Left Termination proof of ../tpdb/LP/terminweb/old-terminweb/quicksort.pl

YES Left Termination proof of ../tpdb/LP/terminweb/old-terminweb/quicksort.pl
Left Termination of the query pattern qs_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

Clauses:

qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys)))).
qs([], []).
part(X, .(Y, Xs), .(Y, Ls), Bs) :- ','(','(=(X1, X), ','(=(X2, Y), isGreater(X1, X2))), part(X, Xs, Ls, Bs)).
part(X, .(Y, Xs), Ls, .(Y, Bs)) :- ','(','(=(X1, X), ','(=(X2, Y), ;(=(X1, X2), isLess(X1, X2)))), part(X, Xs, Ls, Bs)).
part(X, [], [], []).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
app([], Ys, Ys).
isPlus(zero, X, X).
isPlus(succ(X), zero, succ(X)).
isPlus(succ(X), succ(Y), succ(succ(Z))) :- isPlus(X, Y, Z).
isPlus(succ(X), pred(Y), Z) :- isPlus(X, Y, Z).
isPlus(pred(X), zero, pred(X)).
isPlus(pred(X), succ(Y), Z) :- isPlus(X, Y, Z).
isPlus(pred(X), pred(Y), pred(pred(Z))) :- isPlus(X, Y, Z).
isMinus(X, zero, X).
isMinus(zero, succ(Y), pred(Z)) :- isMinus(zero, Y, Z).
isMinus(zero, pred(Y), succ(Z)) :- isMinus(zero, Y, Z).
isMinus(succ(X), succ(Y), Z) :- isMinus(X, Y, Z).
isMinus(succ(X), pred(Y), succ(succ(Z))) :- isMinus(X, Y, Z).
isMinus(pred(X), succ(Y), pred(pred(Z))) :- isMinus(X, Y, Z).
isMinus(pred(X), pred(Y), Z) :- isMinus(X, Y, Z).
isTimes(X, zero, zero).
isTimes(zero, succ(Y), zero).
isTimes(zero, pred(Y), zero).
isTimes(succ(X), succ(Y), Z) :- ','(isTimes(succ(X), Y, A), isPlus(A, succ(X), Z)).
isTimes(succ(X), pred(Y), Z) :- ','(isTimes(succ(X), Y, A), isMinus(A, succ(X), Z)).
isTimes(pred(X), succ(Y), Z) :- ','(isTimes(pred(X), Y, A), isPlus(A, pred(X), Z)).
isTimes(pred(X), pred(Y), Z) :- ','(isTimes(pred(X), Y, A), isMinus(A, pred(X), Z)).
isDiv(zero, succ(Y), zero).
isDiv(zero, pred(Y), zero).
isDiv(succ(X), succ(Y), zero) :- isMinus(succ(X), succ(Y), pred(Z)).
isDiv(succ(X), succ(Y), succ(Z)) :- ','(isMinus(succ(X), succ(Y), A), isDiv(A, succ(Y), Z)).
isDiv(succ(X), pred(Y), Z) :- ','(isMinus(zero, pred(Y), A), ','(isDiv(succ(X), A, B), isMinus(zero, B, Z))).
isDiv(pred(X), pred(Y), zero) :- isMinus(pred(X), pred(Y), succ(Z)).
isDiv(pred(X), pred(Y), succ(Z)) :- ','(isMinus(pred(X), pred(Y), A), isDiv(A, pred(Y), Z)).
isDiv(pred(X), succ(Y), Z) :- ','(isMinus(zero, pred(X), A), ','(isDiv(A, succ(Y), B), isMinus(zero, B, Z))).
isModulo(X, Y, Z) :- ','(isDiv(X, Y, A), ','(isTimes(A, Y, B), isMinus(X, B, Z))).
=(X, X).
isGreater(succ(X), zero).
isGreater(succ(X), pred(Y)).
isGreater(succ(X), succ(Y)) :- isGreater(X, Y).
isGreater(zero, pred(Y)).
isGreater(pred(X), pred(Y)) :- isGreater(X, Y).
isLess(pred(X), zero).
isLess(pred(X), succ(Y)).
isLess(pred(X), pred(Y)) :- isLess(X, Y).
isLess(zero, succ(Y)).
isLess(succ(X), succ(Y)) :- isLess(X, Y).

Queries:

qs(g,a).

Eliminated all or-constructs.

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

Clauses:

qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys)))).
qs([], []).
part(X, .(Y, Xs), .(Y, Ls), Bs) :- ','(=(X1, X), ','(=(X2, Y), ','(isGreater(X1, X2), part(X, Xs, Ls, Bs)))).
part(X, [], [], []).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
app([], Ys, Ys).
isPlus(zero, X, X).
isPlus(succ(X), zero, succ(X)).
isPlus(succ(X), succ(Y), succ(succ(Z))) :- isPlus(X, Y, Z).
isPlus(succ(X), pred(Y), Z) :- isPlus(X, Y, Z).
isPlus(pred(X), zero, pred(X)).
isPlus(pred(X), succ(Y), Z) :- isPlus(X, Y, Z).
isPlus(pred(X), pred(Y), pred(pred(Z))) :- isPlus(X, Y, Z).
isMinus(X, zero, X).
isMinus(zero, succ(Y), pred(Z)) :- isMinus(zero, Y, Z).
isMinus(zero, pred(Y), succ(Z)) :- isMinus(zero, Y, Z).
isMinus(succ(X), succ(Y), Z) :- isMinus(X, Y, Z).
isMinus(succ(X), pred(Y), succ(succ(Z))) :- isMinus(X, Y, Z).
isMinus(pred(X), succ(Y), pred(pred(Z))) :- isMinus(X, Y, Z).
isMinus(pred(X), pred(Y), Z) :- isMinus(X, Y, Z).
isTimes(X, zero, zero).
isTimes(zero, succ(Y), zero).
isTimes(zero, pred(Y), zero).
isTimes(succ(X), succ(Y), Z) :- ','(isTimes(succ(X), Y, A), isPlus(A, succ(X), Z)).
isTimes(succ(X), pred(Y), Z) :- ','(isTimes(succ(X), Y, A), isMinus(A, succ(X), Z)).
isTimes(pred(X), succ(Y), Z) :- ','(isTimes(pred(X), Y, A), isPlus(A, pred(X), Z)).
isTimes(pred(X), pred(Y), Z) :- ','(isTimes(pred(X), Y, A), isMinus(A, pred(X), Z)).
isDiv(zero, succ(Y), zero).
isDiv(zero, pred(Y), zero).
isDiv(succ(X), succ(Y), zero) :- isMinus(succ(X), succ(Y), pred(Z)).
isDiv(succ(X), succ(Y), succ(Z)) :- ','(isMinus(succ(X), succ(Y), A), isDiv(A, succ(Y), Z)).
isDiv(succ(X), pred(Y), Z) :- ','(isMinus(zero, pred(Y), A), ','(isDiv(succ(X), A, B), isMinus(zero, B, Z))).
isDiv(pred(X), pred(Y), zero) :- isMinus(pred(X), pred(Y), succ(Z)).
isDiv(pred(X), pred(Y), succ(Z)) :- ','(isMinus(pred(X), pred(Y), A), isDiv(A, pred(Y), Z)).
isDiv(pred(X), succ(Y), Z) :- ','(isMinus(zero, pred(X), A), ','(isDiv(A, succ(Y), B), isMinus(zero, B, Z))).
isModulo(X, Y, Z) :- ','(isDiv(X, Y, A), ','(isTimes(A, Y, B), isMinus(X, B, Z))).
=(X, X).
isGreater(succ(X), zero).
isGreater(succ(X), pred(Y)).
isGreater(succ(X), succ(Y)) :- isGreater(X, Y).
isGreater(zero, pred(Y)).
isGreater(pred(X), pred(Y)) :- isGreater(X, Y).
isLess(pred(X), zero).
isLess(pred(X), succ(Y)).
isLess(pred(X), pred(Y)) :- isLess(X, Y).
isLess(zero, succ(Y)).
isLess(succ(X), succ(Y)) :- isLess(X, Y).
part(X, .(Y, Xs), Ls, .(Y, Bs)) :- ','(=(X1, X), ','(=(X2, Y), ','(=(X1, X2), part(X, Xs, Ls, Bs)))).
part(X, .(Y, Xs), Ls, .(Y, Bs)) :- ','(=(X1, X), ','(=(X2, Y), ','(isLess(X1, X2), part(X, Xs, Ls, Bs)))).

Queries:

qs(g,a).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
qs_in: (b,f)
part_in: (b,b,f,f)
isGreater_in: (b,b)
isLess_in: (b,b)
app_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
=_in_ag(X, X) → =_out_ag(X, X)
U5_ggaa(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U6_ggaa(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U6_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U7_ggaa(X, Y, Xs, Ls, Bs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, isGreater_out_gg(X1, X2)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U47_ggaa(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
U47_ggaa(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U49_ggaa(X, Y, Xs, Ls, Bs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg(X, X)
U49_ggaa(X, Y, Xs, Ls, Bs, =_out_gg(X1, X2)) → U50_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U50_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U51_ggaa(X, Y, Xs, Ls, Bs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U51_ggaa(X, Y, Xs, Ls, Bs, isLess_out_gg(X1, X2)) → U52_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U52_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

