Left Termination proof of ../tpdb/LP/talp/maria/qsortapp.pl

YES Left Termination proof of ../tpdb/LP/talp/maria/qsortapp.pl
Left Termination of the query pattern qsort_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ CutEliminatorProof

Clauses:

qsort(.(X, L), R) :- ','(partition(L, X, L1, L2), ','(qsort(L2, R2), ','(qsort(L1, R1), append(R1, .(X, R2), R)))).
qsort([], []).
partition([], _, [], []).
partition(.(E, R), C, .(E, Left1), Right) :- ','(','(=(X, E), ','(=(X1, C), isLess(X, X1))), ','(!, partition(R, C, Left1, Right))).
partition(.(E, R), C, Left, .(E, Right1)) :- partition(R, C, Left, Right1).
append([], X, X).
append(.(H, X), Y, .(H, Z)) :- append(X, Y, Z).
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).
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:

qsort(g,a).

Eliminated all cuts by simply ignoring them.

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ CutEliminatorProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

Clauses:

qsort(.(X, L), R) :- ','(partition(L, X, L1, L2), ','(qsort(L2, R2), ','(qsort(L1, R1), append(R1, .(X, R2), R)))).
qsort([], []).
partition([], X, [], []).
partition(.(E, R), C, .(E, Left1), Right) :- ','(=(X, E), ','(=(X1, C), ','(isLess(X, X1), partition(R, C, Left1, Right)))).
partition(.(E, R), C, Left, .(E, Right1)) :- partition(R, C, Left, Right1).
append([], X, X).
append(.(H, X), Y, .(H, Z)) :- append(X, Y, Z).
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).
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:

qsort(g,a).

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

qsort_in_ga(.(X, L), R) → U1_ga(X, L, R, partition_in_ggaa(L, X, L1, L2))
partition_in_ggaa([], X, [], []) → partition_out_ggaa([], X, [], [])
partition_in_ggaa(.(E, R), C, .(E, Left1), Right) → U5_ggaa(E, R, C, Left1, Right, =_in_ag(X, E))
=_in_ag(X, X) → =_out_ag(X, X)
U5_ggaa(E, R, C, Left1, Right, =_out_ag(X, E)) → U6_ggaa(E, R, C, Left1, Right, X, =_in_ag(X1, C))
U6_ggaa(E, R, C, Left1, Right, X, =_out_ag(X1, C)) → U7_ggaa(E, R, C, Left1, Right, isLess_in_gg(X, X1))
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)) → U44_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)) → U45_gg(X, Y, isLess_in_gg(X, Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U44_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U7_ggaa(E, R, C, Left1, Right, isLess_out_gg(X, X1)) → U8_ggaa(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
partition_in_ggaa(.(E, R), C, Left, .(E, Right1)) → U9_ggaa(E, R, C, Left, Right1, partition_in_ggaa(R, C, Left, Right1))
U9_ggaa(E, R, C, Left, Right1, partition_out_ggaa(R, C, Left, Right1)) → partition_out_ggaa(.(E, R), C, Left, .(E, Right1))
U8_ggaa(E, R, C, Left1, Right, partition_out_ggaa(R, C, Left1, Right)) → partition_out_ggaa(.(E, R), C, .(E, Left1), Right)
U1_ga(X, L, R, partition_out_ggaa(L, X, L1, L2)) → U2_ga(X, L, R, L1, qsort_in_ga(L2, R2))
qsort_in_ga([], []) → qsort_out_ga([], [])
U2_ga(X, L, R, L1, qsort_out_ga(L2, R2)) → U3_ga(X, L, R, R2, qsort_in_ga(L1, R1))
U3_ga(X, L, R, R2, qsort_out_ga(L1, R1)) → U4_ga(X, L, R, append_in_gga(R1, .(X, R2), R))
append_in_gga([], X, X) → append_out_gga([], X, X)
append_in_gga(.(H, X), Y, .(H, Z)) → U10_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
U10_gga(H, X, Y, Z, append_out_gga(X, Y, Z)) → append_out_gga(.(H, X), Y, .(H, Z))
U4_ga(X, L, R, append_out_gga(R1, .(X, R2), R)) → qsort_out_ga(.(X, L), R)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ CutEliminatorProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

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

qsort_in_ga(.(X, L), R) → U1_ga(X, L, R, partition_in_ggaa(L, X, L1, L2))
partition_in_ggaa([], X, [], []) → partition_out_ggaa([], X, [], [])
partition_in_ggaa(.(E, R), C, .(E, Left1), Right) → U5_ggaa(E, R, C, Left1, Right, =_in_ag(X, E))
=_in_ag(X, X) → =_out_ag(X, X)
U5_ggaa(E, R, C, Left1, Right, =_out_ag(X, E)) → U6_ggaa(E, R, C, Left1, Right, X, =_in_ag(X1, C))
U6_ggaa(E, R, C, Left1, Right, X, =_out_ag(X1, C)) → U7_ggaa(E, R, C, Left1, Right, isLess_in_gg(X, X1))
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)) → U44_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)) → U45_gg(X, Y, isLess_in_gg(X, Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U44_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U7_ggaa(E, R, C, Left1, Right, isLess_out_gg(X, X1)) → U8_ggaa(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
partition_in_ggaa(.(E, R), C, Left, .(E, Right1)) → U9_ggaa(E, R, C, Left, Right1, partition_in_ggaa(R, C, Left, Right1))
U9_ggaa(E, R, C, Left, Right1, partition_out_ggaa(R, C, Left, Right1)) → partition_out_ggaa(.(E, R), C, Left, .(E, Right1))
U8_ggaa(E, R, C, Left1, Right, partition_out_ggaa(R, C, Left1, Right)) → partition_out_ggaa(.(E, R), C, .(E, Left1), Right)
U1_ga(X, L, R, partition_out_ggaa(L, X, L1, L2)) → U2_ga(X, L, R, L1, qsort_in_ga(L2, R2))
qsort_in_ga([], []) → qsort_out_ga([], [])
U2_ga(X, L, R, L1, qsort_out_ga(L2, R2)) → U3_ga(X, L, R, R2, qsort_in_ga(L1, R1))
U3_ga(X, L, R, R2, qsort_out_ga(L1, R1)) → U4_ga(X, L, R, append_in_gga(R1, .(X, R2), R))
append_in_gga([], X, X) → append_out_gga([], X, X)
append_in_gga(.(H, X), Y, .(H, Z)) → U10_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
U10_gga(H, X, Y, Z, append_out_gga(X, Y, Z)) → append_out_gga(.(H, X), Y, .(H, Z))
U4_ga(X, L, R, append_out_gga(R1, .(X, R2), R)) → qsort_out_ga(.(X, L), R)

