(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, [], []).
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) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

min215(T70, T52, .(T71, T51)) :- le30(T70, T71).
min215(T97, T52, .(T98, T51)) :- gt44(T97, T98).
min215(T48, T52, .(T50, T51)) :- ','(minc25(T48, T50, T55), min215(T55, T52, T51)).
le30(s(T82), s(T83)) :- le30(T82, T83).
gt44(s(T109), s(T110)) :- gt44(T109, T110).
notEq63(s(T165), s(T166)) :- notEq63(T165, T166).
p62(T150, T151, T152, X175) :- notEq63(T150, T151).
p62(T200, T151, .(T201, T202), .(T201, X238)) :- ','(notEqc63(T200, T151), p62(T200, T201, T202, X238)).
minsort1(.(T25, T26), .(T27, T28)) :- min215(T25, T27, T26).
minsort1(.(T151, T152), .(T150, T32)) :- ','(min2c15(T151, T150, T152), p62(T150, T151, T152, X175)).
minsort1(.(T25, T26), .(T31, T32)) :- ','(min2c15(T25, T31, T26), ','(removec53(T31, T25, T26, T124), minsort1(T124, T32))).

Clauses:

min2c15(T39, T39, []).
min2c15(T48, T52, .(T50, T51)) :- ','(minc25(T48, T50, T55), min2c15(T55, T52, T51)).
lec30(s(T82), s(T83)) :- lec30(T82, T83).
lec30(0, s(T90)).
lec30(0, 0).
gtc44(s(T109), s(T110)) :- gtc44(T109, T110).
gtc44(s(T115), 0).
minsortc1([], []).
minsortc1(.(T25, T26), .(T31, T32)) :- ','(min2c15(T25, T31, T26), ','(removec53(T31, T25, T26, T124), minsortc1(T124, T32))).
notEqc63(s(T165), s(T166)) :- notEqc63(T165, T166).
notEqc63(s(T173), 0).
notEqc63(0, s(T176)).
qc62(T183, T151, [], []) :- notEqc63(T183, T151).
qc62(T192, T151, .(T192, T193), T193) :- notEqc63(T192, T151).
qc62(T200, T151, .(T201, T202), .(T201, X238)) :- ','(notEqc63(T200, T151), qc62(T200, T201, T202, X238)).
removec53(T140, T140, T141, T141).
removec53(T150, T151, T152, .(T151, X175)) :- qc62(T150, T151, T152, X175).
minc25(T70, T71, T70) :- lec30(T70, T71).
minc25(T97, T98, T98) :- gtc44(T97, T98).

Afs:

minsort1(x1, x2)  =  minsort1(x1)