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

qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
=_in_ag(X, X) → =_out_ag(X, X)
U5_ggaa(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U6_ggaa(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U6_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U7_ggaa(X, Y, Xs, Ls, Bs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, isGreater_out_gg(X1, X2)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U47_ggaa(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
U47_ggaa(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U49_ggaa(X, Y, Xs, Ls, Bs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg(X, X)
U49_ggaa(X, Y, Xs, Ls, Bs, =_out_gg(X1, X2)) → U50_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U50_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U51_ggaa(X, Y, Xs, Ls, Bs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U51_ggaa(X, Y, Xs, Ls, Bs, isLess_out_gg(X1, X2)) → U52_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U52_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
QS_IN_GA(.(X, Xs), Ys) → PART_IN_GGAA(X, Xs, Littles, Bigs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → =_IN_AG(X1, X)
U5_GGAA(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U6_GGAA(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U5_GGAA(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → =_IN_AG(X2, Y)
U6_GGAA(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U7_GGAA(X, Y, Xs, Ls, Bs, isGreater_in_gg(X1, X2))
U6_GGAA(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → ISGREATER_IN_GG(X1, X2)
ISGREATER_IN_GG(succ(X), succ(Y)) → U43_GG(X, Y, isGreater_in_gg(X, Y))
ISGREATER_IN_GG(succ(X), succ(Y)) → ISGREATER_IN_GG(X, Y)
ISGREATER_IN_GG(pred(X), pred(Y)) → U44_GG(X, Y, isGreater_in_gg(X, Y))
ISGREATER_IN_GG(pred(X), pred(Y)) → ISGREATER_IN_GG(X, Y)
U7_GGAA(X, Y, Xs, Ls, Bs, isGreater_out_gg(X1, X2)) → U8_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U7_GGAA(X, Y, Xs, Ls, Bs, isGreater_out_gg(X1, X2)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U47_GGAA(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → =_IN_AG(X1, X)
U47_GGAA(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U48_GGAA(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U47_GGAA(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → =_IN_AG(X2, Y)
U48_GGAA(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U49_GGAA(X, Y, Xs, Ls, Bs, =_in_gg(X1, X2))
U48_GGAA(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → =_IN_GG(X1, X2)
U49_GGAA(X, Y, Xs, Ls, Bs, =_out_gg(X1, X2)) → U50_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U49_GGAA(X, Y, Xs, Ls, Bs, =_out_gg(X1, X2)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U48_GGAA(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U51_GGAA(X, Y, Xs, Ls, Bs, isLess_in_gg(X1, X2))
U48_GGAA(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → ISLESS_IN_GG(X1, X2)
ISLESS_IN_GG(pred(X), pred(Y)) → U45_GG(X, Y, isLess_in_gg(X, Y))
ISLESS_IN_GG(pred(X), pred(Y)) → ISLESS_IN_GG(X, Y)
ISLESS_IN_GG(succ(X), succ(Y)) → U46_GG(X, Y, isLess_in_gg(X, Y))
ISLESS_IN_GG(succ(X), succ(Y)) → ISLESS_IN_GG(X, Y)
U51_GGAA(X, Y, Xs, Ls, Bs, isLess_out_gg(X1, X2)) → U52_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U51_GGAA(X, Y, Xs, Ls, Bs, isLess_out_gg(X1, X2)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → APP_IN_GGA(Ls, .(X, Bs), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U9_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
=_in_ag(X, X) → =_out_ag(X, X)
U5_ggaa(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U6_ggaa(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U6_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U7_ggaa(X, Y, Xs, Ls, Bs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, isGreater_out_gg(X1, X2)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U47_ggaa(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
U47_ggaa(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U49_ggaa(X, Y, Xs, Ls, Bs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg(X, X)
U49_ggaa(X, Y, Xs, Ls, Bs, =_out_gg(X1, X2)) → U50_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U50_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U51_ggaa(X, Y, Xs, Ls, Bs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U51_ggaa(X, Y, Xs, Ls, Bs, isLess_out_gg(X1, X2)) → U52_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U52_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2) = qs_in_ga(x1)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6)
=_in_ag(x1, x2) = =_in_ag(x2)
=_out_ag(x1, x2) = =_out_ag(x1)
U6_ggaa(x1, x2, x3, x4, x5, x6, x7) = U6_ggaa(x1, x2, x3, x6, x7)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6)
isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2)
succ(x1) = succ(x1)
zero = zero
isGreater_out_gg(x1, x2) = isGreater_out_gg
pred(x1) = pred(x1)
U43_gg(x1, x2, x3) = U43_gg(x3)
U44_gg(x1, x2, x3) = U44_gg(x3)
U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6)
[] = []
part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4)
U47_ggaa(x1, x2, x3, x4, x5, x6) = U47_ggaa(x1, x2, x3, x6)
U48_ggaa(x1, x2, x3, x4, x5, x6, x7) = U48_ggaa(x1, x2, x3, x6, x7)
U49_ggaa(x1, x2, x3, x4, x5, x6) = U49_ggaa(x1, x2, x3, x6)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
U50_ggaa(x1, x2, x3, x4, x5, x6) = U50_ggaa(x2, x6)
U51_ggaa(x1, x2, x3, x4, x5, x6) = U51_ggaa(x1, x2, x3, x6)
isLess_in_gg(x1, x2) = isLess_in_gg(x1, x2)
isLess_out_gg(x1, x2) = isLess_out_gg
U45_gg(x1, x2, x3) = U45_gg(x3)
U46_gg(x1, x2, x3) = U46_gg(x3)
U52_ggaa(x1, x2, x3, x4, x5, x6) = U52_ggaa(x2, x6)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5)
qs_out_ga(x1, x2) = qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4) = U4_ga(x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x5)
U48_GGAA(x1, x2, x3, x4, x5, x6, x7) = U48_GGAA(x1, x2, x3, x6, x7)
ISGREATER_IN_GG(x1, x2) = ISGREATER_IN_GG(x1, x2)
U51_GGAA(x1, x2, x3, x4, x5, x6) = U51_GGAA(x1, x2, x3, x6)
=_IN_GG(x1, x2) = =_IN_GG(x1, x2)
U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4)
U52_GGAA(x1, x2, x3, x4, x5, x6) = U52_GGAA(x2, x6)
ISLESS_IN_GG(x1, x2) = ISLESS_IN_GG(x1, x2)
U43_GG(x1, x2, x3) = U43_GG(x3)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x4, x5)
U4_GA(x1, x2, x3, x4) = U4_GA(x4)
U6_GGAA(x1, x2, x3, x4, x5, x6, x7) = U6_GGAA(x1, x2, x3, x6, x7)
=_IN_AG(x1, x2) = =_IN_AG(x2)
U44_GG(x1, x2, x3) = U44_GG(x3)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x4, x5)
APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2)
PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2)
QS_IN_GA(x1, x2) = QS_IN_GA(x1)
U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6)
U8_GGAA(x1, x2, x3, x4, x5, x6) = U8_GGAA(x2, x6)
U45_GG(x1, x2, x3) = U45_GG(x3)
U46_GG(x1, x2, x3) = U46_GG(x3)
U50_GGAA(x1, x2, x3, x4, x5, x6) = U50_GGAA(x2, x6)
U47_GGAA(x1, x2, x3, x4, x5, x6) = U47_GGAA(x1, x2, x3, x6)
U49_GGAA(x1, x2, x3, x4, x5, x6) = U49_GGAA(x1, x2, x3, x6)