QSORT_IN_GA(.(X, L), R) → U1_GA(X, L, R, partition_in_ggaa(L, X, L1, L2))
QSORT_IN_GA(.(X, L), R) → PARTITION_IN_GGAA(L, X, L1, L2)
PARTITION_IN_GGAA(.(E, R), C, .(E, Left1), Right) → U5_GGAA(E, R, C, Left1, Right, =_in_ag(X, E))
PARTITION_IN_GGAA(.(E, R), C, .(E, Left1), Right) → =_IN_AG(X, E)
U5_GGAA(E, R, C, Left1, Right, =_out_ag(X, E)) → U6_GGAA(E, R, C, Left1, Right, X, =_in_ag(X1, C))
U5_GGAA(E, R, C, Left1, Right, =_out_ag(X, E)) → =_IN_AG(X1, C)
U6_GGAA(E, R, C, Left1, Right, X, =_out_ag(X1, C)) → U7_GGAA(E, R, C, Left1, Right, isLess_in_gg(X, X1))
U6_GGAA(E, R, C, Left1, Right, X, =_out_ag(X1, C)) → ISLESS_IN_GG(X, X1)
ISLESS_IN_GG(pred(X), pred(Y)) → U44_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)) → U45_GG(X, Y, isLess_in_gg(X, Y))
ISLESS_IN_GG(succ(X), succ(Y)) → ISLESS_IN_GG(X, Y)
U7_GGAA(E, R, C, Left1, Right, isLess_out_gg(X, X1)) → U8_GGAA(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
U7_GGAA(E, R, C, Left1, Right, isLess_out_gg(X, X1)) → PARTITION_IN_GGAA(R, C, Left1, Right)
PARTITION_IN_GGAA(.(E, R), C, Left, .(E, Right1)) → U9_GGAA(E, R, C, Left, Right1, partition_in_ggaa(R, C, Left, Right1))
PARTITION_IN_GGAA(.(E, R), C, Left, .(E, Right1)) → PARTITION_IN_GGAA(R, C, Left, Right1)
U1_GA(X, L, R, partition_out_ggaa(L, X, L1, L2)) → U2_GA(X, L, R, L1, qsort_in_ga(L2, R2))
U1_GA(X, L, R, partition_out_ggaa(L, X, L1, L2)) → QSORT_IN_GA(L2, R2)
U2_GA(X, L, R, L1, qsort_out_ga(L2, R2)) → U3_GA(X, L, R, R2, qsort_in_ga(L1, R1))
U2_GA(X, L, R, L1, qsort_out_ga(L2, R2)) → QSORT_IN_GA(L1, R1)
U3_GA(X, L, R, R2, qsort_out_ga(L1, R1)) → U4_GA(X, L, R, append_in_gga(R1, .(X, R2), R))
U3_GA(X, L, R, R2, qsort_out_ga(L1, R1)) → APPEND_IN_GGA(R1, .(X, R2), R)
APPEND_IN_GGA(.(H, X), Y, .(H, Z)) → U10_GGA(H, X, Y, Z, append_in_gga(X, Y, Z))
APPEND_IN_GGA(.(H, X), Y, .(H, Z)) → APPEND_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

qsort_in_ga(.(X, L), R) → U1_ga(X, L, R, partition_in_ggaa(L, X, L1, L2))
partition_in_ggaa([], X, [], []) → partition_out_ggaa([], X, [], [])
partition_in_ggaa(.(E, R), C, .(E, Left1), Right) → U5_ggaa(E, R, C, Left1, Right, =_in_ag(X, E))
=_in_ag(X, X) → =_out_ag(X, X)
U5_ggaa(E, R, C, Left1, Right, =_out_ag(X, E)) → U6_ggaa(E, R, C, Left1, Right, X, =_in_ag(X1, C))
U6_ggaa(E, R, C, Left1, Right, X, =_out_ag(X1, C)) → U7_ggaa(E, R, C, Left1, Right, isLess_in_gg(X, X1))
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)) → U44_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)) → U45_gg(X, Y, isLess_in_gg(X, Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U44_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U7_ggaa(E, R, C, Left1, Right, isLess_out_gg(X, X1)) → U8_ggaa(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
partition_in_ggaa(.(E, R), C, Left, .(E, Right1)) → U9_ggaa(E, R, C, Left, Right1, partition_in_ggaa(R, C, Left, Right1))
U9_ggaa(E, R, C, Left, Right1, partition_out_ggaa(R, C, Left, Right1)) → partition_out_ggaa(.(E, R), C, Left, .(E, Right1))
U8_ggaa(E, R, C, Left1, Right, partition_out_ggaa(R, C, Left1, Right)) → partition_out_ggaa(.(E, R), C, .(E, Left1), Right)
U1_ga(X, L, R, partition_out_ggaa(L, X, L1, L2)) → U2_ga(X, L, R, L1, qsort_in_ga(L2, R2))
qsort_in_ga([], []) → qsort_out_ga([], [])
U2_ga(X, L, R, L1, qsort_out_ga(L2, R2)) → U3_ga(X, L, R, R2, qsort_in_ga(L1, R1))
U3_ga(X, L, R, R2, qsort_out_ga(L1, R1)) → U4_ga(X, L, R, append_in_gga(R1, .(X, R2), R))
append_in_gga([], X, X) → append_out_gga([], X, X)
append_in_gga(.(H, X), Y, .(H, Z)) → U10_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
U10_gga(H, X, Y, Z, append_out_gga(X, Y, Z)) → append_out_gga(.(H, X), Y, .(H, Z))
U4_ga(X, L, R, append_out_gga(R1, .(X, R2), R)) → qsort_out_ga(.(X, L), R)

