Left Termination of the query pattern
qsort(g,a)
w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 CutEliminatorProof (⇒, 0 ms)
↳2 Prolog
↳3 UndefinedPredicateHandlerProof (⇒, 0 ms)
↳4 Prolog
↳5 PrologToPiTRSProof (⇒, 0 ms)
↳6 PiTRS
↳7 DependencyPairsProof (⇔, 78 ms)
↳8 PiDP
↳9 DependencyGraphProof (⇔, 0 ms)
↳10 AND
    ↳11 PiDP
        ↳12 UsableRulesProof (⇔, 0 ms)
        ↳13 PiDP
        ↳14 PiDPToQDPProof (⇒, 0 ms)
        ↳15 QDP
        ↳16 QDPSizeChangeProof (⇔, 0 ms)
        ↳17 YES
    ↳18 PiDP
        ↳19 UsableRulesProof (⇔, 0 ms)
        ↳20 PiDP
        ↳21 PiDPToQDPProof (⇒, 0 ms)
        ↳22 QDP
        ↳23 QDPSizeChangeProof (⇔, 0 ms)
        ↳24 YES
    ↳25 PiDP
        ↳26 PiDPToQDPProof (⇒, 0 ms)
        ↳27 QDP
        ↳28 QDPOrderProof (⇔, 115 ms)
        ↳29 QDP
        ↳30 DependencyGraphProof (⇔, 0 ms)
        ↳31 TRUE

(0) Obligation:

Clauses:

qsort(.(X, L), R) :- ','(partition(L, X, L1, L2), ','(qsort(L2, R2), ','(qsort(L1, R1), append(R1, .(X, R2), R)))).
qsort([], []).
partition([], X1, [], []).
partition(.(E, R), C, .(E, Left1), Right) :- ','(<(E, C), ','(!, 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).

Query: qsort(g,a)

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

qsort(.(X, L), R) :- ','(partition(L, X, L1, L2), ','(qsort(L2, R2), ','(qsort(L1, R1), append(R1, .(X, R2), R)))).
qsort([], []).
partition([], X1, [], []).
partition(.(E, R), C, .(E, Left1), Right) :- ','(<(E, C), 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).

Query: qsort(g,a)

(3) UndefinedPredicateHandlerProof (SOUND transformation)

Added facts for all undefined predicates [PROLOG].

(4) Obligation:

Clauses:

qsort(.(X, L), R) :- ','(partition(L, X, L1, L2), ','(qsort(L2, R2), ','(qsort(L1, R1), append(R1, .(X, R2), R)))).
qsort([], []).
partition([], X1, [], []).
partition(.(E, R), C, .(E, Left1), Right) :- ','(<(E, C), 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).
<(X0, X1).

Query: qsort(g,a)

(5) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. 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)
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([], X1, [], []) → partition_out_ggaa([], X1, [], [])
partition_in_ggaa(.(E, R), C, .(E, Left1), Right) → U5_ggaa(E, R, C, Left1, Right, <_in_gg(E, C))
<_in_gg(X0, X1) → <_out_gg(X0, X1)
U5_ggaa(E, R, C, Left1, Right, <_out_gg(E, C)) → U6_ggaa(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
partition_in_ggaa(.(E, R), C, Left, .(E, Right1)) → U7_ggaa(E, R, C, Left, Right1, partition_in_ggaa(R, C, Left, Right1))
U7_ggaa(E, R, C, Left, Right1, partition_out_ggaa(R, C, Left, Right1)) → partition_out_ggaa(.(E, R), C, Left, .(E, Right1))
U6_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)) → U8_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
U8_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_gg(x1, x2)  =  <_in_gg(x1, x2)
<_out_gg(x1, x2)  =  <_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_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)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(6) Obligation:

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([], X1, [], []) → partition_out_ggaa([], X1, [], [])
partition_in_ggaa(.(E, R), C, .(E, Left1), Right) → U5_ggaa(E, R, C, Left1, Right, <_in_gg(E, C))
<_in_gg(X0, X1) → <_out_gg(X0, X1)
U5_ggaa(E, R, C, Left1, Right, <_out_gg(E, C)) → U6_ggaa(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
partition_in_ggaa(.(E, R), C, Left, .(E, Right1)) → U7_ggaa(E, R, C, Left, Right1, partition_in_ggaa(R, C, Left, Right1))
U7_ggaa(E, R, C, Left, Right1, partition_out_ggaa(R, C, Left, Right1)) → partition_out_ggaa(.(E, R), C, Left, .(E, Right1))
U6_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)) → U8_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
U8_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_gg(x1, x2)  =  <_in_gg(x1, x2)
<_out_gg(x1, x2)  =  <_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_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)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x5)

