IN_GA(x1)

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

(26) UsableRulesProof (EQUIVALENT transformation)

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

(27) Obligation:

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

GT44_IN_GA(s(T109), s(T110)) → GT44_IN_GA(T109, T110)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT44_IN_GA(x1, x2)  =  GT44_IN_GA(x1)

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

(28) PiDPToQDPProof (SOUND transformation)

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

(29) Obligation:

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

GT44_IN_GA(s(T109)) → GT44_IN_GA(T109)

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

(30) 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:

  • GT44_IN_GA(s(T109)) → GT44_IN_GA(T109)
    The graph contains the following edges 1 > 1

(31) YES

(32) Obligation:

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

LE30_IN_GA(s(T82), s(T83)) → LE30_IN_GA(T82, T83)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
LE30_IN_GA(x1, x2)  =  LE30_IN_GA(x1)

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

(33) UsableRulesProof (EQUIVALENT transformation)

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

(34) Obligation:

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

LE30_IN_GA(s(T82), s(T83)) → LE30_IN_GA(T82, T83)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE30_IN_GA(x1, x2)  =  LE30_IN_GA(x1)

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

(35) PiDPToQDPProof (SOUND transformation)

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

(36) Obligation:

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

LE30_IN_GA(s(T82)) → LE30_IN_GA(T82)

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

(37) 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:

  • LE30_IN_GA(s(T82)) → LE30_IN_GA(T82)
    The graph contains the following edges 1 > 1

(38) YES

(39) Obligation:

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

MIN215_IN_GAA(T48, T52, .(T50, T51)) → U2_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MIN215_IN_GAA(x1, x2, x3)  =  MIN215_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x1, x5)

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

(40) UsableRulesProof (EQUIVALENT transformation)

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

(41) Obligation:

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

MIN215_IN_GAA(T48, T52, .(T50, T51)) → U2_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)

The TRS R consists of the following rules:

min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
MIN215_IN_GAA(x1, x2, x3)  =  MIN215_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x1, x5)

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

(42) PiDPToQDPProof (SOUND transformation)

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

(43) Obligation:

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

MIN215_IN_GAA(T48) → U2_GAA(T48, min25_in_gaa(T48))
U2_GAA(T48, min25_out_gaa(T48, T55)) → MIN215_IN_GAA(T55)

The TRS R consists of the following rules:

min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(T97, gt44_in_ga(T97))
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))

The set Q consists of the following terms:

min25_in_gaa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(44) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MIN215_IN_GAA(T48) → U2_GAA(T48, min25_in_gaa(T48)) at position [1] we obtained the following new rules [LPAR04]:

MIN215_IN_GAA(x0) → U2_GAA(x0, U12_gaa(x0, le30_in_ga(x0)))
MIN215_IN_GAA(x0) → U2_GAA(x0, U13_gaa(x0, gt44_in_ga(x0)))

(45) Obligation:

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

U2_GAA(T48, min25_out_gaa(T48, T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(x0, U12_gaa(x0, le30_in_ga(x0)))
MIN215_IN_GAA(x0) → U2_GAA(x0, U13_gaa(x0, gt44_in_ga(x0)))

The TRS R consists of the following rules:

min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(T97, gt44_in_ga(T97))
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))

The set Q consists of the following terms:

min25_in_gaa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(46) 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.

(47) Obligation:

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

U2_GAA(T48, min25_out_gaa(T48, T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(x0, U12_gaa(x0, le30_in_ga(x0)))
MIN215_IN_GAA(x0) → U2_GAA(x0, U13_gaa(x0, gt44_in_ga(x0)))

The TRS R consists of the following rules:

gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))

The set Q consists of the following terms:

min25_in_gaa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(48) 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].

min25_in_gaa(x0)

(49) Obligation:

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

U2_GAA(T48, min25_out_gaa(T48, T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(x0, U12_gaa(x0, le30_in_ga(x0)))
MIN215_IN_GAA(x0) → U2_GAA(x0, U13_gaa(x0, gt44_in_ga(x0)))

The TRS R consists of the following rules:

gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))

The set Q consists of the following terms:

U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(50) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MIN215_IN_GAA(x0) → U2_GAA(x0, U13_gaa(x0, gt44_in_ga(x0)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U12_gaa(x1, x2) ) = x1 + 2


POL( U2_GAA(x1, x2) ) = max{0, 2x2 - 2}


POL( le30_in_ga(x1) ) = 0


POL( s(x1) ) = x1 + 2


POL( U4_ga(x1, x2) ) = x1 + 2


POL( 0 ) = 0


POL( le30_out_ga(x1) ) = max{0, 2x1 - 2}


POL( min25_out_gaa(x1, x2) ) = x2 + 2


POL( U13_gaa(x1, x2) ) = x2


POL( gt44_in_ga(x1) ) = x1


POL( U5_ga(x1, x2) ) = x2 + 2


POL( gt44_out_ga(x1, x2) ) = x2 + 2


POL( MIN215_IN_GAA(x1) ) = 2x1 + 2



The following usable rules [FROCOS05] were oriented:

U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))

(51) Obligation:

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

U2_GAA(T48, min25_out_gaa(T48, T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(x0, U12_gaa(x0, le30_in_ga(x0)))

The TRS R consists of the following rules:

gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))

The set Q consists of the following terms:

U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(52) 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.

(53) Obligation:

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

U2_GAA(T48, min25_out_gaa(T48, T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(x0, U12_gaa(x0, le30_in_ga(x0)))

The TRS R consists of the following rules:

le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))

The set Q consists of the following terms:

U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(54) 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].

U13_gaa(x0, x1)
gt44_in_ga(x0)
U5_ga(x0, x1)

(55) Obligation:

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

U2_GAA(T48, min25_out_gaa(T48, T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(x0, U12_gaa(x0, le30_in_ga(x0)))

The TRS R consists of the following rules:

le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))

The set Q consists of the following terms:

U12_gaa(x0, x1)
le30_in_ga(x0)
U4_ga(x0, x1)

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

(56) NonTerminationProof (EQUIVALENT transformation)

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

s = U2_GAA(0, U12_gaa(0, le30_in_ga(0))) evaluates to t =U2_GAA(0, U12_gaa(0, le30_in_ga(0)))

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




Rewriting sequence

U2_GAA(0, U12_gaa(0, le30_in_ga(0)))U2_GAA(0, U12_gaa(0, le30_out_ga(0)))
with rule le30_in_ga(0) → le30_out_ga(0) at position [1,1] and matcher [ ]

U2_GAA(0, U12_gaa(0, le30_out_ga(0)))U2_GAA(0, min25_out_gaa(0, 0))
with rule U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70) at position [1] and matcher [T70 / 0]

U2_GAA(0, min25_out_gaa(0, 0))MIN215_IN_GAA(0)
with rule U2_GAA(T48, min25_out_gaa(T48, T55)) → MIN215_IN_GAA(T55) at position [] and matcher [T48 / 0, T55 / 0]

MIN215_IN_GAA(0)U2_GAA(0, U12_gaa(0, le30_in_ga(0)))
with rule MIN215_IN_GAA(x0) → U2_GAA(x0, U12_gaa(x0, le30_in_ga(x0)))

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


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



(57) NO

(58) Obligation:

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

GT44_IN_AA(s(T109), s(T110)) → GT44_IN_AA(T109, T110)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
GT44_IN_AA(x1, x2)  =  GT44_IN_AA

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

(59) UsableRulesProof (EQUIVALENT transformation)

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

(60) Obligation:

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

GT44_IN_AA(s(T109), s(T110)) → GT44_IN_AA(T109, T110)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT44_IN_AA(x1, x2)  =  GT44_IN_AA

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

(61) PiDPToQDPProof (SOUND transformation)

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

(62) Obligation:

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

GT44_IN_AAGT44_IN_AA

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

(63) 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 = GT44_IN_AA evaluates to t =GT44_IN_AA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from GT44_IN_AA to GT44_IN_AA.



(64) NO

(65) Obligation:

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

LE30_IN_AA(s(T82), s(T83)) → LE30_IN_AA(T82, T83)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
LE30_IN_AA(x1, x2)  =  LE30_IN_AA

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

(66) UsableRulesProof (EQUIVALENT transformation)

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

(67) Obligation:

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

LE30_IN_AA(s(T82), s(T83)) → LE30_IN_AA(T82, T83)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE30_IN_AA(x1, x2)  =  LE30_IN_AA

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

(68) PiDPToQDPProof (SOUND transformation)

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

(69) Obligation:

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

LE30_IN_AALE30_IN_AA

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

(70) 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 = LE30_IN_AA evaluates to t =LE30_IN_AA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LE30_IN_AA to LE30_IN_AA.



(71) NO

(72) Obligation:

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

MINSORT1_IN_AA(.(T25, T26), .(T31, T32)) → U15_AA(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MINSORT1_IN_AA(x1, x2)  =  MINSORT1_IN_AA
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)

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

(73) UsableRulesProof (EQUIVALENT transformation)

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

(74) Obligation:

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

MINSORT1_IN_AA(.(T25, T26), .(T31, T32)) → U15_AA(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)

The TRS R consists of the following rules:

min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
MINSORT1_IN_AA(x1, x2)  =  MINSORT1_IN_AA
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)

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

(75) PiDPToQDPProof (SOUND transformation)

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

(76) Obligation:

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

MINSORT1_IN_AAU15_AA(min215_in_aaa)
U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa)
U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA

The TRS R consists of the following rules:

min215_in_aaamin215_out_aaa
min215_in_aaaU1_aaa(min25_in_aaa)
min215_in_aaaU2_aaa(min25_in_aaa)
remove53_in_aaaaremove53_out_aaaa
remove53_in_aaaaU11_aaaa(p61_in_aaaa)
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
min25_in_aaaU12_aaa(le30_in_aa)
min25_in_aaaU13_aaa(gt44_in_aa)
U3_aaa(min215_out_gaa(T55)) → min215_out_aaa
p61_in_aaaaU7_aaaa(notEq62_in_aa)
p61_in_aaaaU8_aaaa(notEq62_in_aa)
p61_in_aaaaU9_aaaa(notEq62_in_aa)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
min215_in_gaa(T39) → min215_out_gaa(T39)
min215_in_gaa(T48) → U1_gaa(T48, min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(T48, min25_in_gaa(T48))
U7_aaaa(notEq62_out_aa) → p61_out_aaaa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
le30_in_aaU4_aa(le30_in_aa)
le30_in_aale30_out_aa(0)
gt44_in_aaU5_aa(gt44_in_aa)
gt44_in_aagt44_out_aa(0)
U1_gaa(T48, min25_out_gaa(T48, X59)) → min215_out_gaa(T48)
U2_gaa(T48, min25_out_gaa(T48, T55)) → U3_gaa(T48, min215_in_gaa(T55))
notEq62_in_aaU6_aa(notEq62_in_aa)
notEq62_in_aanotEq62_out_aa
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(T97, gt44_in_ga(T97))
U3_gaa(T48, min215_out_gaa(T55)) → min215_out_gaa(T48)
U6_aa(notEq62_out_aa) → notEq62_out_aa
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))

The set Q consists of the following terms:

min215_in_aaa
remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0, x1)
U2_gaa(x0, x1)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0, x1)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(77) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MINSORT1_IN_AAU15_AA(min215_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

MINSORT1_IN_AAU15_AA(min215_out_aaa)
MINSORT1_IN_AAU15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AAU15_AA(U2_aaa(min25_in_aaa))

(78) Obligation:

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

U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa)
U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AAU15_AA(min215_out_aaa)
MINSORT1_IN_AAU15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AAU15_AA(U2_aaa(min25_in_aaa))

The TRS R consists of the following rules:

min215_in_aaamin215_out_aaa
min215_in_aaaU1_aaa(min25_in_aaa)
min215_in_aaaU2_aaa(min25_in_aaa)
remove53_in_aaaaremove53_out_aaaa
remove53_in_aaaaU11_aaaa(p61_in_aaaa)
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
min25_in_aaaU12_aaa(le30_in_aa)
min25_in_aaaU13_aaa(gt44_in_aa)
U3_aaa(min215_out_gaa(T55)) → min215_out_aaa
p61_in_aaaaU7_aaaa(notEq62_in_aa)
p61_in_aaaaU8_aaaa(notEq62_in_aa)
p61_in_aaaaU9_aaaa(notEq62_in_aa)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
min215_in_gaa(T39) → min215_out_gaa(T39)
min215_in_gaa(T48) → U1_gaa(T48, min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(T48, min25_in_gaa(T48))
U7_aaaa(notEq62_out_aa) → p61_out_aaaa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
le30_in_aaU4_aa(le30_in_aa)
le30_in_aale30_out_aa(0)
gt44_in_aaU5_aa(gt44_in_aa)
gt44_in_aagt44_out_aa(0)
U1_gaa(T48, min25_out_gaa(T48, X59)) → min215_out_gaa(T48)
U2_gaa(T48, min25_out_gaa(T48, T55)) → U3_gaa(T48, min215_in_gaa(T55))
notEq62_in_aaU6_aa(notEq62_in_aa)
notEq62_in_aanotEq62_out_aa
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(T97, gt44_in_ga(T97))
U3_gaa(T48, min215_out_gaa(T55)) → min215_out_gaa(T48)
U6_aa(notEq62_out_aa) → notEq62_out_aa
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))

The set Q consists of the following terms:

min215_in_aaa
remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0, x1)
U2_gaa(x0, x1)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0, x1)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(79) 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.