The argument filtering Pi contains the following mapping:
qsort_in_ga(x1, x2) = qsort_in_ga(x1)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
partition_in_ggaa(x1, x2, x3, x4) = partition_in_ggaa(x1, x2)
[] = []
partition_out_ggaa(x1, x2, x3, x4) = partition_out_ggaa(x3, x4)
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)
isLess_in_gg(x1, x2) = isLess_in_gg(x1, x2)
pred(x1) = pred(x1)
zero = zero
isLess_out_gg(x1, x2) = isLess_out_gg
succ(x1) = succ(x1)
U44_gg(x1, x2, x3) = U44_gg(x3)
U45_gg(x1, x2, x3) = U45_gg(x3)
U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x1, x6)
U9_ggaa(x1, x2, x3, x4, x5, x6) = U9_ggaa(x1, x6)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5)
qsort_out_ga(x1, x2) = qsort_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4) = U4_ga(x4)
append_in_gga(x1, x2, x3) = append_in_gga(x1, x2)
append_out_gga(x1, x2, x3) = append_out_gga(x3)
U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x5)
PARTITION_IN_GGAA(x1, x2, x3, x4) = PARTITION_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6)
QSORT_IN_GA(x1, x2) = QSORT_IN_GA(x1)
U45_GG(x1, x2, x3) = U45_GG(x3)
U8_GGAA(x1, x2, x3, x4, x5, x6) = U8_GGAA(x1, x6)
U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x1, x2, x3, x6)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4)
U9_GGAA(x1, x2, x3, x4, x5, x6) = U9_GGAA(x1, x6)
ISLESS_IN_GG(x1, x2) = ISLESS_IN_GG(x1, x2)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x4, x5)
U10_GGA(x1, x2, x3, x4, x5) = U10_GGA(x1, x5)
U4_GA(x1, x2, x3, x4) = U4_GA(x4)
APPEND_IN_GGA(x1, x2, x3) = APPEND_IN_GGA(x1, x2)
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)

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

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ CutEliminatorProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

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

QSORT_IN_GA(.(X, L), R) → U1_GA(X, L, R, partition_in_ggaa(L, X, L1, L2))
QSORT_IN_GA(.(X, L), R) → PARTITION_IN_GGAA(L, X, L1, L2)
PARTITION_IN_GGAA(.(E, R), C, .(E, Left1), Right) → U5_GGAA(E, R, C, Left1, Right, =_in_ag(X, E))
PARTITION_IN_GGAA(.(E, R), C, .(E, Left1), Right) → =_IN_AG(X, E)
U5_GGAA(E, R, C, Left1, Right, =_out_ag(X, E)) → U6_GGAA(E, R, C, Left1, Right, X, =_in_ag(X1, C))
U5_GGAA(E, R, C, Left1, Right, =_out_ag(X, E)) → =_IN_AG(X1, C)
U6_GGAA(E, R, C, Left1, Right, X, =_out_ag(X1, C)) → U7_GGAA(E, R, C, Left1, Right, isLess_in_gg(X, X1))
U6_GGAA(E, R, C, Left1, Right, X, =_out_ag(X1, C)) → ISLESS_IN_GG(X, X1)
ISLESS_IN_GG(pred(X), pred(Y)) → U44_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)) → U45_GG(X, Y, isLess_in_gg(X, Y))
ISLESS_IN_GG(succ(X), succ(Y)) → ISLESS_IN_GG(X, Y)
U7_GGAA(E, R, C, Left1, Right, isLess_out_gg(X, X1)) → U8_GGAA(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
U7_GGAA(E, R, C, Left1, Right, isLess_out_gg(X, X1)) → PARTITION_IN_GGAA(R, C, Left1, Right)
PARTITION_IN_GGAA(.(E, R), C, Left, .(E, Right1)) → U9_GGAA(E, R, C, Left, Right1, partition_in_ggaa(R, C, Left, Right1))
PARTITION_IN_GGAA(.(E, R), C, Left, .(E, Right1)) → PARTITION_IN_GGAA(R, C, Left, Right1)
U1_GA(X, L, R, partition_out_ggaa(L, X, L1, L2)) → U2_GA(X, L, R, L1, qsort_in_ga(L2, R2))
U1_GA(X, L, R, partition_out_ggaa(L, X, L1, L2)) → QSORT_IN_GA(L2, R2)
U2_GA(X, L, R, L1, qsort_out_ga(L2, R2)) → U3_GA(X, L, R, R2, qsort_in_ga(L1, R1))
U2_GA(X, L, R, L1, qsort_out_ga(L2, R2)) → QSORT_IN_GA(L1, R1)
U3_GA(X, L, R, R2, qsort_out_ga(L1, R1)) → U4_GA(X, L, R, append_in_gga(R1, .(X, R2), R))
U3_GA(X, L, R, R2, qsort_out_ga(L1, R1)) → APPEND_IN_GGA(R1, .(X, R2), R)
APPEND_IN_GGA(.(H, X), Y, .(H, Z)) → U10_GGA(H, X, Y, Z, append_in_gga(X, Y, Z))
APPEND_IN_GGA(.(H, X), Y, .(H, Z)) → APPEND_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