(3) TriplesToPiDPProof (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)
min215_in: (b,f,b)
le30_in: (b,b)
gt44_in: (b,b)
minc25_in: (b,b,f)
lec30_in: (b,b)
gtc44_in: (b,b)
min2c15_in: (b,f,b)
p62_in: (b,b,b,f)
notEq63_in: (b,b)
notEqc63_in: (b,b)
removec53_in: (b,b,b,f)
qc62_in: (b,b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → U11_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(T70, T52, .(T71, T51)) → U1_GAG(T70, T52, T71, T51, le30_in_gg(T70, T71))
MIN215_IN_GAG(T70, T52, .(T71, T51)) → LE30_IN_GG(T70, T71)
LE30_IN_GG(s(T82), s(T83)) → U5_GG(T82, T83, le30_in_gg(T82, T83))
LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)
MIN215_IN_GAG(T97, T52, .(T98, T51)) → U2_GAG(T97, T52, T98, T51, gt44_in_gg(T97, T98))
MIN215_IN_GAG(T97, T52, .(T98, T51)) → GT44_IN_GG(T97, T98)
GT44_IN_GG(s(T109), s(T110)) → U6_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)) → U3_GAG(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U3_GAG(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U4_GAG(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_GAG(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)
MINSORT1_IN_GA(.(T151, T152), .(T150, T32)) → U12_GA(T151, T152, T150, T32, min2c15_in_gag(T151, T150, T152))
U12_GA(T151, T152, T150, T32, min2c15_out_gag(T151, T150, T152)) → U13_GA(T151, T152, T150, T32, p62_in_ggga(T150, T151, T152, X175))
U12_GA(T151, T152, T150, T32, min2c15_out_gag(T151, T150, T152)) → P62_IN_GGGA(T150, T151, T152, X175)
P62_IN_GGGA(T150, T151, T152, X175) → U8_GGGA(T150, T151, T152, X175, notEq63_in_gg(T150, T151))
P62_IN_GGGA(T150, T151, T152, X175) → NOTEQ63_IN_GG(T150, T151)
NOTEQ63_IN_GG(s(T165), s(T166)) → U7_GG(T165, T166, notEq63_in_gg(T165, T166))
NOTEQ63_IN_GG(s(T165), s(T166)) → NOTEQ63_IN_GG(T165, T166)
P62_IN_GGGA(T200, T151, .(T201, T202), .(T201, X238)) → U9_GGGA(T200, T151, T201, T202, X238, notEqc63_in_gg(T200, T151))
U9_GGGA(T200, T151, T201, T202, X238, notEqc63_out_gg(T200, T151)) → U10_GGGA(T200, T151, T201, T202, X238, p62_in_ggga(T200, T201, T202, X238))
U9_GGGA(T200, T151, T201, T202, X238, notEqc63_out_gg(T200, T151)) → P62_IN_GGGA(T200, T201, T202, X238)
MINSORT1_IN_GA(.(T25, T26), .(T31, T32)) → U14_GA(T25, T26, T31, T32, min2c15_in_gag(T25, T31, T26))
U14_GA(T25, T26, T31, T32, min2c15_out_gag(T25, T31, T26)) → U15_GA(T25, T26, T31, T32, removec53_in_ggga(T31, T25, T26, T124))
U15_GA(T25, T26, T31, T32, removec53_out_ggga(T31, T25, T26, T124)) → U16_GA(T25, T26, T31, T32, minsort1_in_ga(T124, T32))
U15_GA(T25, T26, T31, T32, removec53_out_ggga(T31, T25, T26, T124)) → MINSORT1_IN_GA(T124, T32)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71, T70) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, T52, .(T50, T51)) → U18_gag(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U18_gag(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T52, T50, T51, min2c15_in_gag(T55, T52, T51))
U19_gag(T48, T52, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
removec53_in_ggga(T140, T140, T141, T141) → removec53_out_ggga(T140, T140, T141, T141)
removec53_in_ggga(T150, T151, T152, .(T151, X175)) → U30_ggga(T150, T151, T152, X175, qc62_in_ggga(T150, T151, T152, X175))
qc62_in_ggga(T183, T151, [], []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
qc62_in_ggga(T192, T151, .(T192, T193), T193) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
qc62_in_ggga(T200, T151, .(T201, T202), .(T201, X238)) → U28_ggga(T200, T151, T201, T202, X238, notEqc63_in_gg(T200, T151))
U28_ggga(T200, T151, T201, T202, X238, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, X238, qc62_in_ggga(T200, T201, T202, X238))
U29_ggga(T200, T151, T201, T202, X238, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U30_ggga(T150, T151, T152, X175, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
minc25_in_gga(x1, x2, x3)  =  minc25_in_gga(x1, x2)
U31_gga(x1, x2, x3)  =  U31_gga(x1, x2, x3)
lec30_in_gg(x1, x2)  =  lec30_in_gg(x1, x2)
U20_gg(x1, x2, x3)  =  U20_gg(x1, x2, x3)
0  =  0
lec30_out_gg(x1, x2)  =  lec30_out_gg(x1, x2)
minc25_out_gga(x1, x2, x3)  =  minc25_out_gga(x1, x2, x3)
U32_gga(x1, x2, x3)  =  U32_gga(x1, x2, x3)
gtc44_in_gg(x1, x2)  =  gtc44_in_gg(x1, x2)
U21_gg(x1, x2, x3)  =  U21_gg(x1, x2, x3)
gtc44_out_gg(x1, x2)  =  gtc44_out_gg(x1, x2)
min2c15_in_gag(x1, x2, x3)  =  min2c15_in_gag(x1, x3)
[]  =  []
min2c15_out_gag(x1, x2, x3)  =  min2c15_out_gag(x1, x2, x3)
U18_gag(x1, x2, x3, x4, x5)  =  U18_gag(x1, x3, x4, x5)
U19_gag(x1, x2, x3, x4, x5)  =  U19_gag(x1, x3, x4, x5)
p62_in_ggga(x1, x2, x3, x4)  =  p62_in_ggga(x1, x2, x3)
notEq63_in_gg(x1, x2)  =  notEq63_in_gg(x1, x2)
notEqc63_in_gg(x1, x2)  =  notEqc63_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
notEqc63_out_gg(x1, x2)  =  notEqc63_out_gg(x1, x2)
removec53_in_ggga(x1, x2, x3, x4)  =  removec53_in_ggga(x1, x2, x3)
removec53_out_ggga(x1, x2, x3, x4)  =  removec53_out_ggga(x1, x2, x3, x4)
U30_ggga(x1, x2, x3, x4, x5)  =  U30_ggga(x1, x2, x3, x5)
qc62_in_ggga(x1, x2, x3, x4)  =  qc62_in_ggga(x1, x2, x3)
U26_ggga(x1, x2, x3)  =  U26_ggga(x1, x2, x3)
qc62_out_ggga(x1, x2, x3, x4)  =  qc62_out_ggga(x1, x2, x3, x4)
U27_ggga(x1, x2, x3, x4)  =  U27_ggga(x1, x2, x3, x4)
U28_ggga(x1, x2, x3, x4, x5, x6)  =  U28_ggga(x1, x2, x3, x4, x6)
U29_ggga(x1, x2, x3, x4, x5, x6)  =  U29_ggga(x1, x2, x3, x4, x6)
MINSORT1_IN_GA(x1, x2)  =  MINSORT1_IN_GA(x1)
U11_GA(x1, x2, x3, x4, x5)  =  U11_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)
LE30_IN_GG(x1, x2)  =  LE30_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)
GT44_IN_GG(x1, x2)  =  GT44_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x1, x2, x3)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x1, x3, x4, x5)
U4_GAG(x1, x2, x3, x4, x5)  =  U4_GAG(x1, x3, x4, x5)
U12_GA(x1, x2, x3, x4, x5)  =  U12_GA(x1, x2, x5)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x1, x2, x5)
P62_IN_GGGA(x1, x2, x3, x4)  =  P62_IN_GGGA(x1, x2, x3)
U8_GGGA(x1, x2, x3, x4, x5)  =  U8_GGGA(x1, x2, x3, x5)
NOTEQ63_IN_GG(x1, x2)  =  NOTEQ63_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)
U9_GGGA(x1, x2, x3, x4, x5, x6)  =  U9_GGGA(x1, x2, x3, x4, x6)
U10_GGGA(x1, x2, x3, x4, x5, x6)  =  U10_GGGA(x1, x2, x3, x4, x6)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x1, x2, 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)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → U11_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(T70, T52, .(T71, T51)) → U1_GAG(T70, T52, T71, T51, le30_in_gg(T70, T71))
MIN215_IN_GAG(T70, T52, .(T71, T51)) → LE30_IN_GG(T70, T71)
LE30_IN_GG(s(T82), s(T83)) → U5_GG(T82, T83, le30_in_gg(T82, T83))
LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)
MIN215_IN_GAG(T97, T52, .(T98, T51)) → U2_GAG(T97, T52, T98, T51, gt44_in_gg(T97, T98))
MIN215_IN_GAG(T97, T52, .(T98, T51)) → GT44_IN_GG(T97, T98)
GT44_IN_GG(s(T109), s(T110)) → U6_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)) → U3_GAG(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U3_GAG(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U4_GAG(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_GAG(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)
MINSORT1_IN_GA(.(T151, T152), .(T150, T32)) → U12_GA(T151, T152, T150, T32, min2c15_in_gag(T151, T150, T152))
U12_GA(T151, T152, T150, T32, min2c15_out_gag(T151, T150, T152)) → U13_GA(T151, T152, T150, T32, p62_in_ggga(T150, T151, T152, X175))
U12_GA(T151, T152, T150, T32, min2c15_out_gag(T151, T150, T152)) → P62_IN_GGGA(T150, T151, T152, X175)
P62_IN_GGGA(T150, T151, T152, X175) → U8_GGGA(T150, T151, T152, X175, notEq63_in_gg(T150, T151))
P62_IN_GGGA(T150, T151, T152, X175) → NOTEQ63_IN_GG(T150, T151)
NOTEQ63_IN_GG(s(T165), s(T166)) → U7_GG(T165, T166, notEq63_in_gg(T165, T166))
NOTEQ63_IN_GG(s(T165), s(T166)) → NOTEQ63_IN_GG(T165, T166)
P62_IN_GGGA(T200, T151, .(T201, T202), .(T201, X238)) → U9_GGGA(T200, T151, T201, T202, X238, notEqc63_in_gg(T200, T151))
U9_GGGA(T200, T151, T201, T202, X238, notEqc63_out_gg(T200, T151)) → U10_GGGA(T200, T151, T201, T202, X238, p62_in_ggga(T200, T201, T202, X238))
U9_GGGA(T200, T151, T201, T202, X238, notEqc63_out_gg(T200, T151)) → P62_IN_GGGA(T200, T201, T202, X238)
MINSORT1_IN_GA(.(T25, T26), .(T31, T32)) → U14_GA(T25, T26, T31, T32, min2c15_in_gag(T25, T31, T26))
U14_GA(T25, T26, T31, T32, min2c15_out_gag(T25, T31, T26)) → U15_GA(T25, T26, T31, T32, removec53_in_ggga(T31, T25, T26, T124))
U15_GA(T25, T26, T31, T32, removec53_out_ggga(T31, T25, T26, T124)) → U16_GA(T25, T26, T31, T32, minsort1_in_ga(T124, T32))
U15_GA(T25, T26, T31, T32, removec53_out_ggga(T31, T25, T26, T124)) → MINSORT1_IN_GA(T124, T32)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71, T70) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, T52, .(T50, T51)) → U18_gag(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U18_gag(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T52, T50, T51, min2c15_in_gag(T55, T52, T51))
U19_gag(T48, T52, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
removec53_in_ggga(T140, T140, T141, T141) → removec53_out_ggga(T140, T140, T141, T141)
removec53_in_ggga(T150, T151, T152, .(T151, X175)) → U30_ggga(T150, T151, T152, X175, qc62_in_ggga(T150, T151, T152, X175))
qc62_in_ggga(T183, T151, [], []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
qc62_in_ggga(T192, T151, .(T192, T193), T193) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
qc62_in_ggga(T200, T151, .(T201, T202), .(T201, X238)) → U28_ggga(T200, T151, T201, T202, X238, notEqc63_in_gg(T200, T151))
U28_ggga(T200, T151, T201, T202, X238, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, X238, qc62_in_ggga(T200, T201, T202, X238))
U29_ggga(T200, T151, T201, T202, X238, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U30_ggga(T150, T151, T152, X175, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2)  =  minsort1_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
min215_in_gag(x1, x2, x3)  =  min215_in_gag(x1, x3)
le30_in_gg(x1, x2)  =  le30_in_gg(x1, x2)
s(x1)  =  s(x1)
gt44_in_gg(x1, x2)  =  gt44_in_gg(x1, x2)
minc25_in_gga(x1, x2, x3)  =  minc25_in_gga(x1, x2)
U31_gga(x1, x2, x3)  =  U31_gga(x1, x2, x3)
lec30_in_gg(x1, x2)  =  lec30_in_gg(x1, x2)
U20_gg(x1, x2, x3)  =  U20_gg(x1, x2, x3)
0  =  0
lec30_out_gg(x1, x2)  =  lec30_out_gg(x1, x2)
minc25_out_gga(x1, x2, x3)  =  minc25_out_gga(x1, x2, x3)
U32_gga(x1, x2, x3)  =  U32_gga(x1, x2, x3)
gtc44_in_gg(x1, x2)  =  gtc44_in_gg(x1, x2)
U21_gg(x1, x2, x3)  =  U21_gg(x1, x2, x3)
gtc44_out_gg(x1, x2)  =  gtc44_out_gg(x1, x2)
min2c15_in_gag(x1, x2, x3)  =  min2c15_in_gag(x1, x3)
[]  =  []
min2c15_out_gag(x1, x2, x3)  =  min2c15_out_gag(x1, x2, x3)
U18_gag(x1, x2, x3, x4, x5)  =  U18_gag(x1, x3, x4, x5)
U19_gag(x1, x2, x3, x4, x5)  =  U19_gag(x1, x3, x4, x5)
p62_in_ggga(x1, x2, x3, x4)  =  p62_in_ggga(x1, x2, x3)
notEq63_in_gg(x1, x2)  =  notEq63_in_gg(x1, x2)
notEqc63_in_gg(x1, x2)  =  notEqc63_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
notEqc63_out_gg(x1, x2)  =  notEqc63_out_gg(x1, x2)
removec53_in_ggga(x1, x2, x3, x4)  =  removec53_in_ggga(x1, x2, x3)
removec53_out_ggga(x1, x2, x3, x4)  =  removec53_out_ggga(x1, x2, x3, x4)
U30_ggga(x1, x2, x3, x4, x5)  =  U30_ggga(x1, x2, x3, x5)
qc62_in_ggga(x1, x2, x3, x4)  =  qc62_in_ggga(x1, x2, x3)
U26_ggga(x1, x2, x3)  =  U26_ggga(x1, x2, x3)
qc62_out_ggga(x1, x2, x3, x4)  =  qc62_out_ggga(x1, x2, x3, x4)
U27_ggga(x1, x2, x3, x4)  =  U27_ggga(x1, x2, x3, x4)
U28_ggga(x1, x2, x3, x4, x5, x6)  =  U28_ggga(x1, x2, x3, x4, x6)
U29_ggga(x1, x2, x3, x4, x5, x6)  =  U29_ggga(x1, x2, x3, x4, x6)
MINSORT1_IN_GA(x1, x2)  =  MINSORT1_IN_GA(x1)
U11_GA(x1, x2, x3, x4, x5)  =  U11_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)
LE30_IN_GG(x1, x2)  =  LE30_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)
GT44_IN_GG(x1, x2)  =  GT44_IN_GG(x1, x2)
U6_GG(x1, x2, x3)  =  U6_GG(x1, x2, x3)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x1, x3, x4, x5)
U4_GAG(x1, x2, x3, x4, x5)  =  U4_GAG(x1, x3, x4, x5)
U12_GA(x1, x2, x3, x4, x5)  =  U12_GA(x1, x2, x5)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x1, x2, x5)
P62_IN_GGGA(x1, x2, x3, x4)  =  P62_IN_GGGA(x1, x2, x3)
U8_GGGA(x1, x2, x3, x4, x5)  =  U8_GGGA(x1, x2, x3, x5)
NOTEQ63_IN_GG(x1, x2)  =  NOTEQ63_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)
U9_GGGA(x1, x2, x3, x4, x5, x6)  =  U9_GGGA(x1, x2, x3, x4, x6)
U10_GGGA(x1, x2, x3, x4, x5, x6)  =  U10_GGGA(x1, x2, x3, x4, x6)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x1, x2, 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)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 17 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