(80) Obligation:

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

U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa)
U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AAU15_AA(min215_out_aaa)
MINSORT1_IN_AAU15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AAU15_AA(U2_aaa(min25_in_aaa))

The TRS R consists of the following rules:

min25_in_aaaU12_aaa(le30_in_aa)
min25_in_aaaU13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa(T39)
min215_in_gaa(T48) → U1_gaa(T48, min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(T48, min25_in_gaa(T48))
U3_aaa(min215_out_gaa(T55)) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(T97, gt44_in_ga(T97))
U2_gaa(T48, min25_out_gaa(T48, T55)) → U3_gaa(T48, min215_in_gaa(T55))
U3_gaa(T48, min215_out_gaa(T55)) → min215_out_gaa(T48)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))
U1_gaa(T48, min25_out_gaa(T48, X59)) → min215_out_gaa(T48)
gt44_in_aaU5_aa(gt44_in_aa)
gt44_in_aagt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aaU4_aa(le30_in_aa)
le30_in_aale30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
remove53_in_aaaaremove53_out_aaaa
remove53_in_aaaaU11_aaaa(p61_in_aaaa)
p61_in_aaaaU7_aaaa(notEq62_in_aa)
p61_in_aaaaU8_aaaa(notEq62_in_aa)
p61_in_aaaaU9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aaU6_aa(notEq62_in_aa)
notEq62_in_aanotEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa

The set Q consists of the following terms:

min215_in_aaa
remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0, x1)
U2_gaa(x0, x1)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0, x1)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(81) 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].

min215_in_aaa

(82) Obligation:

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

U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa)
U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AAU15_AA(min215_out_aaa)
MINSORT1_IN_AAU15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AAU15_AA(U2_aaa(min25_in_aaa))

The TRS R consists of the following rules:

min25_in_aaaU12_aaa(le30_in_aa)
min25_in_aaaU13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa(T39)
min215_in_gaa(T48) → U1_gaa(T48, min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(T48, min25_in_gaa(T48))
U3_aaa(min215_out_gaa(T55)) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(T97, gt44_in_ga(T97))
U2_gaa(T48, min25_out_gaa(T48, T55)) → U3_gaa(T48, min215_in_gaa(T55))
U3_gaa(T48, min215_out_gaa(T55)) → min215_out_gaa(T48)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))
U1_gaa(T48, min25_out_gaa(T48, X59)) → min215_out_gaa(T48)
gt44_in_aaU5_aa(gt44_in_aa)
gt44_in_aagt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aaU4_aa(le30_in_aa)
le30_in_aale30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
remove53_in_aaaaremove53_out_aaaa
remove53_in_aaaaU11_aaaa(p61_in_aaaa)
p61_in_aaaaU7_aaaa(notEq62_in_aa)
p61_in_aaaaU8_aaaa(notEq62_in_aa)
p61_in_aaaaU9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aaU6_aa(notEq62_in_aa)
notEq62_in_aanotEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa

The set Q consists of the following terms:

remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0, x1)
U2_gaa(x0, x1)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0, x1)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(83) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa)
U15_AA(min215_out_aaa) → U17_AA(U11_aaaa(p61_in_aaaa))

(84) Obligation:

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

U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AAU15_AA(min215_out_aaa)
MINSORT1_IN_AAU15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AAU15_AA(U2_aaa(min25_in_aaa))
U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa)
U15_AA(min215_out_aaa) → U17_AA(U11_aaaa(p61_in_aaaa))

The TRS R consists of the following rules:

min25_in_aaaU12_aaa(le30_in_aa)
min25_in_aaaU13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa(T39)
min215_in_gaa(T48) → U1_gaa(T48, min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(T48, min25_in_gaa(T48))
U3_aaa(min215_out_gaa(T55)) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(T97, gt44_in_ga(T97))
U2_gaa(T48, min25_out_gaa(T48, T55)) → U3_gaa(T48, min215_in_gaa(T55))
U3_gaa(T48, min215_out_gaa(T55)) → min215_out_gaa(T48)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))
U1_gaa(T48, min25_out_gaa(T48, X59)) → min215_out_gaa(T48)
gt44_in_aaU5_aa(gt44_in_aa)
gt44_in_aagt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aaU4_aa(le30_in_aa)
le30_in_aale30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
remove53_in_aaaaremove53_out_aaaa
remove53_in_aaaaU11_aaaa(p61_in_aaaa)
p61_in_aaaaU7_aaaa(notEq62_in_aa)
p61_in_aaaaU8_aaaa(notEq62_in_aa)
p61_in_aaaaU9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aaU6_aa(notEq62_in_aa)
notEq62_in_aanotEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa

The set Q consists of the following terms:

remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0, x1)
U2_gaa(x0, x1)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0, x1)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(85) 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.

(86) Obligation:

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

U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AAU15_AA(min215_out_aaa)
MINSORT1_IN_AAU15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AAU15_AA(U2_aaa(min25_in_aaa))
U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa)
U15_AA(min215_out_aaa) → U17_AA(U11_aaaa(p61_in_aaaa))

The TRS R consists of the following rules:

p61_in_aaaaU7_aaaa(notEq62_in_aa)
p61_in_aaaaU8_aaaa(notEq62_in_aa)
p61_in_aaaaU9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aaU6_aa(notEq62_in_aa)
notEq62_in_aanotEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa
min25_in_aaaU12_aaa(le30_in_aa)
min25_in_aaaU13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa(T39)
min215_in_gaa(T48) → U1_gaa(T48, min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(T48, min25_in_gaa(T48))
U3_aaa(min215_out_gaa(T55)) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(T97, gt44_in_ga(T97))
U2_gaa(T48, min25_out_gaa(T48, T55)) → U3_gaa(T48, min215_in_gaa(T55))
U3_gaa(T48, min215_out_gaa(T55)) → min215_out_gaa(T48)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))
U1_gaa(T48, min25_out_gaa(T48, X59)) → min215_out_gaa(T48)
gt44_in_aaU5_aa(gt44_in_aa)
gt44_in_aagt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aaU4_aa(le30_in_aa)
le30_in_aale30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa

The set Q consists of the following terms:

remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0, x1)
U2_gaa(x0, x1)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0, x1)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(87) 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].

remove53_in_aaaa

(88) Obligation:

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

U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AAU15_AA(min215_out_aaa)
MINSORT1_IN_AAU15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AAU15_AA(U2_aaa(min25_in_aaa))
U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa)
U15_AA(min215_out_aaa) → U17_AA(U11_aaaa(p61_in_aaaa))

The TRS R consists of the following rules:

p61_in_aaaaU7_aaaa(notEq62_in_aa)
p61_in_aaaaU8_aaaa(notEq62_in_aa)
p61_in_aaaaU9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aaU6_aa(notEq62_in_aa)
notEq62_in_aanotEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa
min25_in_aaaU12_aaa(le30_in_aa)
min25_in_aaaU13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa(T39)
min215_in_gaa(T48) → U1_gaa(T48, min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(T48, min25_in_gaa(T48))
U3_aaa(min215_out_gaa(T55)) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(T97, gt44_in_ga(T97))
U2_gaa(T48, min25_out_gaa(T48, T55)) → U3_gaa(T48, min215_in_gaa(T55))
U3_gaa(T48, min215_out_gaa(T55)) → min215_out_gaa(T48)
gt44_in_ga(s(T109)) → U5_ga(T109, gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(s(T115), 0)
U13_gaa(T97, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98)
U5_ga(T109, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
le30_in_ga(s(T82)) → U4_ga(T82, le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga(0)
U12_gaa(T70, le30_out_ga(T70)) → min25_out_gaa(T70, T70)
U4_ga(T82, le30_out_ga(T82)) → le30_out_ga(s(T82))
U1_gaa(T48, min25_out_gaa(T48, X59)) → min215_out_gaa(T48)
gt44_in_aaU5_aa(gt44_in_aa)
gt44_in_aagt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aaU4_aa(le30_in_aa)
le30_in_aale30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa

The set Q consists of the following terms:

U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0, x1)
U2_gaa(x0, x1)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0, x1)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0, x1)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0, x1)
U5_ga(x0, x1)

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

(89) NonTerminationProof (EQUIVALENT transformation)

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

s = MINSORT1_IN_AA evaluates to t =MINSORT1_IN_AA

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




Rewriting sequence

MINSORT1_IN_AAU15_AA(min215_out_aaa)
with rule MINSORT1_IN_AAU15_AA(min215_out_aaa) at position [] and matcher [ ]

U15_AA(min215_out_aaa)U17_AA(remove53_out_aaaa)
with rule U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa) at position [] and matcher [ ]

U17_AA(remove53_out_aaaa)MINSORT1_IN_AA
with rule U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA

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


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



(90) NO

(91) Obligation:

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

NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
NOTEQ62_IN_GG(x1, x2)  =  NOTEQ62_IN_GG(x1, x2)

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

(92) UsableRulesProof (EQUIVALENT transformation)

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

(93) Obligation:

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

NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)

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

(94) PiDPToQDPProof (EQUIVALENT transformation)

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

(95) Obligation:

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

NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)

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

(96) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)
    The graph contains the following edges 1 > 1, 2 > 2

(97) YES

(98) Obligation:

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

NOTEQ62_IN_AG(s(T160), s(T161)) → NOTEQ62_IN_AG(T160, T161)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
NOTEQ62_IN_AG(x1, x2)  =  NOTEQ62_IN_AG(x2)

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

(99) UsableRulesProof (EQUIVALENT transformation)

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

(100) Obligation:

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

NOTEQ62_IN_AG(s(T160), s(T161)) → NOTEQ62_IN_AG(T160, T161)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
NOTEQ62_IN_AG(x1, x2)  =  NOTEQ62_IN_AG(x2)

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

(101) PiDPToQDPProof (SOUND transformation)

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

(102) Obligation:

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

NOTEQ62_IN_AG(s(T161)) → NOTEQ62_IN_AG(T161)

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

(103) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • NOTEQ62_IN_AG(s(T161)) → NOTEQ62_IN_AG(T161)
    The graph contains the following edges 1 > 1

(104) YES

(105) Obligation:

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

P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AGGA(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → P61_IN_AGGA(T192, T193, T194, X229)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
P61_IN_AGGA(x1, x2, x3, x4)  =  P61_IN_AGGA(x2, x3)
U9_AGGA(x1, x2, x3, x4, x5, x6)  =  U9_AGGA(x2, x3, x4, x6)

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

(106) UsableRulesProof (EQUIVALENT transformation)

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

(107) Obligation:

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

P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AGGA(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → P61_IN_AGGA(T192, T193, T194, X229)

The TRS R consists of the following rules:

notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
P61_IN_AGGA(x1, x2, x3, x4)  =  P61_IN_AGGA(x2, x3)
U9_AGGA(x1, x2, x3, x4, x5, x6)  =  U9_AGGA(x2, x3, x4, x6)

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

(108) PiDPToQDPProof (SOUND transformation)

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

(109) Obligation:

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

P61_IN_AGGA(T146, .(T193, T194)) → U9_AGGA(T146, T193, T194, notEq62_in_ag(T146))
U9_AGGA(T146, T193, T194, notEq62_out_ag(T146)) → P61_IN_AGGA(T193, T194)

The TRS R consists of the following rules:

notEq62_in_ag(s(T161)) → U6_ag(T161, notEq62_in_ag(T161))
notEq62_in_ag(0) → notEq62_out_ag(0)
notEq62_in_ag(s(T171)) → notEq62_out_ag(s(T171))
U6_ag(T161, notEq62_out_ag(T161)) → notEq62_out_ag(s(T161))

The set Q consists of the following terms:

notEq62_in_ag(x0)
U6_ag(x0, x1)

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

(110) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • U9_AGGA(T146, T193, T194, notEq62_out_ag(T146)) → P61_IN_AGGA(T193, T194)
    The graph contains the following edges 2 >= 1, 3 >= 2

  • P61_IN_AGGA(T146, .(T193, T194)) → U9_AGGA(T146, T193, T194, notEq62_in_ag(T146))
    The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

(111) YES

(112) Obligation:

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

GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
GT44_IN_GG(x1, x2)  =  GT44_IN_GG(x1, x2)