qsort_in_ga(.(X, L), R) → U1_ga(X, L, R, partition_in_ggaa(L, X, L1, L2))
partition_in_ggaa([], X, [], []) → partition_out_ggaa([], X, [], [])
partition_in_ggaa(.(E, R), C, .(E, Left1), Right) → U5_ggaa(E, R, C, Left1, Right, =_in_ag(X, E))
=_in_ag(X, X) → =_out_ag(X, X)
U5_ggaa(E, R, C, Left1, Right, =_out_ag(X, E)) → U6_ggaa(E, R, C, Left1, Right, X, =_in_ag(X1, C))
U6_ggaa(E, R, C, Left1, Right, X, =_out_ag(X1, C)) → U7_ggaa(E, R, C, Left1, Right, isLess_in_gg(X, X1))
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)) → U44_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)) → U45_gg(X, Y, isLess_in_gg(X, Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U44_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U7_ggaa(E, R, C, Left1, Right, isLess_out_gg(X, X1)) → U8_ggaa(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
partition_in_ggaa(.(E, R), C, Left, .(E, Right1)) → U9_ggaa(E, R, C, Left, Right1, partition_in_ggaa(R, C, Left, Right1))
U9_ggaa(E, R, C, Left, Right1, partition_out_ggaa(R, C, Left, Right1)) → partition_out_ggaa(.(E, R), C, Left, .(E, Right1))
U8_ggaa(E, R, C, Left1, Right, partition_out_ggaa(R, C, Left1, Right)) → partition_out_ggaa(.(E, R), C, .(E, Left1), Right)
U1_ga(X, L, R, partition_out_ggaa(L, X, L1, L2)) → U2_ga(X, L, R, L1, qsort_in_ga(L2, R2))
qsort_in_ga([], []) → qsort_out_ga([], [])
U2_ga(X, L, R, L1, qsort_out_ga(L2, R2)) → U3_ga(X, L, R, R2, qsort_in_ga(L1, R1))
U3_ga(X, L, R, R2, qsort_out_ga(L1, R1)) → U4_ga(X, L, R, append_in_gga(R1, .(X, R2), R))
append_in_gga([], X, X) → append_out_gga([], X, X)
append_in_gga(.(H, X), Y, .(H, Z)) → U10_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
U10_gga(H, X, Y, Z, append_out_gga(X, Y, Z)) → append_out_gga(.(H, X), Y, .(H, Z))
U4_ga(X, L, R, append_out_gga(R1, .(X, R2), R)) → qsort_out_ga(.(X, L), R)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ CutEliminatorProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                        ↳ UsableRulesProof

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

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

APPEND_IN_GGA(.(H, X), Y, .(H, Z)) → APPEND_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

qsort_in_ga(.(X, L), R) → U1_ga(X, L, R, partition_in_ggaa(L, X, L1, L2))
partition_in_ggaa([], X, [], []) → partition_out_ggaa([], X, [], [])
partition_in_ggaa(.(E, R), C, .(E, Left1), Right) → U5_ggaa(E, R, C, Left1, Right, =_in_ag(X, E))
=_in_ag(X, X) → =_out_ag(X, X)
U5_ggaa(E, R, C, Left1, Right, =_out_ag(X, E)) → U6_ggaa(E, R, C, Left1, Right, X, =_in_ag(X1, C))
U6_ggaa(E, R, C, Left1, Right, X, =_out_ag(X1, C)) → U7_ggaa(E, R, C, Left1, Right, isLess_in_gg(X, X1))
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)) → U44_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)) → U45_gg(X, Y, isLess_in_gg(X, Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U44_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U7_ggaa(E, R, C, Left1, Right, isLess_out_gg(X, X1)) → U8_ggaa(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
partition_in_ggaa(.(E, R), C, Left, .(E, Right1)) → U9_ggaa(E, R, C, Left, Right1, partition_in_ggaa(R, C, Left, Right1))
U9_ggaa(E, R, C, Left, Right1, partition_out_ggaa(R, C, Left, Right1)) → partition_out_ggaa(.(E, R), C, Left, .(E, Right1))
U8_ggaa(E, R, C, Left1, Right, partition_out_ggaa(R, C, Left1, Right)) → partition_out_ggaa(.(E, R), C, .(E, Left1), Right)
U1_ga(X, L, R, partition_out_ggaa(L, X, L1, L2)) → U2_ga(X, L, R, L1, qsort_in_ga(L2, R2))
qsort_in_ga([], []) → qsort_out_ga([], [])
U2_ga(X, L, R, L1, qsort_out_ga(L2, R2)) → U3_ga(X, L, R, R2, qsort_in_ga(L1, R1))
U3_ga(X, L, R, R2, qsort_out_ga(L1, R1)) → U4_ga(X, L, R, append_in_gga(R1, .(X, R2), R))
append_in_gga([], X, X) → append_out_gga([], X, X)
append_in_gga(.(H, X), Y, .(H, Z)) → U10_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
U10_gga(H, X, Y, Z, append_out_gga(X, Y, Z)) → append_out_gga(.(H, X), Y, .(H, Z))
U4_ga(X, L, R, append_out_gga(R1, .(X, R2), R)) → qsort_out_ga(.(X, L), R)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ CutEliminatorProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                        ↳ UsableRulesProof

                          ↳ PiDP

                            ↳ PiDPToQDPProof

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

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

APPEND_IN_GGA(.(H, X), Y, .(H, Z)) → APPEND_IN_GGA(X, Y, Z)

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

      ↳ CutEliminatorProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                        ↳ UsableRulesProof

                          ↳ PiDP

                            ↳ PiDPToQDPProof

                              ↳ QDP

                                ↳ QDPSizeChangeProof

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

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

APPEND_IN_GGA(.(H, X), Y) → APPEND_IN_GGA(X, Y)

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:

APPEND_IN_GGA(.(H, X), Y) → APPEND_IN_GGA(X, Y)
The graph contains the following edges 1 > 1, 2 >= 2

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ CutEliminatorProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                        ↳ UsableRulesProof