(7) DependencyPairsProof (EQUIVALENT transformation)

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

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_gg(E, C))
PARTITION_IN_GGAA(.(E, R), C, .(E, Left1), Right) → <_IN_GG(E, C)
U5_GGAA(E, R, C, Left1, Right, <_out_gg(E, C)) → U6_GGAA(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
U5_GGAA(E, R, C, Left1, Right, <_out_gg(E, C)) → PARTITION_IN_GGAA(R, C, Left1, Right)
PARTITION_IN_GGAA(.(E, R), C, Left, .(E, Right1)) → U7_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)) → U8_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([], X1, [], []) → partition_out_ggaa([], X1, [], [])
partition_in_ggaa(.(E, R), C, .(E, Left1), Right) → U5_ggaa(E, R, C, Left1, Right, <_in_gg(E, C))
<_in_gg(X0, X1) → <_out_gg(X0, X1)
U5_ggaa(E, R, C, Left1, Right, <_out_gg(E, C)) → U6_ggaa(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
partition_in_ggaa(.(E, R), C, Left, .(E, Right1)) → U7_ggaa(E, R, C, Left, Right1, partition_in_ggaa(R, C, Left, Right1))
U7_ggaa(E, R, C, Left, Right1, partition_out_ggaa(R, C, Left, Right1)) → partition_out_ggaa(.(E, R), C, Left, .(E, Right1))
U6_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)) → U8_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
U8_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_gg(x1, x2)  =  <_in_gg(x1, x2)
<_out_gg(x1, x2)  =  <_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_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)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x5)
QSORT_IN_GA(x1, x2)  =  QSORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
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)
<_IN_GG(x1, x2)  =  <_IN_GG(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x6)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x6)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x4, x5)
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)
U8_GGA(x1, x2, x3, x4, x5)  =  U8_GGA(x1, x5)

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

(8) Obligation:

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_gg(E, C))
PARTITION_IN_GGAA(.(E, R), C, .(E, Left1), Right) → <_IN_GG(E, C)
U5_GGAA(E, R, C, Left1, Right, <_out_gg(E, C)) → U6_GGAA(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
U5_GGAA(E, R, C, Left1, Right, <_out_gg(E, C)) → PARTITION_IN_GGAA(R, C, Left1, Right)
PARTITION_IN_GGAA(.(E, R), C, Left, .(E, Right1)) → U7_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)) → U8_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([], X1, [], []) → partition_out_ggaa([], X1, [], [])
partition_in_ggaa(.(E, R), C, .(E, Left1), Right) → U5_ggaa(E, R, C, Left1, Right, <_in_gg(E, C))
<_in_gg(X0, X1) → <_out_gg(X0, X1)
U5_ggaa(E, R, C, Left1, Right, <_out_gg(E, C)) → U6_ggaa(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
partition_in_ggaa(.(E, R), C, Left, .(E, Right1)) → U7_ggaa(E, R, C, Left, Right1, partition_in_ggaa(R, C, Left, Right1))
U7_ggaa(E, R, C, Left, Right1, partition_out_ggaa(R, C, Left, Right1)) → partition_out_ggaa(.(E, R), C, Left, .(E, Right1))
U6_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)) → U8_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
U8_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_gg(x1, x2)  =  <_in_gg(x1, x2)
<_out_gg(x1, x2)  =  <_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_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)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x5)
QSORT_IN_GA(x1, x2)  =  QSORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
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)
<_IN_GG(x1, x2)  =  <_IN_GG(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x6)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x6)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x4, x5)
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)
U8_GGA(x1, x2, x3, x4, x5)  =  U8_GGA(x1, x5)

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 8 less nodes.

(10) Complex Obligation (AND)

(11) Obligation:

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([], X1, [], []) → partition_out_ggaa([], X1, [], [])
partition_in_ggaa(.(E, R), C, .(E, Left1), Right) → U5_ggaa(E, R, C, Left1, Right, <_in_gg(E, C))
<_in_gg(X0, X1) → <_out_gg(X0, X1)
U5_ggaa(E, R, C, Left1, Right, <_out_gg(E, C)) → U6_ggaa(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
partition_in_ggaa(.(E, R), C, Left, .(E, Right1)) → U7_ggaa(E, R, C, Left, Right1, partition_in_ggaa(R, C, Left, Right1))
U7_ggaa(E, R, C, Left, Right1, partition_out_ggaa(R, C, Left, Right1)) → partition_out_ggaa(.(E, R), C, Left, .(E, Right1))
U6_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)) → U8_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
U8_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_gg(x1, x2)  =  <_in_gg(x1, x2)
<_out_gg(x1, x2)  =  <_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_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)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x5)
APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)

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

(12) UsableRulesProof (EQUIVALENT transformation)

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

(13) Obligation:

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

(14) PiDPToQDPProof (SOUND transformation)

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

(15) Obligation:

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.