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

(113) UsableRulesProof (EQUIVALENT transformation)

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

(114) Obligation:

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

GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)

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

(115) PiDPToQDPProof (EQUIVALENT transformation)

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

(116) Obligation:

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

GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)

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

(117) 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:

  • GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)
    The graph contains the following edges 1 > 1, 2 > 2

(118) YES

(119) Obligation:

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

LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
LE30_IN_GG(x1, x2)  =  LE30_IN_GG(x1, x2)

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

(120) UsableRulesProof (EQUIVALENT transformation)

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

(121) Obligation:

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

LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)

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

(122) PiDPToQDPProof (EQUIVALENT transformation)

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

(123) Obligation:

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

LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)

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

(124) 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:

  • LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)
    The graph contains the following edges 1 > 1, 2 > 2

(125) YES

(126) Obligation:

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

MIN215_IN_GAG(T48, T52, .(T50, T51)) → U2_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x1, x2, x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x1, x3, x4, x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x1, x3, x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x1, x3, x4, x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x1, x2, x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga(x2, x3)
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x2, x3, x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x2, x3, x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag(x2)
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga(x2, x3)
U8_agga(x1, x2, x3, x4)  =  U8_agga(x1, x2, x3, x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x1, x2, x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg(x1, x2)
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x2, x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x2, x3, x4, x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x1, x2, x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x1, x2, x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
le30_out_ga(x1, x2)  =  le30_out_ga(x1)
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x1, x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x1, x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x1, x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x1, x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x1, x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MIN215_IN_GAG(x1, x2, x3)  =  MIN215_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x1, x3, x4, x5)

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

(127) UsableRulesProof (EQUIVALENT transformation)

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

(128) Obligation:

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

MIN215_IN_GAG(T48, T52, .(T50, T51)) → U2_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)

The TRS R consists of the following rules:

min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x2, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg(x1, x2)
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x1, x2, x3)
U13_gga(x1, x2, x3)  =  U13_gga(x1, x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg(x1, x2)
MIN215_IN_GAG(x1, x2, x3)  =  MIN215_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x1, x3, x4, x5)

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

(129) PiDPToQDPProof (SOUND transformation)

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

(130) Obligation:

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

MIN215_IN_GAG(T48, .(T50, T51)) → U2_GAG(T48, T50, T51, min25_in_gga(T48, T50))
U2_GAG(T48, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T51)

The TRS R consists of the following rules:

min25_in_gga(T70, T71) → U12_gga(T70, T71, le30_in_gg(T70, T71))
min25_in_gga(T97, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))

The set Q consists of the following terms:

min25_in_gga(x0, x1)
U12_gga(x0, x1, x2)
U13_gga(x0, x1, x2)
le30_in_gg(x0, x1)
gt44_in_gg(x0, x1)
U4_gg(x0, x1, x2)
U5_gg(x0, x1, x2)

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

(131) 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:

  • U2_GAG(T48, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T51)
    The graph contains the following edges 4 > 1, 3 >= 2

  • MIN215_IN_GAG(T48, .(T50, T51)) → U2_GAG(T48, T50, T51, min25_in_gga(T48, T50))
    The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

(132) YES

(133) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
minsort1_in: (b,f) (f,f)
min215_in: (b,f,b) (f,f,f) (b,f,f)
min25_in: (b,b,f) (f,f,f) (b,f,f)
le30_in: (b,b) (f,f) (b,f)
gt44_in: (b,b) (f,f) (b,f)
remove53_in: (f,b,b,f) (f,f,f,f)
p61_in: (f,b,b,f) (f,f,f,f)
notEq62_in: (f,b) (b,b) (f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(134) Obligation:

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

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)

(135) 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:

MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → U14_GA(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → MIN215_IN_GAG(T25, T27, T26)
MIN215_IN_GAG(T48, T52, .(T50, T51)) → U1_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
MIN215_IN_GAG(T48, T52, .(T50, T51)) → MIN25_IN_GGA(T48, T50, X59)
MIN25_IN_GGA(T70, T71, T70) → U12_GGA(T70, T71, le30_in_gg(T70, T71))
MIN25_IN_GGA(T70, T71, T70) → LE30_IN_GG(T70, T71)
LE30_IN_GG(s(T82), s(T83)) → U4_GG(T82, T83, le30_in_gg(T82, T83))
LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)
MIN25_IN_GGA(T97, T98, T98) → U13_GGA(T97, T98, gt44_in_gg(T97, T98))
MIN25_IN_GGA(T97, T98, T98) → GT44_IN_GG(T97, T98)
GT44_IN_GG(s(T109), s(T110)) → U5_GG(T109, T110, gt44_in_gg(T109, T110))
GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)
MIN215_IN_GAG(T48, T52, .(T50, T51)) → U2_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_GAG(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)
MINSORT1_IN_GA(.(T25, T26), .(T31, T32)) → U15_GA(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_GA(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → REMOVE53_IN_AGGA(T31, T25, T26, X20)
REMOVE53_IN_AGGA(T145, T146, T147, .(T146, X170)) → U11_AGGA(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
REMOVE53_IN_AGGA(T145, T146, T147, .(T146, X170)) → P61_IN_AGGA(T145, T146, T147, X170)
P61_IN_AGGA(T145, T146, T147, X170) → U7_AGGA(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
P61_IN_AGGA(T145, T146, T147, X170) → NOTEQ62_IN_AG(T145, T146)
NOTEQ62_IN_AG(s(T160), s(T161)) → U6_AG(T160, T161, notEq62_in_ag(T160, T161))
NOTEQ62_IN_AG(s(T160), s(T161)) → NOTEQ62_IN_AG(T160, T161)
P61_IN_AGGA(T184, T146, .(T184, T185), T185) → U8_AGGA(T184, T146, T185, notEq62_in_gg(T184, T146))
P61_IN_AGGA(T184, T146, .(T184, T185), T185) → NOTEQ62_IN_GG(T184, T146)
NOTEQ62_IN_GG(s(T160), s(T161)) → U6_GG(T160, T161, notEq62_in_gg(T160, T161))
NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)
P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AGGA(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → NOTEQ62_IN_AG(T192, T146)
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_AGGA(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → P61_IN_AGGA(T192, T193, T194, X229)
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_GA(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_GA(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_GA(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U17_GA(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)
MINSORT1_IN_AA(.(T25, T26), .(T27, T28)) → U14_AA(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
MINSORT1_IN_AA(.(T25, T26), .(T27, T28)) → MIN215_IN_AAA(T25, T27, T26)
MIN215_IN_AAA(T48, T52, .(T50, T51)) → U1_AAA(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
MIN215_IN_AAA(T48, T52, .(T50, T51)) → MIN25_IN_AAA(T48, T50, X59)
MIN25_IN_AAA(T70, T71, T70) → U12_AAA(T70, T71, le30_in_aa(T70, T71))
MIN25_IN_AAA(T70, T71, T70) → LE30_IN_AA(T70, T71)
LE30_IN_AA(s(T82), s(T83)) → U4_AA(T82, T83, le30_in_aa(T82, T83))
LE30_IN_AA(s(T82), s(T83)) → LE30_IN_AA(T82, T83)
MIN25_IN_AAA(T97, T98, T98) → U13_AAA(T97, T98, gt44_in_aa(T97, T98))
MIN25_IN_AAA(T97, T98, T98) → GT44_IN_AA(T97, T98)
GT44_IN_AA(s(T109), s(T110)) → U5_AA(T109, T110, gt44_in_aa(T109, T110))
GT44_IN_AA(s(T109), s(T110)) → GT44_IN_AA(T109, T110)
MIN215_IN_AAA(T48, T52, .(T50, T51)) → U2_AAA(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_AAA(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_AAA(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U2_AAA(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)
MIN215_IN_GAA(T48, T52, .(T50, T51)) → U1_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
MIN215_IN_GAA(T48, T52, .(T50, T51)) → MIN25_IN_GAA(T48, T50, X59)
MIN25_IN_GAA(T70, T71, T70) → U12_GAA(T70, T71, le30_in_ga(T70, T71))
MIN25_IN_GAA(T70, T71, T70) → LE30_IN_GA(T70, T71)
LE30_IN_GA(s(T82), s(T83)) → U4_GA(T82, T83, le30_in_ga(T82, T83))
LE30_IN_GA(s(T82), s(T83)) → LE30_IN_GA(T82, T83)
MIN25_IN_GAA(T97, T98, T98) → U13_GAA(T97, T98, gt44_in_ga(T97, T98))
MIN25_IN_GAA(T97, T98, T98) → GT44_IN_GA(T97, T98)
GT44_IN_GA(s(T109), s(T110)) → U5_GA(T109, T110, gt44_in_ga(T109, T110))
GT44_IN_GA(s(T109), s(T110)) → GT44_IN_GA(T109, T110)
MIN215_IN_GAA(T48, T52, .(T50, T51)) → U2_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_GAA(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)
MINSORT1_IN_AA(.(T25, T26), .(T31, T32)) → U15_AA(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → REMOVE53_IN_AAAA(T31, T25, T26, X20)
REMOVE53_IN_AAAA(T145, T146, T147, .(T146, X170)) → U11_AAAA(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
REMOVE53_IN_AAAA(T145, T146, T147, .(T146, X170)) → P61_IN_AAAA(T145, T146, T147, X170)
P61_IN_AAAA(T145, T146, T147, X170) → U7_AAAA(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
P61_IN_AAAA(T145, T146, T147, X170) → NOTEQ62_IN_AA(T145, T146)
NOTEQ62_IN_AA(s(T160), s(T161)) → U6_AA(T160, T161, notEq62_in_aa(T160, T161))
NOTEQ62_IN_AA(s(T160), s(T161)) → NOTEQ62_IN_AA(T160, T161)
P61_IN_AAAA(T184, T146, .(T184, T185), T185) → U8_AAAA(T184, T146, T185, notEq62_in_aa(T184, T146))
P61_IN_AAAA(T184, T146, .(T184, T185), T185) → NOTEQ62_IN_AA(T184, T146)
P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AAAA(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → NOTEQ62_IN_AA(T192, T146)
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_AAAA(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → P61_IN_AAAA(T192, T193, T194, X229)
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_AA(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MINSORT1_IN_GA(x1, x2)  =  MINSORT1_IN_GA(x1)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x5)
MIN215_IN_GAG(x1, x2, x3)  =  MIN215_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x5)
MIN25_IN_GGA(x1, x2, x3)  =  MIN25_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3)  =  U12_GGA(x1, x3)
LE30_IN_GG(x1, x2)  =  LE30_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x3)
U13_GGA(x1, x2, x3)  =  U13_GGA(x2, x3)
GT44_IN_GG(x1, x2)  =  GT44_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x4, x5)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x5)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x2, x5)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x5)
REMOVE53_IN_AGGA(x1, x2, x3, x4)  =  REMOVE53_IN_AGGA(x2, x3)
U11_AGGA(x1, x2, x3, x4, x5)  =  U11_AGGA(x5)
P61_IN_AGGA(x1, x2, x3, x4)  =  P61_IN_AGGA(x2, x3)
U7_AGGA(x1, x2, x3, x4, x5)  =  U7_AGGA(x5)
NOTEQ62_IN_AG(x1, x2)  =  NOTEQ62_IN_AG(x2)
U6_AG(x1, x2, x3)  =  U6_AG(x3)
U8_AGGA(x1, x2, x3, x4)  =  U8_AGGA(x4)
NOTEQ62_IN_GG(x1, x2)  =  NOTEQ62_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x3)
U9_AGGA(x1, x2, x3, x4, x5, x6)  =  U9_AGGA(x3, x4, x6)
U10_AGGA(x1, x2, x3, x4, x5, x6)  =  U10_AGGA(x6)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x5)
U18_GA(x1, x2, x3, x4, x5)  =  U18_GA(x5)
MINSORT1_IN_AA(x1, x2)  =  MINSORT1_IN_AA
U14_AA(x1, x2, x3, x4, x5)  =  U14_AA(x5)
MIN215_IN_AAA(x1, x2, x3)  =  MIN215_IN_AAA
U1_AAA(x1, x2, x3, x4, x5)  =  U1_AAA(x5)
MIN25_IN_AAA(x1, x2, x3)  =  MIN25_IN_AAA
U12_AAA(x1, x2, x3)  =  U12_AAA(x3)
LE30_IN_AA(x1, x2)  =  LE30_IN_AA
U4_AA(x1, x2, x3)  =  U4_AA(x3)
U13_AAA(x1, x2, x3)  =  U13_AAA(x3)
GT44_IN_AA(x1, x2)  =  GT44_IN_AA
U5_AA(x1, x2, x3)  =  U5_AA(x3)
U2_AAA(x1, x2, x3, x4, x5)  =  U2_AAA(x5)
U3_AAA(x1, x2, x3, x4, x5)  =  U3_AAA(x5)
MIN215_IN_GAA(x1, x2, x3)  =  MIN215_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x5)
MIN25_IN_GAA(x1, x2, x3)  =  MIN25_IN_GAA(x1)
U12_GAA(x1, x2, x3)  =  U12_GAA(x1, x3)
LE30_IN_GA(x1, x2)  =  LE30_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
U13_GAA(x1, x2, x3)  =  U13_GAA(x3)
GT44_IN_GA(x1, x2)  =  GT44_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x5)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U16_AA(x1, x2, x3, x4, x5)  =  U16_AA(x5)
REMOVE53_IN_AAAA(x1, x2, x3, x4)  =  REMOVE53_IN_AAAA
U11_AAAA(x1, x2, x3, x4, x5)  =  U11_AAAA(x5)
P61_IN_AAAA(x1, x2, x3, x4)  =  P61_IN_AAAA
U7_AAAA(x1, x2, x3, x4, x5)  =  U7_AAAA(x5)
NOTEQ62_IN_AA(x1, x2)  =  NOTEQ62_IN_AA
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U8_AAAA(x1, x2, x3, x4)  =  U8_AAAA(x4)
U9_AAAA(x1, x2, x3, x4, x5, x6)  =  U9_AAAA(x6)
U10_AAAA(x1, x2, x3, x4, x5, x6)  =  U10_AAAA(x6)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
U18_AA(x1, x2, x3, x4, x5)  =  U18_AA(x5)