                      ↳ 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:

qsort_in_ga(.(X, L), R) → U1_ga(X, L, R, partition_in_ggaa(L, X, L1, L2))
partition_in_ggaa([], X, [], []) → partition_out_ggaa([], X, [], [])
partition_in_ggaa(.(E, R), C, .(E, Left1), Right) → U5_ggaa(E, R, C, Left1, Right, =_in_ag(X, E))
=_in_ag(X, X) → =_out_ag(X, X)
U5_ggaa(E, R, C, Left1, Right, =_out_ag(X, E)) → U6_ggaa(E, R, C, Left1, Right, X, =_in_ag(X1, C))
U6_ggaa(E, R, C, Left1, Right, X, =_out_ag(X1, C)) → U7_ggaa(E, R, C, Left1, Right, isLess_in_gg(X, X1))
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)) → U44_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)) → U45_gg(X, Y, isLess_in_gg(X, Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U44_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U7_ggaa(E, R, C, Left1, Right, isLess_out_gg(X, X1)) → U8_ggaa(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
partition_in_ggaa(.(E, R), C, Left, .(E, Right1)) → U9_ggaa(E, R, C, Left, Right1, partition_in_ggaa(R, C, Left, Right1))
U9_ggaa(E, R, C, Left, Right1, partition_out_ggaa(R, C, Left, Right1)) → partition_out_ggaa(.(E, R), C, Left, .(E, Right1))
U8_ggaa(E, R, C, Left1, Right, partition_out_ggaa(R, C, Left1, Right)) → partition_out_ggaa(.(E, R), C, .(E, Left1), Right)
U1_ga(X, L, R, partition_out_ggaa(L, X, L1, L2)) → U2_ga(X, L, R, L1, qsort_in_ga(L2, R2))
qsort_in_ga([], []) → qsort_out_ga([], [])
U2_ga(X, L, R, L1, qsort_out_ga(L2, R2)) → U3_ga(X, L, R, R2, qsort_in_ga(L1, R1))
U3_ga(X, L, R, R2, qsort_out_ga(L1, R1)) → U4_ga(X, L, R, append_in_gga(R1, .(X, R2), R))
append_in_gga([], X, X) → append_out_gga([], X, X)
append_in_gga(.(H, X), Y, .(H, Z)) → U10_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
U10_gga(H, X, Y, Z, append_out_gga(X, Y, Z)) → append_out_gga(.(H, X), Y, .(H, Z))
U4_ga(X, L, R, append_out_gga(R1, .(X, R2), R)) → qsort_out_ga(.(X, L), R)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ CutEliminatorProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                        ↳ UsableRulesProof

                          ↳ PiDP

                            ↳ PiDPToQDPProof

                      ↳ 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

      ↳ CutEliminatorProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                        ↳ UsableRulesProof

                          ↳ PiDP

                            ↳ PiDPToQDPProof

                              ↳ QDP

                                ↳ QDPSizeChangeProof

                      ↳ 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

      ↳ CutEliminatorProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                        ↳ UsableRulesProof

                      ↳ PiDP

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

U5_GGAA(E, R, C, Left1, Right, =_out_ag(X, E)) → U6_GGAA(E, R, C, Left1, Right, X, =_in_ag(X1, C))
PARTITION_IN_GGAA(.(E, R), C, Left, .(E, Right1)) → PARTITION_IN_GGAA(R, C, Left, Right1)
U6_GGAA(E, R, C, Left1, Right, X, =_out_ag(X1, C)) → U7_GGAA(E, R, C, Left1, Right, isLess_in_gg(X, X1))
PARTITION_IN_GGAA(.(E, R), C, .(E, Left1), Right) → U5_GGAA(E, R, C, Left1, Right, =_in_ag(X, E))
U7_GGAA(E, R, C, Left1, Right, isLess_out_gg(X, X1)) → PARTITION_IN_GGAA(R, C, Left1, Right)

The TRS R consists of the following rules:

qsort_in_ga(.(X, L), R) → U1_ga(X, L, R, partition_in_ggaa(L, X, L1, L2))
partition_in_ggaa([], X, [], []) → partition_out_ggaa([], X, [], [])
partition_in_ggaa(.(E, R), C, .(E, Left1), Right) → U5_ggaa(E, R, C, Left1, Right, =_in_ag(X, E))
=_in_ag(X, X) → =_out_ag(X, X)
U5_ggaa(E, R, C, Left1, Right, =_out_ag(X, E)) → U6_ggaa(E, R, C, Left1, Right, X, =_in_ag(X1, C))
U6_ggaa(E, R, C, Left1, Right, X, =_out_ag(X1, C)) → U7_ggaa(E, R, C, Left1, Right, isLess_in_gg(X, X1))
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)) → U44_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)) → U45_gg(X, Y, isLess_in_gg(X, Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U44_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U7_ggaa(E, R, C, Left1, Right, isLess_out_gg(X, X1)) → U8_ggaa(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
partition_in_ggaa(.(E, R), C, Left, .(E, Right1)) → U9_ggaa(E, R, C, Left, Right1, partition_in_ggaa(R, C, Left, Right1))
U9_ggaa(E, R, C, Left, Right1, partition_out_ggaa(R, C, Left, Right1)) → partition_out_ggaa(.(E, R), C, Left, .(E, Right1))
U8_ggaa(E, R, C, Left1, Right, partition_out_ggaa(R, C, Left1, Right)) → partition_out_ggaa(.(E, R), C, .(E, Left1), Right)
U1_ga(X, L, R, partition_out_ggaa(L, X, L1, L2)) → U2_ga(X, L, R, L1, qsort_in_ga(L2, R2))
qsort_in_ga([], []) → qsort_out_ga([], [])
U2_ga(X, L, R, L1, qsort_out_ga(L2, R2)) → U3_ga(X, L, R, R2, qsort_in_ga(L1, R1))
U3_ga(X, L, R, R2, qsort_out_ga(L1, R1)) → U4_ga(X, L, R, append_in_gga(R1, .(X, R2), R))
append_in_gga([], X, X) → append_out_gga([], X, X)
append_in_gga(.(H, X), Y, .(H, Z)) → U10_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
U10_gga(H, X, Y, Z, append_out_gga(X, Y, Z)) → append_out_gga(.(H, X), Y, .(H, Z))
U4_ga(X, L, R, append_out_gga(R1, .(X, R2), R)) → qsort_out_ga(.(X, L), R)