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

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

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

QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
QS_IN_GA(.(X, Xs), Ys) → PART_IN_GGAA(X, Xs, Littles, Bigs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → =_IN_AG(X1, X)
U5_GGAA(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U6_GGAA(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U5_GGAA(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → =_IN_AG(X2, Y)
U6_GGAA(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U7_GGAA(X, Y, Xs, Ls, Bs, isGreater_in_gg(X1, X2))
U6_GGAA(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → ISGREATER_IN_GG(X1, X2)
ISGREATER_IN_GG(succ(X), succ(Y)) → U43_GG(X, Y, isGreater_in_gg(X, Y))
ISGREATER_IN_GG(succ(X), succ(Y)) → ISGREATER_IN_GG(X, Y)
ISGREATER_IN_GG(pred(X), pred(Y)) → U44_GG(X, Y, isGreater_in_gg(X, Y))
ISGREATER_IN_GG(pred(X), pred(Y)) → ISGREATER_IN_GG(X, Y)
U7_GGAA(X, Y, Xs, Ls, Bs, isGreater_out_gg(X1, X2)) → U8_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U7_GGAA(X, Y, Xs, Ls, Bs, isGreater_out_gg(X1, X2)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U47_GGAA(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → =_IN_AG(X1, X)
U47_GGAA(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U48_GGAA(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U47_GGAA(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → =_IN_AG(X2, Y)
U48_GGAA(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U49_GGAA(X, Y, Xs, Ls, Bs, =_in_gg(X1, X2))
U48_GGAA(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → =_IN_GG(X1, X2)
U49_GGAA(X, Y, Xs, Ls, Bs, =_out_gg(X1, X2)) → U50_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U49_GGAA(X, Y, Xs, Ls, Bs, =_out_gg(X1, X2)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U48_GGAA(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U51_GGAA(X, Y, Xs, Ls, Bs, isLess_in_gg(X1, X2))
U48_GGAA(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → ISLESS_IN_GG(X1, X2)
ISLESS_IN_GG(pred(X), pred(Y)) → U45_GG(X, Y, isLess_in_gg(X, Y))
ISLESS_IN_GG(pred(X), pred(Y)) → ISLESS_IN_GG(X, Y)
ISLESS_IN_GG(succ(X), succ(Y)) → U46_GG(X, Y, isLess_in_gg(X, Y))
ISLESS_IN_GG(succ(X), succ(Y)) → ISLESS_IN_GG(X, Y)
U51_GGAA(X, Y, Xs, Ls, Bs, isLess_out_gg(X1, X2)) → U52_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U51_GGAA(X, Y, Xs, Ls, Bs, isLess_out_gg(X1, X2)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → APP_IN_GGA(Ls, .(X, Bs), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U9_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
=_in_ag(X, X) → =_out_ag(X, X)
U5_ggaa(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U6_ggaa(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U6_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U7_ggaa(X, Y, Xs, Ls, Bs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, isGreater_out_gg(X1, X2)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U47_ggaa(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
U47_ggaa(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U49_ggaa(X, Y, Xs, Ls, Bs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg(X, X)
U49_ggaa(X, Y, Xs, Ls, Bs, =_out_gg(X1, X2)) → U50_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U50_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U51_ggaa(X, Y, Xs, Ls, Bs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U51_ggaa(X, Y, Xs, Ls, Bs, isLess_out_gg(X1, X2)) → U52_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U52_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                        ↳ UsableRulesProof

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

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

APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
=_in_ag(X, X) → =_out_ag(X, X)
U5_ggaa(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U6_ggaa(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U6_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U7_ggaa(X, Y, Xs, Ls, Bs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, isGreater_out_gg(X1, X2)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U47_ggaa(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
U47_ggaa(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U49_ggaa(X, Y, Xs, Ls, Bs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg(X, X)
U49_ggaa(X, Y, Xs, Ls, Bs, =_out_gg(X1, X2)) → U50_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U50_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U51_ggaa(X, Y, Xs, Ls, Bs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U51_ggaa(X, Y, Xs, Ls, Bs, isLess_out_gg(X1, X2)) → U52_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U52_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                        ↳ UsableRulesProof

                          ↳ PiDP

                            ↳ PiDPToQDPProof

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

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

APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                        ↳ UsableRulesProof

                          ↳ PiDP

                            ↳ PiDPToQDPProof

                              ↳ QDP

                                ↳ QDPSizeChangeProof

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

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

APP_IN_GGA(.(X, Xs), Ys) → APP_IN_GGA(Xs, Ys)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:

APP_IN_GGA(.(X, Xs), Ys) → APP_IN_GGA(Xs, Ys)
The graph contains the following edges 1 > 1, 2 >= 2

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                        ↳ UsableRulesProof

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

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

ISLESS_IN_GG(pred(X), pred(Y)) → ISLESS_IN_GG(X, Y)
ISLESS_IN_GG(succ(X), succ(Y)) → ISLESS_IN_GG(X, Y)

The TRS R consists of the following rules:

qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
=_in_ag(X, X) → =_out_ag(X, X)
U5_ggaa(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U6_ggaa(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U6_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U7_ggaa(X, Y, Xs, Ls, Bs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, isGreater_out_gg(X1, X2)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U47_ggaa(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
U47_ggaa(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U49_ggaa(X, Y, Xs, Ls, Bs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg(X, X)
U49_ggaa(X, Y, Xs, Ls, Bs, =_out_gg(X1, X2)) → U50_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U50_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U51_ggaa(X, Y, Xs, Ls, Bs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U51_ggaa(X, Y, Xs, Ls, Bs, isLess_out_gg(X1, X2)) → U52_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U52_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                        ↳ UsableRulesProof

                          ↳ PiDP

                            ↳ PiDPToQDPProof

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

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

ISLESS_IN_GG(pred(X), pred(Y)) → ISLESS_IN_GG(X, Y)
ISLESS_IN_GG(succ(X), succ(Y)) → ISLESS_IN_GG(X, Y)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                        ↳ UsableRulesProof

                          ↳ PiDP

                            ↳ PiDPToQDPProof

                              ↳ QDP

                                ↳ QDPSizeChangeProof

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

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

ISLESS_IN_GG(pred(X), pred(Y)) → ISLESS_IN_GG(X, Y)
ISLESS_IN_GG(succ(X), succ(Y)) → ISLESS_IN_GG(X, Y)

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

ISLESS_IN_GG(pred(X), pred(Y)) → ISLESS_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2
ISLESS_IN_GG(succ(X), succ(Y)) → ISLESS_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                        ↳ UsableRulesProof

