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

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇒, 82 ms)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇒, 100 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇒, 0 ms)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔, 0 ms)
        ↳13 YES
    ↳14 PiDP
        ↳15 UsableRulesProof (⇔, 0 ms)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇒, 0 ms)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔, 0 ms)
        ↳20 YES
    ↳21 PiDP
        ↳22 PiDPToQDPProof (⇒, 0 ms)
        ↳23 QDP
        ↳24 Rewriting (⇔, 0 ms)
        ↳25 QDP
        ↳26 UsableRulesProof (⇔, 0 ms)
        ↳27 QDP
        ↳28 QReductionProof (⇔, 0 ms)
        ↳29 QDP
        ↳30 Rewriting (⇔, 0 ms)
        ↳31 QDP
        ↳32 UsableRulesProof (⇔, 0 ms)
        ↳33 QDP
        ↳34 QReductionProof (⇔, 0 ms)
        ↳35 QDP
        ↳36 Instantiation (⇔, 0 ms)
        ↳37 QDP
        ↳38 QDPOrderProof (⇔, 79 ms)
        ↳39 QDP
        ↳40 DependencyGraphProof (⇔, 0 ms)
        ↳41 TRUE

(0) Obligation:

Clauses:

perm(Xs, .(X, Ys)) :- ','(app(X1s, .(X, X2s), Xs), ','(app(X1s, X2s, Zs), perm(Zs, Ys))).
perm([], []).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
app([], Ys, Ys).

Query: perm(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

appA(.(X1, X2), X3, X4, .(X1, X5)) :- appA(X2, X3, X4, X5).
appE(.(X1, X2), X3, .(X1, X4)) :- appE(X2, X3, X4).
permB(.(X1, X2), .(X3, X4)) :- appA(X5, X3, X6, X2).
permB(.(X1, X2), .(X3, X4)) :- ','(appcA(X5, X3, X6, X2), appE(X5, X6, X7)).
permB(.(X1, X2), .(X3, X4)) :- ','(appcA(X5, X3, X6, X2), ','(appcC(X1, X5, X6, X7), permB(X7, X4))).
permB(.(X1, X2), .(X1, X3)) :- ','(appcD(X2, X4), permB(X4, X3)).

Clauses:

appcA(.(X1, X2), X3, X4, .(X1, X5)) :- appcA(X2, X3, X4, X5).
appcA([], X1, X2, .(X1, X2)).
permcB(.(X1, X2), .(X3, X4)) :- ','(appcA(X5, X3, X6, X2), ','(appcC(X1, X5, X6, X7), permcB(X7, X4))).
permcB(.(X1, X2), .(X1, X3)) :- ','(appcD(X2, X4), permcB(X4, X3)).
permcB([], []).
permcB([], []).
appcE(.(X1, X2), X3, .(X1, X4)) :- appcE(X2, X3, X4).
appcE([], X1, X1).
appcC(X1, X2, X3, .(X1, X4)) :- appcE(X2, X3, X4).
appcD(X1, X1).

Afs:

permB(x1, x2)  =  permB(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
permB_in: (b,f)
appA_in: (f,f,f,b)
appcA_in: (f,f,f,b)
appE_in: (b,b,f)
appcC_in: (b,b,b,f)
appcE_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

PERMB_IN_GA(.(X1, X2), .(X3, X4)) → U3_GA(X1, X2, X3, X4, appA_in_aaag(X5, X3, X6, X2))
PERMB_IN_GA(.(X1, X2), .(X3, X4)) → APPA_IN_AAAG(X5, X3, X6, X2)
APPA_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) → U1_AAAG(X1, X2, X3, X4, X5, appA_in_aaag(X2, X3, X4, X5))
APPA_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) → APPA_IN_AAAG(X2, X3, X4, X5)
PERMB_IN_GA(.(X1, X2), .(X3, X4)) → U4_GA(X1, X2, X3, X4, appcA_in_aaag(X5, X3, X6, X2))
U4_GA(X1, X2, X3, X4, appcA_out_aaag(X5, X3, X6, X2)) → U5_GA(X1, X2, X3, X4, appE_in_gga(X5, X6, X7))
U4_GA(X1, X2, X3, X4, appcA_out_aaag(X5, X3, X6, X2)) → APPE_IN_GGA(X5, X6, X7)
APPE_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U2_GGA(X1, X2, X3, X4, appE_in_gga(X2, X3, X4))
APPE_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPE_IN_GGA(X2, X3, X4)
U4_GA(X1, X2, X3, X4, appcA_out_aaag(X5, X3, X6, X2)) → U6_GA(X1, X2, X3, X4, appcC_in_ggga(X1, X5, X6, X7))
U6_GA(X1, X2, X3, X4, appcC_out_ggga(X1, X5, X6, X7)) → U7_GA(X1, X2, X3, X4, permB_in_ga(X7, X4))
U6_GA(X1, X2, X3, X4, appcC_out_ggga(X1, X5, X6, X7)) → PERMB_IN_GA(X7, X4)
PERMB_IN_GA(.(X1, X2), .(X1, X3)) → U8_GA(X1, X2, X3, appcD_in_ga(X2, X4))
U8_GA(X1, X2, X3, appcD_out_ga(X2, X4)) → U9_GA(X1, X2, X3, permB_in_ga(X4, X3))
U8_GA(X1, X2, X3, appcD_out_ga(X2, X4)) → PERMB_IN_GA(X4, X3)