The argument filtering Pi contains the following mapping:
qsort_in_ga(x1, x2) = qsort_in_ga(x1)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
partition_in_ggaa(x1, x2, x3, x4) = partition_in_ggaa(x1, x2)
[] = []
partition_out_ggaa(x1, x2, x3, x4) = partition_out_ggaa(x3, x4)
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)
isLess_in_gg(x1, x2) = isLess_in_gg(x1, x2)
pred(x1) = pred(x1)
zero = zero
isLess_out_gg(x1, x2) = isLess_out_gg
succ(x1) = succ(x1)
U44_gg(x1, x2, x3) = U44_gg(x3)
U45_gg(x1, x2, x3) = U45_gg(x3)
U8_ggaa(x1, x2, x3, x4, x5, x6) = U8_ggaa(x1, x6)
U9_ggaa(x1, x2, x3, x4, x5, x6) = U9_ggaa(x1, x6)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5)
qsort_out_ga(x1, x2) = qsort_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4) = U4_ga(x4)
append_in_gga(x1, x2, x3) = append_in_gga(x1, x2)
append_out_gga(x1, x2, x3) = append_out_gga(x3)
U10_gga(x1, x2, x3, x4, x5) = U10_gga(x1, x5)
PARTITION_IN_GGAA(x1, x2, x3, x4) = PARTITION_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_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)

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

      ↳ CutEliminatorProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                        ↳ UsableRulesProof

                          ↳ PiDP

                            ↳ PiDPToQDPProof

                      ↳ PiDP

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

U5_GGAA(E, R, C, Left1, Right, =_out_ag(X, E)) → U6_GGAA(E, R, C, Left1, Right, X, =_in_ag(X1, C))
PARTITION_IN_GGAA(.(E, R), C, Left, .(E, Right1)) → PARTITION_IN_GGAA(R, C, Left, Right1)
U6_GGAA(E, R, C, Left1, Right, X, =_out_ag(X1, C)) → U7_GGAA(E, R, C, Left1, Right, isLess_in_gg(X, X1))
PARTITION_IN_GGAA(.(E, R), C, .(E, Left1), Right) → U5_GGAA(E, R, C, Left1, Right, =_in_ag(X, E))
U7_GGAA(E, R, C, Left1, Right, isLess_out_gg(X, X1)) → PARTITION_IN_GGAA(R, C, Left1, Right)

The TRS R consists of the following rules:

=_in_ag(X, X) → =_out_ag(X, X)
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)) → U44_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)) → U45_gg(X, Y, isLess_in_gg(X, Y))
U44_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U45_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)
isLess_in_gg(x1, x2) = isLess_in_gg(x1, x2)
pred(x1) = pred(x1)
zero = zero
isLess_out_gg(x1, x2) = isLess_out_gg
succ(x1) = succ(x1)
U44_gg(x1, x2, x3) = U44_gg(x3)
U45_gg(x1, x2, x3) = U45_gg(x3)
PARTITION_IN_GGAA(x1, x2, x3, x4) = PARTITION_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_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)

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

      ↳ CutEliminatorProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                        ↳ UsableRulesProof

                          ↳ PiDP

                            ↳ PiDPToQDPProof

                              ↳ QDP

                                ↳ QDPSizeChangeProof

                      ↳ PiDP

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

U5_GGAA(E, R, C, =_out_ag(X)) → U6_GGAA(E, R, C, X, =_in_ag(C))
PARTITION_IN_GGAA(.(E, R), C) → PARTITION_IN_GGAA(R, C)
U7_GGAA(E, R, C, isLess_out_gg) → PARTITION_IN_GGAA(R, C)
U6_GGAA(E, R, C, X, =_out_ag(X1)) → U7_GGAA(E, R, C, isLess_in_gg(X, X1))
PARTITION_IN_GGAA(.(E, R), C) → U5_GGAA(E, R, C, =_in_ag(E))

The TRS R consists of the following rules:

=_in_ag(X) → =_out_ag(X)
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)) → U44_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U45_gg(isLess_in_gg(X, Y))
U44_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg

The set Q consists of the following terms:

=_in_ag(x0)
isLess_in_gg(x0, x1)
U44_gg(x0)
U45_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:

U6_GGAA(E, R, C, X, =_out_ag(X1)) → U7_GGAA(E, R, C, isLess_in_gg(X, X1))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3
PARTITION_IN_GGAA(.(E, R), C) → U5_GGAA(E, R, C, =_in_ag(E))
The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3
PARTITION_IN_GGAA(.(E, R), C) → PARTITION_IN_GGAA(R, C)
The graph contains the following edges 1 > 1, 2 >= 2
U7_GGAA(E, R, C, isLess_out_gg) → PARTITION_IN_GGAA(R, C)
The graph contains the following edges 2 >= 1, 3 >= 2
U5_GGAA(E, R, C, =_out_ag(X)) → U6_GGAA(E, R, C, X, =_in_ag(C))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ CutEliminatorProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                        ↳ PiDPToQDPProof

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

U1_GA(X, L, R, partition_out_ggaa(L, X, L1, L2)) → QSORT_IN_GA(L2, R2)
U1_GA(X, L, R, partition_out_ggaa(L, X, L1, L2)) → U2_GA(X, L, R, L1, qsort_in_ga(L2, R2))
QSORT_IN_GA(.(X, L), R) → U1_GA(X, L, R, partition_in_ggaa(L, X, L1, L2))
U2_GA(X, L, R, L1, qsort_out_ga(L2, R2)) → QSORT_IN_GA(L1, R1)