                      ↳ PiDP

                      ↳ PiDP

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

ISGREATER_IN_GG(succ(X), succ(Y)) → ISGREATER_IN_GG(X, Y)
ISGREATER_IN_GG(pred(X), pred(Y)) → ISGREATER_IN_GG(X, Y)

The TRS R consists of the following rules:

qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
=_in_ag(X, X) → =_out_ag(X, X)
U5_ggaa(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U6_ggaa(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U6_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U7_ggaa(X, Y, Xs, Ls, Bs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, isGreater_out_gg(X1, X2)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U47_ggaa(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
U47_ggaa(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U49_ggaa(X, Y, Xs, Ls, Bs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg(X, X)
U49_ggaa(X, Y, Xs, Ls, Bs, =_out_gg(X1, X2)) → U50_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U50_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U51_ggaa(X, Y, Xs, Ls, Bs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U51_ggaa(X, Y, Xs, Ls, Bs, isLess_out_gg(X1, X2)) → U52_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U52_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                        ↳ UsableRulesProof

                          ↳ PiDP

                            ↳ PiDPToQDPProof

                      ↳ PiDP

                      ↳ PiDP

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

ISGREATER_IN_GG(succ(X), succ(Y)) → ISGREATER_IN_GG(X, Y)
ISGREATER_IN_GG(pred(X), pred(Y)) → ISGREATER_IN_GG(X, Y)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                        ↳ UsableRulesProof

                          ↳ PiDP

                            ↳ PiDPToQDPProof

                              ↳ QDP

                                ↳ QDPSizeChangeProof

                      ↳ PiDP

                      ↳ PiDP

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

ISGREATER_IN_GG(succ(X), succ(Y)) → ISGREATER_IN_GG(X, Y)
ISGREATER_IN_GG(pred(X), pred(Y)) → ISGREATER_IN_GG(X, Y)

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

ISGREATER_IN_GG(succ(X), succ(Y)) → ISGREATER_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2
ISGREATER_IN_GG(pred(X), pred(Y)) → ISGREATER_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                        ↳ UsableRulesProof

                      ↳ PiDP

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

U48_GGAA(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U49_GGAA(X, Y, Xs, Ls, Bs, =_in_gg(X1, X2))
U47_GGAA(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U48_GGAA(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U49_GGAA(X, Y, Xs, Ls, Bs, =_out_gg(X1, X2)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
U7_GGAA(X, Y, Xs, Ls, Bs, isGreater_out_gg(X1, X2)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U51_GGAA(X, Y, Xs, Ls, Bs, isLess_out_gg(X1, X2)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U47_GGAA(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
U6_GGAA(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U7_GGAA(X, Y, Xs, Ls, Bs, isGreater_in_gg(X1, X2))
U5_GGAA(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U6_GGAA(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U48_GGAA(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U51_GGAA(X, Y, Xs, Ls, Bs, isLess_in_gg(X1, X2))

The TRS R consists of the following rules:

qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
=_in_ag(X, X) → =_out_ag(X, X)
U5_ggaa(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U6_ggaa(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U6_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U7_ggaa(X, Y, Xs, Ls, Bs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, isGreater_out_gg(X1, X2)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U47_ggaa(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
U47_ggaa(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U49_ggaa(X, Y, Xs, Ls, Bs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg(X, X)
U49_ggaa(X, Y, Xs, Ls, Bs, =_out_gg(X1, X2)) → U50_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U50_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U51_ggaa(X, Y, Xs, Ls, Bs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U51_ggaa(X, Y, Xs, Ls, Bs, isLess_out_gg(X1, X2)) → U52_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U52_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2) = qs_in_ga(x1)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6)
=_in_ag(x1, x2) = =_in_ag(x2)
=_out_ag(x1, x2) = =_out_ag(x1)
U6_ggaa(x1, x2, x3, x4, x5, x6, x7) = U6_ggaa(x1, x2, x3, x6, x7)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x1, x2, x3, x6)
isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2)
succ(x1) = succ(x1)
zero = zero
isGreater_out_gg(x1, x2) = isGreater_out_gg
pred(x1) = pred(x1)
U43_gg(x1, x2, x3) = U43_gg(x3)
U44_gg(x1, x2, x3) = U44_gg(x3)
U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x2, x6)
[] = []
part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4)
U47_ggaa(x1, x2, x3, x4, x5, x6) = U47_ggaa(x1, x2, x3, x6)
U48_ggaa(x1, x2, x3, x4, x5, x6, x7) = U48_ggaa(x1, x2, x3, x6, x7)
U49_ggaa(x1, x2, x3, x4, x5, x6) = U49_ggaa(x1, x2, x3, x6)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
U50_ggaa(x1, x2, x3, x4, x5, x6) = U50_ggaa(x2, x6)
U51_ggaa(x1, x2, x3, x4, x5, x6) = U51_ggaa(x1, x2, x3, x6)
isLess_in_gg(x1, x2) = isLess_in_gg(x1, x2)
isLess_out_gg(x1, x2) = isLess_out_gg
U45_gg(x1, x2, x3) = U45_gg(x3)
U46_gg(x1, x2, x3) = U46_gg(x3)
U52_ggaa(x1, x2, x3, x4, x5, x6) = U52_ggaa(x2, x6)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5)
qs_out_ga(x1, x2) = qs_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4) = U4_ga(x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x5)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
U48_GGAA(x1, x2, x3, x4, x5, x6, x7) = U48_GGAA(x1, x2, x3, x6, x7)
U51_GGAA(x1, x2, x3, x4, x5, x6) = U51_GGAA(x1, x2, x3, x6)
U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6)
U6_GGAA(x1, x2, x3, x4, x5, x6, x7) = U6_GGAA(x1, x2, x3, x6, x7)
PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6)
U47_GGAA(x1, x2, x3, x4, x5, x6) = U47_GGAA(x1, x2, x3, x6)
U49_GGAA(x1, x2, x3, x4, x5, x6) = U49_GGAA(x1, x2, x3, x6)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                        ↳ UsableRulesProof