The TRS R consists of the following rules:

appcA_in_aaag(.(X1, X2), X3, X4, .(X1, X5)) → U11_aaag(X1, X2, X3, X4, X5, appcA_in_aaag(X2, X3, X4, X5))
appcA_in_aaag([], X1, X2, .(X1, X2)) → appcA_out_aaag([], X1, X2, .(X1, X2))
U11_aaag(X1, X2, X3, X4, X5, appcA_out_aaag(X2, X3, X4, X5)) → appcA_out_aaag(.(X1, X2), X3, X4, .(X1, X5))
appcC_in_ggga(X1, X2, X3, .(X1, X4)) → U18_ggga(X1, X2, X3, X4, appcE_in_gga(X2, X3, X4))
appcE_in_gga(.(X1, X2), X3, .(X1, X4)) → U17_gga(X1, X2, X3, X4, appcE_in_gga(X2, X3, X4))
appcE_in_gga([], X1, X1) → appcE_out_gga([], X1, X1)
U17_gga(X1, X2, X3, X4, appcE_out_gga(X2, X3, X4)) → appcE_out_gga(.(X1, X2), X3, .(X1, X4))
U18_ggga(X1, X2, X3, X4, appcE_out_gga(X2, X3, X4)) → appcC_out_ggga(X1, X2, X3, .(X1, X4))
appcD_in_ga(X1, X1) → appcD_out_ga(X1, X1)

The argument filtering Pi contains the following mapping:
permB_in_ga(x1, x2)  =  permB_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
appA_in_aaag(x1, x2, x3, x4)  =  appA_in_aaag(x4)
appcA_in_aaag(x1, x2, x3, x4)  =  appcA_in_aaag(x4)
U11_aaag(x1, x2, x3, x4, x5, x6)  =  U11_aaag(x1, x5, x6)
appcA_out_aaag(x1, x2, x3, x4)  =  appcA_out_aaag(x1, x2, x3, x4)
appE_in_gga(x1, x2, x3)  =  appE_in_gga(x1, x2)
appcC_in_ggga(x1, x2, x3, x4)  =  appcC_in_ggga(x1, x2, x3)
U18_ggga(x1, x2, x3, x4, x5)  =  U18_ggga(x1, x2, x3, x5)
appcE_in_gga(x1, x2, x3)  =  appcE_in_gga(x1, x2)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
[]  =  []
appcE_out_gga(x1, x2, x3)  =  appcE_out_gga(x1, x2, x3)
appcC_out_ggga(x1, x2, x3, x4)  =  appcC_out_ggga(x1, x2, x3, x4)
appcD_in_ga(x1, x2)  =  appcD_in_ga(x1)
appcD_out_ga(x1, x2)  =  appcD_out_ga(x1, x2)
PERMB_IN_GA(x1, x2)  =  PERMB_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x5)
APPA_IN_AAAG(x1, x2, x3, x4)  =  APPA_IN_AAAG(x4)
U1_AAAG(x1, x2, x3, x4, x5, x6)  =  U1_AAAG(x1, x5, x6)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x5)
APPE_IN_GGA(x1, x2, x3)  =  APPE_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x5)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x5)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x2, x4)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x1, x2, x4)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