The TRS R consists of the following rules:

qsort_in_ga(.(X, L), R) → U1_ga(X, L, R, partition_in_ggaa(L, X, L1, L2))
partition_in_ggaa([], X, [], []) → partition_out_ggaa([], X, [], [])
partition_in_ggaa(.(E, R), C, .(E, Left1), Right) → U5_ggaa(E, R, C, Left1, Right, =_in_ag(X, E))
=_in_ag(X, X) → =_out_ag(X, X)
U5_ggaa(E, R, C, Left1, Right, =_out_ag(X, E)) → U6_ggaa(E, R, C, Left1, Right, X, =_in_ag(X1, C))
U6_ggaa(E, R, C, Left1, Right, X, =_out_ag(X1, C)) → U7_ggaa(E, R, C, Left1, Right, isLess_in_gg(X, X1))
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)) → U44_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)) → U45_gg(X, Y, isLess_in_gg(X, Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U44_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U7_ggaa(E, R, C, Left1, Right, isLess_out_gg(X, X1)) → U8_ggaa(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
partition_in_ggaa(.(E, R), C, Left, .(E, Right1)) → U9_ggaa(E, R, C, Left, Right1, partition_in_ggaa(R, C, Left, Right1))
U9_ggaa(E, R, C, Left, Right1, partition_out_ggaa(R, C, Left, Right1)) → partition_out_ggaa(.(E, R), C, Left, .(E, Right1))
U8_ggaa(E, R, C, Left1, Right, partition_out_ggaa(R, C, Left1, Right)) → partition_out_ggaa(.(E, R), C, .(E, Left1), Right)
U1_ga(X, L, R, partition_out_ggaa(L, X, L1, L2)) → U2_ga(X, L, R, L1, qsort_in_ga(L2, R2))
qsort_in_ga([], []) → qsort_out_ga([], [])
U2_ga(X, L, R, L1, qsort_out_ga(L2, R2)) → U3_ga(X, L, R, R2, qsort_in_ga(L1, R1))
U3_ga(X, L, R, R2, qsort_out_ga(L1, R1)) → U4_ga(X, L, R, append_in_gga(R1, .(X, R2), R))
append_in_gga([], X, X) → append_out_gga([], X, X)
append_in_gga(.(H, X), Y, .(H, Z)) → U10_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
U10_gga(H, X, Y, Z, append_out_gga(X, Y, Z)) → append_out_gga(.(H, X), Y, .(H, Z))
U4_ga(X, L, R, append_out_gga(R1, .(X, R2), R)) → qsort_out_ga(.(X, L), R)

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ CutEliminatorProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ QDPOrderProof

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

U2_GA(X, L1, qsort_out_ga(R2)) → QSORT_IN_GA(L1)
U1_GA(X, partition_out_ggaa(L1, L2)) → U2_GA(X, L1, qsort_in_ga(L2))
QSORT_IN_GA(.(X, L)) → U1_GA(X, partition_in_ggaa(L, X))
U1_GA(X, partition_out_ggaa(L1, L2)) → QSORT_IN_GA(L2)

The TRS R consists of the following rules:

qsort_in_ga(.(X, L)) → U1_ga(X, partition_in_ggaa(L, X))
partition_in_ggaa([], X) → partition_out_ggaa([], [])
partition_in_ggaa(.(E, R), C) → U5_ggaa(E, R, C, =_in_ag(E))
=_in_ag(X) → =_out_ag(X)
U5_ggaa(E, R, C, =_out_ag(X)) → U6_ggaa(E, R, C, X, =_in_ag(C))
U6_ggaa(E, R, C, X, =_out_ag(X1)) → U7_ggaa(E, R, C, isLess_in_gg(X, X1))
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)) → U44_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U45_gg(isLess_in_gg(X, Y))
U45_gg(isLess_out_gg) → isLess_out_gg
U44_gg(isLess_out_gg) → isLess_out_gg
U7_ggaa(E, R, C, isLess_out_gg) → U8_ggaa(E, partition_in_ggaa(R, C))
partition_in_ggaa(.(E, R), C) → U9_ggaa(E, partition_in_ggaa(R, C))
U9_ggaa(E, partition_out_ggaa(Left, Right1)) → partition_out_ggaa(Left, .(E, Right1))
U8_ggaa(E, partition_out_ggaa(Left1, Right)) → partition_out_ggaa(.(E, Left1), Right)
U1_ga(X, partition_out_ggaa(L1, L2)) → U2_ga(X, L1, qsort_in_ga(L2))
qsort_in_ga([]) → qsort_out_ga([])
U2_ga(X, L1, qsort_out_ga(R2)) → U3_ga(X, R2, qsort_in_ga(L1))
U3_ga(X, R2, qsort_out_ga(R1)) → U4_ga(append_in_gga(R1, .(X, R2)))
append_in_gga([], X) → append_out_gga(X)
append_in_gga(.(H, X), Y) → U10_gga(H, append_in_gga(X, Y))
U10_gga(H, append_out_gga(Z)) → append_out_gga(.(H, Z))
U4_ga(append_out_gga(R)) → qsort_out_ga(R)

The set Q consists of the following terms:

qsort_in_ga(x0)
partition_in_ggaa(x0, x1)
=_in_ag(x0)
U5_ggaa(x0, x1, x2, x3)
U6_ggaa(x0, x1, x2, x3, x4)
isLess_in_gg(x0, x1)
U45_gg(x0)
U44_gg(x0)
U7_ggaa(x0, x1, x2, x3)
U9_ggaa(x0, x1)
U8_ggaa(x0, x1)
U1_ga(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
append_in_gga(x0, x1)
U10_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.

QSORT_IN_GA(.(X, L)) → U1_GA(X, partition_in_ggaa(L, X))

The remaining pairs can at least be oriented weakly.