NOTEQ63_IN_GG(s(T165), s(T166)) → NOTEQ63_IN_GG(T165, T166)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71, T70) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, T52, .(T50, T51)) → U18_gag(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U18_gag(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T52, T50, T51, min2c15_in_gag(T55, T52, T51))
U19_gag(T48, T52, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
removec53_in_ggga(T140, T140, T141, T141) → removec53_out_ggga(T140, T140, T141, T141)
removec53_in_ggga(T150, T151, T152, .(T151, X175)) → U30_ggga(T150, T151, T152, X175, qc62_in_ggga(T150, T151, T152, X175))
qc62_in_ggga(T183, T151, [], []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
qc62_in_ggga(T192, T151, .(T192, T193), T193) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
qc62_in_ggga(T200, T151, .(T201, T202), .(T201, X238)) → U28_ggga(T200, T151, T201, T202, X238, notEqc63_in_gg(T200, T151))
U28_ggga(T200, T151, T201, T202, X238, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, X238, qc62_in_ggga(T200, T201, T202, X238))
U29_ggga(T200, T151, T201, T202, X238, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U30_ggga(T150, T151, T152, X175, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
minc25_in_gga(x1, x2, x3)  =  minc25_in_gga(x1, x2)
U31_gga(x1, x2, x3)  =  U31_gga(x1, x2, x3)
lec30_in_gg(x1, x2)  =  lec30_in_gg(x1, x2)
U20_gg(x1, x2, x3)  =  U20_gg(x1, x2, x3)
0  =  0
lec30_out_gg(x1, x2)  =  lec30_out_gg(x1, x2)
minc25_out_gga(x1, x2, x3)  =  minc25_out_gga(x1, x2, x3)
U32_gga(x1, x2, x3)  =  U32_gga(x1, x2, x3)
gtc44_in_gg(x1, x2)  =  gtc44_in_gg(x1, x2)
U21_gg(x1, x2, x3)  =  U21_gg(x1, x2, x3)
gtc44_out_gg(x1, x2)  =  gtc44_out_gg(x1, x2)
min2c15_in_gag(x1, x2, x3)  =  min2c15_in_gag(x1, x3)
[]  =  []
min2c15_out_gag(x1, x2, x3)  =  min2c15_out_gag(x1, x2, x3)
U18_gag(x1, x2, x3, x4, x5)  =  U18_gag(x1, x3, x4, x5)
U19_gag(x1, x2, x3, x4, x5)  =  U19_gag(x1, x3, x4, x5)
notEqc63_in_gg(x1, x2)  =  notEqc63_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
notEqc63_out_gg(x1, x2)  =  notEqc63_out_gg(x1, x2)
removec53_in_ggga(x1, x2, x3, x4)  =  removec53_in_ggga(x1, x2, x3)
removec53_out_ggga(x1, x2, x3, x4)  =  removec53_out_ggga(x1, x2, x3, x4)
U30_ggga(x1, x2, x3, x4, x5)  =  U30_ggga(x1, x2, x3, x5)
qc62_in_ggga(x1, x2, x3, x4)  =  qc62_in_ggga(x1, x2, x3)
U26_ggga(x1, x2, x3)  =  U26_ggga(x1, x2, x3)
qc62_out_ggga(x1, x2, x3, x4)  =  qc62_out_ggga(x1, x2, x3, x4)
U27_ggga(x1, x2, x3, x4)  =  U27_ggga(x1, x2, x3, x4)
U28_ggga(x1, x2, x3, x4, x5, x6)  =  U28_ggga(x1, x2, x3, x4, x6)
U29_ggga(x1, x2, x3, x4, x5, x6)  =  U29_ggga(x1, x2, x3, x4, x6)
NOTEQ63_IN_GG(x1, x2)  =  NOTEQ63_IN_GG(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

NOTEQ63_IN_GG(s(T165), s(T166)) → NOTEQ63_IN_GG(T165, T166)

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

(10) PiDPToQDPProof (EQUIVALENT transformation)

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

(11) Obligation:

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

NOTEQ63_IN_GG(s(T165), s(T166)) → NOTEQ63_IN_GG(T165, T166)

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

(12) 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:

  • NOTEQ63_IN_GG(s(T165), s(T166)) → NOTEQ63_IN_GG(T165, T166)
    The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

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

P62_IN_GGGA(T200, T151, .(T201, T202), .(T201, X238)) → U9_GGGA(T200, T151, T201, T202, X238, notEqc63_in_gg(T200, T151))
U9_GGGA(T200, T151, T201, T202, X238, notEqc63_out_gg(T200, T151)) → P62_IN_GGGA(T200, T201, T202, X238)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71, T70) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, T52, .(T50, T51)) → U18_gag(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U18_gag(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T52, T50, T51, min2c15_in_gag(T55, T52, T51))
U19_gag(T48, T52, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
removec53_in_ggga(T140, T140, T141, T141) → removec53_out_ggga(T140, T140, T141, T141)
removec53_in_ggga(T150, T151, T152, .(T151, X175)) → U30_ggga(T150, T151, T152, X175, qc62_in_ggga(T150, T151, T152, X175))
qc62_in_ggga(T183, T151, [], []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
qc62_in_ggga(T192, T151, .(T192, T193), T193) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
qc62_in_ggga(T200, T151, .(T201, T202), .(T201, X238)) → U28_ggga(T200, T151, T201, T202, X238, notEqc63_in_gg(T200, T151))
U28_ggga(T200, T151, T201, T202, X238, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, X238, qc62_in_ggga(T200, T201, T202, X238))
U29_ggga(T200, T151, T201, T202, X238, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U30_ggga(T150, T151, T152, X175, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
minc25_in_gga(x1, x2, x3)  =  minc25_in_gga(x1, x2)
U31_gga(x1, x2, x3)  =  U31_gga(x1, x2, x3)
lec30_in_gg(x1, x2)  =  lec30_in_gg(x1, x2)
U20_gg(x1, x2, x3)  =  U20_gg(x1, x2, x3)
0  =  0
lec30_out_gg(x1, x2)  =  lec30_out_gg(x1, x2)
minc25_out_gga(x1, x2, x3)  =  minc25_out_gga(x1, x2, x3)
U32_gga(x1, x2, x3)  =  U32_gga(x1, x2, x3)
gtc44_in_gg(x1, x2)  =  gtc44_in_gg(x1, x2)
U21_gg(x1, x2, x3)  =  U21_gg(x1, x2, x3)
gtc44_out_gg(x1, x2)  =  gtc44_out_gg(x1, x2)
min2c15_in_gag(x1, x2, x3)  =  min2c15_in_gag(x1, x3)
[]  =  []
min2c15_out_gag(x1, x2, x3)  =  min2c15_out_gag(x1, x2, x3)
U18_gag(x1, x2, x3, x4, x5)  =  U18_gag(x1, x3, x4, x5)
U19_gag(x1, x2, x3, x4, x5)  =  U19_gag(x1, x3, x4, x5)
notEqc63_in_gg(x1, x2)  =  notEqc63_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
notEqc63_out_gg(x1, x2)  =  notEqc63_out_gg(x1, x2)
removec53_in_ggga(x1, x2, x3, x4)  =  removec53_in_ggga(x1, x2, x3)
removec53_out_ggga(x1, x2, x3, x4)  =  removec53_out_ggga(x1, x2, x3, x4)
U30_ggga(x1, x2, x3, x4, x5)  =  U30_ggga(x1, x2, x3, x5)
qc62_in_ggga(x1, x2, x3, x4)  =  qc62_in_ggga(x1, x2, x3)
U26_ggga(x1, x2, x3)  =  U26_ggga(x1, x2, x3)
qc62_out_ggga(x1, x2, x3, x4)  =  qc62_out_ggga(x1, x2, x3, x4)
U27_ggga(x1, x2, x3, x4)  =  U27_ggga(x1, x2, x3, x4)
U28_ggga(x1, x2, x3, x4, x5, x6)  =  U28_ggga(x1, x2, x3, x4, x6)
U29_ggga(x1, x2, x3, x4, x5, x6)  =  U29_ggga(x1, x2, x3, x4, x6)
P62_IN_GGGA(x1, x2, x3, x4)  =  P62_IN_GGGA(x1, x2, x3)
U9_GGGA(x1, x2, x3, x4, x5, x6)  =  U9_GGGA(x1, x2, x3, x4, x6)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

P62_IN_GGGA(T200, T151, .(T201, T202), .(T201, X238)) → U9_GGGA(T200, T151, T201, T202, X238, notEqc63_in_gg(T200, T151))
U9_GGGA(T200, T151, T201, T202, X238, notEqc63_out_gg(T200, T151)) → P62_IN_GGGA(T200, T201, T202, X238)

The TRS R consists of the following rules:

notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
0  =  0
notEqc63_in_gg(x1, x2)  =  notEqc63_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
notEqc63_out_gg(x1, x2)  =  notEqc63_out_gg(x1, x2)
P62_IN_GGGA(x1, x2, x3, x4)  =  P62_IN_GGGA(x1, x2, x3)
U9_GGGA(x1, x2, x3, x4, x5, x6)  =  U9_GGGA(x1, x2, x3, x4, x6)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

P62_IN_GGGA(T200, T151, .(T201, T202)) → U9_GGGA(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U9_GGGA(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → P62_IN_GGGA(T200, T201, T202)

The TRS R consists of the following rules:

notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))

The set Q consists of the following terms:

notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)

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

(19) 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_GGGA(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → P62_IN_GGGA(T200, T201, T202)
    The graph contains the following edges 1 >= 1, 5 > 1, 3 >= 2, 4 >= 3

  • P62_IN_GGGA(T200, T151, .(T201, T202)) → U9_GGGA(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
    The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 3 > 4

(20) YES

(21) 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:

minc25_in_gga(T70, T71, T70) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, T52, .(T50, T51)) → U18_gag(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U18_gag(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T52, T50, T51, min2c15_in_gag(T55, T52, T51))
U19_gag(T48, T52, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
removec53_in_ggga(T140, T140, T141, T141) → removec53_out_ggga(T140, T140, T141, T141)
removec53_in_ggga(T150, T151, T152, .(T151, X175)) → U30_ggga(T150, T151, T152, X175, qc62_in_ggga(T150, T151, T152, X175))
qc62_in_ggga(T183, T151, [], []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
qc62_in_ggga(T192, T151, .(T192, T193), T193) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
qc62_in_ggga(T200, T151, .(T201, T202), .(T201, X238)) → U28_ggga(T200, T151, T201, T202, X238, notEqc63_in_gg(T200, T151))
U28_ggga(T200, T151, T201, T202, X238, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, X238, qc62_in_ggga(T200, T201, T202, X238))
U29_ggga(T200, T151, T201, T202, X238, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U30_ggga(T150, T151, T152, X175, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
minc25_in_gga(x1, x2, x3)  =  minc25_in_gga(x1, x2)
U31_gga(x1, x2, x3)  =  U31_gga(x1, x2, x3)
lec30_in_gg(x1, x2)  =  lec30_in_gg(x1, x2)
U20_gg(x1, x2, x3)  =  U20_gg(x1, x2, x3)
0  =  0
lec30_out_gg(x1, x2)  =  lec30_out_gg(x1, x2)
minc25_out_gga(x1, x2, x3)  =  minc25_out_gga(x1, x2, x3)
U32_gga(x1, x2, x3)  =  U32_gga(x1, x2, x3)
gtc44_in_gg(x1, x2)  =  gtc44_in_gg(x1, x2)
U21_gg(x1, x2, x3)  =  U21_gg(x1, x2, x3)
gtc44_out_gg(x1, x2)  =  gtc44_out_gg(x1, x2)
min2c15_in_gag(x1, x2, x3)  =  min2c15_in_gag(x1, x3)
[]  =  []
min2c15_out_gag(x1, x2, x3)  =  min2c15_out_gag(x1, x2, x3)
U18_gag(x1, x2, x3, x4, x5)  =  U18_gag(x1, x3, x4, x5)
U19_gag(x1, x2, x3, x4, x5)  =  U19_gag(x1, x3, x4, x5)
notEqc63_in_gg(x1, x2)  =  notEqc63_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
notEqc63_out_gg(x1, x2)  =  notEqc63_out_gg(x1, x2)
removec53_in_ggga(x1, x2, x3, x4)  =  removec53_in_ggga(x1, x2, x3)
removec53_out_ggga(x1, x2, x3, x4)  =  removec53_out_ggga(x1, x2, x3, x4)
U30_ggga(x1, x2, x3, x4, x5)  =  U30_ggga(x1, x2, x3, x5)
qc62_in_ggga(x1, x2, x3, x4)  =  qc62_in_ggga(x1, x2, x3)
U26_ggga(x1, x2, x3)  =  U26_ggga(x1, x2, x3)
qc62_out_ggga(x1, x2, x3, x4)  =  qc62_out_ggga(x1, x2, x3, x4)
U27_ggga(x1, x2, x3, x4)  =  U27_ggga(x1, x2, x3, x4)
U28_ggga(x1, x2, x3, x4, x5, x6)  =  U28_ggga(x1, x2, x3, x4, x6)
U29_ggga(x1, x2, x3, x4, x5, x6)  =  U29_ggga(x1, x2, x3, x4, x6)
GT44_IN_GG(x1, x2)  =  GT44_IN_GG(x1, x2)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) 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

(24) PiDPToQDPProof (EQUIVALENT transformation)

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

(25) 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.