PERMB_IN_GA(.(X1, X2), .(X3, X4)) → U3_GA(X1, X2, X3, X4, appA_in_aaag(X5, X3, X6, X2))
PERMB_IN_GA(.(X1, X2), .(X3, X4)) → APPA_IN_AAAG(X5, X3, X6, X2)
APPA_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) → U1_AAAG(X1, X2, X3, X4, X5, appA_in_aaag(X2, X3, X4, X5))
APPA_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) → APPA_IN_AAAG(X2, X3, X4, X5)
PERMB_IN_GA(.(X1, X2), .(X3, X4)) → U4_GA(X1, X2, X3, X4, appcA_in_aaag(X5, X3, X6, X2))
U4_GA(X1, X2, X3, X4, appcA_out_aaag(X5, X3, X6, X2)) → U5_GA(X1, X2, X3, X4, appE_in_gga(X5, X6, X7))
U4_GA(X1, X2, X3, X4, appcA_out_aaag(X5, X3, X6, X2)) → APPE_IN_GGA(X5, X6, X7)
APPE_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U2_GGA(X1, X2, X3, X4, appE_in_gga(X2, X3, X4))
APPE_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPE_IN_GGA(X2, X3, X4)
U4_GA(X1, X2, X3, X4, appcA_out_aaag(X5, X3, X6, X2)) → U6_GA(X1, X2, X3, X4, appcC_in_ggga(X1, X5, X6, X7))
U6_GA(X1, X2, X3, X4, appcC_out_ggga(X1, X5, X6, X7)) → U7_GA(X1, X2, X3, X4, permB_in_ga(X7, X4))
U6_GA(X1, X2, X3, X4, appcC_out_ggga(X1, X5, X6, X7)) → PERMB_IN_GA(X7, X4)
PERMB_IN_GA(.(X1, X2), .(X1, X3)) → U8_GA(X1, X2, X3, appcD_in_ga(X2, X4))
U8_GA(X1, X2, X3, appcD_out_ga(X2, X4)) → U9_GA(X1, X2, X3, permB_in_ga(X4, X3))
U8_GA(X1, X2, X3, appcD_out_ga(X2, X4)) → PERMB_IN_GA(X4, X3)

The TRS R consists of the following rules:

appcA_in_aaag(.(X1, X2), X3, X4, .(X1, X5)) → U11_aaag(X1, X2, X3, X4, X5, appcA_in_aaag(X2, X3, X4, X5))
appcA_in_aaag([], X1, X2, .(X1, X2)) → appcA_out_aaag([], X1, X2, .(X1, X2))
U11_aaag(X1, X2, X3, X4, X5, appcA_out_aaag(X2, X3, X4, X5)) → appcA_out_aaag(.(X1, X2), X3, X4, .(X1, X5))
appcC_in_ggga(X1, X2, X3, .(X1, X4)) → U18_ggga(X1, X2, X3, X4, appcE_in_gga(X2, X3, X4))
appcE_in_gga(.(X1, X2), X3, .(X1, X4)) → U17_gga(X1, X2, X3, X4, appcE_in_gga(X2, X3, X4))
appcE_in_gga([], X1, X1) → appcE_out_gga([], X1, X1)
U17_gga(X1, X2, X3, X4, appcE_out_gga(X2, X3, X4)) → appcE_out_gga(.(X1, X2), X3, .(X1, X4))
U18_ggga(X1, X2, X3, X4, appcE_out_gga(X2, X3, X4)) → appcC_out_ggga(X1, X2, X3, .(X1, X4))
appcD_in_ga(X1, X1) → appcD_out_ga(X1, X1)

The argument filtering Pi contains the following mapping:
permB_in_ga(x1, x2)  =  permB_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
appA_in_aaag(x1, x2, x3, x4)  =  appA_in_aaag(x4)
appcA_in_aaag(x1, x2, x3, x4)  =  appcA_in_aaag(x4)
U11_aaag(x1, x2, x3, x4, x5, x6)  =  U11_aaag(x1, x5, x6)
appcA_out_aaag(x1, x2, x3, x4)  =  appcA_out_aaag(x1, x2, x3, x4)
appE_in_gga(x1, x2, x3)  =  appE_in_gga(x1, x2)
appcC_in_ggga(x1, x2, x3, x4)  =  appcC_in_ggga(x1, x2, x3)
U18_ggga(x1, x2, x3, x4, x5)  =  U18_ggga(x1, x2, x3, x5)
appcE_in_gga(x1, x2, x3)  =  appcE_in_gga(x1, x2)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
[]  =  []
appcE_out_gga(x1, x2, x3)  =  appcE_out_gga(x1, x2, x3)
appcC_out_ggga(x1, x2, x3, x4)  =  appcC_out_ggga(x1, x2, x3, x4)
appcD_in_ga(x1, x2)  =  appcD_in_ga(x1)
appcD_out_ga(x1, x2)  =  appcD_out_ga(x1, x2)
PERMB_IN_GA(x1, x2)  =  PERMB_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x5)
APPA_IN_AAAG(x1, x2, x3, x4)  =  APPA_IN_AAAG(x4)
U1_AAAG(x1, x2, x3, x4, x5, x6)  =  U1_AAAG(x1, x5, x6)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x5)
APPE_IN_GGA(x1, x2, x3)  =  APPE_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x5)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x5)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x2, x4)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x1, x2, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Complex Obligation (AND)