U2_GA(X, L1, qsort_out_ga(R2)) → QSORT_IN_GA(L1)
U1_GA(X, partition_out_ggaa(L1, L2)) → U2_GA(X, L1, qsort_in_ga(L2))
U1_GA(X, partition_out_ggaa(L1, L2)) → QSORT_IN_GA(L2)

Used ordering: Polynomial interpretation [25]:

POL(.(x₁, x₂)) = 1 + x₂
POL(=_in_ag(x₁)) = 0
POL(=_out_ag(x₁)) = 0
POL(QSORT_IN_GA(x₁)) = x₁
POL(U10_gga(x₁, x₂)) = x₂
POL(U1_GA(x₁, x₂)) = 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(U44_gg(x₁)) = 0
POL(U45_gg(x₁)) = 0
POL(U4_ga(x₁)) = 0
POL(U5_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₂
POL(U6_ggaa(x₁, x₂, x₃, x₄, x₅)) = 1 + x₂
POL(U7_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₂
POL(U8_ggaa(x₁, x₂)) = 1 + x₂
POL(U9_ggaa(x₁, x₂)) = 1 + x₂
POL([]) = 0
POL(append_in_gga(x₁, x₂)) = 1 + x₁
POL(append_out_gga(x₁)) = 0
POL(isLess_in_gg(x₁, x₂)) = 0
POL(isLess_out_gg) = 0
POL(partition_in_ggaa(x₁, x₂)) = x₁
POL(partition_out_ggaa(x₁, x₂)) = x₁ + x₂
POL(pred(x₁)) = 0
POL(qsort_in_ga(x₁)) = 0
POL(qsort_out_ga(x₁)) = 0
POL(succ(x₁)) = 0
POL(zero) = 0

The following usable rules [17] were oriented:

partition_in_ggaa([], X) → partition_out_ggaa([], [])
partition_in_ggaa(.(E, R), C) → U9_ggaa(E, partition_in_ggaa(R, C))
partition_in_ggaa(.(E, R), C) → U5_ggaa(E, R, C, =_in_ag(E))
U9_ggaa(E, partition_out_ggaa(Left, Right1)) → partition_out_ggaa(Left, .(E, Right1))
U8_ggaa(E, partition_out_ggaa(Left1, Right)) → partition_out_ggaa(.(E, Left1), Right)
U5_ggaa(E, R, C, =_out_ag(X)) → U6_ggaa(E, R, C, X, =_in_ag(C))
U6_ggaa(E, R, C, X, =_out_ag(X1)) → U7_ggaa(E, R, C, isLess_in_gg(X, X1))
U7_ggaa(E, R, C, isLess_out_gg) → U8_ggaa(E, partition_in_ggaa(R, C))

↳ Prolog

  ↳ PredefinedPredicateTransformerProof

    ↳ Prolog

      ↳ CutEliminatorProof

        ↳ Prolog

          ↳ PrologToPiTRSProof

            ↳ PiTRS

              ↳ DependencyPairsProof

                ↳ PiDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                      ↳ PiDP

                        ↳ PiDPToQDPProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

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

U2_GA(X, L1, qsort_out_ga(R2)) → QSORT_IN_GA(L1)
U1_GA(X, partition_out_ggaa(L1, L2)) → U2_GA(X, L1, qsort_in_ga(L2))
U1_GA(X, partition_out_ggaa(L1, L2)) → QSORT_IN_GA(L2)

The TRS R consists of the following rules:

qsort_in_ga(.(X, L)) → U1_ga(X, partition_in_ggaa(L, X))
partition_in_ggaa([], X) → partition_out_ggaa([], [])
partition_in_ggaa(.(E, R), C) → U5_ggaa(E, R, C, =_in_ag(E))
=_in_ag(X) → =_out_ag(X)
U5_ggaa(E, R, C, =_out_ag(X)) → U6_ggaa(E, R, C, X, =_in_ag(C))
U6_ggaa(E, R, C, X, =_out_ag(X1)) → U7_ggaa(E, R, C, isLess_in_gg(X, X1))
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)) → U44_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U45_gg(isLess_in_gg(X, Y))
U45_gg(isLess_out_gg) → isLess_out_gg
U44_gg(isLess_out_gg) → isLess_out_gg
U7_ggaa(E, R, C, isLess_out_gg) → U8_ggaa(E, partition_in_ggaa(R, C))
partition_in_ggaa(.(E, R), C) → U9_ggaa(E, partition_in_ggaa(R, C))
U9_ggaa(E, partition_out_ggaa(Left, Right1)) → partition_out_ggaa(Left, .(E, Right1))
U8_ggaa(E, partition_out_ggaa(Left1, Right)) → partition_out_ggaa(.(E, Left1), Right)
U1_ga(X, partition_out_ggaa(L1, L2)) → U2_ga(X, L1, qsort_in_ga(L2))
qsort_in_ga([]) → qsort_out_ga([])
U2_ga(X, L1, qsort_out_ga(R2)) → U3_ga(X, R2, qsort_in_ga(L1))
U3_ga(X, R2, qsort_out_ga(R1)) → U4_ga(append_in_gga(R1, .(X, R2)))
append_in_gga([], X) → append_out_gga(X)
append_in_gga(.(H, X), Y) → U10_gga(H, append_in_gga(X, Y))
U10_gga(H, append_out_gga(Z)) → append_out_gga(.(H, Z))
U4_ga(append_out_gga(R)) → qsort_out_ga(R)

The set Q consists of the following terms:

qsort_in_ga(x0)
partition_in_ggaa(x0, x1)
=_in_ag(x0)
U5_ggaa(x0, x1, x2, x3)
U6_ggaa(x0, x1, x2, x3, x4)
isLess_in_gg(x0, x1)
U45_gg(x0)
U44_gg(x0)
U7_ggaa(x0, x1, x2, x3)
U9_ggaa(x0, x1)
U8_ggaa(x0, x1)
U1_ga(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
append_in_gga(x0, x1)
U10_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 3 less nodes.