(16) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(17) YES

(18) Obligation:

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

U5_GGAA(E, R, C, Left1, Right, <_out_gg(E, C)) → PARTITION_IN_GGAA(R, C, Left1, Right)
PARTITION_IN_GGAA(.(E, R), C, .(E, Left1), Right) → U5_GGAA(E, R, C, Left1, Right, <_in_gg(E, C))
PARTITION_IN_GGAA(.(E, R), C, Left, .(E, Right1)) → PARTITION_IN_GGAA(R, C, Left, Right1)

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([], X1, [], []) → partition_out_ggaa([], X1, [], [])
partition_in_ggaa(.(E, R), C, .(E, Left1), Right) → U5_ggaa(E, R, C, Left1, Right, <_in_gg(E, C))
<_in_gg(X0, X1) → <_out_gg(X0, X1)
U5_ggaa(E, R, C, Left1, Right, <_out_gg(E, C)) → U6_ggaa(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
partition_in_ggaa(.(E, R), C, Left, .(E, Right1)) → U7_ggaa(E, R, C, Left, Right1, partition_in_ggaa(R, C, Left, Right1))
U7_ggaa(E, R, C, Left, Right1, partition_out_ggaa(R, C, Left, Right1)) → partition_out_ggaa(.(E, R), C, Left, .(E, Right1))
U6_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)) → U8_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
U8_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_gg(x1, x2)  =  <_in_gg(x1, x2)
<_out_gg(x1, x2)  =  <_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_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)
U8_gga(x1, x2, x3, x4, x5)  =  U8_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)

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

(19) UsableRulesProof (EQUIVALENT transformation)

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

(20) Obligation:

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

U5_GGAA(E, R, C, Left1, Right, <_out_gg(E, C)) → PARTITION_IN_GGAA(R, C, Left1, Right)
PARTITION_IN_GGAA(.(E, R), C, .(E, Left1), Right) → U5_GGAA(E, R, C, Left1, Right, <_in_gg(E, C))
PARTITION_IN_GGAA(.(E, R), C, Left, .(E, Right1)) → PARTITION_IN_GGAA(R, C, Left, Right1)

The TRS R consists of the following rules:

<_in_gg(X0, X1) → <_out_gg(X0, X1)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
<_in_gg(x1, x2)  =  <_in_gg(x1, x2)
<_out_gg(x1, x2)  =  <_out_gg
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)

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

(21) PiDPToQDPProof (SOUND transformation)

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

(22) Obligation:

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

U5_GGAA(E, R, C, <_out_gg) → PARTITION_IN_GGAA(R, C)
PARTITION_IN_GGAA(.(E, R), C) → U5_GGAA(E, R, C, <_in_gg(E, C))
PARTITION_IN_GGAA(.(E, R), C) → PARTITION_IN_GGAA(R, C)

The TRS R consists of the following rules:

<_in_gg(X0, X1) → <_out_gg

The set Q consists of the following terms:

<_in_gg(x0, x1)

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

(23) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • PARTITION_IN_GGAA(.(E, R), C) → U5_GGAA(E, R, C, <_in_gg(E, C))
    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

  • U5_GGAA(E, R, C, <_out_gg) → PARTITION_IN_GGAA(R, C)
    The graph contains the following edges 2 >= 1, 3 >= 2

(24) YES

(25) Obligation:

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

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

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([], X1, [], []) → partition_out_ggaa([], X1, [], [])
partition_in_ggaa(.(E, R), C, .(E, Left1), Right) → U5_ggaa(E, R, C, Left1, Right, <_in_gg(E, C))
<_in_gg(X0, X1) → <_out_gg(X0, X1)
U5_ggaa(E, R, C, Left1, Right, <_out_gg(E, C)) → U6_ggaa(E, R, C, Left1, Right, partition_in_ggaa(R, C, Left1, Right))
partition_in_ggaa(.(E, R), C, Left, .(E, Right1)) → U7_ggaa(E, R, C, Left, Right1, partition_in_ggaa(R, C, Left, Right1))
U7_ggaa(E, R, C, Left, Right1, partition_out_ggaa(R, C, Left, Right1)) → partition_out_ggaa(.(E, R), C, Left, .(E, Right1))
U6_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)) → U8_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
U8_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_gg(x1, x2)  =  <_in_gg(x1, x2)
<_out_gg(x1, x2)  =  <_out_gg
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_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)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x5)
QSORT_IN_GA(x1, x2)  =  QSORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x4, x5)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

