(0) Obligation:

Clauses:

pred(0, 0).
pred(s(0), 0).
pred(s(s(X)), s(Y)) :- pred(s(X), Y).
double(0, 0).
double(s(X), s(s(Y))) :- ','(pred(s(X), Z), double(Z, Y)).
half(0, 0).
half(s(s(X)), s(U)) :- ','(pred(s(s(X)), Y), ','(pred(Y, Z), half(Z, U))).
f(s(X)) :- ','(half(s(X), Y), ','(double(Y, Z), f(Z))).

Queries:

f(g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

pred14(s(T15), s(X44)) :- pred14(T15, X44).
pred24(s(s(T19)), s(X55)) :- pred24(s(T19), X55).
pred9(T12, s(X35)) :- pred14(T12, X35).
half39(s(s(T23)), s(X70)) :- pred9(T23, X68).
half39(s(s(T23)), s(X70)) :- ','(predc9(T23, T25), pred24(T25, X69)).
half39(s(s(T23)), s(X70)) :- ','(predc9(T23, T25), ','(predc24(T25, T27), half39(T27, X70))).
p57(T33, X96, X97) :- pred24(s(T33), X96).
p57(T33, s(T37), s(s(X108))) :- ','(predc24(s(T33), s(T37)), p57(T37, X107, X108)).
f1(s(s(T6))) :- pred9(T6, X16).
f1(s(s(T6))) :- ','(predc9(T6, T8), pred24(T8, X17)).
f1(s(s(T6))) :- ','(predc9(T6, T8), ','(predc24(T8, T16), half39(T16, X18))).
f1(s(s(T6))) :- ','(predc9(T6, T8), ','(predc24(T8, T16), ','(halfc39(T16, T33), p57(T33, X96, X97)))).
f1(s(s(T6))) :- ','(predc9(T6, T8), ','(predc24(T8, T16), ','(halfc39(T16, T20), ','(doublec53(T20, T29), f1(T29))))).

Clauses:

predc14(0, 0).
predc14(s(T15), s(X44)) :- predc14(T15, X44).
predc24(0, 0).
predc24(s(0), 0).
predc24(s(s(T19)), s(X55)) :- predc24(s(T19), X55).
predc9(T12, s(X35)) :- predc14(T12, X35).
halfc39(0, 0).
halfc39(s(s(T23)), s(X70)) :- ','(predc9(T23, T25), ','(predc24(T25, T27), halfc39(T27, X70))).
fc1(s(s(T6))) :- ','(predc9(T6, T8), ','(predc24(T8, T16), ','(halfc39(T16, T20), ','(doublec53(T20, T29), fc1(T29))))).
qc57(T33, 0, 0) :- predc24(s(T33), 0).
qc57(T33, s(T37), s(s(X108))) :- ','(predc24(s(T33), s(T37)), qc57(T37, X107, X108)).
doublec53(T33, s(s(X97))) :- qc57(T33, X96, X97).

Afs:

f1(x1)  =  f1(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:
f1_in: (b)
pred9_in: (b,f)
pred14_in: (b,f)
predc9_in: (b,f)
predc14_in: (b,f)
pred24_in: (b,f)
predc24_in: (b,f) (b,b)
half39_in: (b,f)
halfc39_in: (b,f)
p57_in: (b,f,f)
doublec53_in: (b,f)
qc57_in: (b,f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

F1_IN_G(s(s(T6))) → U12_G(T6, pred9_in_ga(T6, X16))
F1_IN_G(s(s(T6))) → PRED9_IN_GA(T6, X16)
PRED9_IN_GA(T12, s(X35)) → U3_GA(T12, X35, pred14_in_ga(T12, X35))
PRED9_IN_GA(T12, s(X35)) → PRED14_IN_GA(T12, X35)
PRED14_IN_GA(s(T15), s(X44)) → U1_GA(T15, X44, pred14_in_ga(T15, X44))
PRED14_IN_GA(s(T15), s(X44)) → PRED14_IN_GA(T15, X44)
F1_IN_G(s(s(T6))) → U13_G(T6, predc9_in_ga(T6, T8))
U13_G(T6, predc9_out_ga(T6, T8)) → U14_G(T6, pred24_in_ga(T8, X17))
U13_G(T6, predc9_out_ga(T6, T8)) → PRED24_IN_GA(T8, X17)
PRED24_IN_GA(s(s(T19)), s(X55)) → U2_GA(T19, X55, pred24_in_ga(s(T19), X55))
PRED24_IN_GA(s(s(T19)), s(X55)) → PRED24_IN_GA(s(T19), X55)
U13_G(T6, predc9_out_ga(T6, T8)) → U15_G(T6, predc24_in_ga(T8, T16))
U15_G(T6, predc24_out_ga(T8, T16)) → U16_G(T6, half39_in_ga(T16, X18))
U15_G(T6, predc24_out_ga(T8, T16)) → HALF39_IN_GA(T16, X18)
HALF39_IN_GA(s(s(T23)), s(X70)) → U4_GA(T23, X70, pred9_in_ga(T23, X68))
HALF39_IN_GA(s(s(T23)), s(X70)) → PRED9_IN_GA(T23, X68)
HALF39_IN_GA(s(s(T23)), s(X70)) → U5_GA(T23, X70, predc9_in_ga(T23, T25))
U5_GA(T23, X70, predc9_out_ga(T23, T25)) → U6_GA(T23, X70, pred24_in_ga(T25, X69))
U5_GA(T23, X70, predc9_out_ga(T23, T25)) → PRED24_IN_GA(T25, X69)
U5_GA(T23, X70, predc9_out_ga(T23, T25)) → U7_GA(T23, X70, predc24_in_ga(T25, T27))
U7_GA(T23, X70, predc24_out_ga(T25, T27)) → U8_GA(T23, X70, half39_in_ga(T27, X70))
U7_GA(T23, X70, predc24_out_ga(T25, T27)) → HALF39_IN_GA(T27, X70)
U15_G(T6, predc24_out_ga(T8, T16)) → U17_G(T6, halfc39_in_ga(T16, T33))
U17_G(T6, halfc39_out_ga(T16, T33)) → U18_G(T6, p57_in_gaa(T33, X96, X97))
U17_G(T6, halfc39_out_ga(T16, T33)) → P57_IN_GAA(T33, X96, X97)
P57_IN_GAA(T33, X96, X97) → U9_GAA(T33, X96, X97, pred24_in_ga(s(T33), X96))
P57_IN_GAA(T33, X96, X97) → PRED24_IN_GA(s(T33), X96)
P57_IN_GAA(T33, s(T37), s(s(X108))) → U10_GAA(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U10_GAA(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U11_GAA(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U10_GAA(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16, T20))
U19_G(T6, halfc39_out_ga(T16, T20)) → U20_G(T6, doublec53_in_ga(T20, T29))
U20_G(T6, doublec53_out_ga(T20, T29)) → U21_G(T6, f1_in_g(T29))
U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)

The TRS R consists of the following rules:

predc9_in_ga(T12, s(X35)) → U25_ga(T12, X35, predc14_in_ga(T12, X35))
predc14_in_ga(0, 0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15), s(X44)) → U23_ga(T15, X44, predc14_in_ga(T15, X44))
U23_ga(T15, X44, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U25_ga(T12, X35, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
predc24_in_ga(0, 0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
halfc39_in_ga(0, 0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23)), s(X70)) → U26_ga(T23, X70, predc9_in_ga(T23, T25))
U26_ga(T23, X70, predc9_out_ga(T23, T25)) → U27_ga(T23, X70, predc24_in_ga(T25, T27))
U27_ga(T23, X70, predc24_out_ga(T25, T27)) → U28_ga(T23, X70, halfc39_in_ga(T27, X70))
U28_ga(T23, X70, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
doublec53_in_ga(T33, s(s(X97))) → U37_ga(T33, X97, qc57_in_gaa(T33, X96, X97))
qc57_in_gaa(T33, 0, 0) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
predc24_in_gg(0, 0) → predc24_out_gg(0, 0)
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
predc24_in_gg(s(s(T19)), s(X55)) → U24_gg(T19, X55, predc24_in_gg(s(T19), X55))
U24_gg(T19, X55, predc24_out_gg(s(T19), X55)) → predc24_out_gg(s(s(T19)), s(X55))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
qc57_in_gaa(T33, s(T37), s(s(X108))) → U35_gaa(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U35_gaa(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, X108, qc57_in_gaa(T37, X107, X108))
U36_gaa(T33, T37, X108, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U37_ga(T33, X97, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
predc9_in_ga(x1, x2)  =  predc9_in_ga(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
predc14_in_ga(x1, x2)  =  predc14_in_ga(x1)
0  =  0
predc14_out_ga(x1, x2)  =  predc14_out_ga(x1, x2)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
predc9_out_ga(x1, x2)  =  predc9_out_ga(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
halfc39_in_ga(x1, x2)  =  halfc39_in_ga(x1)
halfc39_out_ga(x1, x2)  =  halfc39_out_ga(x1, x2)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
doublec53_in_ga(x1, x2)  =  doublec53_in_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
qc57_in_gaa(x1, x2, x3)  =  qc57_in_gaa(x1)
U34_gaa(x1, x2)  =  U34_gaa(x1, x2)
predc24_in_gg(x1, x2)  =  predc24_in_gg(x1, x2)
predc24_out_gg(x1, x2)  =  predc24_out_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
qc57_out_gaa(x1, x2, x3)  =  qc57_out_gaa(x1, x2, x3)
U35_gaa(x1, x2, x3, x4)  =  U35_gaa(x1, x4)
U36_gaa(x1, x2, x3, x4)  =  U36_gaa(x1, x2, x4)
doublec53_out_ga(x1, x2)  =  doublec53_out_ga(x1, x2)
F1_IN_G(x1)  =  F1_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x1, x2)
PRED9_IN_GA(x1, x2)  =  PRED9_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
PRED14_IN_GA(x1, x2)  =  PRED14_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U13_G(x1, x2)  =  U13_G(x1, x2)
U14_G(x1, x2)  =  U14_G(x1, x2)
PRED24_IN_GA(x1, x2)  =  PRED24_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U15_G(x1, x2)  =  U15_G(x1, x2)
U16_G(x1, x2)  =  U16_G(x1, x2)
HALF39_IN_GA(x1, x2)  =  HALF39_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U8_GA(x1, x2, x3)  =  U8_GA(x1, x3)
U17_G(x1, x2)  =  U17_G(x1, x2)
U18_G(x1, x2)  =  U18_G(x1, x2)
P57_IN_GAA(x1, x2, x3)  =  P57_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x4)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x1, x2, x4)
U19_G(x1, x2)  =  U19_G(x1, x2)
U20_G(x1, x2)  =  U20_G(x1, x2)
U21_G(x1, x2)  =  U21_G(x1, x2)

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:

F1_IN_G(s(s(T6))) → U12_G(T6, pred9_in_ga(T6, X16))
F1_IN_G(s(s(T6))) → PRED9_IN_GA(T6, X16)
PRED9_IN_GA(T12, s(X35)) → U3_GA(T12, X35, pred14_in_ga(T12, X35))
PRED9_IN_GA(T12, s(X35)) → PRED14_IN_GA(T12, X35)
PRED14_IN_GA(s(T15), s(X44)) → U1_GA(T15, X44, pred14_in_ga(T15, X44))
PRED14_IN_GA(s(T15), s(X44)) → PRED14_IN_GA(T15, X44)
F1_IN_G(s(s(T6))) → U13_G(T6, predc9_in_ga(T6, T8))
U13_G(T6, predc9_out_ga(T6, T8)) → U14_G(T6, pred24_in_ga(T8, X17))
U13_G(T6, predc9_out_ga(T6, T8)) → PRED24_IN_GA(T8, X17)
PRED24_IN_GA(s(s(T19)), s(X55)) → U2_GA(T19, X55, pred24_in_ga(s(T19), X55))
PRED24_IN_GA(s(s(T19)), s(X55)) → PRED24_IN_GA(s(T19), X55)
U13_G(T6, predc9_out_ga(T6, T8)) → U15_G(T6, predc24_in_ga(T8, T16))
U15_G(T6, predc24_out_ga(T8, T16)) → U16_G(T6, half39_in_ga(T16, X18))
U15_G(T6, predc24_out_ga(T8, T16)) → HALF39_IN_GA(T16, X18)
HALF39_IN_GA(s(s(T23)), s(X70)) → U4_GA(T23, X70, pred9_in_ga(T23, X68))
HALF39_IN_GA(s(s(T23)), s(X70)) → PRED9_IN_GA(T23, X68)
HALF39_IN_GA(s(s(T23)), s(X70)) → U5_GA(T23, X70, predc9_in_ga(T23, T25))
U5_GA(T23, X70, predc9_out_ga(T23, T25)) → U6_GA(T23, X70, pred24_in_ga(T25, X69))
U5_GA(T23, X70, predc9_out_ga(T23, T25)) → PRED24_IN_GA(T25, X69)
U5_GA(T23, X70, predc9_out_ga(T23, T25)) → U7_GA(T23, X70, predc24_in_ga(T25, T27))
U7_GA(T23, X70, predc24_out_ga(T25, T27)) → U8_GA(T23, X70, half39_in_ga(T27, X70))
U7_GA(T23, X70, predc24_out_ga(T25, T27)) → HALF39_IN_GA(T27, X70)
U15_G(T6, predc24_out_ga(T8, T16)) → U17_G(T6, halfc39_in_ga(T16, T33))
U17_G(T6, halfc39_out_ga(T16, T33)) → U18_G(T6, p57_in_gaa(T33, X96, X97))
U17_G(T6, halfc39_out_ga(T16, T33)) → P57_IN_GAA(T33, X96, X97)
P57_IN_GAA(T33, X96, X97) → U9_GAA(T33, X96, X97, pred24_in_ga(s(T33), X96))
P57_IN_GAA(T33, X96, X97) → PRED24_IN_GA(s(T33), X96)
P57_IN_GAA(T33, s(T37), s(s(X108))) → U10_GAA(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U10_GAA(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U11_GAA(T33, T37, X108, p57_in_gaa(T37, X107, X108))
U10_GAA(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16, T20))
U19_G(T6, halfc39_out_ga(T16, T20)) → U20_G(T6, doublec53_in_ga(T20, T29))
U20_G(T6, doublec53_out_ga(T20, T29)) → U21_G(T6, f1_in_g(T29))
U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)

The TRS R consists of the following rules:

predc9_in_ga(T12, s(X35)) → U25_ga(T12, X35, predc14_in_ga(T12, X35))
predc14_in_ga(0, 0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15), s(X44)) → U23_ga(T15, X44, predc14_in_ga(T15, X44))
U23_ga(T15, X44, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U25_ga(T12, X35, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
predc24_in_ga(0, 0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
halfc39_in_ga(0, 0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23)), s(X70)) → U26_ga(T23, X70, predc9_in_ga(T23, T25))
U26_ga(T23, X70, predc9_out_ga(T23, T25)) → U27_ga(T23, X70, predc24_in_ga(T25, T27))
U27_ga(T23, X70, predc24_out_ga(T25, T27)) → U28_ga(T23, X70, halfc39_in_ga(T27, X70))
U28_ga(T23, X70, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
doublec53_in_ga(T33, s(s(X97))) → U37_ga(T33, X97, qc57_in_gaa(T33, X96, X97))
qc57_in_gaa(T33, 0, 0) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
predc24_in_gg(0, 0) → predc24_out_gg(0, 0)
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
predc24_in_gg(s(s(T19)), s(X55)) → U24_gg(T19, X55, predc24_in_gg(s(T19), X55))
U24_gg(T19, X55, predc24_out_gg(s(T19), X55)) → predc24_out_gg(s(s(T19)), s(X55))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
qc57_in_gaa(T33, s(T37), s(s(X108))) → U35_gaa(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U35_gaa(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, X108, qc57_in_gaa(T37, X107, X108))
U36_gaa(T33, T37, X108, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U37_ga(T33, X97, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
pred9_in_ga(x1, x2)  =  pred9_in_ga(x1)
pred14_in_ga(x1, x2)  =  pred14_in_ga(x1)
predc9_in_ga(x1, x2)  =  predc9_in_ga(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
predc14_in_ga(x1, x2)  =  predc14_in_ga(x1)
0  =  0
predc14_out_ga(x1, x2)  =  predc14_out_ga(x1, x2)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
predc9_out_ga(x1, x2)  =  predc9_out_ga(x1, x2)
pred24_in_ga(x1, x2)  =  pred24_in_ga(x1)
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
half39_in_ga(x1, x2)  =  half39_in_ga(x1)
halfc39_in_ga(x1, x2)  =  halfc39_in_ga(x1)
halfc39_out_ga(x1, x2)  =  halfc39_out_ga(x1, x2)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
p57_in_gaa(x1, x2, x3)  =  p57_in_gaa(x1)
doublec53_in_ga(x1, x2)  =  doublec53_in_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
qc57_in_gaa(x1, x2, x3)  =  qc57_in_gaa(x1)
U34_gaa(x1, x2)  =  U34_gaa(x1, x2)
predc24_in_gg(x1, x2)  =  predc24_in_gg(x1, x2)
predc24_out_gg(x1, x2)  =  predc24_out_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
qc57_out_gaa(x1, x2, x3)  =  qc57_out_gaa(x1, x2, x3)
U35_gaa(x1, x2, x3, x4)  =  U35_gaa(x1, x4)
U36_gaa(x1, x2, x3, x4)  =  U36_gaa(x1, x2, x4)
doublec53_out_ga(x1, x2)  =  doublec53_out_ga(x1, x2)
F1_IN_G(x1)  =  F1_IN_G(x1)
U12_G(x1, x2)  =  U12_G(x1, x2)
PRED9_IN_GA(x1, x2)  =  PRED9_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
PRED14_IN_GA(x1, x2)  =  PRED14_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U13_G(x1, x2)  =  U13_G(x1, x2)
U14_G(x1, x2)  =  U14_G(x1, x2)
PRED24_IN_GA(x1, x2)  =  PRED24_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U15_G(x1, x2)  =  U15_G(x1, x2)
U16_G(x1, x2)  =  U16_G(x1, x2)
HALF39_IN_GA(x1, x2)  =  HALF39_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U8_GA(x1, x2, x3)  =  U8_GA(x1, x3)
U17_G(x1, x2)  =  U17_G(x1, x2)
U18_G(x1, x2)  =  U18_G(x1, x2)
P57_IN_GAA(x1, x2, x3)  =  P57_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x4)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x1, x2, x4)
U19_G(x1, x2)  =  U19_G(x1, x2)
U20_G(x1, x2)  =  U20_G(x1, x2)
U21_G(x1, x2)  =  U21_G(x1, x2)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 22 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