(7) Obligation:

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

APPE_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPE_IN_GGA(X2, X3, X4)

The TRS R consists of the following rules:

appcA_in_aaag(.(X1, X2), X3, X4, .(X1, X5)) → U11_aaag(X1, X2, X3, X4, X5, appcA_in_aaag(X2, X3, X4, X5))
appcA_in_aaag([], X1, X2, .(X1, X2)) → appcA_out_aaag([], X1, X2, .(X1, X2))
U11_aaag(X1, X2, X3, X4, X5, appcA_out_aaag(X2, X3, X4, X5)) → appcA_out_aaag(.(X1, X2), X3, X4, .(X1, X5))
appcC_in_ggga(X1, X2, X3, .(X1, X4)) → U18_ggga(X1, X2, X3, X4, appcE_in_gga(X2, X3, X4))
appcE_in_gga(.(X1, X2), X3, .(X1, X4)) → U17_gga(X1, X2, X3, X4, appcE_in_gga(X2, X3, X4))
appcE_in_gga([], X1, X1) → appcE_out_gga([], X1, X1)
U17_gga(X1, X2, X3, X4, appcE_out_gga(X2, X3, X4)) → appcE_out_gga(.(X1, X2), X3, .(X1, X4))
U18_ggga(X1, X2, X3, X4, appcE_out_gga(X2, X3, X4)) → appcC_out_ggga(X1, X2, X3, .(X1, X4))
appcD_in_ga(X1, X1) → appcD_out_ga(X1, X1)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appcA_in_aaag(x1, x2, x3, x4)  =  appcA_in_aaag(x4)
U11_aaag(x1, x2, x3, x4, x5, x6)  =  U11_aaag(x1, x5, x6)
appcA_out_aaag(x1, x2, x3, x4)  =  appcA_out_aaag(x1, x2, x3, x4)
appcC_in_ggga(x1, x2, x3, x4)  =  appcC_in_ggga(x1, x2, x3)
U18_ggga(x1, x2, x3, x4, x5)  =  U18_ggga(x1, x2, x3, x5)
appcE_in_gga(x1, x2, x3)  =  appcE_in_gga(x1, x2)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
[]  =  []
appcE_out_gga(x1, x2, x3)  =  appcE_out_gga(x1, x2, x3)
appcC_out_ggga(x1, x2, x3, x4)  =  appcC_out_ggga(x1, x2, x3, x4)
appcD_in_ga(x1, x2)  =  appcD_in_ga(x1)
appcD_out_ga(x1, x2)  =  appcD_out_ga(x1, x2)
APPE_IN_GGA(x1, x2, x3)  =  APPE_IN_GGA(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

APPE_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPE_IN_GGA(X2, X3, X4)

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

APPE_IN_GGA(.(X1, X2), X3) → APPE_IN_GGA(X2, X3)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • APPE_IN_GGA(.(X1, X2), X3) → APPE_IN_GGA(X2, X3)
    The graph contains the following edges 1 > 1, 2 >= 2

(13) YES

(14) Obligation:

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

APPA_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) → APPA_IN_AAAG(X2, X3, X4, X5)

The TRS R consists of the following rules:

appcA_in_aaag(.(X1, X2), X3, X4, .(X1, X5)) → U11_aaag(X1, X2, X3, X4, X5, appcA_in_aaag(X2, X3, X4, X5))
appcA_in_aaag([], X1, X2, .(X1, X2)) → appcA_out_aaag([], X1, X2, .(X1, X2))
U11_aaag(X1, X2, X3, X4, X5, appcA_out_aaag(X2, X3, X4, X5)) → appcA_out_aaag(.(X1, X2), X3, X4, .(X1, X5))
appcC_in_ggga(X1, X2, X3, .(X1, X4)) → U18_ggga(X1, X2, X3, X4, appcE_in_gga(X2, X3, X4))
appcE_in_gga(.(X1, X2), X3, .(X1, X4)) → U17_gga(X1, X2, X3, X4, appcE_in_gga(X2, X3, X4))
appcE_in_gga([], X1, X1) → appcE_out_gga([], X1, X1)
U17_gga(X1, X2, X3, X4, appcE_out_gga(X2, X3, X4)) → appcE_out_gga(.(X1, X2), X3, .(X1, X4))
U18_ggga(X1, X2, X3, X4, appcE_out_gga(X2, X3, X4)) → appcC_out_ggga(X1, X2, X3, .(X1, X4))
appcD_in_ga(X1, X1) → appcD_out_ga(X1, X1)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appcA_in_aaag(x1, x2, x3, x4)  =  appcA_in_aaag(x4)
U11_aaag(x1, x2, x3, x4, x5, x6)  =  U11_aaag(x1, x5, x6)
appcA_out_aaag(x1, x2, x3, x4)  =  appcA_out_aaag(x1, x2, x3, x4)
appcC_in_ggga(x1, x2, x3, x4)  =  appcC_in_ggga(x1, x2, x3)
U18_ggga(x1, x2, x3, x4, x5)  =  U18_ggga(x1, x2, x3, x5)
appcE_in_gga(x1, x2, x3)  =  appcE_in_gga(x1, x2)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
[]  =  []
appcE_out_gga(x1, x2, x3)  =  appcE_out_gga(x1, x2, x3)
appcC_out_ggga(x1, x2, x3, x4)  =  appcC_out_ggga(x1, x2, x3, x4)
appcD_in_ga(x1, x2)  =  appcD_in_ga(x1)
appcD_out_ga(x1, x2)  =  appcD_out_ga(x1, x2)
APPA_IN_AAAG(x1, x2, x3, x4)  =  APPA_IN_AAAG(x4)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

APPA_IN_AAAG(.(X1, X2), X3, X4, .(X1, X5)) → APPA_IN_AAAG(X2, X3, X4, X5)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPA_IN_AAAG(x1, x2, x3, x4)  =  APPA_IN_AAAG(x4)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

APPA_IN_AAAG(.(X1, X5)) → APPA_IN_AAAG(X5)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • APPA_IN_AAAG(.(X1, X5)) → APPA_IN_AAAG(X5)
    The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

PERMB_IN_GA(.(X1, X2), .(X3, X4)) → U4_GA(X1, X2, X3, X4, appcA_in_aaag(X5, X3, X6, X2))
U4_GA(X1, X2, X3, X4, appcA_out_aaag(X5, X3, X6, X2)) → U6_GA(X1, X2, X3, X4, appcC_in_ggga(X1, X5, X6, X7))
U6_GA(X1, X2, X3, X4, appcC_out_ggga(X1, X5, X6, X7)) → PERMB_IN_GA(X7, X4)
PERMB_IN_GA(.(X1, X2), .(X1, X3)) → U8_GA(X1, X2, X3, appcD_in_ga(X2, X4))
U8_GA(X1, X2, X3, appcD_out_ga(X2, X4)) → PERMB_IN_GA(X4, X3)

The TRS R consists of the following rules:

appcA_in_aaag(.(X1, X2), X3, X4, .(X1, X5)) → U11_aaag(X1, X2, X3, X4, X5, appcA_in_aaag(X2, X3, X4, X5))
appcA_in_aaag([], X1, X2, .(X1, X2)) → appcA_out_aaag([], X1, X2, .(X1, X2))
U11_aaag(X1, X2, X3, X4, X5, appcA_out_aaag(X2, X3, X4, X5)) → appcA_out_aaag(.(X1, X2), X3, X4, .(X1, X5))
appcC_in_ggga(X1, X2, X3, .(X1, X4)) → U18_ggga(X1, X2, X3, X4, appcE_in_gga(X2, X3, X4))
appcE_in_gga(.(X1, X2), X3, .(X1, X4)) → U17_gga(X1, X2, X3, X4, appcE_in_gga(X2, X3, X4))
appcE_in_gga([], X1, X1) → appcE_out_gga([], X1, X1)
U17_gga(X1, X2, X3, X4, appcE_out_gga(X2, X3, X4)) → appcE_out_gga(.(X1, X2), X3, .(X1, X4))
U18_ggga(X1, X2, X3, X4, appcE_out_gga(X2, X3, X4)) → appcC_out_ggga(X1, X2, X3, .(X1, X4))
appcD_in_ga(X1, X1) → appcD_out_ga(X1, X1)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appcA_in_aaag(x1, x2, x3, x4)  =  appcA_in_aaag(x4)
U11_aaag(x1, x2, x3, x4, x5, x6)  =  U11_aaag(x1, x5, x6)
appcA_out_aaag(x1, x2, x3, x4)  =  appcA_out_aaag(x1, x2, x3, x4)
appcC_in_ggga(x1, x2, x3, x4)  =  appcC_in_ggga(x1, x2, x3)
U18_ggga(x1, x2, x3, x4, x5)  =  U18_ggga(x1, x2, x3, x5)
appcE_in_gga(x1, x2, x3)  =  appcE_in_gga(x1, x2)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
[]  =  []
appcE_out_gga(x1, x2, x3)  =  appcE_out_gga(x1, x2, x3)
appcC_out_ggga(x1, x2, x3, x4)  =  appcC_out_ggga(x1, x2, x3, x4)
appcD_in_ga(x1, x2)  =  appcD_in_ga(x1)
appcD_out_ga(x1, x2)  =  appcD_out_ga(x1, x2)
PERMB_IN_GA(x1, x2)  =  PERMB_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x5)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x5)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x2, x4)