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

(136) Obligation:

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

MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → U14_GA(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → MIN215_IN_GAG(T25, T27, T26)
MIN215_IN_GAG(T48, T52, .(T50, T51)) → U1_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
MIN215_IN_GAG(T48, T52, .(T50, T51)) → MIN25_IN_GGA(T48, T50, X59)
MIN25_IN_GGA(T70, T71, T70) → U12_GGA(T70, T71, le30_in_gg(T70, T71))
MIN25_IN_GGA(T70, T71, T70) → LE30_IN_GG(T70, T71)
LE30_IN_GG(s(T82), s(T83)) → U4_GG(T82, T83, le30_in_gg(T82, T83))
LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)
MIN25_IN_GGA(T97, T98, T98) → U13_GGA(T97, T98, gt44_in_gg(T97, T98))
MIN25_IN_GGA(T97, T98, T98) → GT44_IN_GG(T97, T98)
GT44_IN_GG(s(T109), s(T110)) → U5_GG(T109, T110, gt44_in_gg(T109, T110))
GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)
MIN215_IN_GAG(T48, T52, .(T50, T51)) → U2_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_GAG(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)
MINSORT1_IN_GA(.(T25, T26), .(T31, T32)) → U15_GA(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_GA(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → REMOVE53_IN_AGGA(T31, T25, T26, X20)
REMOVE53_IN_AGGA(T145, T146, T147, .(T146, X170)) → U11_AGGA(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
REMOVE53_IN_AGGA(T145, T146, T147, .(T146, X170)) → P61_IN_AGGA(T145, T146, T147, X170)
P61_IN_AGGA(T145, T146, T147, X170) → U7_AGGA(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
P61_IN_AGGA(T145, T146, T147, X170) → NOTEQ62_IN_AG(T145, T146)
NOTEQ62_IN_AG(s(T160), s(T161)) → U6_AG(T160, T161, notEq62_in_ag(T160, T161))
NOTEQ62_IN_AG(s(T160), s(T161)) → NOTEQ62_IN_AG(T160, T161)
P61_IN_AGGA(T184, T146, .(T184, T185), T185) → U8_AGGA(T184, T146, T185, notEq62_in_gg(T184, T146))
P61_IN_AGGA(T184, T146, .(T184, T185), T185) → NOTEQ62_IN_GG(T184, T146)
NOTEQ62_IN_GG(s(T160), s(T161)) → U6_GG(T160, T161, notEq62_in_gg(T160, T161))
NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)
P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AGGA(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → NOTEQ62_IN_AG(T192, T146)
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_AGGA(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → P61_IN_AGGA(T192, T193, T194, X229)
U15_GA(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_GA(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_GA(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_GA(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U17_GA(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)
MINSORT1_IN_AA(.(T25, T26), .(T27, T28)) → U14_AA(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
MINSORT1_IN_AA(.(T25, T26), .(T27, T28)) → MIN215_IN_AAA(T25, T27, T26)
MIN215_IN_AAA(T48, T52, .(T50, T51)) → U1_AAA(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
MIN215_IN_AAA(T48, T52, .(T50, T51)) → MIN25_IN_AAA(T48, T50, X59)
MIN25_IN_AAA(T70, T71, T70) → U12_AAA(T70, T71, le30_in_aa(T70, T71))
MIN25_IN_AAA(T70, T71, T70) → LE30_IN_AA(T70, T71)
LE30_IN_AA(s(T82), s(T83)) → U4_AA(T82, T83, le30_in_aa(T82, T83))
LE30_IN_AA(s(T82), s(T83)) → LE30_IN_AA(T82, T83)
MIN25_IN_AAA(T97, T98, T98) → U13_AAA(T97, T98, gt44_in_aa(T97, T98))
MIN25_IN_AAA(T97, T98, T98) → GT44_IN_AA(T97, T98)
GT44_IN_AA(s(T109), s(T110)) → U5_AA(T109, T110, gt44_in_aa(T109, T110))
GT44_IN_AA(s(T109), s(T110)) → GT44_IN_AA(T109, T110)
MIN215_IN_AAA(T48, T52, .(T50, T51)) → U2_AAA(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_AAA(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_AAA(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U2_AAA(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)
MIN215_IN_GAA(T48, T52, .(T50, T51)) → U1_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
MIN215_IN_GAA(T48, T52, .(T50, T51)) → MIN25_IN_GAA(T48, T50, X59)
MIN25_IN_GAA(T70, T71, T70) → U12_GAA(T70, T71, le30_in_ga(T70, T71))
MIN25_IN_GAA(T70, T71, T70) → LE30_IN_GA(T70, T71)
LE30_IN_GA(s(T82), s(T83)) → U4_GA(T82, T83, le30_in_ga(T82, T83))
LE30_IN_GA(s(T82), s(T83)) → LE30_IN_GA(T82, T83)
MIN25_IN_GAA(T97, T98, T98) → U13_GAA(T97, T98, gt44_in_ga(T97, T98))
MIN25_IN_GAA(T97, T98, T98) → GT44_IN_GA(T97, T98)
GT44_IN_GA(s(T109), s(T110)) → U5_GA(T109, T110, gt44_in_ga(T109, T110))
GT44_IN_GA(s(T109), s(T110)) → GT44_IN_GA(T109, T110)
MIN215_IN_GAA(T48, T52, .(T50, T51)) → U2_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_GAA(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)
MINSORT1_IN_AA(.(T25, T26), .(T31, T32)) → U15_AA(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → REMOVE53_IN_AAAA(T31, T25, T26, X20)
REMOVE53_IN_AAAA(T145, T146, T147, .(T146, X170)) → U11_AAAA(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
REMOVE53_IN_AAAA(T145, T146, T147, .(T146, X170)) → P61_IN_AAAA(T145, T146, T147, X170)
P61_IN_AAAA(T145, T146, T147, X170) → U7_AAAA(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
P61_IN_AAAA(T145, T146, T147, X170) → NOTEQ62_IN_AA(T145, T146)
NOTEQ62_IN_AA(s(T160), s(T161)) → U6_AA(T160, T161, notEq62_in_aa(T160, T161))
NOTEQ62_IN_AA(s(T160), s(T161)) → NOTEQ62_IN_AA(T160, T161)
P61_IN_AAAA(T184, T146, .(T184, T185), T185) → U8_AAAA(T184, T146, T185, notEq62_in_aa(T184, T146))
P61_IN_AAAA(T184, T146, .(T184, T185), T185) → NOTEQ62_IN_AA(T184, T146)
P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AAAA(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → NOTEQ62_IN_AA(T192, T146)
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_AAAA(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → P61_IN_AAAA(T192, T193, T194, X229)
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_AA(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MINSORT1_IN_GA(x1, x2)  =  MINSORT1_IN_GA(x1)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x5)
MIN215_IN_GAG(x1, x2, x3)  =  MIN215_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x5)
MIN25_IN_GGA(x1, x2, x3)  =  MIN25_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3)  =  U12_GGA(x1, x3)
LE30_IN_GG(x1, x2)  =  LE30_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x3)
U13_GGA(x1, x2, x3)  =  U13_GGA(x2, x3)
GT44_IN_GG(x1, x2)  =  GT44_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x4, x5)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x5)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x2, x5)
U16_GA(x1, x2, x3, x4, x5)  =  U16_GA(x5)
REMOVE53_IN_AGGA(x1, x2, x3, x4)  =  REMOVE53_IN_AGGA(x2, x3)
U11_AGGA(x1, x2, x3, x4, x5)  =  U11_AGGA(x5)
P61_IN_AGGA(x1, x2, x3, x4)  =  P61_IN_AGGA(x2, x3)
U7_AGGA(x1, x2, x3, x4, x5)  =  U7_AGGA(x5)
NOTEQ62_IN_AG(x1, x2)  =  NOTEQ62_IN_AG(x2)
U6_AG(x1, x2, x3)  =  U6_AG(x3)
U8_AGGA(x1, x2, x3, x4)  =  U8_AGGA(x4)
NOTEQ62_IN_GG(x1, x2)  =  NOTEQ62_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x3)
U9_AGGA(x1, x2, x3, x4, x5, x6)  =  U9_AGGA(x3, x4, x6)
U10_AGGA(x1, x2, x3, x4, x5, x6)  =  U10_AGGA(x6)
U17_GA(x1, x2, x3, x4, x5)  =  U17_GA(x5)
U18_GA(x1, x2, x3, x4, x5)  =  U18_GA(x5)
MINSORT1_IN_AA(x1, x2)  =  MINSORT1_IN_AA
U14_AA(x1, x2, x3, x4, x5)  =  U14_AA(x5)
MIN215_IN_AAA(x1, x2, x3)  =  MIN215_IN_AAA
U1_AAA(x1, x2, x3, x4, x5)  =  U1_AAA(x5)
MIN25_IN_AAA(x1, x2, x3)  =  MIN25_IN_AAA
U12_AAA(x1, x2, x3)  =  U12_AAA(x3)
LE30_IN_AA(x1, x2)  =  LE30_IN_AA
U4_AA(x1, x2, x3)  =  U4_AA(x3)
U13_AAA(x1, x2, x3)  =  U13_AAA(x3)
GT44_IN_AA(x1, x2)  =  GT44_IN_AA
U5_AA(x1, x2, x3)  =  U5_AA(x3)
U2_AAA(x1, x2, x3, x4, x5)  =  U2_AAA(x5)
U3_AAA(x1, x2, x3, x4, x5)  =  U3_AAA(x5)
MIN215_IN_GAA(x1, x2, x3)  =  MIN215_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x5)
MIN25_IN_GAA(x1, x2, x3)  =  MIN25_IN_GAA(x1)
U12_GAA(x1, x2, x3)  =  U12_GAA(x1, x3)
LE30_IN_GA(x1, x2)  =  LE30_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
U13_GAA(x1, x2, x3)  =  U13_GAA(x3)
GT44_IN_GA(x1, x2)  =  GT44_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x5)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U16_AA(x1, x2, x3, x4, x5)  =  U16_AA(x5)
REMOVE53_IN_AAAA(x1, x2, x3, x4)  =  REMOVE53_IN_AAAA
U11_AAAA(x1, x2, x3, x4, x5)  =  U11_AAAA(x5)
P61_IN_AAAA(x1, x2, x3, x4)  =  P61_IN_AAAA
U7_AAAA(x1, x2, x3, x4, x5)  =  U7_AAAA(x5)
NOTEQ62_IN_AA(x1, x2)  =  NOTEQ62_IN_AA
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U8_AAAA(x1, x2, x3, x4)  =  U8_AAAA(x4)
U9_AAAA(x1, x2, x3, x4, x5, x6)  =  U9_AAAA(x6)
U10_AAAA(x1, x2, x3, x4, x5, x6)  =  U10_AAAA(x6)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)
U18_AA(x1, x2, x3, x4, x5)  =  U18_AA(x5)

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

(137) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 14 SCCs with 61 less nodes.