U1_GA(X, partition_out_ggaa(L1, L2)) → U2_GA(X, L1, qsort_in_ga(L2))
U2_GA(X, L1, qsort_out_ga(R2)) → QSORT_IN_GA(L1)
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([], X1) → partition_out_ggaa([], [])
partition_in_ggaa(.(E, R), C) → U5_ggaa(E, R, C, <_in_gg(E, C))
<_in_gg(X0, X1) → <_out_gg
U5_ggaa(E, R, C, <_out_gg) → U6_ggaa(E, partition_in_ggaa(R, C))
partition_in_ggaa(.(E, R), C) → U7_ggaa(E, partition_in_ggaa(R, C))
U7_ggaa(E, partition_out_ggaa(Left, Right1)) → partition_out_ggaa(Left, .(E, Right1))
U6_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) → U8_gga(H, append_in_gga(X, Y))
U8_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_gg(x0, x1)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1)
U6_ggaa(x0, x1)
U1_ga(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
append_in_gga(x0, x1)
U8_gga(x0, x1)
U4_ga(x0)

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

(28) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


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 remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x2   
POL(<_in_gg(x1, x2)) = 1   
POL(<_out_gg) = 1   
POL(QSORT_IN_GA(x1)) = 1 + x1   
POL(U1_GA(x1, x2)) = 1 + x2   
POL(U1_ga(x1, x2)) = 0   
POL(U2_GA(x1, x2, x3)) = 1 + x2   
POL(U2_ga(x1, x2, x3)) = 0   
POL(U3_ga(x1, x2, x3)) = 0   
POL(U4_ga(x1)) = 0   
POL(U5_ggaa(x1, x2, x3, x4)) = 1 + x2 + x4   
POL(U6_ggaa(x1, x2)) = 1 + x2   
POL(U7_ggaa(x1, x2)) = 1 + x2   
POL(U8_gga(x1, x2)) = 0   
POL([]) = 0   
POL(append_in_gga(x1, x2)) = 0   
POL(append_out_gga(x1)) = 0   
POL(partition_in_ggaa(x1, x2)) = 1 + x1   
POL(partition_out_ggaa(x1, x2)) = 1 + x1 + x2   
POL(qsort_in_ga(x1)) = 0   
POL(qsort_out_ga(x1)) = 0   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

partition_in_ggaa([], X1) → partition_out_ggaa([], [])
partition_in_ggaa(.(E, R), C) → U5_ggaa(E, R, C, <_in_gg(E, C))
partition_in_ggaa(.(E, R), C) → U7_ggaa(E, partition_in_ggaa(R, C))
<_in_gg(X0, X1) → <_out_gg
U5_ggaa(E, R, C, <_out_gg) → U6_ggaa(E, partition_in_ggaa(R, C))
U6_ggaa(E, partition_out_ggaa(Left1, Right)) → partition_out_ggaa(.(E, Left1), Right)
U7_ggaa(E, partition_out_ggaa(Left, Right1)) → partition_out_ggaa(Left, .(E, Right1))

(29) Obligation:

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

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

The TRS R consists of the following rules:

qsort_in_ga(.(X, L)) → U1_ga(X, partition_in_ggaa(L, X))
partition_in_ggaa([], X1) → partition_out_ggaa([], [])
partition_in_ggaa(.(E, R), C) → U5_ggaa(E, R, C, <_in_gg(E, C))
<_in_gg(X0, X1) → <_out_gg
U5_ggaa(E, R, C, <_out_gg) → U6_ggaa(E, partition_in_ggaa(R, C))
partition_in_ggaa(.(E, R), C) → U7_ggaa(E, partition_in_ggaa(R, C))
U7_ggaa(E, partition_out_ggaa(Left, Right1)) → partition_out_ggaa(Left, .(E, Right1))
U6_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) → U8_gga(H, append_in_gga(X, Y))
U8_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_gg(x0, x1)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1)
U6_ggaa(x0, x1)
U1_ga(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
append_in_gga(x0, x1)
U8_gga(x0, x1)
U4_ga(x0)

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

(30) DependencyGraphProof (EQUIVALENT transformation)

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

(31) TRUE