(26) 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

(27) YES

(28) 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:

minc25_in_gga(T70, T71, T70) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, T52, .(T50, T51)) → U18_gag(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U18_gag(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T52, T50, T51, min2c15_in_gag(T55, T52, T51))
U19_gag(T48, T52, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
removec53_in_ggga(T140, T140, T141, T141) → removec53_out_ggga(T140, T140, T141, T141)
removec53_in_ggga(T150, T151, T152, .(T151, X175)) → U30_ggga(T150, T151, T152, X175, qc62_in_ggga(T150, T151, T152, X175))
qc62_in_ggga(T183, T151, [], []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
qc62_in_ggga(T192, T151, .(T192, T193), T193) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
qc62_in_ggga(T200, T151, .(T201, T202), .(T201, X238)) → U28_ggga(T200, T151, T201, T202, X238, notEqc63_in_gg(T200, T151))
U28_ggga(T200, T151, T201, T202, X238, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, X238, qc62_in_ggga(T200, T201, T202, X238))
U29_ggga(T200, T151, T201, T202, X238, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U30_ggga(T150, T151, T152, X175, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
minc25_in_gga(x1, x2, x3)  =  minc25_in_gga(x1, x2)
U31_gga(x1, x2, x3)  =  U31_gga(x1, x2, x3)
lec30_in_gg(x1, x2)  =  lec30_in_gg(x1, x2)
U20_gg(x1, x2, x3)  =  U20_gg(x1, x2, x3)
0  =  0
lec30_out_gg(x1, x2)  =  lec30_out_gg(x1, x2)
minc25_out_gga(x1, x2, x3)  =  minc25_out_gga(x1, x2, x3)
U32_gga(x1, x2, x3)  =  U32_gga(x1, x2, x3)
gtc44_in_gg(x1, x2)  =  gtc44_in_gg(x1, x2)
U21_gg(x1, x2, x3)  =  U21_gg(x1, x2, x3)
gtc44_out_gg(x1, x2)  =  gtc44_out_gg(x1, x2)
min2c15_in_gag(x1, x2, x3)  =  min2c15_in_gag(x1, x3)
[]  =  []
min2c15_out_gag(x1, x2, x3)  =  min2c15_out_gag(x1, x2, x3)
U18_gag(x1, x2, x3, x4, x5)  =  U18_gag(x1, x3, x4, x5)
U19_gag(x1, x2, x3, x4, x5)  =  U19_gag(x1, x3, x4, x5)
notEqc63_in_gg(x1, x2)  =  notEqc63_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
notEqc63_out_gg(x1, x2)  =  notEqc63_out_gg(x1, x2)
removec53_in_ggga(x1, x2, x3, x4)  =  removec53_in_ggga(x1, x2, x3)
removec53_out_ggga(x1, x2, x3, x4)  =  removec53_out_ggga(x1, x2, x3, x4)
U30_ggga(x1, x2, x3, x4, x5)  =  U30_ggga(x1, x2, x3, x5)
qc62_in_ggga(x1, x2, x3, x4)  =  qc62_in_ggga(x1, x2, x3)
U26_ggga(x1, x2, x3)  =  U26_ggga(x1, x2, x3)
qc62_out_ggga(x1, x2, x3, x4)  =  qc62_out_ggga(x1, x2, x3, x4)
U27_ggga(x1, x2, x3, x4)  =  U27_ggga(x1, x2, x3, x4)
U28_ggga(x1, x2, x3, x4, x5, x6)  =  U28_ggga(x1, x2, x3, x4, x6)
U29_ggga(x1, x2, x3, x4, x5, x6)  =  U29_ggga(x1, x2, x3, x4, x6)
LE30_IN_GG(x1, x2)  =  LE30_IN_GG(x1, x2)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) 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

(31) PiDPToQDPProof (EQUIVALENT transformation)

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

(32) 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.

(33) 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

(34) YES

(35) Obligation:

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

MIN215_IN_GAG(T48, T52, .(T50, T51)) → U3_GAG(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U3_GAG(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71, T70) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, T52, .(T50, T51)) → U18_gag(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U18_gag(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T52, T50, T51, min2c15_in_gag(T55, T52, T51))
U19_gag(T48, T52, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
removec53_in_ggga(T140, T140, T141, T141) → removec53_out_ggga(T140, T140, T141, T141)
removec53_in_ggga(T150, T151, T152, .(T151, X175)) → U30_ggga(T150, T151, T152, X175, qc62_in_ggga(T150, T151, T152, X175))
qc62_in_ggga(T183, T151, [], []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
qc62_in_ggga(T192, T151, .(T192, T193), T193) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
qc62_in_ggga(T200, T151, .(T201, T202), .(T201, X238)) → U28_ggga(T200, T151, T201, T202, X238, notEqc63_in_gg(T200, T151))
U28_ggga(T200, T151, T201, T202, X238, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, X238, qc62_in_ggga(T200, T201, T202, X238))
U29_ggga(T200, T151, T201, T202, X238, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U30_ggga(T150, T151, T152, X175, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
minc25_in_gga(x1, x2, x3)  =  minc25_in_gga(x1, x2)
U31_gga(x1, x2, x3)  =  U31_gga(x1, x2, x3)
lec30_in_gg(x1, x2)  =  lec30_in_gg(x1, x2)
U20_gg(x1, x2, x3)  =  U20_gg(x1, x2, x3)
0  =  0
lec30_out_gg(x1, x2)  =  lec30_out_gg(x1, x2)
minc25_out_gga(x1, x2, x3)  =  minc25_out_gga(x1, x2, x3)
U32_gga(x1, x2, x3)  =  U32_gga(x1, x2, x3)
gtc44_in_gg(x1, x2)  =  gtc44_in_gg(x1, x2)
U21_gg(x1, x2, x3)  =  U21_gg(x1, x2, x3)
gtc44_out_gg(x1, x2)  =  gtc44_out_gg(x1, x2)
min2c15_in_gag(x1, x2, x3)  =  min2c15_in_gag(x1, x3)
[]  =  []
min2c15_out_gag(x1, x2, x3)  =  min2c15_out_gag(x1, x2, x3)
U18_gag(x1, x2, x3, x4, x5)  =  U18_gag(x1, x3, x4, x5)
U19_gag(x1, x2, x3, x4, x5)  =  U19_gag(x1, x3, x4, x5)
notEqc63_in_gg(x1, x2)  =  notEqc63_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
notEqc63_out_gg(x1, x2)  =  notEqc63_out_gg(x1, x2)
removec53_in_ggga(x1, x2, x3, x4)  =  removec53_in_ggga(x1, x2, x3)
removec53_out_ggga(x1, x2, x3, x4)  =  removec53_out_ggga(x1, x2, x3, x4)
U30_ggga(x1, x2, x3, x4, x5)  =  U30_ggga(x1, x2, x3, x5)
qc62_in_ggga(x1, x2, x3, x4)  =  qc62_in_ggga(x1, x2, x3)
U26_ggga(x1, x2, x3)  =  U26_ggga(x1, x2, x3)
qc62_out_ggga(x1, x2, x3, x4)  =  qc62_out_ggga(x1, x2, x3, x4)
U27_ggga(x1, x2, x3, x4)  =  U27_ggga(x1, x2, x3, x4)
U28_ggga(x1, x2, x3, x4, x5, x6)  =  U28_ggga(x1, x2, x3, x4, x6)
U29_ggga(x1, x2, x3, x4, x5, x6)  =  U29_ggga(x1, x2, x3, x4, x6)
MIN215_IN_GAG(x1, x2, x3)  =  MIN215_IN_GAG(x1, x3)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x1, x3, x4, x5)

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

(36) UsableRulesProof (EQUIVALENT transformation)

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

(37) Obligation:

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

MIN215_IN_GAG(T48, T52, .(T50, T51)) → U3_GAG(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U3_GAG(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71, T70) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
minc25_in_gga(x1, x2, x3)  =  minc25_in_gga(x1, x2)
U31_gga(x1, x2, x3)  =  U31_gga(x1, x2, x3)
lec30_in_gg(x1, x2)  =  lec30_in_gg(x1, x2)
U20_gg(x1, x2, x3)  =  U20_gg(x1, x2, x3)
0  =  0
lec30_out_gg(x1, x2)  =  lec30_out_gg(x1, x2)
minc25_out_gga(x1, x2, x3)  =  minc25_out_gga(x1, x2, x3)
U32_gga(x1, x2, x3)  =  U32_gga(x1, x2, x3)
gtc44_in_gg(x1, x2)  =  gtc44_in_gg(x1, x2)
U21_gg(x1, x2, x3)  =  U21_gg(x1, x2, x3)
gtc44_out_gg(x1, x2)  =  gtc44_out_gg(x1, x2)
MIN215_IN_GAG(x1, x2, x3)  =  MIN215_IN_GAG(x1, x3)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x1, x3, x4, x5)

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

(38) PiDPToQDPProof (SOUND transformation)