(138) Complex Obligation (AND)

(139) Obligation:

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

NOTEQ62_IN_AA(s(T160), s(T161)) → NOTEQ62_IN_AA(T160, T161)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
NOTEQ62_IN_AA(x1, x2)  =  NOTEQ62_IN_AA

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

(140) UsableRulesProof (EQUIVALENT transformation)

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

(141) Obligation:

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

NOTEQ62_IN_AA(s(T160), s(T161)) → NOTEQ62_IN_AA(T160, T161)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
NOTEQ62_IN_AA(x1, x2)  =  NOTEQ62_IN_AA

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

(142) PiDPToQDPProof (SOUND transformation)

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

(143) Obligation:

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

NOTEQ62_IN_AANOTEQ62_IN_AA

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

(144) NonTerminationProof (EQUIVALENT transformation)

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

s = NOTEQ62_IN_AA evaluates to t =NOTEQ62_IN_AA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from NOTEQ62_IN_AA to NOTEQ62_IN_AA.



(145) NO

(146) Obligation:

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

P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AAAA(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → P61_IN_AAAA(T192, T193, T194, X229)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
P61_IN_AAAA(x1, x2, x3, x4)  =  P61_IN_AAAA
U9_AAAA(x1, x2, x3, x4, x5, x6)  =  U9_AAAA(x6)

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

(147) UsableRulesProof (EQUIVALENT transformation)

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

(148) Obligation:

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

P61_IN_AAAA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AAAA(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_AAAA(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → P61_IN_AAAA(T192, T193, T194, X229)

The TRS R consists of the following rules:

notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
P61_IN_AAAA(x1, x2, x3, x4)  =  P61_IN_AAAA
U9_AAAA(x1, x2, x3, x4, x5, x6)  =  U9_AAAA(x6)

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

(149) PiDPToQDPProof (SOUND transformation)

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

(150) Obligation:

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

P61_IN_AAAAU9_AAAA(notEq62_in_aa)
U9_AAAA(notEq62_out_aa) → P61_IN_AAAA

The TRS R consists of the following rules:

notEq62_in_aaU6_aa(notEq62_in_aa)
notEq62_in_aanotEq62_out_aa
U6_aa(notEq62_out_aa) → notEq62_out_aa

The set Q consists of the following terms:

notEq62_in_aa
U6_aa(x0)

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

(151) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P61_IN_AAAAU9_AAAA(notEq62_in_aa) at position [0] we obtained the following new rules [LPAR04]:

P61_IN_AAAAU9_AAAA(U6_aa(notEq62_in_aa))
P61_IN_AAAAU9_AAAA(notEq62_out_aa)

(152) Obligation:

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

U9_AAAA(notEq62_out_aa) → P61_IN_AAAA
P61_IN_AAAAU9_AAAA(U6_aa(notEq62_in_aa))
P61_IN_AAAAU9_AAAA(notEq62_out_aa)

The TRS R consists of the following rules:

notEq62_in_aaU6_aa(notEq62_in_aa)
notEq62_in_aanotEq62_out_aa
U6_aa(notEq62_out_aa) → notEq62_out_aa

The set Q consists of the following terms:

notEq62_in_aa
U6_aa(x0)

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

(153) NonTerminationProof (EQUIVALENT transformation)

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

s = P61_IN_AAAA evaluates to t =P61_IN_AAAA

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




Rewriting sequence

P61_IN_AAAAU9_AAAA(notEq62_out_aa)
with rule P61_IN_AAAAU9_AAAA(notEq62_out_aa) at position [] and matcher [ ]

U9_AAAA(notEq62_out_aa)P61_IN_AAAA
with rule U9_AAAA(notEq62_out_aa) → P61_IN_AAAA

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


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



(154) NO

(155) Obligation:

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

GT44_IN_GA(s(T109), s(T110)) → GT44_IN_GA(T109, T110)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
GT44_IN_GA(x1, x2)  =  GT44_IN_GA(x1)

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

(156) UsableRulesProof (EQUIVALENT transformation)

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

(157) Obligation:

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

GT44_IN_GA(s(T109), s(T110)) → GT44_IN_GA(T109, T110)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT44_IN_GA(x1, x2)  =  GT44_IN_GA(x1)

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

(158) PiDPToQDPProof (SOUND transformation)

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

(159) Obligation:

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

GT44_IN_GA(s(T109)) → GT44_IN_GA(T109)

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

(160) 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:

  • GT44_IN_GA(s(T109)) → GT44_IN_GA(T109)
    The graph contains the following edges 1 > 1

(161) YES

(162) Obligation:

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

LE30_IN_GA(s(T82), s(T83)) → LE30_IN_GA(T82, T83)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
LE30_IN_GA(x1, x2)  =  LE30_IN_GA(x1)

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

(163) UsableRulesProof (EQUIVALENT transformation)

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

(164) Obligation:

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

LE30_IN_GA(s(T82), s(T83)) → LE30_IN_GA(T82, T83)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE30_IN_GA(x1, x2)  =  LE30_IN_GA(x1)

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

(165) PiDPToQDPProof (SOUND transformation)

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

(166) Obligation:

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

LE30_IN_GA(s(T82)) → LE30_IN_GA(T82)

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

(167) 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:

  • LE30_IN_GA(s(T82)) → LE30_IN_GA(T82)
    The graph contains the following edges 1 > 1

(168) YES

(169) Obligation:

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

MIN215_IN_GAA(T48, T52, .(T50, T51)) → U2_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MIN215_IN_GAA(x1, x2, x3)  =  MIN215_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x5)

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

(170) UsableRulesProof (EQUIVALENT transformation)

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

(171) Obligation:

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

MIN215_IN_GAA(T48, T52, .(T50, T51)) → U2_GAA(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_GAA(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → MIN215_IN_GAA(T55, T52, T51)

The TRS R consists of the following rules:

min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
MIN215_IN_GAA(x1, x2, x3)  =  MIN215_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x5)

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

(172) PiDPToQDPProof (SOUND transformation)

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

(173) Obligation:

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

MIN215_IN_GAA(T48) → U2_GAA(min25_in_gaa(T48))
U2_GAA(min25_out_gaa(T55)) → MIN215_IN_GAA(T55)

The TRS R consists of the following rules:

min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(gt44_in_ga(T97))
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U4_ga(le30_out_ga) → le30_out_ga
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))

The set Q consists of the following terms:

min25_in_gaa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(174) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MIN215_IN_GAA(T48) → U2_GAA(min25_in_gaa(T48)) at position [0] we obtained the following new rules [LPAR04]:

MIN215_IN_GAA(x0) → U2_GAA(U12_gaa(x0, le30_in_ga(x0)))
MIN215_IN_GAA(x0) → U2_GAA(U13_gaa(gt44_in_ga(x0)))

(175) Obligation:

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

U2_GAA(min25_out_gaa(T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(U12_gaa(x0, le30_in_ga(x0)))
MIN215_IN_GAA(x0) → U2_GAA(U13_gaa(gt44_in_ga(x0)))

The TRS R consists of the following rules:

min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(gt44_in_ga(T97))
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U4_ga(le30_out_ga) → le30_out_ga
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))

The set Q consists of the following terms:

min25_in_gaa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(176) 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.

(177) Obligation:

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

U2_GAA(min25_out_gaa(T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(U12_gaa(x0, le30_in_ga(x0)))
MIN215_IN_GAA(x0) → U2_GAA(U13_gaa(gt44_in_ga(x0)))

The TRS R consists of the following rules:

gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga

The set Q consists of the following terms:

min25_in_gaa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(178) 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].

min25_in_gaa(x0)

(179) Obligation:

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

U2_GAA(min25_out_gaa(T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(U12_gaa(x0, le30_in_ga(x0)))
MIN215_IN_GAA(x0) → U2_GAA(U13_gaa(gt44_in_ga(x0)))

The TRS R consists of the following rules:

gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga

The set Q consists of the following terms:

U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(180) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MIN215_IN_GAA(x0) → U2_GAA(U13_gaa(gt44_in_ga(x0)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(MIN215_IN_GAA(x1)) = 1 + x1   
POL(U12_gaa(x1, x2)) = 1 + x1   
POL(U13_gaa(x1)) = x1   
POL(U2_GAA(x1)) = x1   
POL(U4_ga(x1)) = 0   
POL(U5_ga(x1)) = 1 + x1   
POL(gt44_in_ga(x1)) = x1   
POL(gt44_out_ga(x1)) = 1 + x1   
POL(le30_in_ga(x1)) = 0   
POL(le30_out_ga) = 0   
POL(min25_out_gaa(x1)) = 1 + x1   
POL(s(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented:

U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))

(181) Obligation:

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

U2_GAA(min25_out_gaa(T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(U12_gaa(x0, le30_in_ga(x0)))

The TRS R consists of the following rules:

gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga

The set Q consists of the following terms:

U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(182) 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.

(183) Obligation:

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

U2_GAA(min25_out_gaa(T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(U12_gaa(x0, le30_in_ga(x0)))

The TRS R consists of the following rules:

le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga

The set Q consists of the following terms:

U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(184) 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].

U13_gaa(x0)
gt44_in_ga(x0)
U5_ga(x0)

(185) Obligation:

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

U2_GAA(min25_out_gaa(T55)) → MIN215_IN_GAA(T55)
MIN215_IN_GAA(x0) → U2_GAA(U12_gaa(x0, le30_in_ga(x0)))

The TRS R consists of the following rules:

le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga

The set Q consists of the following terms:

U12_gaa(x0, x1)
le30_in_ga(x0)
U4_ga(x0)

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

(186) NonTerminationProof (EQUIVALENT transformation)

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

s = U2_GAA(U12_gaa(T70, le30_in_ga(0))) evaluates to t =U2_GAA(U12_gaa(T70, le30_in_ga(T70)))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [T70 / 0]
  • Matcher: [ ]




Rewriting sequence

U2_GAA(U12_gaa(0, le30_in_ga(0)))U2_GAA(U12_gaa(0, le30_out_ga))
with rule le30_in_ga(0) → le30_out_ga at position [0,1] and matcher [ ]

U2_GAA(U12_gaa(0, le30_out_ga))U2_GAA(min25_out_gaa(0))
with rule U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70) at position [0] and matcher [T70 / 0]

U2_GAA(min25_out_gaa(0))MIN215_IN_GAA(0)
with rule U2_GAA(min25_out_gaa(T55)) → MIN215_IN_GAA(T55) at position [] and matcher [T55 / 0]

MIN215_IN_GAA(0)U2_GAA(U12_gaa(0, le30_in_ga(0)))
with rule MIN215_IN_GAA(x0) → U2_GAA(U12_gaa(x0, le30_in_ga(x0)))

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


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



(187) NO

(188) Obligation:

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

GT44_IN_AA(s(T109), s(T110)) → GT44_IN_AA(T109, T110)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
GT44_IN_AA(x1, x2)  =  GT44_IN_AA

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

(189) UsableRulesProof (EQUIVALENT transformation)

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

(190) Obligation:

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

GT44_IN_AA(s(T109), s(T110)) → GT44_IN_AA(T109, T110)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
GT44_IN_AA(x1, x2)  =  GT44_IN_AA

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

(191) PiDPToQDPProof (SOUND transformation)

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

(192) Obligation:

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

GT44_IN_AAGT44_IN_AA

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

(193) 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 = GT44_IN_AA evaluates to t =GT44_IN_AA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from GT44_IN_AA to GT44_IN_AA.



(194) NO

(195) Obligation:

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

LE30_IN_AA(s(T82), s(T83)) → LE30_IN_AA(T82, T83)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
LE30_IN_AA(x1, x2)  =  LE30_IN_AA

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

(196) UsableRulesProof (EQUIVALENT transformation)

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

(197) Obligation:

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

LE30_IN_AA(s(T82), s(T83)) → LE30_IN_AA(T82, T83)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE30_IN_AA(x1, x2)  =  LE30_IN_AA

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

(198) PiDPToQDPProof (SOUND transformation)

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

(199) Obligation:

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

LE30_IN_AALE30_IN_AA

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

(200) 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 = LE30_IN_AA evaluates to t =LE30_IN_AA

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LE30_IN_AA to LE30_IN_AA.