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

(22) PiDPToQDPProof (SOUND transformation)

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

(23) Obligation:

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

PERMB_IN_GA(.(X1, X2)) → U4_GA(X1, X2, appcA_in_aaag(X2))
U4_GA(X1, X2, appcA_out_aaag(X5, X3, X6, X2)) → U6_GA(X1, X2, appcC_in_ggga(X1, X5, X6))
U6_GA(X1, X2, appcC_out_ggga(X1, X5, X6, X7)) → PERMB_IN_GA(X7)
PERMB_IN_GA(.(X1, X2)) → U8_GA(X1, X2, appcD_in_ga(X2))
U8_GA(X1, X2, appcD_out_ga(X2, X4)) → PERMB_IN_GA(X4)

The TRS R consists of the following rules:

appcA_in_aaag(.(X1, X5)) → U11_aaag(X1, X5, appcA_in_aaag(X5))
appcA_in_aaag(.(X1, X2)) → appcA_out_aaag([], X1, X2, .(X1, X2))
U11_aaag(X1, X5, appcA_out_aaag(X2, X3, X4, X5)) → appcA_out_aaag(.(X1, X2), X3, X4, .(X1, X5))
appcC_in_ggga(X1, X2, X3) → U18_ggga(X1, X2, X3, appcE_in_gga(X2, X3))
appcE_in_gga(.(X1, X2), X3) → U17_gga(X1, X2, X3, appcE_in_gga(X2, X3))
appcE_in_gga([], X1) → appcE_out_gga([], X1, X1)
U17_gga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcE_out_gga(.(X1, X2), X3, .(X1, X4))
U18_ggga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcC_out_ggga(X1, X2, X3, .(X1, X4))
appcD_in_ga(X1) → appcD_out_ga(X1, X1)

The set Q consists of the following terms:

appcA_in_aaag(x0)
U11_aaag(x0, x1, x2)
appcC_in_ggga(x0, x1, x2)
appcE_in_gga(x0, x1)
U17_gga(x0, x1, x2, x3)
U18_ggga(x0, x1, x2, x3)
appcD_in_ga(x0)

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

(24) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U4_GA(X1, X2, appcA_out_aaag(X5, X3, X6, X2)) → U6_GA(X1, X2, appcC_in_ggga(X1, X5, X6)) at position [2] we obtained the following new rules [LPAR04]:

U4_GA(X1, X2, appcA_out_aaag(X5, X3, X6, X2)) → U6_GA(X1, X2, U18_ggga(X1, X5, X6, appcE_in_gga(X5, X6)))

(25) Obligation:

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

PERMB_IN_GA(.(X1, X2)) → U4_GA(X1, X2, appcA_in_aaag(X2))
U6_GA(X1, X2, appcC_out_ggga(X1, X5, X6, X7)) → PERMB_IN_GA(X7)
PERMB_IN_GA(.(X1, X2)) → U8_GA(X1, X2, appcD_in_ga(X2))
U8_GA(X1, X2, appcD_out_ga(X2, X4)) → PERMB_IN_GA(X4)
U4_GA(X1, X2, appcA_out_aaag(X5, X3, X6, X2)) → U6_GA(X1, X2, U18_ggga(X1, X5, X6, appcE_in_gga(X5, X6)))