PRED24_IN_GA(s(s(T19)), s(X55)) → PRED24_IN_GA(s(T19), X55)

The TRS R consists of the following rules:

predc9_in_ga(T12, s(X35)) → U25_ga(T12, X35, predc14_in_ga(T12, X35))
predc14_in_ga(0, 0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15), s(X44)) → U23_ga(T15, X44, predc14_in_ga(T15, X44))
U23_ga(T15, X44, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U25_ga(T12, X35, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
predc24_in_ga(0, 0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
halfc39_in_ga(0, 0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23)), s(X70)) → U26_ga(T23, X70, predc9_in_ga(T23, T25))
U26_ga(T23, X70, predc9_out_ga(T23, T25)) → U27_ga(T23, X70, predc24_in_ga(T25, T27))
U27_ga(T23, X70, predc24_out_ga(T25, T27)) → U28_ga(T23, X70, halfc39_in_ga(T27, X70))
U28_ga(T23, X70, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
doublec53_in_ga(T33, s(s(X97))) → U37_ga(T33, X97, qc57_in_gaa(T33, X96, X97))
qc57_in_gaa(T33, 0, 0) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
predc24_in_gg(0, 0) → predc24_out_gg(0, 0)
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
predc24_in_gg(s(s(T19)), s(X55)) → U24_gg(T19, X55, predc24_in_gg(s(T19), X55))
U24_gg(T19, X55, predc24_out_gg(s(T19), X55)) → predc24_out_gg(s(s(T19)), s(X55))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
qc57_in_gaa(T33, s(T37), s(s(X108))) → U35_gaa(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U35_gaa(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, X108, qc57_in_gaa(T37, X107, X108))
U36_gaa(T33, T37, X108, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U37_ga(T33, X97, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
predc9_in_ga(x1, x2)  =  predc9_in_ga(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
predc14_in_ga(x1, x2)  =  predc14_in_ga(x1)
0  =  0
predc14_out_ga(x1, x2)  =  predc14_out_ga(x1, x2)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
predc9_out_ga(x1, x2)  =  predc9_out_ga(x1, x2)
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
halfc39_in_ga(x1, x2)  =  halfc39_in_ga(x1)
halfc39_out_ga(x1, x2)  =  halfc39_out_ga(x1, x2)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
doublec53_in_ga(x1, x2)  =  doublec53_in_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
qc57_in_gaa(x1, x2, x3)  =  qc57_in_gaa(x1)
U34_gaa(x1, x2)  =  U34_gaa(x1, x2)
predc24_in_gg(x1, x2)  =  predc24_in_gg(x1, x2)
predc24_out_gg(x1, x2)  =  predc24_out_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
qc57_out_gaa(x1, x2, x3)  =  qc57_out_gaa(x1, x2, x3)
U35_gaa(x1, x2, x3, x4)  =  U35_gaa(x1, x4)
U36_gaa(x1, x2, x3, x4)  =  U36_gaa(x1, x2, x4)
doublec53_out_ga(x1, x2)  =  doublec53_out_ga(x1, x2)
PRED24_IN_GA(x1, x2)  =  PRED24_IN_GA(x1)

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:

PRED24_IN_GA(s(s(T19)), s(X55)) → PRED24_IN_GA(s(T19), X55)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PRED24_IN_GA(x1, x2)  =  PRED24_IN_GA(x1)

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

(10) PiDPToQDPProof (SOUND 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:

PRED24_IN_GA(s(s(T19))) → PRED24_IN_GA(s(T19))

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:

  • PRED24_IN_GA(s(s(T19))) → PRED24_IN_GA(s(T19))
    The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

P57_IN_GAA(T33, s(T37), s(s(X108))) → U10_GAA(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U10_GAA(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)

The TRS R consists of the following rules:

predc9_in_ga(T12, s(X35)) → U25_ga(T12, X35, predc14_in_ga(T12, X35))
predc14_in_ga(0, 0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15), s(X44)) → U23_ga(T15, X44, predc14_in_ga(T15, X44))
U23_ga(T15, X44, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U25_ga(T12, X35, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
predc24_in_ga(0, 0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
halfc39_in_ga(0, 0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23)), s(X70)) → U26_ga(T23, X70, predc9_in_ga(T23, T25))
U26_ga(T23, X70, predc9_out_ga(T23, T25)) → U27_ga(T23, X70, predc24_in_ga(T25, T27))
U27_ga(T23, X70, predc24_out_ga(T25, T27)) → U28_ga(T23, X70, halfc39_in_ga(T27, X70))
U28_ga(T23, X70, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
doublec53_in_ga(T33, s(s(X97))) → U37_ga(T33, X97, qc57_in_gaa(T33, X96, X97))
qc57_in_gaa(T33, 0, 0) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
predc24_in_gg(0, 0) → predc24_out_gg(0, 0)
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
predc24_in_gg(s(s(T19)), s(X55)) → U24_gg(T19, X55, predc24_in_gg(s(T19), X55))
U24_gg(T19, X55, predc24_out_gg(s(T19), X55)) → predc24_out_gg(s(s(T19)), s(X55))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
qc57_in_gaa(T33, s(T37), s(s(X108))) → U35_gaa(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U35_gaa(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, X108, qc57_in_gaa(T37, X107, X108))
U36_gaa(T33, T37, X108, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U37_ga(T33, X97, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
predc9_in_ga(x1, x2)  =  predc9_in_ga(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
predc14_in_ga(x1, x2)  =  predc14_in_ga(x1)
0  =  0
predc14_out_ga(x1, x2)  =  predc14_out_ga(x1, x2)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
predc9_out_ga(x1, x2)  =  predc9_out_ga(x1, x2)
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
halfc39_in_ga(x1, x2)  =  halfc39_in_ga(x1)
halfc39_out_ga(x1, x2)  =  halfc39_out_ga(x1, x2)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
doublec53_in_ga(x1, x2)  =  doublec53_in_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
qc57_in_gaa(x1, x2, x3)  =  qc57_in_gaa(x1)
U34_gaa(x1, x2)  =  U34_gaa(x1, x2)
predc24_in_gg(x1, x2)  =  predc24_in_gg(x1, x2)
predc24_out_gg(x1, x2)  =  predc24_out_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
qc57_out_gaa(x1, x2, x3)  =  qc57_out_gaa(x1, x2, x3)
U35_gaa(x1, x2, x3, x4)  =  U35_gaa(x1, x4)
U36_gaa(x1, x2, x3, x4)  =  U36_gaa(x1, x2, x4)
doublec53_out_ga(x1, x2)  =  doublec53_out_ga(x1, x2)
P57_IN_GAA(x1, x2, x3)  =  P57_IN_GAA(x1)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)