                          ↳ PiDP

                            ↳ PiDPToQDPProof

                      ↳ PiDP

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

U48_GGAA(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U49_GGAA(X, Y, Xs, Ls, Bs, =_in_gg(X1, X2))
U47_GGAA(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U48_GGAA(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U49_GGAA(X, Y, Xs, Ls, Bs, =_out_gg(X1, X2)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
U7_GGAA(X, Y, Xs, Ls, Bs, isGreater_out_gg(X1, X2)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U51_GGAA(X, Y, Xs, Ls, Bs, isLess_out_gg(X1, X2)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U47_GGAA(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
U6_GGAA(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U7_GGAA(X, Y, Xs, Ls, Bs, isGreater_in_gg(X1, X2))
U5_GGAA(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U6_GGAA(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U48_GGAA(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U51_GGAA(X, Y, Xs, Ls, Bs, isLess_in_gg(X1, X2))

The TRS R consists of the following rules:

=_in_gg(X, X) → =_out_gg(X, X)
=_in_ag(X, X) → =_out_ag(X, X)
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
=_in_ag(x1, x2) = =_in_ag(x2)
=_out_ag(x1, x2) = =_out_ag(x1)
isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2)
succ(x1) = succ(x1)
zero = zero
isGreater_out_gg(x1, x2) = isGreater_out_gg
pred(x1) = pred(x1)
U43_gg(x1, x2, x3) = U43_gg(x3)
U44_gg(x1, x2, x3) = U44_gg(x3)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
isLess_in_gg(x1, x2) = isLess_in_gg(x1, x2)
isLess_out_gg(x1, x2) = isLess_out_gg
U45_gg(x1, x2, x3) = U45_gg(x3)
U46_gg(x1, x2, x3) = U46_gg(x3)
U48_GGAA(x1, x2, x3, x4, x5, x6, x7) = U48_GGAA(x1, x2, x3, x6, x7)
U51_GGAA(x1, x2, x3, x4, x5, x6) = U51_GGAA(x1, x2, x3, x6)
U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6)
U6_GGAA(x1, x2, x3, x4, x5, x6, x7) = U6_GGAA(x1, x2, x3, x6, x7)
PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6)
U47_GGAA(x1, x2, x3, x4, x5, x6) = U47_GGAA(x1, x2, x3, x6)
U49_GGAA(x1, x2, x3, x4, x5, x6) = U49_GGAA(x1, x2, x3, x6)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                        ↳ UsableRulesProof

                          ↳ PiDP

                            ↳ PiDPToQDPProof

                              ↳ QDP

                                ↳ QDPSizeChangeProof

                      ↳ PiDP

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

U51_GGAA(X, Y, Xs, isLess_out_gg) → PART_IN_GGAA(X, Xs)
U5_GGAA(X, Y, Xs, =_out_ag(X1)) → U6_GGAA(X, Y, Xs, X1, =_in_ag(Y))
PART_IN_GGAA(X, .(Y, Xs)) → U47_GGAA(X, Y, Xs, =_in_ag(X))
U48_GGAA(X, Y, Xs, X1, =_out_ag(X2)) → U51_GGAA(X, Y, Xs, isLess_in_gg(X1, X2))
U48_GGAA(X, Y, Xs, X1, =_out_ag(X2)) → U49_GGAA(X, Y, Xs, =_in_gg(X1, X2))
U7_GGAA(X, Y, Xs, isGreater_out_gg) → PART_IN_GGAA(X, Xs)
U6_GGAA(X, Y, Xs, X1, =_out_ag(X2)) → U7_GGAA(X, Y, Xs, isGreater_in_gg(X1, X2))
PART_IN_GGAA(X, .(Y, Xs)) → U5_GGAA(X, Y, Xs, =_in_ag(X))
U49_GGAA(X, Y, Xs, =_out_gg) → PART_IN_GGAA(X, Xs)
U47_GGAA(X, Y, Xs, =_out_ag(X1)) → U48_GGAA(X, Y, Xs, X1, =_in_ag(Y))

The TRS R consists of the following rules:

=_in_gg(X, X) → =_out_gg
=_in_ag(X) → =_out_ag(X)
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U43_gg(isGreater_out_gg) → isGreater_out_gg
U44_gg(isGreater_out_gg) → isGreater_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U46_gg(isLess_out_gg) → isLess_out_gg

The set Q consists of the following terms:

=_in_gg(x0, x1)
=_in_ag(x0)
isGreater_in_gg(x0, x1)
isLess_in_gg(x0, x1)
U43_gg(x0)
U44_gg(x0)
U45_gg(x0)
U46_gg(x0)

We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:

U49_GGAA(X, Y, Xs, =_out_gg) → PART_IN_GGAA(X, Xs)
The graph contains the following edges 1 >= 1, 3 >= 2
U47_GGAA(X, Y, Xs, =_out_ag(X1)) → U48_GGAA(X, Y, Xs, X1, =_in_ag(Y))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4
PART_IN_GGAA(X, .(Y, Xs)) → U47_GGAA(X, Y, Xs, =_in_ag(X))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3
U48_GGAA(X, Y, Xs, X1, =_out_ag(X2)) → U49_GGAA(X, Y, Xs, =_in_gg(X1, X2))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3
U48_GGAA(X, Y, Xs, X1, =_out_ag(X2)) → U51_GGAA(X, Y, Xs, isLess_in_gg(X1, X2))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3
PART_IN_GGAA(X, .(Y, Xs)) → U5_GGAA(X, Y, Xs, =_in_ag(X))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3
U5_GGAA(X, Y, Xs, =_out_ag(X1)) → U6_GGAA(X, Y, Xs, X1, =_in_ag(Y))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4
U6_GGAA(X, Y, Xs, X1, =_out_ag(X2)) → U7_GGAA(X, Y, Xs, isGreater_in_gg(X1, X2))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3
U7_GGAA(X, Y, Xs, isGreater_out_gg) → PART_IN_GGAA(X, Xs)
The graph contains the following edges 1 >= 1, 3 >= 2
U51_GGAA(X, Y, Xs, isLess_out_gg) → PART_IN_GGAA(X, Xs)
The graph contains the following edges 1 >= 1, 3 >= 2