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

(39) Obligation:

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

MIN215_IN_GAG(T48, .(T50, T51)) → U3_GAG(T48, T50, T51, minc25_in_gga(T48, T50))
U3_GAG(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T51)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
U31_gga(x0, x1, x2)
U32_gga(x0, x1, x2)
lec30_in_gg(x0, x1)
gtc44_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U21_gg(x0, x1, x2)

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

(40) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • U3_GAG(T48, T50, T51, minc25_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)) → U3_GAG(T48, T50, T51, minc25_in_gga(T48, T50))
    The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

(41) YES

(42) Obligation:

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

MINSORT1_IN_GA(.(T25, T26), .(T31, T32)) → U14_GA(T25, T26, T31, T32, min2c15_in_gag(T25, T31, T26))
U14_GA(T25, T26, T31, T32, min2c15_out_gag(T25, T31, T26)) → U15_GA(T25, T26, T31, T32, removec53_in_ggga(T31, T25, T26, T124))
U15_GA(T25, T26, T31, T32, removec53_out_ggga(T31, T25, T26, T124)) → MINSORT1_IN_GA(T124, T32)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71, T70) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, T52, .(T50, T51)) → U18_gag(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U18_gag(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T52, T50, T51, min2c15_in_gag(T55, T52, T51))
U19_gag(T48, T52, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
removec53_in_ggga(T140, T140, T141, T141) → removec53_out_ggga(T140, T140, T141, T141)
removec53_in_ggga(T150, T151, T152, .(T151, X175)) → U30_ggga(T150, T151, T152, X175, qc62_in_ggga(T150, T151, T152, X175))
qc62_in_ggga(T183, T151, [], []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
qc62_in_ggga(T192, T151, .(T192, T193), T193) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
qc62_in_ggga(T200, T151, .(T201, T202), .(T201, X238)) → U28_ggga(T200, T151, T201, T202, X238, notEqc63_in_gg(T200, T151))
U28_ggga(T200, T151, T201, T202, X238, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, X238, qc62_in_ggga(T200, T201, T202, X238))
U29_ggga(T200, T151, T201, T202, X238, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U30_ggga(T150, T151, T152, X175, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
minc25_in_gga(x1, x2, x3)  =  minc25_in_gga(x1, x2)
U31_gga(x1, x2, x3)  =  U31_gga(x1, x2, x3)
lec30_in_gg(x1, x2)  =  lec30_in_gg(x1, x2)
U20_gg(x1, x2, x3)  =  U20_gg(x1, x2, x3)
0  =  0
lec30_out_gg(x1, x2)  =  lec30_out_gg(x1, x2)
minc25_out_gga(x1, x2, x3)  =  minc25_out_gga(x1, x2, x3)
U32_gga(x1, x2, x3)  =  U32_gga(x1, x2, x3)
gtc44_in_gg(x1, x2)  =  gtc44_in_gg(x1, x2)
U21_gg(x1, x2, x3)  =  U21_gg(x1, x2, x3)
gtc44_out_gg(x1, x2)  =  gtc44_out_gg(x1, x2)
min2c15_in_gag(x1, x2, x3)  =  min2c15_in_gag(x1, x3)
[]  =  []
min2c15_out_gag(x1, x2, x3)  =  min2c15_out_gag(x1, x2, x3)
U18_gag(x1, x2, x3, x4, x5)  =  U18_gag(x1, x3, x4, x5)
U19_gag(x1, x2, x3, x4, x5)  =  U19_gag(x1, x3, x4, x5)
notEqc63_in_gg(x1, x2)  =  notEqc63_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
notEqc63_out_gg(x1, x2)  =  notEqc63_out_gg(x1, x2)
removec53_in_ggga(x1, x2, x3, x4)  =  removec53_in_ggga(x1, x2, x3)
removec53_out_ggga(x1, x2, x3, x4)  =  removec53_out_ggga(x1, x2, x3, x4)
U30_ggga(x1, x2, x3, x4, x5)  =  U30_ggga(x1, x2, x3, x5)
qc62_in_ggga(x1, x2, x3, x4)  =  qc62_in_ggga(x1, x2, x3)
U26_ggga(x1, x2, x3)  =  U26_ggga(x1, x2, x3)
qc62_out_ggga(x1, x2, x3, x4)  =  qc62_out_ggga(x1, x2, x3, x4)
U27_ggga(x1, x2, x3, x4)  =  U27_ggga(x1, x2, x3, x4)
U28_ggga(x1, x2, x3, x4, x5, x6)  =  U28_ggga(x1, x2, x3, x4, x6)
U29_ggga(x1, x2, x3, x4, x5, x6)  =  U29_ggga(x1, x2, x3, x4, x6)
MINSORT1_IN_GA(x1, x2)  =  MINSORT1_IN_GA(x1)
U14_GA(x1, x2, x3, x4, x5)  =  U14_GA(x1, x2, x5)
U15_GA(x1, x2, x3, x4, x5)  =  U15_GA(x1, x2, x5)

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

(43) PiDPToQDPProof (SOUND transformation)

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

(44) Obligation:

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

MINSORT1_IN_GA(.(T25, T26)) → U14_GA(T25, T26, min2c15_in_gag(T25, T26))
U14_GA(T25, T26, min2c15_out_gag(T25, T31, T26)) → U15_GA(T25, T26, removec53_in_ggga(T31, T25, T26))
U15_GA(T25, T26, removec53_out_ggga(T31, T25, T26, T124)) → MINSORT1_IN_GA(T124)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
removec53_in_ggga(T140, T140, T141) → removec53_out_ggga(T140, T140, T141, T141)
removec53_in_ggga(T150, T151, T152) → U30_ggga(T150, T151, T152, qc62_in_ggga(T150, T151, T152))
qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
removec53_in_ggga(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(45) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MINSORT1_IN_GA(.(T25, T26)) → U14_GA(T25, T26, min2c15_in_gag(T25, T26)) at position [2] we obtained the following new rules [LPAR04]:

MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))

(46) Obligation:

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

U14_GA(T25, T26, min2c15_out_gag(T25, T31, T26)) → U15_GA(T25, T26, removec53_in_ggga(T31, T25, T26))
U15_GA(T25, T26, removec53_out_ggga(T31, T25, T26, T124)) → MINSORT1_IN_GA(T124)
MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))

The TRS R consists of the following rules:

minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
removec53_in_ggga(T140, T140, T141) → removec53_out_ggga(T140, T140, T141, T141)
removec53_in_ggga(T150, T151, T152) → U30_ggga(T150, T151, T152, qc62_in_ggga(T150, T151, T152))
qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
removec53_in_ggga(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(47) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U14_GA(T25, T26, min2c15_out_gag(T25, T31, T26)) → U15_GA(T25, T26, removec53_in_ggga(T31, T25, T26)) at position [2] we obtained the following new rules [LPAR04]:

U14_GA(x0, x1, min2c15_out_gag(x0, x0, x1)) → U15_GA(x0, x1, removec53_out_ggga(x0, x0, x1, x1))
U14_GA(x1, x2, min2c15_out_gag(x1, x0, x2)) → U15_GA(x1, x2, U30_ggga(x0, x1, x2, qc62_in_ggga(x0, x1, x2)))

(48) Obligation:

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

U15_GA(T25, T26, removec53_out_ggga(T31, T25, T26, T124)) → MINSORT1_IN_GA(T124)
MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))
U14_GA(x0, x1, min2c15_out_gag(x0, x0, x1)) → U15_GA(x0, x1, removec53_out_ggga(x0, x0, x1, x1))
U14_GA(x1, x2, min2c15_out_gag(x1, x0, x2)) → U15_GA(x1, x2, U30_ggga(x0, x1, x2, qc62_in_ggga(x0, x1, x2)))

The TRS R consists of the following rules:

minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
removec53_in_ggga(T140, T140, T141) → removec53_out_ggga(T140, T140, T141, T141)
removec53_in_ggga(T150, T151, T152) → U30_ggga(T150, T151, T152, qc62_in_ggga(T150, T151, T152))
qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
removec53_in_ggga(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(49) 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.

(50) Obligation:

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

U15_GA(T25, T26, removec53_out_ggga(T31, T25, T26, T124)) → MINSORT1_IN_GA(T124)
MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))
U14_GA(x0, x1, min2c15_out_gag(x0, x0, x1)) → U15_GA(x0, x1, removec53_out_ggga(x0, x0, x1, x1))
U14_GA(x1, x2, min2c15_out_gag(x1, x0, x2)) → U15_GA(x1, x2, U30_ggga(x0, x1, x2, qc62_in_ggga(x0, x1, x2)))

The TRS R consists of the following rules:

qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
removec53_in_ggga(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(51) 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].

removec53_in_ggga(x0, x1, x2)

(52) Obligation:

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

U15_GA(T25, T26, removec53_out_ggga(T31, T25, T26, T124)) → MINSORT1_IN_GA(T124)
MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))
U14_GA(x0, x1, min2c15_out_gag(x0, x0, x1)) → U15_GA(x0, x1, removec53_out_ggga(x0, x0, x1, x1))
U14_GA(x1, x2, min2c15_out_gag(x1, x0, x2)) → U15_GA(x1, x2, U30_ggga(x0, x1, x2, qc62_in_ggga(x0, x1, x2)))

The TRS R consists of the following rules:

qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(53) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U14_GA(x0, x1, min2c15_out_gag(x0, x0, x1)) → U15_GA(x0, x1, removec53_out_ggga(x0, x0, x1, x1)) we obtained the following new rules [LPAR04]:

U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], removec53_out_ggga(z0, z0, [], []))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, z0, .(z1, z2))) → U15_GA(z0, .(z1, z2), removec53_out_ggga(z0, z0, .(z1, z2), .(z1, z2)))

(54) Obligation:

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

U15_GA(T25, T26, removec53_out_ggga(T31, T25, T26, T124)) → MINSORT1_IN_GA(T124)
MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))
U14_GA(x1, x2, min2c15_out_gag(x1, x0, x2)) → U15_GA(x1, x2, U30_ggga(x0, x1, x2, qc62_in_ggga(x0, x1, x2)))
U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], removec53_out_ggga(z0, z0, [], []))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, z0, .(z1, z2))) → U15_GA(z0, .(z1, z2), removec53_out_ggga(z0, z0, .(z1, z2), .(z1, z2)))