The TRS R consists of the following rules:

appcA_in_aaag(.(X1, X5)) → U11_aaag(X1, X5, appcA_in_aaag(X5))
appcA_in_aaag(.(X1, X2)) → appcA_out_aaag([], X1, X2, .(X1, X2))
U11_aaag(X1, X5, appcA_out_aaag(X2, X3, X4, X5)) → appcA_out_aaag(.(X1, X2), X3, X4, .(X1, X5))
appcC_in_ggga(X1, X2, X3) → U18_ggga(X1, X2, X3, appcE_in_gga(X2, X3))
appcE_in_gga(.(X1, X2), X3) → U17_gga(X1, X2, X3, appcE_in_gga(X2, X3))
appcE_in_gga([], X1) → appcE_out_gga([], X1, X1)
U17_gga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcE_out_gga(.(X1, X2), X3, .(X1, X4))
U18_ggga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcC_out_ggga(X1, X2, X3, .(X1, X4))
appcD_in_ga(X1) → appcD_out_ga(X1, X1)

The set Q consists of the following terms:

appcA_in_aaag(x0)
U11_aaag(x0, x1, x2)
appcC_in_ggga(x0, x1, x2)
appcE_in_gga(x0, x1)
U17_gga(x0, x1, x2, x3)
U18_ggga(x0, x1, x2, x3)
appcD_in_ga(x0)

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

(26) UsableRulesProof (EQUIVALENT transformation)

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

(27) Obligation:

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

PERMB_IN_GA(.(X1, X2)) → U4_GA(X1, X2, appcA_in_aaag(X2))
U6_GA(X1, X2, appcC_out_ggga(X1, X5, X6, X7)) → PERMB_IN_GA(X7)
PERMB_IN_GA(.(X1, X2)) → U8_GA(X1, X2, appcD_in_ga(X2))
U8_GA(X1, X2, appcD_out_ga(X2, X4)) → PERMB_IN_GA(X4)
U4_GA(X1, X2, appcA_out_aaag(X5, X3, X6, X2)) → U6_GA(X1, X2, U18_ggga(X1, X5, X6, appcE_in_gga(X5, X6)))

The TRS R consists of the following rules:

appcE_in_gga(.(X1, X2), X3) → U17_gga(X1, X2, X3, appcE_in_gga(X2, X3))
appcE_in_gga([], X1) → appcE_out_gga([], X1, X1)
U18_ggga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcC_out_ggga(X1, X2, X3, .(X1, X4))
U17_gga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcE_out_gga(.(X1, X2), X3, .(X1, X4))
appcD_in_ga(X1) → appcD_out_ga(X1, X1)
appcA_in_aaag(.(X1, X5)) → U11_aaag(X1, X5, appcA_in_aaag(X5))
appcA_in_aaag(.(X1, X2)) → appcA_out_aaag([], X1, X2, .(X1, X2))
U11_aaag(X1, X5, appcA_out_aaag(X2, X3, X4, X5)) → appcA_out_aaag(.(X1, X2), X3, X4, .(X1, X5))

The set Q consists of the following terms:

appcA_in_aaag(x0)
U11_aaag(x0, x1, x2)
appcC_in_ggga(x0, x1, x2)
appcE_in_gga(x0, x1)
U17_gga(x0, x1, x2, x3)
U18_ggga(x0, x1, x2, x3)
appcD_in_ga(x0)

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

(28) QReductionProof (EQUIVALENT transformation)

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

appcC_in_ggga(x0, x1, x2)

(29) Obligation:

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

PERMB_IN_GA(.(X1, X2)) → U4_GA(X1, X2, appcA_in_aaag(X2))
U6_GA(X1, X2, appcC_out_ggga(X1, X5, X6, X7)) → PERMB_IN_GA(X7)
PERMB_IN_GA(.(X1, X2)) → U8_GA(X1, X2, appcD_in_ga(X2))
U8_GA(X1, X2, appcD_out_ga(X2, X4)) → PERMB_IN_GA(X4)
U4_GA(X1, X2, appcA_out_aaag(X5, X3, X6, X2)) → U6_GA(X1, X2, U18_ggga(X1, X5, X6, appcE_in_gga(X5, X6)))

The TRS R consists of the following rules:

appcE_in_gga(.(X1, X2), X3) → U17_gga(X1, X2, X3, appcE_in_gga(X2, X3))
appcE_in_gga([], X1) → appcE_out_gga([], X1, X1)
U18_ggga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcC_out_ggga(X1, X2, X3, .(X1, X4))
U17_gga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcE_out_gga(.(X1, X2), X3, .(X1, X4))
appcD_in_ga(X1) → appcD_out_ga(X1, X1)
appcA_in_aaag(.(X1, X5)) → U11_aaag(X1, X5, appcA_in_aaag(X5))
appcA_in_aaag(.(X1, X2)) → appcA_out_aaag([], X1, X2, .(X1, X2))
U11_aaag(X1, X5, appcA_out_aaag(X2, X3, X4, X5)) → appcA_out_aaag(.(X1, X2), X3, X4, .(X1, X5))