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:

P57_IN_GAA(T33, s(T37), s(s(X108))) → U10_GAA(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U10_GAA(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37, X107, X108)

The TRS R consists of the following rules:

predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
P57_IN_GAA(x1, x2, x3)  =  P57_IN_GAA(x1)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)

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:

P57_IN_GAA(T33) → U10_GAA(T33, predc24_in_ga(s(T33)))
U10_GAA(T33, predc24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37)

The TRS R consists of the following rules:

predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)

The set Q consists of the following terms:

predc24_in_ga(x0)
U24_ga(x0, x1)

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U10_GAA(T33, predc24_out_ga(s(T33), s(T37))) → P57_IN_GAA(T37)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1   
POL(P57_IN_GAA(x1)) = 1 + x1   
POL(U10_GAA(x1, x2)) = x2   
POL(U24_ga(x1, x2)) = 1 + x2   
POL(predc24_in_ga(x1)) = x1   
POL(predc24_out_ga(x1, x2)) = 1 + x2   
POL(s(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented:

predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))

(20) Obligation:

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

P57_IN_GAA(T33) → U10_GAA(T33, predc24_in_ga(s(T33)))

The TRS R consists of the following rules:

predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)

The set Q consists of the following terms:

predc24_in_ga(x0)
U24_ga(x0, x1)

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

(21) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(22) TRUE

(23) Obligation:

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

PRED14_IN_GA(s(T15), s(X44)) → PRED14_IN_GA(T15, X44)

The TRS R consists of the following rules:

predc9_in_ga(T12, s(X35)) → U25_ga(T12, X35, predc14_in_ga(T12, X35))
predc14_in_ga(0, 0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15), s(X44)) → U23_ga(T15, X44, predc14_in_ga(T15, X44))
U23_ga(T15, X44, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U25_ga(T12, X35, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
predc24_in_ga(0, 0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
halfc39_in_ga(0, 0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23)), s(X70)) → U26_ga(T23, X70, predc9_in_ga(T23, T25))
U26_ga(T23, X70, predc9_out_ga(T23, T25)) → U27_ga(T23, X70, predc24_in_ga(T25, T27))
U27_ga(T23, X70, predc24_out_ga(T25, T27)) → U28_ga(T23, X70, halfc39_in_ga(T27, X70))
U28_ga(T23, X70, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
doublec53_in_ga(T33, s(s(X97))) → U37_ga(T33, X97, qc57_in_gaa(T33, X96, X97))
qc57_in_gaa(T33, 0, 0) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
predc24_in_gg(0, 0) → predc24_out_gg(0, 0)
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
predc24_in_gg(s(s(T19)), s(X55)) → U24_gg(T19, X55, predc24_in_gg(s(T19), X55))
U24_gg(T19, X55, predc24_out_gg(s(T19), X55)) → predc24_out_gg(s(s(T19)), s(X55))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
qc57_in_gaa(T33, s(T37), s(s(X108))) → U35_gaa(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U35_gaa(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, X108, qc57_in_gaa(T37, X107, X108))
U36_gaa(T33, T37, X108, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U37_ga(T33, X97, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
predc9_in_ga(x1, x2)  =  predc9_in_ga(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
predc14_in_ga(x1, x2)  =  predc14_in_ga(x1)
0  =  0
predc14_out_ga(x1, x2)  =  predc14_out_ga(x1, x2)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
predc9_out_ga(x1, x2)  =  predc9_out_ga(x1, x2)
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
halfc39_in_ga(x1, x2)  =  halfc39_in_ga(x1)
halfc39_out_ga(x1, x2)  =  halfc39_out_ga(x1, x2)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
doublec53_in_ga(x1, x2)  =  doublec53_in_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
qc57_in_gaa(x1, x2, x3)  =  qc57_in_gaa(x1)
U34_gaa(x1, x2)  =  U34_gaa(x1, x2)
predc24_in_gg(x1, x2)  =  predc24_in_gg(x1, x2)
predc24_out_gg(x1, x2)  =  predc24_out_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
qc57_out_gaa(x1, x2, x3)  =  qc57_out_gaa(x1, x2, x3)
U35_gaa(x1, x2, x3, x4)  =  U35_gaa(x1, x4)
U36_gaa(x1, x2, x3, x4)  =  U36_gaa(x1, x2, x4)
doublec53_out_ga(x1, x2)  =  doublec53_out_ga(x1, x2)
PRED14_IN_GA(x1, x2)  =  PRED14_IN_GA(x1)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

PRED14_IN_GA(s(T15), s(X44)) → PRED14_IN_GA(T15, X44)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
PRED14_IN_GA(x1, x2)  =  PRED14_IN_GA(x1)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

PRED14_IN_GA(s(T15)) → PRED14_IN_GA(T15)

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

(28) 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:

  • PRED14_IN_GA(s(T15)) → PRED14_IN_GA(T15)
    The graph contains the following edges 1 > 1

(29) YES

(30) Obligation:

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

HALF39_IN_GA(s(s(T23)), s(X70)) → U5_GA(T23, X70, predc9_in_ga(T23, T25))
U5_GA(T23, X70, predc9_out_ga(T23, T25)) → U7_GA(T23, X70, predc24_in_ga(T25, T27))
U7_GA(T23, X70, predc24_out_ga(T25, T27)) → HALF39_IN_GA(T27, X70)

The TRS R consists of the following rules:

predc9_in_ga(T12, s(X35)) → U25_ga(T12, X35, predc14_in_ga(T12, X35))
predc14_in_ga(0, 0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15), s(X44)) → U23_ga(T15, X44, predc14_in_ga(T15, X44))
U23_ga(T15, X44, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U25_ga(T12, X35, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
predc24_in_ga(0, 0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
halfc39_in_ga(0, 0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23)), s(X70)) → U26_ga(T23, X70, predc9_in_ga(T23, T25))
U26_ga(T23, X70, predc9_out_ga(T23, T25)) → U27_ga(T23, X70, predc24_in_ga(T25, T27))
U27_ga(T23, X70, predc24_out_ga(T25, T27)) → U28_ga(T23, X70, halfc39_in_ga(T27, X70))
U28_ga(T23, X70, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
doublec53_in_ga(T33, s(s(X97))) → U37_ga(T33, X97, qc57_in_gaa(T33, X96, X97))
qc57_in_gaa(T33, 0, 0) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
predc24_in_gg(0, 0) → predc24_out_gg(0, 0)
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
predc24_in_gg(s(s(T19)), s(X55)) → U24_gg(T19, X55, predc24_in_gg(s(T19), X55))
U24_gg(T19, X55, predc24_out_gg(s(T19), X55)) → predc24_out_gg(s(s(T19)), s(X55))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
qc57_in_gaa(T33, s(T37), s(s(X108))) → U35_gaa(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U35_gaa(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, X108, qc57_in_gaa(T37, X107, X108))
U36_gaa(T33, T37, X108, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U37_ga(T33, X97, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
predc9_in_ga(x1, x2)  =  predc9_in_ga(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
predc14_in_ga(x1, x2)  =  predc14_in_ga(x1)
0  =  0
predc14_out_ga(x1, x2)  =  predc14_out_ga(x1, x2)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
predc9_out_ga(x1, x2)  =  predc9_out_ga(x1, x2)
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
halfc39_in_ga(x1, x2)  =  halfc39_in_ga(x1)
halfc39_out_ga(x1, x2)  =  halfc39_out_ga(x1, x2)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
doublec53_in_ga(x1, x2)  =  doublec53_in_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
qc57_in_gaa(x1, x2, x3)  =  qc57_in_gaa(x1)
U34_gaa(x1, x2)  =  U34_gaa(x1, x2)
predc24_in_gg(x1, x2)  =  predc24_in_gg(x1, x2)
predc24_out_gg(x1, x2)  =  predc24_out_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
qc57_out_gaa(x1, x2, x3)  =  qc57_out_gaa(x1, x2, x3)
U35_gaa(x1, x2, x3, x4)  =  U35_gaa(x1, x4)
U36_gaa(x1, x2, x3, x4)  =  U36_gaa(x1, x2, x4)
doublec53_out_ga(x1, x2)  =  doublec53_out_ga(x1, x2)
HALF39_IN_GA(x1, x2)  =  HALF39_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