(201) NO

(202) Obligation:

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

MINSORT1_IN_AA(.(T25, T26), .(T31, T32)) → U15_AA(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MINSORT1_IN_AA(x1, x2)  =  MINSORT1_IN_AA
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)

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

(203) UsableRulesProof (EQUIVALENT transformation)

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

(204) Obligation:

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

MINSORT1_IN_AA(.(T25, T26), .(T31, T32)) → U15_AA(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_AA(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_AA(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_AA(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → MINSORT1_IN_AA(T122, T32)

The TRS R consists of the following rules:

min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
MINSORT1_IN_AA(x1, x2)  =  MINSORT1_IN_AA
U15_AA(x1, x2, x3, x4, x5)  =  U15_AA(x5)
U17_AA(x1, x2, x3, x4, x5)  =  U17_AA(x5)

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

(205) PiDPToQDPProof (SOUND transformation)

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

(206) Obligation:

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

MINSORT1_IN_AAU15_AA(min215_in_aaa)
U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa)
U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA

The TRS R consists of the following rules:

min215_in_aaamin215_out_aaa
min215_in_aaaU1_aaa(min25_in_aaa)
min215_in_aaaU2_aaa(min25_in_aaa)
remove53_in_aaaaremove53_out_aaaa
remove53_in_aaaaU11_aaaa(p61_in_aaaa)
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
min25_in_aaaU12_aaa(le30_in_aa)
min25_in_aaaU13_aaa(gt44_in_aa)
U3_aaa(min215_out_gaa) → min215_out_aaa
p61_in_aaaaU7_aaaa(notEq62_in_aa)
p61_in_aaaaU8_aaaa(notEq62_in_aa)
p61_in_aaaaU9_aaaa(notEq62_in_aa)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
min215_in_gaa(T39) → min215_out_gaa
min215_in_gaa(T48) → U1_gaa(min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(min25_in_gaa(T48))
U7_aaaa(notEq62_out_aa) → p61_out_aaaa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
le30_in_aaU4_aa(le30_in_aa)
le30_in_aale30_out_aa(0)
gt44_in_aaU5_aa(gt44_in_aa)
gt44_in_aagt44_out_aa(0)
U1_gaa(min25_out_gaa(X59)) → min215_out_gaa
U2_gaa(min25_out_gaa(T55)) → U3_gaa(min215_in_gaa(T55))
notEq62_in_aaU6_aa(notEq62_in_aa)
notEq62_in_aanotEq62_out_aa
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(gt44_in_ga(T97))
U3_gaa(min215_out_gaa) → min215_out_gaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U4_ga(le30_out_ga) → le30_out_ga
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))

The set Q consists of the following terms:

min215_in_aaa
remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0)
U2_gaa(x0)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(207) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MINSORT1_IN_AAU15_AA(min215_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

MINSORT1_IN_AAU15_AA(min215_out_aaa)
MINSORT1_IN_AAU15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AAU15_AA(U2_aaa(min25_in_aaa))

(208) Obligation:

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

U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa)
U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AAU15_AA(min215_out_aaa)
MINSORT1_IN_AAU15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AAU15_AA(U2_aaa(min25_in_aaa))

The TRS R consists of the following rules:

min215_in_aaamin215_out_aaa
min215_in_aaaU1_aaa(min25_in_aaa)
min215_in_aaaU2_aaa(min25_in_aaa)
remove53_in_aaaaremove53_out_aaaa
remove53_in_aaaaU11_aaaa(p61_in_aaaa)
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
min25_in_aaaU12_aaa(le30_in_aa)
min25_in_aaaU13_aaa(gt44_in_aa)
U3_aaa(min215_out_gaa) → min215_out_aaa
p61_in_aaaaU7_aaaa(notEq62_in_aa)
p61_in_aaaaU8_aaaa(notEq62_in_aa)
p61_in_aaaaU9_aaaa(notEq62_in_aa)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
min215_in_gaa(T39) → min215_out_gaa
min215_in_gaa(T48) → U1_gaa(min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(min25_in_gaa(T48))
U7_aaaa(notEq62_out_aa) → p61_out_aaaa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
le30_in_aaU4_aa(le30_in_aa)
le30_in_aale30_out_aa(0)
gt44_in_aaU5_aa(gt44_in_aa)
gt44_in_aagt44_out_aa(0)
U1_gaa(min25_out_gaa(X59)) → min215_out_gaa
U2_gaa(min25_out_gaa(T55)) → U3_gaa(min215_in_gaa(T55))
notEq62_in_aaU6_aa(notEq62_in_aa)
notEq62_in_aanotEq62_out_aa
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(gt44_in_ga(T97))
U3_gaa(min215_out_gaa) → min215_out_gaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U4_ga(le30_out_ga) → le30_out_ga
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))

The set Q consists of the following terms:

min215_in_aaa
remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0)
U2_gaa(x0)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(209) 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.

(210) Obligation:

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

U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa)
U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AAU15_AA(min215_out_aaa)
MINSORT1_IN_AAU15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AAU15_AA(U2_aaa(min25_in_aaa))

The TRS R consists of the following rules:

min25_in_aaaU12_aaa(le30_in_aa)
min25_in_aaaU13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa
min215_in_gaa(T48) → U1_gaa(min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(min25_in_gaa(T48))
U3_aaa(min215_out_gaa) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(gt44_in_ga(T97))
U2_gaa(min25_out_gaa(T55)) → U3_gaa(min215_in_gaa(T55))
U3_gaa(min215_out_gaa) → min215_out_gaa
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga
U1_gaa(min25_out_gaa(X59)) → min215_out_gaa
gt44_in_aaU5_aa(gt44_in_aa)
gt44_in_aagt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aaU4_aa(le30_in_aa)
le30_in_aale30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
remove53_in_aaaaremove53_out_aaaa
remove53_in_aaaaU11_aaaa(p61_in_aaaa)
p61_in_aaaaU7_aaaa(notEq62_in_aa)
p61_in_aaaaU8_aaaa(notEq62_in_aa)
p61_in_aaaaU9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aaU6_aa(notEq62_in_aa)
notEq62_in_aanotEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa

The set Q consists of the following terms:

min215_in_aaa
remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0)
U2_gaa(x0)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(211) 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].

min215_in_aaa

(212) Obligation:

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

U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa)
U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AAU15_AA(min215_out_aaa)
MINSORT1_IN_AAU15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AAU15_AA(U2_aaa(min25_in_aaa))

The TRS R consists of the following rules:

min25_in_aaaU12_aaa(le30_in_aa)
min25_in_aaaU13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa
min215_in_gaa(T48) → U1_gaa(min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(min25_in_gaa(T48))
U3_aaa(min215_out_gaa) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(gt44_in_ga(T97))
U2_gaa(min25_out_gaa(T55)) → U3_gaa(min215_in_gaa(T55))
U3_gaa(min215_out_gaa) → min215_out_gaa
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga
U1_gaa(min25_out_gaa(X59)) → min215_out_gaa
gt44_in_aaU5_aa(gt44_in_aa)
gt44_in_aagt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aaU4_aa(le30_in_aa)
le30_in_aale30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
remove53_in_aaaaremove53_out_aaaa
remove53_in_aaaaU11_aaaa(p61_in_aaaa)
p61_in_aaaaU7_aaaa(notEq62_in_aa)
p61_in_aaaaU8_aaaa(notEq62_in_aa)
p61_in_aaaaU9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aaU6_aa(notEq62_in_aa)
notEq62_in_aanotEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa

The set Q consists of the following terms:

remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0)
U2_gaa(x0)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(213) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U15_AA(min215_out_aaa) → U17_AA(remove53_in_aaaa) at position [0] we obtained the following new rules [LPAR04]:

U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa)
U15_AA(min215_out_aaa) → U17_AA(U11_aaaa(p61_in_aaaa))

(214) Obligation:

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

U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AAU15_AA(min215_out_aaa)
MINSORT1_IN_AAU15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AAU15_AA(U2_aaa(min25_in_aaa))
U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa)
U15_AA(min215_out_aaa) → U17_AA(U11_aaaa(p61_in_aaaa))

The TRS R consists of the following rules:

min25_in_aaaU12_aaa(le30_in_aa)
min25_in_aaaU13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa
min215_in_gaa(T48) → U1_gaa(min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(min25_in_gaa(T48))
U3_aaa(min215_out_gaa) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(gt44_in_ga(T97))
U2_gaa(min25_out_gaa(T55)) → U3_gaa(min215_in_gaa(T55))
U3_gaa(min215_out_gaa) → min215_out_gaa
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga
U1_gaa(min25_out_gaa(X59)) → min215_out_gaa
gt44_in_aaU5_aa(gt44_in_aa)
gt44_in_aagt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aaU4_aa(le30_in_aa)
le30_in_aale30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa
remove53_in_aaaaremove53_out_aaaa
remove53_in_aaaaU11_aaaa(p61_in_aaaa)
p61_in_aaaaU7_aaaa(notEq62_in_aa)
p61_in_aaaaU8_aaaa(notEq62_in_aa)
p61_in_aaaaU9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aaU6_aa(notEq62_in_aa)
notEq62_in_aanotEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa

The set Q consists of the following terms:

remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0)
U2_gaa(x0)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(215) 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.

(216) Obligation:

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

U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AAU15_AA(min215_out_aaa)
MINSORT1_IN_AAU15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AAU15_AA(U2_aaa(min25_in_aaa))
U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa)
U15_AA(min215_out_aaa) → U17_AA(U11_aaaa(p61_in_aaaa))

The TRS R consists of the following rules:

p61_in_aaaaU7_aaaa(notEq62_in_aa)
p61_in_aaaaU8_aaaa(notEq62_in_aa)
p61_in_aaaaU9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aaU6_aa(notEq62_in_aa)
notEq62_in_aanotEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa
min25_in_aaaU12_aaa(le30_in_aa)
min25_in_aaaU13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa
min215_in_gaa(T48) → U1_gaa(min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(min25_in_gaa(T48))
U3_aaa(min215_out_gaa) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(gt44_in_ga(T97))
U2_gaa(min25_out_gaa(T55)) → U3_gaa(min215_in_gaa(T55))
U3_gaa(min215_out_gaa) → min215_out_gaa
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga
U1_gaa(min25_out_gaa(X59)) → min215_out_gaa
gt44_in_aaU5_aa(gt44_in_aa)
gt44_in_aagt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aaU4_aa(le30_in_aa)
le30_in_aale30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa

The set Q consists of the following terms:

remove53_in_aaaa
U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0)
U2_gaa(x0)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(217) 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].

remove53_in_aaaa

(218) Obligation:

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

U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA
MINSORT1_IN_AAU15_AA(min215_out_aaa)
MINSORT1_IN_AAU15_AA(U1_aaa(min25_in_aaa))
MINSORT1_IN_AAU15_AA(U2_aaa(min25_in_aaa))
U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa)
U15_AA(min215_out_aaa) → U17_AA(U11_aaaa(p61_in_aaaa))

The TRS R consists of the following rules:

p61_in_aaaaU7_aaaa(notEq62_in_aa)
p61_in_aaaaU8_aaaa(notEq62_in_aa)
p61_in_aaaaU9_aaaa(notEq62_in_aa)
U11_aaaa(p61_out_aaaa) → remove53_out_aaaa
notEq62_in_aaU6_aa(notEq62_in_aa)
notEq62_in_aanotEq62_out_aa
U9_aaaa(notEq62_out_aa) → U10_aaaa(p61_in_aaaa)
U10_aaaa(p61_out_aaaa) → p61_out_aaaa
U6_aa(notEq62_out_aa) → notEq62_out_aa
U8_aaaa(notEq62_out_aa) → p61_out_aaaa
U7_aaaa(notEq62_out_aa) → p61_out_aaaa
min25_in_aaaU12_aaa(le30_in_aa)
min25_in_aaaU13_aaa(gt44_in_aa)
U2_aaa(min25_out_aaa(T55)) → U3_aaa(min215_in_gaa(T55))
min215_in_gaa(T39) → min215_out_gaa
min215_in_gaa(T48) → U1_gaa(min25_in_gaa(T48))
min215_in_gaa(T48) → U2_gaa(min25_in_gaa(T48))
U3_aaa(min215_out_gaa) → min215_out_aaa
min25_in_gaa(T70) → U12_gaa(T70, le30_in_ga(T70))
min25_in_gaa(T97) → U13_gaa(gt44_in_ga(T97))
U2_gaa(min25_out_gaa(T55)) → U3_gaa(min215_in_gaa(T55))
U3_gaa(min215_out_gaa) → min215_out_gaa
gt44_in_ga(s(T109)) → U5_ga(gt44_in_ga(T109))
gt44_in_ga(s(T115)) → gt44_out_ga(0)
U13_gaa(gt44_out_ga(T98)) → min25_out_gaa(T98)
U5_ga(gt44_out_ga(T110)) → gt44_out_ga(s(T110))
le30_in_ga(s(T82)) → U4_ga(le30_in_ga(T82))
le30_in_ga(0) → le30_out_ga
U12_gaa(T70, le30_out_ga) → min25_out_gaa(T70)
U4_ga(le30_out_ga) → le30_out_ga
U1_gaa(min25_out_gaa(X59)) → min215_out_gaa
gt44_in_aaU5_aa(gt44_in_aa)
gt44_in_aagt44_out_aa(0)
U13_aaa(gt44_out_aa(T98)) → min25_out_aaa(T98)
U5_aa(gt44_out_aa(T110)) → gt44_out_aa(s(T110))
le30_in_aaU4_aa(le30_in_aa)
le30_in_aale30_out_aa(0)
U12_aaa(le30_out_aa(T70)) → min25_out_aaa(T70)
U4_aa(le30_out_aa(T82)) → le30_out_aa(s(T82))
U1_aaa(min25_out_aaa(X59)) → min215_out_aaa