The TRS R consists of the following rules:

qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(55) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U14_GA(x1, x2, min2c15_out_gag(x1, x0, x2)) → U15_GA(x1, x2, U30_ggga(x0, x1, x2, qc62_in_ggga(x0, x1, x2))) we obtained the following new rules [LPAR04]:

U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], U30_ggga(z0, z0, [], qc62_in_ggga(z0, z0, [])))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, x2, .(z1, z2))) → U15_GA(z0, .(z1, z2), U30_ggga(x2, z0, .(z1, z2), qc62_in_ggga(x2, z0, .(z1, z2))))

(56) Obligation:

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

U15_GA(T25, T26, removec53_out_ggga(T31, T25, T26, T124)) → MINSORT1_IN_GA(T124)
MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))
U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], removec53_out_ggga(z0, z0, [], []))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, z0, .(z1, z2))) → U15_GA(z0, .(z1, z2), removec53_out_ggga(z0, z0, .(z1, z2), .(z1, z2)))
U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], U30_ggga(z0, z0, [], qc62_in_ggga(z0, z0, [])))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, x2, .(z1, z2))) → U15_GA(z0, .(z1, z2), U30_ggga(x2, z0, .(z1, z2), qc62_in_ggga(x2, z0, .(z1, z2))))

The TRS R consists of the following rules:

qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(57) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], U30_ggga(z0, z0, [], qc62_in_ggga(z0, z0, []))) at position [2,3] we obtained the following new rules [LPAR04]:

U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], U30_ggga(z0, z0, [], U26_ggga(z0, z0, notEqc63_in_gg(z0, z0))))

(58) Obligation:

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

U15_GA(T25, T26, removec53_out_ggga(T31, T25, T26, T124)) → MINSORT1_IN_GA(T124)
MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))
U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], removec53_out_ggga(z0, z0, [], []))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, z0, .(z1, z2))) → U15_GA(z0, .(z1, z2), removec53_out_ggga(z0, z0, .(z1, z2), .(z1, z2)))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, x2, .(z1, z2))) → U15_GA(z0, .(z1, z2), U30_ggga(x2, z0, .(z1, z2), qc62_in_ggga(x2, z0, .(z1, z2))))
U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], U30_ggga(z0, z0, [], U26_ggga(z0, z0, notEqc63_in_gg(z0, z0))))

The TRS R consists of the following rules:

qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(59) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U15_GA(T25, T26, removec53_out_ggga(T31, T25, T26, T124)) → MINSORT1_IN_GA(T124) we obtained the following new rules [LPAR04]:

U15_GA(z0, [], removec53_out_ggga(z0, z0, [], [])) → MINSORT1_IN_GA([])
U15_GA(z0, .(z1, z2), removec53_out_ggga(z0, z0, .(z1, z2), .(z1, z2))) → MINSORT1_IN_GA(.(z1, z2))
U15_GA(z0, .(z1, z2), removec53_out_ggga(x2, z0, .(z1, z2), x3)) → MINSORT1_IN_GA(x3)
U15_GA(z0, [], removec53_out_ggga(x2, z0, [], x3)) → MINSORT1_IN_GA(x3)

(60) Obligation:

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

MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))
U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], removec53_out_ggga(z0, z0, [], []))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, z0, .(z1, z2))) → U15_GA(z0, .(z1, z2), removec53_out_ggga(z0, z0, .(z1, z2), .(z1, z2)))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, x2, .(z1, z2))) → U15_GA(z0, .(z1, z2), U30_ggga(x2, z0, .(z1, z2), qc62_in_ggga(x2, z0, .(z1, z2))))
U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], U30_ggga(z0, z0, [], U26_ggga(z0, z0, notEqc63_in_gg(z0, z0))))
U15_GA(z0, [], removec53_out_ggga(z0, z0, [], [])) → MINSORT1_IN_GA([])
U15_GA(z0, .(z1, z2), removec53_out_ggga(z0, z0, .(z1, z2), .(z1, z2))) → MINSORT1_IN_GA(.(z1, z2))
U15_GA(z0, .(z1, z2), removec53_out_ggga(x2, z0, .(z1, z2), x3)) → MINSORT1_IN_GA(x3)
U15_GA(z0, [], removec53_out_ggga(x2, z0, [], x3)) → MINSORT1_IN_GA(x3)

The TRS R consists of the following rules:

qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(61) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(62) Obligation:

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

U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], removec53_out_ggga(z0, z0, [], []))
U15_GA(z0, [], removec53_out_ggga(x2, z0, [], x3)) → MINSORT1_IN_GA(x3)
MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], U30_ggga(z0, z0, [], U26_ggga(z0, z0, notEqc63_in_gg(z0, z0))))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, z0, .(z1, z2))) → U15_GA(z0, .(z1, z2), removec53_out_ggga(z0, z0, .(z1, z2), .(z1, z2)))
U15_GA(z0, .(z1, z2), removec53_out_ggga(z0, z0, .(z1, z2), .(z1, z2))) → MINSORT1_IN_GA(.(z1, z2))
U15_GA(z0, .(z1, z2), removec53_out_ggga(x2, z0, .(z1, z2), x3)) → MINSORT1_IN_GA(x3)
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, x2, .(z1, z2))) → U15_GA(z0, .(z1, z2), U30_ggga(x2, z0, .(z1, z2), qc62_in_ggga(x2, z0, .(z1, z2))))

The TRS R consists of the following rules:

qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(63) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U15_GA(z0, [], removec53_out_ggga(x2, z0, [], x3)) → MINSORT1_IN_GA(x3) we obtained the following new rules [LPAR04]:

U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, []))) → MINSORT1_IN_GA(.(y_0, []))
U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, .(y_1, y_2)))) → MINSORT1_IN_GA(.(y_0, .(y_1, y_2)))

(64) Obligation:

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

U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], removec53_out_ggga(z0, z0, [], []))
MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], U30_ggga(z0, z0, [], U26_ggga(z0, z0, notEqc63_in_gg(z0, z0))))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, z0, .(z1, z2))) → U15_GA(z0, .(z1, z2), removec53_out_ggga(z0, z0, .(z1, z2), .(z1, z2)))
U15_GA(z0, .(z1, z2), removec53_out_ggga(z0, z0, .(z1, z2), .(z1, z2))) → MINSORT1_IN_GA(.(z1, z2))
U15_GA(z0, .(z1, z2), removec53_out_ggga(x2, z0, .(z1, z2), x3)) → MINSORT1_IN_GA(x3)
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, x2, .(z1, z2))) → U15_GA(z0, .(z1, z2), U30_ggga(x2, z0, .(z1, z2), qc62_in_ggga(x2, z0, .(z1, z2))))
U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, []))) → MINSORT1_IN_GA(.(y_0, []))
U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, .(y_1, y_2)))) → MINSORT1_IN_GA(.(y_0, .(y_1, y_2)))

The TRS R consists of the following rules:

qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(65) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(66) Obligation:

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

U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], U30_ggga(z0, z0, [], U26_ggga(z0, z0, notEqc63_in_gg(z0, z0))))
U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, []))) → MINSORT1_IN_GA(.(y_0, []))
MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, .(y_1, y_2)))) → MINSORT1_IN_GA(.(y_0, .(y_1, y_2)))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, z0, .(z1, z2))) → U15_GA(z0, .(z1, z2), removec53_out_ggga(z0, z0, .(z1, z2), .(z1, z2)))
U15_GA(z0, .(z1, z2), removec53_out_ggga(z0, z0, .(z1, z2), .(z1, z2))) → MINSORT1_IN_GA(.(z1, z2))
U15_GA(z0, .(z1, z2), removec53_out_ggga(x2, z0, .(z1, z2), x3)) → MINSORT1_IN_GA(x3)
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, x2, .(z1, z2))) → U15_GA(z0, .(z1, z2), U30_ggga(x2, z0, .(z1, z2), qc62_in_ggga(x2, z0, .(z1, z2))))

The TRS R consists of the following rules:

qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(67) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U15_GA(z0, .(z1, z2), removec53_out_ggga(z0, z0, .(z1, z2), .(z1, z2))) → MINSORT1_IN_GA(.(z1, z2)) we obtained the following new rules [LPAR04]:

U15_GA(x0, .(x1, []), removec53_out_ggga(x0, x0, .(x1, []), .(x1, []))) → MINSORT1_IN_GA(.(x1, []))
U15_GA(x0, .(x1, .(y_1, y_2)), removec53_out_ggga(x0, x0, .(x1, .(y_1, y_2)), .(x1, .(y_1, y_2)))) → MINSORT1_IN_GA(.(x1, .(y_1, y_2)))

(68) Obligation:

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

U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], U30_ggga(z0, z0, [], U26_ggga(z0, z0, notEqc63_in_gg(z0, z0))))
U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, []))) → MINSORT1_IN_GA(.(y_0, []))
MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, .(y_1, y_2)))) → MINSORT1_IN_GA(.(y_0, .(y_1, y_2)))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, z0, .(z1, z2))) → U15_GA(z0, .(z1, z2), removec53_out_ggga(z0, z0, .(z1, z2), .(z1, z2)))
U15_GA(z0, .(z1, z2), removec53_out_ggga(x2, z0, .(z1, z2), x3)) → MINSORT1_IN_GA(x3)
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, x2, .(z1, z2))) → U15_GA(z0, .(z1, z2), U30_ggga(x2, z0, .(z1, z2), qc62_in_ggga(x2, z0, .(z1, z2))))
U15_GA(x0, .(x1, []), removec53_out_ggga(x0, x0, .(x1, []), .(x1, []))) → MINSORT1_IN_GA(.(x1, []))
U15_GA(x0, .(x1, .(y_1, y_2)), removec53_out_ggga(x0, x0, .(x1, .(y_1, y_2)), .(x1, .(y_1, y_2)))) → MINSORT1_IN_GA(.(x1, .(y_1, y_2)))

The TRS R consists of the following rules:

qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(69) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U15_GA(z0, .(z1, z2), removec53_out_ggga(x2, z0, .(z1, z2), x3)) → MINSORT1_IN_GA(x3) we obtained the following new rules [LPAR04]:

U15_GA(x0, .(x1, x2), removec53_out_ggga(x3, x0, .(x1, x2), .(y_0, []))) → MINSORT1_IN_GA(.(y_0, []))
U15_GA(x0, .(x1, x2), removec53_out_ggga(x3, x0, .(x1, x2), .(y_0, .(y_1, y_2)))) → MINSORT1_IN_GA(.(y_0, .(y_1, y_2)))

(70) Obligation:

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

U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], U30_ggga(z0, z0, [], U26_ggga(z0, z0, notEqc63_in_gg(z0, z0))))
U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, []))) → MINSORT1_IN_GA(.(y_0, []))
MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, .(y_1, y_2)))) → MINSORT1_IN_GA(.(y_0, .(y_1, y_2)))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, z0, .(z1, z2))) → U15_GA(z0, .(z1, z2), removec53_out_ggga(z0, z0, .(z1, z2), .(z1, z2)))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, x2, .(z1, z2))) → U15_GA(z0, .(z1, z2), U30_ggga(x2, z0, .(z1, z2), qc62_in_ggga(x2, z0, .(z1, z2))))
U15_GA(x0, .(x1, []), removec53_out_ggga(x0, x0, .(x1, []), .(x1, []))) → MINSORT1_IN_GA(.(x1, []))
U15_GA(x0, .(x1, .(y_1, y_2)), removec53_out_ggga(x0, x0, .(x1, .(y_1, y_2)), .(x1, .(y_1, y_2)))) → MINSORT1_IN_GA(.(x1, .(y_1, y_2)))
U15_GA(x0, .(x1, x2), removec53_out_ggga(x3, x0, .(x1, x2), .(y_0, []))) → MINSORT1_IN_GA(.(y_0, []))
U15_GA(x0, .(x1, x2), removec53_out_ggga(x3, x0, .(x1, x2), .(y_0, .(y_1, y_2)))) → MINSORT1_IN_GA(.(y_0, .(y_1, y_2)))

The TRS R consists of the following rules:

qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(71) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, z0, .(z1, z2))) → U15_GA(z0, .(z1, z2), removec53_out_ggga(z0, z0, .(z1, z2), .(z1, z2))) we obtained the following new rules [LPAR04]:

U14_GA(x0, .(x1, []), min2c15_out_gag(x0, x0, .(x1, []))) → U15_GA(x0, .(x1, []), removec53_out_ggga(x0, x0, .(x1, []), .(x1, [])))
U14_GA(x0, .(x1, .(y_2, y_3)), min2c15_out_gag(x0, x0, .(x1, .(y_2, y_3)))) → U15_GA(x0, .(x1, .(y_2, y_3)), removec53_out_ggga(x0, x0, .(x1, .(y_2, y_3)), .(x1, .(y_2, y_3))))

(72) Obligation:

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

U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], U30_ggga(z0, z0, [], U26_ggga(z0, z0, notEqc63_in_gg(z0, z0))))
U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, []))) → MINSORT1_IN_GA(.(y_0, []))
MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, .(y_1, y_2)))) → MINSORT1_IN_GA(.(y_0, .(y_1, y_2)))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, x2, .(z1, z2))) → U15_GA(z0, .(z1, z2), U30_ggga(x2, z0, .(z1, z2), qc62_in_ggga(x2, z0, .(z1, z2))))
U15_GA(x0, .(x1, []), removec53_out_ggga(x0, x0, .(x1, []), .(x1, []))) → MINSORT1_IN_GA(.(x1, []))
U15_GA(x0, .(x1, .(y_1, y_2)), removec53_out_ggga(x0, x0, .(x1, .(y_1, y_2)), .(x1, .(y_1, y_2)))) → MINSORT1_IN_GA(.(x1, .(y_1, y_2)))
U15_GA(x0, .(x1, x2), removec53_out_ggga(x3, x0, .(x1, x2), .(y_0, []))) → MINSORT1_IN_GA(.(y_0, []))
U15_GA(x0, .(x1, x2), removec53_out_ggga(x3, x0, .(x1, x2), .(y_0, .(y_1, y_2)))) → MINSORT1_IN_GA(.(y_0, .(y_1, y_2)))
U14_GA(x0, .(x1, []), min2c15_out_gag(x0, x0, .(x1, []))) → U15_GA(x0, .(x1, []), removec53_out_ggga(x0, x0, .(x1, []), .(x1, [])))
U14_GA(x0, .(x1, .(y_2, y_3)), min2c15_out_gag(x0, x0, .(x1, .(y_2, y_3)))) → U15_GA(x0, .(x1, .(y_2, y_3)), removec53_out_ggga(x0, x0, .(x1, .(y_2, y_3)), .(x1, .(y_2, y_3))))

The TRS R consists of the following rules:

qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(73) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U14_GA(x0, .(x1, []), min2c15_out_gag(x0, x0, .(x1, []))) → U15_GA(x0, .(x1, []), removec53_out_ggga(x0, x0, .(x1, []), .(x1, [])))
U14_GA(x0, .(x1, .(y_2, y_3)), min2c15_out_gag(x0, x0, .(x1, .(y_2, y_3)))) → U15_GA(x0, .(x1, .(y_2, y_3)), removec53_out_ggga(x0, x0, .(x1, .(y_2, y_3)), .(x1, .(y_2, y_3))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x2   
POL(0) = 0   
POL(MINSORT1_IN_GA(x1)) = x1   
POL(U14_GA(x1, x2, x3)) = 1 + x2   
POL(U15_GA(x1, x2, x3)) = x3   
POL(U18_gag(x1, x2, x3, x4)) = 0   
POL(U19_gag(x1, x2, x3, x4)) = 0   
POL(U20_gg(x1, x2, x3)) = 0   
POL(U21_gg(x1, x2, x3)) = 0   
POL(U25_gg(x1, x2, x3)) = 0   
POL(U26_ggga(x1, x2, x3)) = 0   
POL(U27_ggga(x1, x2, x3, x4)) = x3   
POL(U28_ggga(x1, x2, x3, x4, x5)) = 1 + x4   
POL(U29_ggga(x1, x2, x3, x4, x5)) = 1 + x5   
POL(U30_ggga(x1, x2, x3, x4)) = 1 + x4   
POL(U31_gga(x1, x2, x3)) = 0   
POL(U32_gga(x1, x2, x3)) = 0   
POL([]) = 0   
POL(gtc44_in_gg(x1, x2)) = 0   
POL(gtc44_out_gg(x1, x2)) = 0   
POL(lec30_in_gg(x1, x2)) = 0   
POL(lec30_out_gg(x1, x2)) = 0   
POL(min2c15_in_gag(x1, x2)) = 0   
POL(min2c15_out_gag(x1, x2, x3)) = 0   
POL(minc25_in_gga(x1, x2)) = 0   
POL(minc25_out_gga(x1, x2, x3)) = 0   
POL(notEqc63_in_gg(x1, x2)) = 0   
POL(notEqc63_out_gg(x1, x2)) = 0   
POL(qc62_in_ggga(x1, x2, x3)) = x3   
POL(qc62_out_ggga(x1, x2, x3, x4)) = x4   
POL(removec53_out_ggga(x1, x2, x3, x4)) = x4   
POL(s(x1)) = 0   

The following usable rules [FROCOS05] were oriented:

U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)

(74) Obligation:

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

U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], U30_ggga(z0, z0, [], U26_ggga(z0, z0, notEqc63_in_gg(z0, z0))))
U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, []))) → MINSORT1_IN_GA(.(y_0, []))
MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, .(y_1, y_2)))) → MINSORT1_IN_GA(.(y_0, .(y_1, y_2)))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, x2, .(z1, z2))) → U15_GA(z0, .(z1, z2), U30_ggga(x2, z0, .(z1, z2), qc62_in_ggga(x2, z0, .(z1, z2))))
U15_GA(x0, .(x1, []), removec53_out_ggga(x0, x0, .(x1, []), .(x1, []))) → MINSORT1_IN_GA(.(x1, []))
U15_GA(x0, .(x1, .(y_1, y_2)), removec53_out_ggga(x0, x0, .(x1, .(y_1, y_2)), .(x1, .(y_1, y_2)))) → MINSORT1_IN_GA(.(x1, .(y_1, y_2)))
U15_GA(x0, .(x1, x2), removec53_out_ggga(x3, x0, .(x1, x2), .(y_0, []))) → MINSORT1_IN_GA(.(y_0, []))
U15_GA(x0, .(x1, x2), removec53_out_ggga(x3, x0, .(x1, x2), .(y_0, .(y_1, y_2)))) → MINSORT1_IN_GA(.(y_0, .(y_1, y_2)))

The TRS R consists of the following rules:

qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(75) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U15_GA(x0, .(x1, []), removec53_out_ggga(x0, x0, .(x1, []), .(x1, []))) → MINSORT1_IN_GA(.(x1, []))
U15_GA(x0, .(x1, .(y_1, y_2)), removec53_out_ggga(x0, x0, .(x1, .(y_1, y_2)), .(x1, .(y_1, y_2)))) → MINSORT1_IN_GA(.(x1, .(y_1, y_2)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(U14_GA(x1, x2, x3)) = -I + 0A·x1 + -I·x2 + 1A·x3

POL([]) = 0A

POL(min2c15_out_gag(x1, x2, x3)) = 0A + 0A·x1 + 0A·x2 + 0A·x3

POL(U15_GA(x1, x2, x3)) = -I + 0A·x1 + -I·x2 + 0A·x3

POL(U30_ggga(x1, x2, x3, x4)) = -I + 0A·x1 + 1A·x2 + -I·x3 + 1A·x4

POL(U26_ggga(x1, x2, x3)) = -I + -I·x1 + 0A·x2 + 0A·x3

POL(notEqc63_in_gg(x1, x2)) = -I + -I·x1 + 0A·x2

POL(removec53_out_ggga(x1, x2, x3, x4)) = 0A + 0A·x1 + -I·x2 + 1A·x3 + 0A·x4

POL(.(x1, x2)) = 0A + 1A·x1 + 1A·x2

POL(MINSORT1_IN_GA(x1)) = -I + 0A·x1

POL(U18_gag(x1, x2, x3, x4)) = -I + -I·x1 + 1A·x2 + 1A·x3 + 0A·x4

POL(minc25_in_gga(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(qc62_in_ggga(x1, x2, x3)) = -I + -I·x1 + 0A·x2 + 0A·x3