HALF39_IN_GA(s(s(T23)), s(X70)) → U5_GA(T23, X70, predc9_in_ga(T23, T25))
U5_GA(T23, X70, predc9_out_ga(T23, T25)) → U7_GA(T23, X70, predc24_in_ga(T25, T27))
U7_GA(T23, X70, predc24_out_ga(T25, T27)) → HALF39_IN_GA(T27, X70)

The TRS R consists of the following rules:

predc9_in_ga(T12, s(X35)) → U25_ga(T12, X35, predc14_in_ga(T12, X35))
predc24_in_ga(0, 0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
U25_ga(T12, X35, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc14_in_ga(0, 0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15), s(X44)) → U23_ga(T15, X44, predc14_in_ga(T15, X44))
U23_ga(T15, X44, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
predc9_in_ga(x1, x2)  =  predc9_in_ga(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
predc14_in_ga(x1, x2)  =  predc14_in_ga(x1)
0  =  0
predc14_out_ga(x1, x2)  =  predc14_out_ga(x1, x2)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
predc9_out_ga(x1, x2)  =  predc9_out_ga(x1, x2)
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
HALF39_IN_GA(x1, x2)  =  HALF39_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

HALF39_IN_GA(s(s(T23))) → U5_GA(T23, predc9_in_ga(T23))
U5_GA(T23, predc9_out_ga(T23, T25)) → U7_GA(T23, predc24_in_ga(T25))
U7_GA(T23, predc24_out_ga(T25, T27)) → HALF39_IN_GA(T27)

The TRS R consists of the following rules:

predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
predc14_in_ga(x0)
U23_ga(x0, x1)

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

(35) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule HALF39_IN_GA(s(s(T23))) → U5_GA(T23, predc9_in_ga(T23)) at position [1] we obtained the following new rules [LPAR04]:

HALF39_IN_GA(s(s(T23))) → U5_GA(T23, U25_ga(T23, predc14_in_ga(T23)))

(36) Obligation:

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

U5_GA(T23, predc9_out_ga(T23, T25)) → U7_GA(T23, predc24_in_ga(T25))
U7_GA(T23, predc24_out_ga(T25, T27)) → HALF39_IN_GA(T27)
HALF39_IN_GA(s(s(T23))) → U5_GA(T23, U25_ga(T23, predc14_in_ga(T23)))

The TRS R consists of the following rules:

predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
predc14_in_ga(x0)
U23_ga(x0, x1)

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

(37) 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.

(38) Obligation:

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

U5_GA(T23, predc9_out_ga(T23, T25)) → U7_GA(T23, predc24_in_ga(T25))
U7_GA(T23, predc24_out_ga(T25, T27)) → HALF39_IN_GA(T27)
HALF39_IN_GA(s(s(T23))) → U5_GA(T23, U25_ga(T23, predc14_in_ga(T23)))

The TRS R consists of the following rules:

predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
predc14_in_ga(x0)
U23_ga(x0, x1)

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

(39) 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].

predc9_in_ga(x0)

(40) Obligation:

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

U5_GA(T23, predc9_out_ga(T23, T25)) → U7_GA(T23, predc24_in_ga(T25))
U7_GA(T23, predc24_out_ga(T25, T27)) → HALF39_IN_GA(T27)
HALF39_IN_GA(s(s(T23))) → U5_GA(T23, U25_ga(T23, predc14_in_ga(T23)))

The TRS R consists of the following rules:

predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))

The set Q consists of the following terms:

predc24_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
predc14_in_ga(x0)
U23_ga(x0, x1)

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

(41) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


HALF39_IN_GA(s(s(T23))) → U5_GA(T23, U25_ga(T23, predc14_in_ga(T23)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(HALF39_IN_GA(x1)) = x1   
POL(U23_ga(x1, x2)) = 1 + x2   
POL(U24_ga(x1, x2)) = 1 + x2   
POL(U25_ga(x1, x2)) = 1 + x2   
POL(U5_GA(x1, x2)) = x2   
POL(U7_GA(x1, x2)) = x2   
POL(predc14_in_ga(x1)) = x1   
POL(predc14_out_ga(x1, x2)) = x2   
POL(predc24_in_ga(x1)) = x1   
POL(predc24_out_ga(x1, x2)) = x2   
POL(predc9_out_ga(x1, x2)) = x2   
POL(s(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented:

predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))

(42) Obligation:

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

U5_GA(T23, predc9_out_ga(T23, T25)) → U7_GA(T23, predc24_in_ga(T25))
U7_GA(T23, predc24_out_ga(T25, T27)) → HALF39_IN_GA(T27)

The TRS R consists of the following rules:

predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))

The set Q consists of the following terms:

predc24_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
predc14_in_ga(x0)
U23_ga(x0, x1)

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

(43) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(44) TRUE

(45) Obligation:

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

F1_IN_G(s(s(T6))) → U13_G(T6, predc9_in_ga(T6, T8))
U13_G(T6, predc9_out_ga(T6, T8)) → U15_G(T6, predc24_in_ga(T8, T16))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16, T20))
U19_G(T6, halfc39_out_ga(T16, T20)) → U20_G(T6, doublec53_in_ga(T20, T29))
U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)

The TRS R consists of the following rules:

predc9_in_ga(T12, s(X35)) → U25_ga(T12, X35, predc14_in_ga(T12, X35))
predc14_in_ga(0, 0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15), s(X44)) → U23_ga(T15, X44, predc14_in_ga(T15, X44))
U23_ga(T15, X44, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U25_ga(T12, X35, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
predc24_in_ga(0, 0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
halfc39_in_ga(0, 0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23)), s(X70)) → U26_ga(T23, X70, predc9_in_ga(T23, T25))
U26_ga(T23, X70, predc9_out_ga(T23, T25)) → U27_ga(T23, X70, predc24_in_ga(T25, T27))
U27_ga(T23, X70, predc24_out_ga(T25, T27)) → U28_ga(T23, X70, halfc39_in_ga(T27, X70))
U28_ga(T23, X70, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
doublec53_in_ga(T33, s(s(X97))) → U37_ga(T33, X97, qc57_in_gaa(T33, X96, X97))
qc57_in_gaa(T33, 0, 0) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
predc24_in_gg(0, 0) → predc24_out_gg(0, 0)
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
predc24_in_gg(s(s(T19)), s(X55)) → U24_gg(T19, X55, predc24_in_gg(s(T19), X55))
U24_gg(T19, X55, predc24_out_gg(s(T19), X55)) → predc24_out_gg(s(s(T19)), s(X55))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
qc57_in_gaa(T33, s(T37), s(s(X108))) → U35_gaa(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U35_gaa(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, X108, qc57_in_gaa(T37, X107, X108))
U36_gaa(T33, T37, X108, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U37_ga(T33, X97, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
predc9_in_ga(x1, x2)  =  predc9_in_ga(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
predc14_in_ga(x1, x2)  =  predc14_in_ga(x1)
0  =  0
predc14_out_ga(x1, x2)  =  predc14_out_ga(x1, x2)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
predc9_out_ga(x1, x2)  =  predc9_out_ga(x1, x2)
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
halfc39_in_ga(x1, x2)  =  halfc39_in_ga(x1)
halfc39_out_ga(x1, x2)  =  halfc39_out_ga(x1, x2)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
doublec53_in_ga(x1, x2)  =  doublec53_in_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
qc57_in_gaa(x1, x2, x3)  =  qc57_in_gaa(x1)
U34_gaa(x1, x2)  =  U34_gaa(x1, x2)
predc24_in_gg(x1, x2)  =  predc24_in_gg(x1, x2)
predc24_out_gg(x1, x2)  =  predc24_out_gg(x1, x2)
U24_gg(x1, x2, x3)  =  U24_gg(x1, x2, x3)
qc57_out_gaa(x1, x2, x3)  =  qc57_out_gaa(x1, x2, x3)
U35_gaa(x1, x2, x3, x4)  =  U35_gaa(x1, x4)
U36_gaa(x1, x2, x3, x4)  =  U36_gaa(x1, x2, x4)
doublec53_out_ga(x1, x2)  =  doublec53_out_ga(x1, x2)
F1_IN_G(x1)  =  F1_IN_G(x1)
U13_G(x1, x2)  =  U13_G(x1, x2)
U15_G(x1, x2)  =  U15_G(x1, x2)
U19_G(x1, x2)  =  U19_G(x1, x2)
U20_G(x1, x2)  =  U20_G(x1, x2)