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                        ↳ PiDPToQDPProof

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

QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)

The TRS R consists of the following rules:

qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
=_in_ag(X, X) → =_out_ag(X, X)
U5_ggaa(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U6_ggaa(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U6_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U7_ggaa(X, Y, Xs, Ls, Bs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U7_ggaa(X, Y, Xs, Ls, Bs, isGreater_out_gg(X1, X2)) → U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U47_ggaa(X, Y, Xs, Ls, Bs, =_in_ag(X1, X))
U47_ggaa(X, Y, Xs, Ls, Bs, =_out_ag(X1, X)) → U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_in_ag(X2, Y))
U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U49_ggaa(X, Y, Xs, Ls, Bs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg(X, X)
U49_ggaa(X, Y, Xs, Ls, Bs, =_out_gg(X1, X2)) → U50_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U50_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U48_ggaa(X, Y, Xs, Ls, Bs, X1, =_out_ag(X2, Y)) → U51_ggaa(X, Y, Xs, Ls, Bs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U51_ggaa(X, Y, Xs, Ls, Bs, isLess_out_gg(X1, X2)) → U52_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U52_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
qs_in_ga([], []) → qs_out_ga([], [])
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ QDPOrderProof

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

QS_IN_GA(.(X, Xs)) → U1_GA(X, part_in_ggaa(X, Xs))
U2_GA(X, Bigs, qs_out_ga(Ls)) → QS_IN_GA(Bigs)
U1_GA(X, part_out_ggaa(Littles, Bigs)) → U2_GA(X, Bigs, qs_in_ga(Littles))
U1_GA(X, part_out_ggaa(Littles, Bigs)) → QS_IN_GA(Littles)

The TRS R consists of the following rules:

qs_in_ga(.(X, Xs)) → U1_ga(X, part_in_ggaa(X, Xs))
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U5_ggaa(X, Y, Xs, =_out_ag(X1)) → U6_ggaa(X, Y, Xs, X1, =_in_ag(Y))
U6_ggaa(X, Y, Xs, X1, =_out_ag(X2)) → U7_ggaa(X, Y, Xs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U7_ggaa(X, Y, Xs, isGreater_out_gg) → U8_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X, []) → part_out_ggaa([], [])
part_in_ggaa(X, .(Y, Xs)) → U47_ggaa(X, Y, Xs, =_in_ag(X))
U47_ggaa(X, Y, Xs, =_out_ag(X1)) → U48_ggaa(X, Y, Xs, X1, =_in_ag(Y))
U48_ggaa(X, Y, Xs, X1, =_out_ag(X2)) → U49_ggaa(X, Y, Xs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_ggaa(X, Y, Xs, =_out_gg) → U50_ggaa(Y, part_in_ggaa(X, Xs))
U50_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
U48_ggaa(X, Y, Xs, X1, =_out_ag(X2)) → U51_ggaa(X, Y, Xs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_ggaa(X, Y, Xs, isLess_out_gg) → U52_ggaa(Y, part_in_ggaa(X, Xs))
U52_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
U8_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
U1_ga(X, part_out_ggaa(Littles, Bigs)) → U2_ga(X, Bigs, qs_in_ga(Littles))
qs_in_ga([]) → qs_out_ga([])
U2_ga(X, Bigs, qs_out_ga(Ls)) → U3_ga(X, Ls, qs_in_ga(Bigs))
U3_ga(X, Ls, qs_out_ga(Bs)) → U4_ga(app_in_gga(Ls, .(X, Bs)))
app_in_gga(.(X, Xs), Ys) → U9_gga(X, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga(Ys)
U9_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))
U4_ga(app_out_gga(Ys)) → qs_out_ga(Ys)

The set Q consists of the following terms:

qs_in_ga(x0)
part_in_ggaa(x0, x1)
=_in_ag(x0)
U5_ggaa(x0, x1, x2, x3)
U6_ggaa(x0, x1, x2, x3, x4)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U7_ggaa(x0, x1, x2, x3)
U47_ggaa(x0, x1, x2, x3)
U48_ggaa(x0, x1, x2, x3, x4)
=_in_gg(x0, x1)
U49_ggaa(x0, x1, x2, x3)
U50_ggaa(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_ggaa(x0, x1, x2, x3)
U52_ggaa(x0, x1)
U8_ggaa(x0, x1)
U1_ga(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
app_in_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(x0)

We have to consider all (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

U1_GA(X, part_out_ggaa(Littles, Bigs)) → U2_GA(X, Bigs, qs_in_ga(Littles))
U1_GA(X, part_out_ggaa(Littles, Bigs)) → QS_IN_GA(Littles)

The remaining pairs can at least be oriented weakly.