POL(s(x1)) = 0A + -I·x1

POL(U25_gg(x1, x2, x3)) = 0A + -I·x1 + -I·x2 + -I·x3

POL(notEqc63_out_gg(x1, x2)) = 0A + 0A·x1 + -I·x2

POL(qc62_out_ggga(x1, x2, x3, x4)) = 0A + -I·x1 + -I·x2 + 0A·x3 + 0A·x4

POL(U31_gga(x1, x2, x3)) = -I + 0A·x1 + 0A·x2 + -I·x3

POL(lec30_in_gg(x1, x2)) = 0A + -I·x1 + 0A·x2

POL(U32_gga(x1, x2, x3)) = -I + 0A·x1 + 0A·x2 + -I·x3

POL(gtc44_in_gg(x1, x2)) = 0A + -I·x1 + -I·x2

POL(minc25_out_gga(x1, x2, x3)) = -I + 0A·x1 + -I·x2 + 0A·x3

POL(U19_gag(x1, x2, x3, x4)) = -I + 0A·x1 + 1A·x2 + 1A·x3 + 0A·x4

POL(min2c15_in_gag(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(U27_ggga(x1, x2, x3, x4)) = 0A + 1A·x1 + 0A·x2 + 1A·x3 + 0A·x4

POL(U28_ggga(x1, x2, x3, x4, x5)) = -I + -I·x1 + -I·x2 + 1A·x3 + 1A·x4 + 0A·x5

POL(0) = 0A

POL(U29_ggga(x1, x2, x3, x4, x5)) = -I + 0A·x1 + -I·x2 + 1A·x3 + -I·x4 + 1A·x5

POL(U20_gg(x1, x2, x3)) = -I + 0A·x1 + -I·x2 + -I·x3

POL(lec30_out_gg(x1, x2)) = -I + -I·x1 + 5A·x2

POL(U21_gg(x1, x2, x3)) = 0A + -I·x1 + 3A·x2 + 1A·x3

POL(gtc44_out_gg(x1, x2)) = -I + 0A·x1 + 2A·x2

The following usable rules [FROCOS05] were oriented:

notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)

(76) Obligation:

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

U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], U30_ggga(z0, z0, [], U26_ggga(z0, z0, notEqc63_in_gg(z0, z0))))
U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, []))) → MINSORT1_IN_GA(.(y_0, []))
MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, .(y_1, y_2)))) → MINSORT1_IN_GA(.(y_0, .(y_1, y_2)))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, x2, .(z1, z2))) → U15_GA(z0, .(z1, z2), U30_ggga(x2, z0, .(z1, z2), qc62_in_ggga(x2, z0, .(z1, z2))))
U15_GA(x0, .(x1, x2), removec53_out_ggga(x3, x0, .(x1, x2), .(y_0, []))) → MINSORT1_IN_GA(.(y_0, []))
U15_GA(x0, .(x1, x2), removec53_out_ggga(x3, x0, .(x1, x2), .(y_0, .(y_1, y_2)))) → MINSORT1_IN_GA(.(y_0, .(y_1, y_2)))

The TRS R consists of the following rules:

qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(77) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U15_GA(x0, .(x1, x2), removec53_out_ggga(x3, x0, .(x1, x2), .(y_0, []))) → MINSORT1_IN_GA(.(y_0, []))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(U14_GA(x1, x2, x3)) = 0A + -I·x1 + 1A·x2 + 0A·x3

POL([]) = 0A

POL(min2c15_out_gag(x1, x2, x3)) = 0A + -I·x1 + -I·x2 + -I·x3

POL(U15_GA(x1, x2, x3)) = -I + -I·x1 + 1A·x2 + 0A·x3

POL(U30_ggga(x1, x2, x3, x4)) = 1A + -I·x1 + -I·x2 + 0A·x3 + 0A·x4

POL(U26_ggga(x1, x2, x3)) = 1A + -I·x1 + -I·x2 + 0A·x3

POL(notEqc63_in_gg(x1, x2)) = 0A + -I·x1 + -I·x2

POL(removec53_out_ggga(x1, x2, x3, x4)) = 0A + -I·x1 + 0A·x2 + 0A·x3 + 0A·x4

POL(.(x1, x2)) = 1A + -I·x1 + 1A·x2

POL(MINSORT1_IN_GA(x1)) = -I + 0A·x1

POL(U18_gag(x1, x2, x3, x4)) = -I + -I·x1 + -I·x2 + 0A·x3 + -I·x4

POL(minc25_in_gga(x1, x2)) = -I + -I·x1 + 0A·x2

POL(qc62_in_ggga(x1, x2, x3)) = -I + -I·x1 + -I·x2 + 1A·x3

POL(s(x1)) = 0A + -I·x1

POL(U25_gg(x1, x2, x3)) = -I + -I·x1 + -I·x2 + 0A·x3

POL(notEqc63_out_gg(x1, x2)) = 0A + -I·x1 + 0A·x2

POL(qc62_out_ggga(x1, x2, x3, x4)) = 0A + -I·x1 + 0A·x2 + 0A·x3 + 1A·x4

POL(U31_gga(x1, x2, x3)) = 0A + 5A·x1 + 0A·x2 + 0A·x3

POL(lec30_in_gg(x1, x2)) = 0A + 0A·x1 + -I·x2

POL(U32_gga(x1, x2, x3)) = -I + 3A·x1 + -I·x2 + 0A·x3

POL(gtc44_in_gg(x1, x2)) = 1A + 0A·x1 + 0A·x2

POL(minc25_out_gga(x1, x2, x3)) = 0A + 5A·x1 + -I·x2 + -I·x3

POL(U19_gag(x1, x2, x3, x4)) = -I + -I·x1 + -I·x2 + 0A·x3 + 0A·x4

POL(min2c15_in_gag(x1, x2)) = -I + -I·x1 + 0A·x2

POL(U27_ggga(x1, x2, x3, x4)) = 0A + -I·x1 + -I·x2 + 2A·x3 + 2A·x4

POL(U28_ggga(x1, x2, x3, x4, x5)) = 2A + -I·x1 + -I·x2 + -I·x3 + 2A·x4 + 0A·x5

POL(0) = 0A

POL(U29_ggga(x1, x2, x3, x4, x5)) = 2A + -I·x1 + 0A·x2 + -I·x3 + -I·x4 + 1A·x5

POL(U20_gg(x1, x2, x3)) = -I + 5A·x1 + -I·x2 + -I·x3

POL(lec30_out_gg(x1, x2)) = 0A + -I·x1 + -I·x2

POL(U21_gg(x1, x2, x3)) = -I + -I·x1 + -I·x2 + 4A·x3

POL(gtc44_out_gg(x1, x2)) = -I + 0A·x1 + 4A·x2

The following usable rules [FROCOS05] were oriented:

notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))

(78) Obligation:

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

U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], U30_ggga(z0, z0, [], U26_ggga(z0, z0, notEqc63_in_gg(z0, z0))))
U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, []))) → MINSORT1_IN_GA(.(y_0, []))
MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, .(y_1, y_2)))) → MINSORT1_IN_GA(.(y_0, .(y_1, y_2)))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))
U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, x2, .(z1, z2))) → U15_GA(z0, .(z1, z2), U30_ggga(x2, z0, .(z1, z2), qc62_in_ggga(x2, z0, .(z1, z2))))
U15_GA(x0, .(x1, x2), removec53_out_ggga(x3, x0, .(x1, x2), .(y_0, .(y_1, y_2)))) → MINSORT1_IN_GA(.(y_0, .(y_1, y_2)))

The TRS R consists of the following rules:

qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(79) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.

(80) Complex Obligation (AND)

(81) Obligation:

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

U14_GA(z0, .(z1, z2), min2c15_out_gag(z0, x2, .(z1, z2))) → U15_GA(z0, .(z1, z2), U30_ggga(x2, z0, .(z1, z2), qc62_in_ggga(x2, z0, .(z1, z2))))
U15_GA(x0, .(x1, x2), removec53_out_ggga(x3, x0, .(x1, x2), .(y_0, .(y_1, y_2)))) → MINSORT1_IN_GA(.(y_0, .(y_1, y_2)))
MINSORT1_IN_GA(.(x0, .(x1, x2))) → U14_GA(x0, .(x1, x2), U18_gag(x0, x1, x2, minc25_in_gga(x0, x1)))

The TRS R consists of the following rules:

qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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

(82) Obligation:

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

U15_GA(x0, [], removec53_out_ggga(x1, x0, [], .(y_0, []))) → MINSORT1_IN_GA(.(y_0, []))
MINSORT1_IN_GA(.(x0, [])) → U14_GA(x0, [], min2c15_out_gag(x0, x0, []))
U14_GA(z0, [], min2c15_out_gag(z0, z0, [])) → U15_GA(z0, [], U30_ggga(z0, z0, [], U26_ggga(z0, z0, notEqc63_in_gg(z0, z0))))

The TRS R consists of the following rules:

qc62_in_ggga(T183, T151, []) → U26_ggga(T183, T151, notEqc63_in_gg(T183, T151))
qc62_in_ggga(T192, T151, .(T192, T193)) → U27_ggga(T192, T151, T193, notEqc63_in_gg(T192, T151))
qc62_in_ggga(T200, T151, .(T201, T202)) → U28_ggga(T200, T151, T201, T202, notEqc63_in_gg(T200, T151))
U30_ggga(T150, T151, T152, qc62_out_ggga(T150, T151, T152, X175)) → removec53_out_ggga(T150, T151, T152, .(T151, X175))
notEqc63_in_gg(s(T165), s(T166)) → U25_gg(T165, T166, notEqc63_in_gg(T165, T166))
notEqc63_in_gg(s(T173), 0) → notEqc63_out_gg(s(T173), 0)
notEqc63_in_gg(0, s(T176)) → notEqc63_out_gg(0, s(T176))
U28_ggga(T200, T151, T201, T202, notEqc63_out_gg(T200, T151)) → U29_ggga(T200, T151, T201, T202, qc62_in_ggga(T200, T201, T202))
U29_ggga(T200, T151, T201, T202, qc62_out_ggga(T200, T201, T202, X238)) → qc62_out_ggga(T200, T151, .(T201, T202), .(T201, X238))
U25_gg(T165, T166, notEqc63_out_gg(T165, T166)) → notEqc63_out_gg(s(T165), s(T166))
U27_ggga(T192, T151, T193, notEqc63_out_gg(T192, T151)) → qc62_out_ggga(T192, T151, .(T192, T193), T193)
U26_ggga(T183, T151, notEqc63_out_gg(T183, T151)) → qc62_out_ggga(T183, T151, [], [])
minc25_in_gga(T70, T71) → U31_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U32_gga(T97, T98, gtc44_in_gg(T97, T98))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U32_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U31_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U32_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc63_in_gg(x0, x1)
U25_gg(x0, x1, x2)
qc62_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2)
U27_ggga(x0, x1, x2, x3)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3, x4)
U30_ggga(x0, x1, x2, x3)

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