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

(46) UsableRulesProof (EQUIVALENT transformation)

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

(47) Obligation:

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

F1_IN_G(s(s(T6))) → U13_G(T6, predc9_in_ga(T6, T8))
U13_G(T6, predc9_out_ga(T6, T8)) → U15_G(T6, predc24_in_ga(T8, T16))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16, T20))
U19_G(T6, halfc39_out_ga(T16, T20)) → U20_G(T6, doublec53_in_ga(T20, T29))
U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)

The TRS R consists of the following rules:

predc9_in_ga(T12, s(X35)) → U25_ga(T12, X35, predc14_in_ga(T12, X35))
predc24_in_ga(0, 0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0), 0) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19)), s(X55)) → U24_ga(T19, X55, predc24_in_ga(s(T19), X55))
halfc39_in_ga(0, 0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23)), s(X70)) → U26_ga(T23, X70, predc9_in_ga(T23, T25))
doublec53_in_ga(T33, s(s(X97))) → U37_ga(T33, X97, qc57_in_gaa(T33, X96, X97))
U25_ga(T12, X35, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U24_ga(T19, X55, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
U26_ga(T23, X70, predc9_out_ga(T23, T25)) → U27_ga(T23, X70, predc24_in_ga(T25, T27))
U37_ga(T33, X97, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc14_in_ga(0, 0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15), s(X44)) → U23_ga(T15, X44, predc14_in_ga(T15, X44))
U27_ga(T23, X70, predc24_out_ga(T25, T27)) → U28_ga(T23, X70, halfc39_in_ga(T27, X70))
qc57_in_gaa(T33, 0, 0) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33, s(T37), s(s(X108))) → U35_gaa(T33, T37, X108, predc24_in_ga(s(T33), s(T37)))
U23_ga(T15, X44, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U28_ga(T23, X70, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
U35_gaa(T33, T37, X108, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, X108, qc57_in_gaa(T37, X107, X108))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U36_gaa(T33, T37, X108, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
predc9_in_ga(x1, x2)  =  predc9_in_ga(x1)
U25_ga(x1, x2, x3)  =  U25_ga(x1, x3)
predc14_in_ga(x1, x2)  =  predc14_in_ga(x1)
0  =  0
predc14_out_ga(x1, x2)  =  predc14_out_ga(x1, x2)
U23_ga(x1, x2, x3)  =  U23_ga(x1, x3)
predc9_out_ga(x1, x2)  =  predc9_out_ga(x1, x2)
predc24_in_ga(x1, x2)  =  predc24_in_ga(x1)
predc24_out_ga(x1, x2)  =  predc24_out_ga(x1, x2)
U24_ga(x1, x2, x3)  =  U24_ga(x1, x3)
halfc39_in_ga(x1, x2)  =  halfc39_in_ga(x1)
halfc39_out_ga(x1, x2)  =  halfc39_out_ga(x1, x2)
U26_ga(x1, x2, x3)  =  U26_ga(x1, x3)
U27_ga(x1, x2, x3)  =  U27_ga(x1, x3)
U28_ga(x1, x2, x3)  =  U28_ga(x1, x3)
doublec53_in_ga(x1, x2)  =  doublec53_in_ga(x1)
U37_ga(x1, x2, x3)  =  U37_ga(x1, x3)
qc57_in_gaa(x1, x2, x3)  =  qc57_in_gaa(x1)
U34_gaa(x1, x2)  =  U34_gaa(x1, x2)
predc24_in_gg(x1, x2)  =  predc24_in_gg(x1, x2)
predc24_out_gg(x1, x2)  =  predc24_out_gg(x1, x2)
qc57_out_gaa(x1, x2, x3)  =  qc57_out_gaa(x1, x2, x3)
U35_gaa(x1, x2, x3, x4)  =  U35_gaa(x1, x4)
U36_gaa(x1, x2, x3, x4)  =  U36_gaa(x1, x2, x4)
doublec53_out_ga(x1, x2)  =  doublec53_out_ga(x1, x2)
F1_IN_G(x1)  =  F1_IN_G(x1)
U13_G(x1, x2)  =  U13_G(x1, x2)
U15_G(x1, x2)  =  U15_G(x1, x2)
U19_G(x1, x2)  =  U19_G(x1, x2)
U20_G(x1, x2)  =  U20_G(x1, x2)

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

(48) PiDPToQDPProof (SOUND transformation)

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

(49) Obligation:

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

F1_IN_G(s(s(T6))) → U13_G(T6, predc9_in_ga(T6))
U13_G(T6, predc9_out_ga(T6, T8)) → U15_G(T6, predc24_in_ga(T8))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16))
U19_G(T6, halfc39_out_ga(T16, T20)) → U20_G(T6, doublec53_in_ga(T20))
U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)

The TRS R consists of the following rules:

predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
doublec53_in_ga(T33) → U37_ga(T33, qc57_in_gaa(T33))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
doublec53_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(50) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F1_IN_G(s(s(T6))) → U13_G(T6, predc9_in_ga(T6)) at position [1] we obtained the following new rules [LPAR04]:

F1_IN_G(s(s(T6))) → U13_G(T6, U25_ga(T6, predc14_in_ga(T6)))

(51) Obligation:

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

U13_G(T6, predc9_out_ga(T6, T8)) → U15_G(T6, predc24_in_ga(T8))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16))
U19_G(T6, halfc39_out_ga(T16, T20)) → U20_G(T6, doublec53_in_ga(T20))
U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
F1_IN_G(s(s(T6))) → U13_G(T6, U25_ga(T6, predc14_in_ga(T6)))

The TRS R consists of the following rules:

predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
doublec53_in_ga(T33) → U37_ga(T33, qc57_in_gaa(T33))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
doublec53_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(52) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U13_G(T6, predc9_out_ga(T6, T8)) → U15_G(T6, predc24_in_ga(T8)) at position [1] we obtained the following new rules [LPAR04]:

U13_G(y0, predc9_out_ga(y0, 0)) → U15_G(y0, predc24_out_ga(0, 0))
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))

(53) Obligation:

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

U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16))
U19_G(T6, halfc39_out_ga(T16, T20)) → U20_G(T6, doublec53_in_ga(T20))
U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
F1_IN_G(s(s(T6))) → U13_G(T6, U25_ga(T6, predc14_in_ga(T6)))
U13_G(y0, predc9_out_ga(y0, 0)) → U15_G(y0, predc24_out_ga(0, 0))
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))

The TRS R consists of the following rules:

predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
doublec53_in_ga(T33) → U37_ga(T33, qc57_in_gaa(T33))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
doublec53_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(54) DependencyGraphProof (EQUIVALENT transformation)

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

(55) Obligation:

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

U19_G(T6, halfc39_out_ga(T16, T20)) → U20_G(T6, doublec53_in_ga(T20))
U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
F1_IN_G(s(s(T6))) → U13_G(T6, U25_ga(T6, predc14_in_ga(T6)))
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))

The TRS R consists of the following rules:

predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
doublec53_in_ga(T33) → U37_ga(T33, qc57_in_gaa(T33))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
doublec53_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(56) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U19_G(T6, halfc39_out_ga(T16, T20)) → U20_G(T6, doublec53_in_ga(T20)) at position [1] we obtained the following new rules [LPAR04]:

U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))

(57) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
F1_IN_G(s(s(T6))) → U13_G(T6, U25_ga(T6, predc14_in_ga(T6)))
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))

The TRS R consists of the following rules:

predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
predc24_in_ga(0) → predc24_out_ga(0, 0)
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
doublec53_in_ga(T33) → U37_ga(T33, qc57_in_gaa(T33))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
doublec53_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(58) UsableRulesProof (EQUIVALENT transformation)

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

(59) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
F1_IN_G(s(s(T6))) → U13_G(T6, U25_ga(T6, predc14_in_ga(T6)))
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
doublec53_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(60) QReductionProof (EQUIVALENT transformation)

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

doublec53_in_ga(x0)

(61) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
F1_IN_G(s(s(T6))) → U13_G(T6, U25_ga(T6, predc14_in_ga(T6)))
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(62) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule F1_IN_G(s(s(T6))) → U13_G(T6, U25_ga(T6, predc14_in_ga(T6))) at position [1,1] we obtained the following new rules [LPAR04]:

F1_IN_G(s(s(0))) → U13_G(0, U25_ga(0, predc14_out_ga(0, 0)))
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))

(63) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))
F1_IN_G(s(s(0))) → U13_G(0, U25_ga(0, predc14_out_ga(0, 0)))
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(64) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule F1_IN_G(s(s(0))) → U13_G(0, U25_ga(0, predc14_out_ga(0, 0))) at position [1] we obtained the following new rules [LPAR04]:

F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))

(65) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(66) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U15_G(T6, predc24_out_ga(T8, T16)) → U19_G(T6, halfc39_in_ga(T16)) at position [1] we obtained the following new rules [LPAR04]:

U15_G(y0, predc24_out_ga(y1, 0)) → U19_G(y0, halfc39_out_ga(0, 0))
U15_G(y0, predc24_out_ga(y1, s(s(x0)))) → U19_G(y0, U26_ga(x0, predc9_in_ga(x0)))

(67) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U15_G(y0, predc24_out_ga(y1, 0)) → U19_G(y0, halfc39_out_ga(0, 0))
U15_G(y0, predc24_out_ga(y1, s(s(x0)))) → U19_G(y0, U26_ga(x0, predc9_in_ga(x0)))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(68) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U15_G(y0, predc24_out_ga(y1, s(s(x0)))) → U19_G(y0, U26_ga(x0, predc9_in_ga(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

U15_G(y0, predc24_out_ga(y1, s(s(x0)))) → U19_G(y0, U26_ga(x0, U25_ga(x0, predc14_in_ga(x0))))

(69) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0))
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U15_G(y0, predc24_out_ga(y1, 0)) → U19_G(y0, halfc39_out_ga(0, 0))
U15_G(y0, predc24_out_ga(y1, s(s(x0)))) → U19_G(y0, U26_ga(x0, U25_ga(x0, predc14_in_ga(x0))))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(70) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U13_G(y0, predc9_out_ga(y0, s(0))) → U15_G(y0, predc24_out_ga(s(0), 0)) we obtained the following new rules [LPAR04]:

U13_G(s(z0), predc9_out_ga(s(z0), s(0))) → U15_G(s(z0), predc24_out_ga(s(0), 0))
U13_G(0, predc9_out_ga(0, s(0))) → U15_G(0, predc24_out_ga(s(0), 0))

(71) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0))))
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U15_G(y0, predc24_out_ga(y1, 0)) → U19_G(y0, halfc39_out_ga(0, 0))
U15_G(y0, predc24_out_ga(y1, s(s(x0)))) → U19_G(y0, U26_ga(x0, U25_ga(x0, predc14_in_ga(x0))))
U13_G(s(z0), predc9_out_ga(s(z0), s(0))) → U15_G(s(z0), predc24_out_ga(s(0), 0))
U13_G(0, predc9_out_ga(0, s(0))) → U15_G(0, predc24_out_ga(s(0), 0))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(72) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U13_G(y0, predc9_out_ga(y0, s(s(x0)))) → U15_G(y0, U24_ga(x0, predc24_in_ga(s(x0)))) we obtained the following new rules [LPAR04]:

U13_G(s(z0), predc9_out_ga(s(z0), s(s(x1)))) → U15_G(s(z0), U24_ga(x1, predc24_in_ga(s(x1))))

(73) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U15_G(y0, predc24_out_ga(y1, 0)) → U19_G(y0, halfc39_out_ga(0, 0))
U15_G(y0, predc24_out_ga(y1, s(s(x0)))) → U19_G(y0, U26_ga(x0, U25_ga(x0, predc14_in_ga(x0))))
U13_G(s(z0), predc9_out_ga(s(z0), s(0))) → U15_G(s(z0), predc24_out_ga(s(0), 0))
U13_G(0, predc9_out_ga(0, s(0))) → U15_G(0, predc24_out_ga(s(0), 0))
U13_G(s(z0), predc9_out_ga(s(z0), s(s(x1)))) → U15_G(s(z0), U24_ga(x1, predc24_in_ga(s(x1))))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(74) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U15_G(y0, predc24_out_ga(y1, 0)) → U19_G(y0, halfc39_out_ga(0, 0)) we obtained the following new rules [LPAR04]:

U15_G(s(z0), predc24_out_ga(s(0), 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(0, predc24_out_ga(s(0), 0)) → U19_G(0, halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))

(75) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U15_G(y0, predc24_out_ga(y1, s(s(x0)))) → U19_G(y0, U26_ga(x0, U25_ga(x0, predc14_in_ga(x0))))
U13_G(s(z0), predc9_out_ga(s(z0), s(0))) → U15_G(s(z0), predc24_out_ga(s(0), 0))
U13_G(0, predc9_out_ga(0, s(0))) → U15_G(0, predc24_out_ga(s(0), 0))
U13_G(s(z0), predc9_out_ga(s(z0), s(s(x1)))) → U15_G(s(z0), U24_ga(x1, predc24_in_ga(s(x1))))
U15_G(s(z0), predc24_out_ga(s(0), 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(0, predc24_out_ga(s(0), 0)) → U19_G(0, halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(76) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U15_G(y0, predc24_out_ga(y1, s(s(x0)))) → U19_G(y0, U26_ga(x0, U25_ga(x0, predc14_in_ga(x0)))) we obtained the following new rules [LPAR04]:

U15_G(s(z0), predc24_out_ga(x1, s(s(x2)))) → U19_G(s(z0), U26_ga(x2, U25_ga(x2, predc14_in_ga(x2))))

(77) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0)))
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U13_G(s(z0), predc9_out_ga(s(z0), s(0))) → U15_G(s(z0), predc24_out_ga(s(0), 0))
U13_G(0, predc9_out_ga(0, s(0))) → U15_G(0, predc24_out_ga(s(0), 0))
U13_G(s(z0), predc9_out_ga(s(z0), s(s(x1)))) → U15_G(s(z0), U24_ga(x1, predc24_in_ga(s(x1))))
U15_G(s(z0), predc24_out_ga(s(0), 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(0, predc24_out_ga(s(0), 0)) → U19_G(0, halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, s(s(x2)))) → U19_G(s(z0), U26_ga(x2, U25_ga(x2, predc14_in_ga(x2))))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(78) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U19_G(y0, halfc39_out_ga(y1, x0)) → U20_G(y0, U37_ga(x0, qc57_in_gaa(x0))) we obtained the following new rules [LPAR04]:

U19_G(s(z0), halfc39_out_ga(0, 0)) → U20_G(s(z0), U37_ga(0, qc57_in_gaa(0)))
U19_G(0, halfc39_out_ga(0, 0)) → U20_G(0, U37_ga(0, qc57_in_gaa(0)))
U19_G(s(z0), halfc39_out_ga(x1, x2)) → U20_G(s(z0), U37_ga(x2, qc57_in_gaa(x2)))

(79) Obligation:

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

U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29)
F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U13_G(s(z0), predc9_out_ga(s(z0), s(0))) → U15_G(s(z0), predc24_out_ga(s(0), 0))
U13_G(0, predc9_out_ga(0, s(0))) → U15_G(0, predc24_out_ga(s(0), 0))
U13_G(s(z0), predc9_out_ga(s(z0), s(s(x1)))) → U15_G(s(z0), U24_ga(x1, predc24_in_ga(s(x1))))
U15_G(s(z0), predc24_out_ga(s(0), 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(0, predc24_out_ga(s(0), 0)) → U19_G(0, halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, s(s(x2)))) → U19_G(s(z0), U26_ga(x2, U25_ga(x2, predc14_in_ga(x2))))
U19_G(s(z0), halfc39_out_ga(0, 0)) → U20_G(s(z0), U37_ga(0, qc57_in_gaa(0)))
U19_G(0, halfc39_out_ga(0, 0)) → U20_G(0, U37_ga(0, qc57_in_gaa(0)))
U19_G(s(z0), halfc39_out_ga(x1, x2)) → U20_G(s(z0), U37_ga(x2, qc57_in_gaa(x2)))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(80) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U20_G(T6, doublec53_out_ga(T20, T29)) → F1_IN_G(T29) we obtained the following new rules [LPAR04]:

U20_G(s(z0), doublec53_out_ga(x1, x2)) → F1_IN_G(x2)
U20_G(0, doublec53_out_ga(x1, x2)) → F1_IN_G(x2)

(81) Obligation:

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

F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U13_G(s(z0), predc9_out_ga(s(z0), s(0))) → U15_G(s(z0), predc24_out_ga(s(0), 0))
U13_G(0, predc9_out_ga(0, s(0))) → U15_G(0, predc24_out_ga(s(0), 0))
U13_G(s(z0), predc9_out_ga(s(z0), s(s(x1)))) → U15_G(s(z0), U24_ga(x1, predc24_in_ga(s(x1))))
U15_G(s(z0), predc24_out_ga(s(0), 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(0, predc24_out_ga(s(0), 0)) → U19_G(0, halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, s(s(x2)))) → U19_G(s(z0), U26_ga(x2, U25_ga(x2, predc14_in_ga(x2))))
U19_G(s(z0), halfc39_out_ga(0, 0)) → U20_G(s(z0), U37_ga(0, qc57_in_gaa(0)))
U19_G(0, halfc39_out_ga(0, 0)) → U20_G(0, U37_ga(0, qc57_in_gaa(0)))
U19_G(s(z0), halfc39_out_ga(x1, x2)) → U20_G(s(z0), U37_ga(x2, qc57_in_gaa(x2)))
U20_G(s(z0), doublec53_out_ga(x1, x2)) → F1_IN_G(x2)
U20_G(0, doublec53_out_ga(x1, x2)) → F1_IN_G(x2)

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(82) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U20_G(s(z0), doublec53_out_ga(x1, x2)) → F1_IN_G(x2) we obtained the following new rules [LPAR04]:

U20_G(s(x0), doublec53_out_ga(x1, s(s(s(y_0))))) → F1_IN_G(s(s(s(y_0))))
U20_G(s(x0), doublec53_out_ga(x1, s(s(0)))) → F1_IN_G(s(s(0)))

(83) Obligation:

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

F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U13_G(s(z0), predc9_out_ga(s(z0), s(0))) → U15_G(s(z0), predc24_out_ga(s(0), 0))
U13_G(0, predc9_out_ga(0, s(0))) → U15_G(0, predc24_out_ga(s(0), 0))
U13_G(s(z0), predc9_out_ga(s(z0), s(s(x1)))) → U15_G(s(z0), U24_ga(x1, predc24_in_ga(s(x1))))
U15_G(s(z0), predc24_out_ga(s(0), 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(0, predc24_out_ga(s(0), 0)) → U19_G(0, halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, s(s(x2)))) → U19_G(s(z0), U26_ga(x2, U25_ga(x2, predc14_in_ga(x2))))
U19_G(s(z0), halfc39_out_ga(0, 0)) → U20_G(s(z0), U37_ga(0, qc57_in_gaa(0)))
U19_G(0, halfc39_out_ga(0, 0)) → U20_G(0, U37_ga(0, qc57_in_gaa(0)))
U19_G(s(z0), halfc39_out_ga(x1, x2)) → U20_G(s(z0), U37_ga(x2, qc57_in_gaa(x2)))
U20_G(0, doublec53_out_ga(x1, x2)) → F1_IN_G(x2)
U20_G(s(x0), doublec53_out_ga(x1, s(s(s(y_0))))) → F1_IN_G(s(s(s(y_0))))
U20_G(s(x0), doublec53_out_ga(x1, s(s(0)))) → F1_IN_G(s(s(0)))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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

(84) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U20_G(0, doublec53_out_ga(x1, x2)) → F1_IN_G(x2) we obtained the following new rules [LPAR04]:

U20_G(0, doublec53_out_ga(x0, s(s(s(y_0))))) → F1_IN_G(s(s(s(y_0))))
U20_G(0, doublec53_out_ga(x0, s(s(0)))) → F1_IN_G(s(s(0)))

(85) Obligation:

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

F1_IN_G(s(s(s(x0)))) → U13_G(s(x0), U25_ga(s(x0), U23_ga(x0, predc14_in_ga(x0))))
F1_IN_G(s(s(0))) → U13_G(0, predc9_out_ga(0, s(0)))
U13_G(s(z0), predc9_out_ga(s(z0), s(0))) → U15_G(s(z0), predc24_out_ga(s(0), 0))
U13_G(0, predc9_out_ga(0, s(0))) → U15_G(0, predc24_out_ga(s(0), 0))
U13_G(s(z0), predc9_out_ga(s(z0), s(s(x1)))) → U15_G(s(z0), U24_ga(x1, predc24_in_ga(s(x1))))
U15_G(s(z0), predc24_out_ga(s(0), 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(0, predc24_out_ga(s(0), 0)) → U19_G(0, halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, 0)) → U19_G(s(z0), halfc39_out_ga(0, 0))
U15_G(s(z0), predc24_out_ga(x1, s(s(x2)))) → U19_G(s(z0), U26_ga(x2, U25_ga(x2, predc14_in_ga(x2))))
U19_G(s(z0), halfc39_out_ga(0, 0)) → U20_G(s(z0), U37_ga(0, qc57_in_gaa(0)))
U19_G(0, halfc39_out_ga(0, 0)) → U20_G(0, U37_ga(0, qc57_in_gaa(0)))
U19_G(s(z0), halfc39_out_ga(x1, x2)) → U20_G(s(z0), U37_ga(x2, qc57_in_gaa(x2)))
U20_G(s(x0), doublec53_out_ga(x1, s(s(s(y_0))))) → F1_IN_G(s(s(s(y_0))))
U20_G(s(x0), doublec53_out_ga(x1, s(s(0)))) → F1_IN_G(s(s(0)))
U20_G(0, doublec53_out_ga(x0, s(s(s(y_0))))) → F1_IN_G(s(s(s(y_0))))
U20_G(0, doublec53_out_ga(x0, s(s(0)))) → F1_IN_G(s(s(0)))

The TRS R consists of the following rules:

qc57_in_gaa(T33) → U34_gaa(T33, predc24_in_gg(s(T33), 0))
qc57_in_gaa(T33) → U35_gaa(T33, predc24_in_ga(s(T33)))
U37_ga(T33, qc57_out_gaa(T33, X96, X97)) → doublec53_out_ga(T33, s(s(X97)))
predc24_in_ga(s(0)) → predc24_out_ga(s(0), 0)
predc24_in_ga(s(s(T19))) → U24_ga(T19, predc24_in_ga(s(T19)))
U35_gaa(T33, predc24_out_ga(s(T33), s(T37))) → U36_gaa(T33, T37, qc57_in_gaa(T37))
U36_gaa(T33, T37, qc57_out_gaa(T37, X107, X108)) → qc57_out_gaa(T33, s(T37), s(s(X108)))
U24_ga(T19, predc24_out_ga(s(T19), X55)) → predc24_out_ga(s(s(T19)), s(X55))
predc24_in_gg(s(0), 0) → predc24_out_gg(s(0), 0)
U34_gaa(T33, predc24_out_gg(s(T33), 0)) → qc57_out_gaa(T33, 0, 0)
halfc39_in_ga(0) → halfc39_out_ga(0, 0)
halfc39_in_ga(s(s(T23))) → U26_ga(T23, predc9_in_ga(T23))
predc9_in_ga(T12) → U25_ga(T12, predc14_in_ga(T12))
U26_ga(T23, predc9_out_ga(T23, T25)) → U27_ga(T23, predc24_in_ga(T25))
predc24_in_ga(0) → predc24_out_ga(0, 0)
U27_ga(T23, predc24_out_ga(T25, T27)) → U28_ga(T23, halfc39_in_ga(T27))
U28_ga(T23, halfc39_out_ga(T27, X70)) → halfc39_out_ga(s(s(T23)), s(X70))
predc14_in_ga(0) → predc14_out_ga(0, 0)
predc14_in_ga(s(T15)) → U23_ga(T15, predc14_in_ga(T15))
U25_ga(T12, predc14_out_ga(T12, X35)) → predc9_out_ga(T12, s(X35))
U23_ga(T15, predc14_out_ga(T15, X44)) → predc14_out_ga(s(T15), s(X44))

The set Q consists of the following terms:

predc9_in_ga(x0)
predc24_in_ga(x0)
halfc39_in_ga(x0)
U25_ga(x0, x1)
U24_ga(x0, x1)
U26_ga(x0, x1)
U37_ga(x0, x1)
predc14_in_ga(x0)
U27_ga(x0, x1)
qc57_in_gaa(x0)
U23_ga(x0, x1)
U28_ga(x0, x1)
U34_gaa(x0, x1)
U35_gaa(x0, x1)
predc24_in_gg(x0, x1)
U36_gaa(x0, x1, x2)

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