QS_IN_GA(.(X, Xs)) → U1_GA(X, part_in_ggaa(X, Xs))
U2_GA(X, Bigs, qs_out_ga(Ls)) → QS_IN_GA(Bigs)

Used ordering: Polynomial interpretation [25]:

POL(.(x₁, x₂)) = 1 + x₂
POL(=_in_ag(x₁)) = 0
POL(=_in_gg(x₁, x₂)) = 1
POL(=_out_ag(x₁)) = 0
POL(=_out_gg) = 0
POL(QS_IN_GA(x₁)) = x₁
POL(U1_GA(x₁, x₂)) = 1 + x₂
POL(U1_ga(x₁, x₂)) = 0
POL(U2_GA(x₁, x₂, x₃)) = x₂
POL(U2_ga(x₁, x₂, x₃)) = 0
POL(U3_ga(x₁, x₂, x₃)) = 0
POL(U43_gg(x₁)) = 0
POL(U44_gg(x₁)) = 0
POL(U45_gg(x₁)) = 0
POL(U46_gg(x₁)) = 0
POL(U47_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U48_ggaa(x₁, x₂, x₃, x₄, x₅)) = 1 + x₃
POL(U49_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U4_ga(x₁)) = 0
POL(U50_ggaa(x₁, x₂)) = 1 + x₂
POL(U51_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U52_ggaa(x₁, x₂)) = 1 + x₂
POL(U5_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U6_ggaa(x₁, x₂, x₃, x₄, x₅)) = 1 + x₃
POL(U7_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U8_ggaa(x₁, x₂)) = 1 + x₂
POL(U9_gga(x₁, x₂)) = 0
POL([]) = 0
POL(app_in_gga(x₁, x₂)) = 0
POL(app_out_gga(x₁)) = 0
POL(isGreater_in_gg(x₁, x₂)) = 0
POL(isGreater_out_gg) = 0
POL(isLess_in_gg(x₁, x₂)) = 0
POL(isLess_out_gg) = 0
POL(part_in_ggaa(x₁, x₂)) = x₂
POL(part_out_ggaa(x₁, x₂)) = x₁ + x₂
POL(pred(x₁)) = 0
POL(qs_in_ga(x₁)) = 0
POL(qs_out_ga(x₁)) = 0
POL(succ(x₁)) = 0
POL(zero) = 0

The following usable rules [17] were oriented:

U8_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
part_in_ggaa(X, .(Y, Xs)) → U47_ggaa(X, Y, Xs, =_in_ag(X))
U48_ggaa(X, Y, Xs, X1, =_out_ag(X2)) → U51_ggaa(X, Y, Xs, isLess_in_gg(X1, X2))
U50_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, X1, =_out_ag(X2)) → U7_ggaa(X, Y, Xs, isGreater_in_gg(X1, X2))
U7_ggaa(X, Y, Xs, isGreater_out_gg) → U8_ggaa(Y, part_in_ggaa(X, Xs))
U51_ggaa(X, Y, Xs, isLess_out_gg) → U52_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X, []) → part_out_ggaa([], [])
U49_ggaa(X, Y, Xs, =_out_gg) → U50_ggaa(Y, part_in_ggaa(X, Xs))
U5_ggaa(X, Y, Xs, =_out_ag(X1)) → U6_ggaa(X, Y, Xs, X1, =_in_ag(Y))
U47_ggaa(X, Y, Xs, =_out_ag(X1)) → U48_ggaa(X, Y, Xs, X1, =_in_ag(Y))
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, =_in_ag(X))
U48_ggaa(X, Y, Xs, X1, =_out_ag(X2)) → U49_ggaa(X, Y, Xs, =_in_gg(X1, X2))
U52_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ OrTransformerProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

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

QS_IN_GA(.(X, Xs)) → U1_GA(X, part_in_ggaa(X, Xs))
U2_GA(X, Bigs, qs_out_ga(Ls)) → QS_IN_GA(Bigs)

The TRS R consists of the following rules:

qs_in_ga(.(X, Xs)) → U1_ga(X, part_in_ggaa(X, Xs))
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U5_ggaa(X, Y, Xs, =_out_ag(X1)) → U6_ggaa(X, Y, Xs, X1, =_in_ag(Y))
U6_ggaa(X, Y, Xs, X1, =_out_ag(X2)) → U7_ggaa(X, Y, Xs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U7_ggaa(X, Y, Xs, isGreater_out_gg) → U8_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X, []) → part_out_ggaa([], [])
part_in_ggaa(X, .(Y, Xs)) → U47_ggaa(X, Y, Xs, =_in_ag(X))
U47_ggaa(X, Y, Xs, =_out_ag(X1)) → U48_ggaa(X, Y, Xs, X1, =_in_ag(Y))
U48_ggaa(X, Y, Xs, X1, =_out_ag(X2)) → U49_ggaa(X, Y, Xs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_ggaa(X, Y, Xs, =_out_gg) → U50_ggaa(Y, part_in_ggaa(X, Xs))
U50_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
U48_ggaa(X, Y, Xs, X1, =_out_ag(X2)) → U51_ggaa(X, Y, Xs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_ggaa(X, Y, Xs, isLess_out_gg) → U52_ggaa(Y, part_in_ggaa(X, Xs))
U52_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
U8_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
U1_ga(X, part_out_ggaa(Littles, Bigs)) → U2_ga(X, Bigs, qs_in_ga(Littles))
qs_in_ga([]) → qs_out_ga([])
U2_ga(X, Bigs, qs_out_ga(Ls)) → U3_ga(X, Ls, qs_in_ga(Bigs))
U3_ga(X, Ls, qs_out_ga(Bs)) → U4_ga(app_in_gga(Ls, .(X, Bs)))
app_in_gga(.(X, Xs), Ys) → U9_gga(X, app_in_gga(Xs, Ys))
app_in_gga([], Ys) → app_out_gga(Ys)
U9_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))
U4_ga(app_out_gga(Ys)) → qs_out_ga(Ys)

The set Q consists of the following terms:

qs_in_ga(x0)
part_in_ggaa(x0, x1)
=_in_ag(x0)
U5_ggaa(x0, x1, x2, x3)
U6_ggaa(x0, x1, x2, x3, x4)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U7_ggaa(x0, x1, x2, x3)
U47_ggaa(x0, x1, x2, x3)
U48_ggaa(x0, x1, x2, x3, x4)
=_in_gg(x0, x1)
U49_ggaa(x0, x1, x2, x3)
U50_ggaa(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_ggaa(x0, x1, x2, x3)
U52_ggaa(x0, x1)
U8_ggaa(x0, x1)
U1_ga(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
app_in_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.