The set Q consists of the following terms:

U1_aaa(x0)
U2_aaa(x0)
U11_aaaa(x0)
min25_in_aaa
U3_aaa(x0)
p61_in_aaaa
U12_aaa(x0)
U13_aaa(x0)
min215_in_gaa(x0)
U7_aaaa(x0)
U8_aaaa(x0)
U9_aaaa(x0)
le30_in_aa
gt44_in_aa
U1_gaa(x0)
U2_gaa(x0)
notEq62_in_aa
U10_aaaa(x0)
U4_aa(x0)
U5_aa(x0)
min25_in_gaa(x0)
U3_gaa(x0)
U6_aa(x0)
U12_gaa(x0, x1)
U13_gaa(x0)
le30_in_ga(x0)
gt44_in_ga(x0)
U4_ga(x0)
U5_ga(x0)

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

(219) NonTerminationProof (EQUIVALENT transformation)

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

s = MINSORT1_IN_AA evaluates to t =MINSORT1_IN_AA

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




Rewriting sequence

MINSORT1_IN_AAU15_AA(min215_out_aaa)
with rule MINSORT1_IN_AAU15_AA(min215_out_aaa) at position [] and matcher [ ]

U15_AA(min215_out_aaa)U17_AA(remove53_out_aaaa)
with rule U15_AA(min215_out_aaa) → U17_AA(remove53_out_aaaa) at position [] and matcher [ ]

U17_AA(remove53_out_aaaa)MINSORT1_IN_AA
with rule U17_AA(remove53_out_aaaa) → MINSORT1_IN_AA

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


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



(220) NO

(221) Obligation:

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

NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
NOTEQ62_IN_GG(x1, x2)  =  NOTEQ62_IN_GG(x1, x2)

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

(222) UsableRulesProof (EQUIVALENT transformation)

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

(223) Obligation:

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

NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)

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

(224) PiDPToQDPProof (EQUIVALENT transformation)

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

(225) Obligation:

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

NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)

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

(226) 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:

  • NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)
    The graph contains the following edges 1 > 1, 2 > 2

(227) YES

(228) Obligation:

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

NOTEQ62_IN_AG(s(T160), s(T161)) → NOTEQ62_IN_AG(T160, T161)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
NOTEQ62_IN_AG(x1, x2)  =  NOTEQ62_IN_AG(x2)

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

(229) UsableRulesProof (EQUIVALENT transformation)

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

(230) Obligation:

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

NOTEQ62_IN_AG(s(T160), s(T161)) → NOTEQ62_IN_AG(T160, T161)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
NOTEQ62_IN_AG(x1, x2)  =  NOTEQ62_IN_AG(x2)

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

(231) PiDPToQDPProof (SOUND transformation)

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

(232) Obligation:

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

NOTEQ62_IN_AG(s(T161)) → NOTEQ62_IN_AG(T161)

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

(233) 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:

  • NOTEQ62_IN_AG(s(T161)) → NOTEQ62_IN_AG(T161)
    The graph contains the following edges 1 > 1

(234) YES

(235) Obligation:

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

P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AGGA(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → P61_IN_AGGA(T192, T193, T194, X229)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
P61_IN_AGGA(x1, x2, x3, x4)  =  P61_IN_AGGA(x2, x3)
U9_AGGA(x1, x2, x3, x4, x5, x6)  =  U9_AGGA(x3, x4, x6)

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

(236) UsableRulesProof (EQUIVALENT transformation)

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

(237) Obligation:

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

P61_IN_AGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_AGGA(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_AGGA(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → P61_IN_AGGA(T192, T193, T194, X229)

The TRS R consists of the following rules:

notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
P61_IN_AGGA(x1, x2, x3, x4)  =  P61_IN_AGGA(x2, x3)
U9_AGGA(x1, x2, x3, x4, x5, x6)  =  U9_AGGA(x3, x4, x6)

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

(238) PiDPToQDPProof (SOUND transformation)

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

(239) Obligation:

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

P61_IN_AGGA(T146, .(T193, T194)) → U9_AGGA(T193, T194, notEq62_in_ag(T146))
U9_AGGA(T193, T194, notEq62_out_ag) → P61_IN_AGGA(T193, T194)

The TRS R consists of the following rules:

notEq62_in_ag(s(T161)) → U6_ag(notEq62_in_ag(T161))
notEq62_in_ag(0) → notEq62_out_ag
notEq62_in_ag(s(T171)) → notEq62_out_ag
U6_ag(notEq62_out_ag) → notEq62_out_ag

The set Q consists of the following terms:

notEq62_in_ag(x0)
U6_ag(x0)

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

(240) 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:

  • U9_AGGA(T193, T194, notEq62_out_ag) → P61_IN_AGGA(T193, T194)
    The graph contains the following edges 1 >= 1, 2 >= 2

  • P61_IN_AGGA(T146, .(T193, T194)) → U9_AGGA(T193, T194, notEq62_in_ag(T146))
    The graph contains the following edges 2 > 1, 2 > 2

(241) YES

(242) Obligation:

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

GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
GT44_IN_GG(x1, x2)  =  GT44_IN_GG(x1, x2)

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

(243) UsableRulesProof (EQUIVALENT transformation)

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

(244) Obligation:

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

GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)

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

(245) PiDPToQDPProof (EQUIVALENT transformation)

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

(246) Obligation:

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

GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)

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

(247) 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:

  • GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)
    The graph contains the following edges 1 > 1, 2 > 2

(248) YES

(249) Obligation:

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

LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
LE30_IN_GG(x1, x2)  =  LE30_IN_GG(x1, x2)

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

(250) UsableRulesProof (EQUIVALENT transformation)

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

(251) Obligation:

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

LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)

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

(252) PiDPToQDPProof (EQUIVALENT transformation)

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

(253) Obligation:

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

LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)

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

(254) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)
    The graph contains the following edges 1 > 1, 2 > 2

(255) YES

(256) Obligation:

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

MIN215_IN_GAG(T48, T52, .(T50, T51)) → U2_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)

The TRS R consists of the following rules:

minsort1_in_ga([], []) → minsort1_out_ga([], [])
minsort1_in_ga(.(T25, T26), .(T27, T28)) → U14_ga(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
min215_in_gag(T39, T39, []) → min215_out_gag(T39, T39, [])
min215_in_gag(T48, T52, .(T50, T51)) → U1_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, X59))
min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
U1_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, X59)) → min215_out_gag(T48, T52, .(T50, T51))
min215_in_gag(T48, T52, .(T50, T51)) → U2_gag(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_gag(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → U3_gag(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_gag(T48, T52, T50, T51, min215_out_gag(T55, T52, T51)) → min215_out_gag(T48, T52, .(T50, T51))
U14_ga(T25, T26, T27, T28, min215_out_gag(T25, T27, T26)) → minsort1_out_ga(.(T25, T26), .(T27, T28))
minsort1_in_ga(.(T25, T26), .(T31, T32)) → U15_ga(T25, T26, T31, T32, min215_in_gag(T25, T31, T26))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U16_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, X20))
remove53_in_agga(T135, T135, T136, T136) → remove53_out_agga(T135, T135, T136, T136)
remove53_in_agga(T145, T146, T147, .(T146, X170)) → U11_agga(T145, T146, T147, X170, p61_in_agga(T145, T146, T147, X170))
p61_in_agga(T145, T146, T147, X170) → U7_agga(T145, T146, T147, X170, notEq62_in_ag(T145, T146))
notEq62_in_ag(s(T160), s(T161)) → U6_ag(T160, T161, notEq62_in_ag(T160, T161))
notEq62_in_ag(s(T168), 0) → notEq62_out_ag(s(T168), 0)
notEq62_in_ag(0, s(T171)) → notEq62_out_ag(0, s(T171))
U6_ag(T160, T161, notEq62_out_ag(T160, T161)) → notEq62_out_ag(s(T160), s(T161))
U7_agga(T145, T146, T147, X170, notEq62_out_ag(T145, T146)) → p61_out_agga(T145, T146, T147, X170)
p61_in_agga(T184, T146, .(T184, T185), T185) → U8_agga(T184, T146, T185, notEq62_in_gg(T184, T146))
notEq62_in_gg(s(T160), s(T161)) → U6_gg(T160, T161, notEq62_in_gg(T160, T161))
notEq62_in_gg(s(T168), 0) → notEq62_out_gg(s(T168), 0)
notEq62_in_gg(0, s(T171)) → notEq62_out_gg(0, s(T171))
U6_gg(T160, T161, notEq62_out_gg(T160, T161)) → notEq62_out_gg(s(T160), s(T161))
U8_agga(T184, T146, T185, notEq62_out_gg(T184, T146)) → p61_out_agga(T184, T146, .(T184, T185), T185)
p61_in_agga(T192, T146, .(T193, T194), .(T193, X229)) → U9_agga(T192, T146, T193, T194, X229, notEq62_in_ag(T192, T146))
U9_agga(T192, T146, T193, T194, X229, notEq62_out_ag(T192, T146)) → U10_agga(T192, T146, T193, T194, X229, p61_in_agga(T192, T193, T194, X229))
U10_agga(T192, T146, T193, T194, X229, p61_out_agga(T192, T193, T194, X229)) → p61_out_agga(T192, T146, .(T193, T194), .(T193, X229))
U11_agga(T145, T146, T147, X170, p61_out_agga(T145, T146, T147, X170)) → remove53_out_agga(T145, T146, T147, .(T146, X170))
U16_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, X20)) → minsort1_out_ga(.(T25, T26), .(T31, T32))
U15_ga(T25, T26, T31, T32, min215_out_gag(T25, T31, T26)) → U17_ga(T25, T26, T31, T32, remove53_in_agga(T31, T25, T26, T122))
U17_ga(T25, T26, T31, T32, remove53_out_agga(T31, T25, T26, T122)) → U18_ga(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
minsort1_in_aa([], []) → minsort1_out_aa([], [])
minsort1_in_aa(.(T25, T26), .(T27, T28)) → U14_aa(T25, T26, T27, T28, min215_in_aaa(T25, T27, T26))
min215_in_aaa(T39, T39, []) → min215_out_aaa(T39, T39, [])
min215_in_aaa(T48, T52, .(T50, T51)) → U1_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, X59))
min25_in_aaa(T70, T71, T70) → U12_aaa(T70, T71, le30_in_aa(T70, T71))
le30_in_aa(s(T82), s(T83)) → U4_aa(T82, T83, le30_in_aa(T82, T83))
le30_in_aa(0, s(T90)) → le30_out_aa(0, s(T90))
le30_in_aa(0, 0) → le30_out_aa(0, 0)
U4_aa(T82, T83, le30_out_aa(T82, T83)) → le30_out_aa(s(T82), s(T83))
U12_aaa(T70, T71, le30_out_aa(T70, T71)) → min25_out_aaa(T70, T71, T70)
min25_in_aaa(T97, T98, T98) → U13_aaa(T97, T98, gt44_in_aa(T97, T98))
gt44_in_aa(s(T109), s(T110)) → U5_aa(T109, T110, gt44_in_aa(T109, T110))
gt44_in_aa(s(T115), 0) → gt44_out_aa(s(T115), 0)
U5_aa(T109, T110, gt44_out_aa(T109, T110)) → gt44_out_aa(s(T109), s(T110))
U13_aaa(T97, T98, gt44_out_aa(T97, T98)) → min25_out_aaa(T97, T98, T98)
U1_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, X59)) → min215_out_aaa(T48, T52, .(T50, T51))
min215_in_aaa(T48, T52, .(T50, T51)) → U2_aaa(T48, T52, T50, T51, min25_in_aaa(T48, T50, T55))
U2_aaa(T48, T52, T50, T51, min25_out_aaa(T48, T50, T55)) → U3_aaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
min215_in_gaa(T39, T39, []) → min215_out_gaa(T39, T39, [])
min215_in_gaa(T48, T52, .(T50, T51)) → U1_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, X59))
min25_in_gaa(T70, T71, T70) → U12_gaa(T70, T71, le30_in_ga(T70, T71))
le30_in_ga(s(T82), s(T83)) → U4_ga(T82, T83, le30_in_ga(T82, T83))
le30_in_ga(0, s(T90)) → le30_out_ga(0, s(T90))
le30_in_ga(0, 0) → le30_out_ga(0, 0)
U4_ga(T82, T83, le30_out_ga(T82, T83)) → le30_out_ga(s(T82), s(T83))
U12_gaa(T70, T71, le30_out_ga(T70, T71)) → min25_out_gaa(T70, T71, T70)
min25_in_gaa(T97, T98, T98) → U13_gaa(T97, T98, gt44_in_ga(T97, T98))
gt44_in_ga(s(T109), s(T110)) → U5_ga(T109, T110, gt44_in_ga(T109, T110))
gt44_in_ga(s(T115), 0) → gt44_out_ga(s(T115), 0)
U5_ga(T109, T110, gt44_out_ga(T109, T110)) → gt44_out_ga(s(T109), s(T110))
U13_gaa(T97, T98, gt44_out_ga(T97, T98)) → min25_out_gaa(T97, T98, T98)
U1_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, X59)) → min215_out_gaa(T48, T52, .(T50, T51))
min215_in_gaa(T48, T52, .(T50, T51)) → U2_gaa(T48, T52, T50, T51, min25_in_gaa(T48, T50, T55))
U2_gaa(T48, T52, T50, T51, min25_out_gaa(T48, T50, T55)) → U3_gaa(T48, T52, T50, T51, min215_in_gaa(T55, T52, T51))
U3_gaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_gaa(T48, T52, .(T50, T51))
U3_aaa(T48, T52, T50, T51, min215_out_gaa(T55, T52, T51)) → min215_out_aaa(T48, T52, .(T50, T51))
U14_aa(T25, T26, T27, T28, min215_out_aaa(T25, T27, T26)) → minsort1_out_aa(.(T25, T26), .(T27, T28))
minsort1_in_aa(.(T25, T26), .(T31, T32)) → U15_aa(T25, T26, T31, T32, min215_in_aaa(T25, T31, T26))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U16_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, X20))
remove53_in_aaaa(T135, T135, T136, T136) → remove53_out_aaaa(T135, T135, T136, T136)
remove53_in_aaaa(T145, T146, T147, .(T146, X170)) → U11_aaaa(T145, T146, T147, X170, p61_in_aaaa(T145, T146, T147, X170))
p61_in_aaaa(T145, T146, T147, X170) → U7_aaaa(T145, T146, T147, X170, notEq62_in_aa(T145, T146))
notEq62_in_aa(s(T160), s(T161)) → U6_aa(T160, T161, notEq62_in_aa(T160, T161))
notEq62_in_aa(s(T168), 0) → notEq62_out_aa(s(T168), 0)
notEq62_in_aa(0, s(T171)) → notEq62_out_aa(0, s(T171))
U6_aa(T160, T161, notEq62_out_aa(T160, T161)) → notEq62_out_aa(s(T160), s(T161))
U7_aaaa(T145, T146, T147, X170, notEq62_out_aa(T145, T146)) → p61_out_aaaa(T145, T146, T147, X170)
p61_in_aaaa(T184, T146, .(T184, T185), T185) → U8_aaaa(T184, T146, T185, notEq62_in_aa(T184, T146))
U8_aaaa(T184, T146, T185, notEq62_out_aa(T184, T146)) → p61_out_aaaa(T184, T146, .(T184, T185), T185)
p61_in_aaaa(T192, T146, .(T193, T194), .(T193, X229)) → U9_aaaa(T192, T146, T193, T194, X229, notEq62_in_aa(T192, T146))
U9_aaaa(T192, T146, T193, T194, X229, notEq62_out_aa(T192, T146)) → U10_aaaa(T192, T146, T193, T194, X229, p61_in_aaaa(T192, T193, T194, X229))
U10_aaaa(T192, T146, T193, T194, X229, p61_out_aaaa(T192, T193, T194, X229)) → p61_out_aaaa(T192, T146, .(T193, T194), .(T193, X229))
U11_aaaa(T145, T146, T147, X170, p61_out_aaaa(T145, T146, T147, X170)) → remove53_out_aaaa(T145, T146, T147, .(T146, X170))
U16_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, X20)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U15_aa(T25, T26, T31, T32, min215_out_aaa(T25, T31, T26)) → U17_aa(T25, T26, T31, T32, remove53_in_aaaa(T31, T25, T26, T122))
U17_aa(T25, T26, T31, T32, remove53_out_aaaa(T31, T25, T26, T122)) → U18_aa(T25, T26, T31, T32, minsort1_in_aa(T122, T32))
U18_aa(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_aa(.(T25, T26), .(T31, T32))
U18_ga(T25, T26, T31, T32, minsort1_out_aa(T122, T32)) → minsort1_out_ga(.(T25, T26), .(T31, T32))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
[]  =  []
minsort1_out_ga(x1, x2)  =  minsort1_out_ga
.(x1, x2)  =  .(x1, x2)
U14_ga(x1, x2, x3, x4, x5)  =  U14_ga(x5)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
min215_out_gag(x1, x2, x3)  =  min215_out_gag
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x4, x5)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
U15_ga(x1, x2, x3, x4, x5)  =  U15_ga(x1, x2, x5)
U16_ga(x1, x2, x3, x4, x5)  =  U16_ga(x5)
remove53_in_agga(x1, x2, x3, x4)  =  remove53_in_agga(x2, x3)
remove53_out_agga(x1, x2, x3, x4)  =  remove53_out_agga
U11_agga(x1, x2, x3, x4, x5)  =  U11_agga(x5)
p61_in_agga(x1, x2, x3, x4)  =  p61_in_agga(x2, x3)
U7_agga(x1, x2, x3, x4, x5)  =  U7_agga(x5)
notEq62_in_ag(x1, x2)  =  notEq62_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x3)
notEq62_out_ag(x1, x2)  =  notEq62_out_ag
p61_out_agga(x1, x2, x3, x4)  =  p61_out_agga
U8_agga(x1, x2, x3, x4)  =  U8_agga(x4)
notEq62_in_gg(x1, x2)  =  notEq62_in_gg(x1, x2)
U6_gg(x1, x2, x3)  =  U6_gg(x3)
notEq62_out_gg(x1, x2)  =  notEq62_out_gg
U9_agga(x1, x2, x3, x4, x5, x6)  =  U9_agga(x3, x4, x6)
U10_agga(x1, x2, x3, x4, x5, x6)  =  U10_agga(x6)
U17_ga(x1, x2, x3, x4, x5)  =  U17_ga(x5)
U18_ga(x1, x2, x3, x4, x5)  =  U18_ga(x5)
minsort1_in_aa(x1, x2)  =  minsort1_in_aa
minsort1_out_aa(x1, x2)  =  minsort1_out_aa
U14_aa(x1, x2, x3, x4, x5)  =  U14_aa(x5)
min215_in_aaa(x1, x2, x3)  =  min215_in_aaa
min215_out_aaa(x1, x2, x3)  =  min215_out_aaa
U1_aaa(x1, x2, x3, x4, x5)  =  U1_aaa(x5)
min25_in_aaa(x1, x2, x3)  =  min25_in_aaa
U12_aaa(x1, x2, x3)  =  U12_aaa(x3)
le30_in_aa(x1, x2)  =  le30_in_aa
U4_aa(x1, x2, x3)  =  U4_aa(x3)
le30_out_aa(x1, x2)  =  le30_out_aa(x1)
min25_out_aaa(x1, x2, x3)  =  min25_out_aaa(x3)
U13_aaa(x1, x2, x3)  =  U13_aaa(x3)
gt44_in_aa(x1, x2)  =  gt44_in_aa
U5_aa(x1, x2, x3)  =  U5_aa(x3)
gt44_out_aa(x1, x2)  =  gt44_out_aa(x2)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
min215_in_gaa(x1, x2, x3)  =  min215_in_gaa(x1)
min215_out_gaa(x1, x2, x3)  =  min215_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
min25_in_gaa(x1, x2, x3)  =  min25_in_gaa(x1)
U12_gaa(x1, x2, x3)  =  U12_gaa(x1, x3)
le30_in_ga(x1, x2)  =  le30_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
le30_out_ga(x1, x2)  =  le30_out_ga
min25_out_gaa(x1, x2, x3)  =  min25_out_gaa(x3)
U13_gaa(x1, x2, x3)  =  U13_gaa(x3)
gt44_in_ga(x1, x2)  =  gt44_in_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
gt44_out_ga(x1, x2)  =  gt44_out_ga(x2)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
U15_aa(x1, x2, x3, x4, x5)  =  U15_aa(x5)
U16_aa(x1, x2, x3, x4, x5)  =  U16_aa(x5)
remove53_in_aaaa(x1, x2, x3, x4)  =  remove53_in_aaaa
remove53_out_aaaa(x1, x2, x3, x4)  =  remove53_out_aaaa
U11_aaaa(x1, x2, x3, x4, x5)  =  U11_aaaa(x5)
p61_in_aaaa(x1, x2, x3, x4)  =  p61_in_aaaa
U7_aaaa(x1, x2, x3, x4, x5)  =  U7_aaaa(x5)
notEq62_in_aa(x1, x2)  =  notEq62_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
notEq62_out_aa(x1, x2)  =  notEq62_out_aa
p61_out_aaaa(x1, x2, x3, x4)  =  p61_out_aaaa
U8_aaaa(x1, x2, x3, x4)  =  U8_aaaa(x4)
U9_aaaa(x1, x2, x3, x4, x5, x6)  =  U9_aaaa(x6)
U10_aaaa(x1, x2, x3, x4, x5, x6)  =  U10_aaaa(x6)
U17_aa(x1, x2, x3, x4, x5)  =  U17_aa(x5)
U18_aa(x1, x2, x3, x4, x5)  =  U18_aa(x5)
MIN215_IN_GAG(x1, x2, x3)  =  MIN215_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x4, x5)

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

(257) UsableRulesProof (EQUIVALENT transformation)

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

(258) Obligation:

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

MIN215_IN_GAG(T48, T52, .(T50, T51)) → U2_GAG(T48, T52, T50, T51, min25_in_gga(T48, T50, T55))
U2_GAG(T48, T52, T50, T51, min25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)

The TRS R consists of the following rules:

min25_in_gga(T70, T71, T70) → U12_gga(T70, T71, le30_in_gg(T70, T71))
min25_in_gga(T97, T98, T98) → U13_gga(T97, T98, gt44_in_gg(T97, T98))
U12_gga(T70, T71, le30_out_gg(T70, T71)) → min25_out_gga(T70, T71, T70)
U13_gga(T97, T98, gt44_out_gg(T97, T98)) → min25_out_gga(T97, T98, T98)
le30_in_gg(s(T82), s(T83)) → U4_gg(T82, T83, le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg(0, s(T90))
le30_in_gg(0, 0) → le30_out_gg(0, 0)
gt44_in_gg(s(T109), s(T110)) → U5_gg(T109, T110, gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg(s(T115), 0)
U4_gg(T82, T83, le30_out_gg(T82, T83)) → le30_out_gg(s(T82), s(T83))
U5_gg(T109, T110, gt44_out_gg(T109, T110)) → gt44_out_gg(s(T109), s(T110))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
min25_in_gga(x1, x2, x3)  =  min25_in_gga(x1, x2)
U12_gga(x1, x2, x3)  =  U12_gga(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
0  =  0
le30_out_gg(x1, x2)  =  le30_out_gg
min25_out_gga(x1, x2, x3)  =  min25_out_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x2, x3)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
gt44_out_gg(x1, x2)  =  gt44_out_gg
MIN215_IN_GAG(x1, x2, x3)  =  MIN215_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x4, x5)

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

(259) PiDPToQDPProof (SOUND transformation)

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

(260) Obligation:

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

MIN215_IN_GAG(T48, .(T50, T51)) → U2_GAG(T51, min25_in_gga(T48, T50))
U2_GAG(T51, min25_out_gga(T55)) → MIN215_IN_GAG(T55, T51)

The TRS R consists of the following rules:

min25_in_gga(T70, T71) → U12_gga(T70, le30_in_gg(T70, T71))
min25_in_gga(T97, T98) → U13_gga(T98, gt44_in_gg(T97, T98))
U12_gga(T70, le30_out_gg) → min25_out_gga(T70)
U13_gga(T98, gt44_out_gg) → min25_out_gga(T98)
le30_in_gg(s(T82), s(T83)) → U4_gg(le30_in_gg(T82, T83))
le30_in_gg(0, s(T90)) → le30_out_gg
le30_in_gg(0, 0) → le30_out_gg
gt44_in_gg(s(T109), s(T110)) → U5_gg(gt44_in_gg(T109, T110))
gt44_in_gg(s(T115), 0) → gt44_out_gg
U4_gg(le30_out_gg) → le30_out_gg
U5_gg(gt44_out_gg) → gt44_out_gg

The set Q consists of the following terms:

min25_in_gga(x0, x1)
U12_gga(x0, x1)
U13_gga(x0, x1)
le30_in_gg(x0, x1)
gt44_in_gg(x0, x1)
U4_gg(x0)
U5_gg(x0)

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

(261) 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:

  • U2_GAG(T51, min25_out_gga(T55)) → MIN215_IN_GAG(T55, T51)
    The graph contains the following edges 2 > 1, 1 >= 2

  • MIN215_IN_GAG(T48, .(T50, T51)) → U2_GAG(T51, min25_in_gga(T48, T50))
    The graph contains the following edges 2 > 1

(262) YES