The set Q consists of the following terms:

appcA_in_aaag(x0)
U11_aaag(x0, x1, x2)
appcE_in_gga(x0, x1)
U17_gga(x0, x1, x2, x3)
U18_ggga(x0, x1, x2, x3)
appcD_in_ga(x0)

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

(30) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule PERMB_IN_GA(.(X1, X2)) → U8_GA(X1, X2, appcD_in_ga(X2)) at position [2] we obtained the following new rules [LPAR04]:

PERMB_IN_GA(.(X1, X2)) → U8_GA(X1, X2, appcD_out_ga(X2, X2))

(31) Obligation:

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

PERMB_IN_GA(.(X1, X2)) → U4_GA(X1, X2, appcA_in_aaag(X2))
U6_GA(X1, X2, appcC_out_ggga(X1, X5, X6, X7)) → PERMB_IN_GA(X7)
U8_GA(X1, X2, appcD_out_ga(X2, X4)) → PERMB_IN_GA(X4)
U4_GA(X1, X2, appcA_out_aaag(X5, X3, X6, X2)) → U6_GA(X1, X2, U18_ggga(X1, X5, X6, appcE_in_gga(X5, X6)))
PERMB_IN_GA(.(X1, X2)) → U8_GA(X1, X2, appcD_out_ga(X2, X2))

The TRS R consists of the following rules:

appcE_in_gga(.(X1, X2), X3) → U17_gga(X1, X2, X3, appcE_in_gga(X2, X3))
appcE_in_gga([], X1) → appcE_out_gga([], X1, X1)
U18_ggga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcC_out_ggga(X1, X2, X3, .(X1, X4))
U17_gga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcE_out_gga(.(X1, X2), X3, .(X1, X4))
appcD_in_ga(X1) → appcD_out_ga(X1, X1)
appcA_in_aaag(.(X1, X5)) → U11_aaag(X1, X5, appcA_in_aaag(X5))
appcA_in_aaag(.(X1, X2)) → appcA_out_aaag([], X1, X2, .(X1, X2))
U11_aaag(X1, X5, appcA_out_aaag(X2, X3, X4, X5)) → appcA_out_aaag(.(X1, X2), X3, X4, .(X1, X5))

The set Q consists of the following terms:

appcA_in_aaag(x0)
U11_aaag(x0, x1, x2)
appcE_in_gga(x0, x1)
U17_gga(x0, x1, x2, x3)
U18_ggga(x0, x1, x2, x3)
appcD_in_ga(x0)

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

(32) UsableRulesProof (EQUIVALENT transformation)

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

(33) Obligation:

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

PERMB_IN_GA(.(X1, X2)) → U4_GA(X1, X2, appcA_in_aaag(X2))
U6_GA(X1, X2, appcC_out_ggga(X1, X5, X6, X7)) → PERMB_IN_GA(X7)
U8_GA(X1, X2, appcD_out_ga(X2, X4)) → PERMB_IN_GA(X4)
U4_GA(X1, X2, appcA_out_aaag(X5, X3, X6, X2)) → U6_GA(X1, X2, U18_ggga(X1, X5, X6, appcE_in_gga(X5, X6)))
PERMB_IN_GA(.(X1, X2)) → U8_GA(X1, X2, appcD_out_ga(X2, X2))

The TRS R consists of the following rules:

appcE_in_gga(.(X1, X2), X3) → U17_gga(X1, X2, X3, appcE_in_gga(X2, X3))
appcE_in_gga([], X1) → appcE_out_gga([], X1, X1)
U18_ggga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcC_out_ggga(X1, X2, X3, .(X1, X4))
U17_gga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcE_out_gga(.(X1, X2), X3, .(X1, X4))
appcA_in_aaag(.(X1, X5)) → U11_aaag(X1, X5, appcA_in_aaag(X5))
appcA_in_aaag(.(X1, X2)) → appcA_out_aaag([], X1, X2, .(X1, X2))
U11_aaag(X1, X5, appcA_out_aaag(X2, X3, X4, X5)) → appcA_out_aaag(.(X1, X2), X3, X4, .(X1, X5))

The set Q consists of the following terms:

appcA_in_aaag(x0)
U11_aaag(x0, x1, x2)
appcE_in_gga(x0, x1)
U17_gga(x0, x1, x2, x3)
U18_ggga(x0, x1, x2, x3)
appcD_in_ga(x0)

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

(34) QReductionProof (EQUIVALENT transformation)

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

appcD_in_ga(x0)

(35) Obligation:

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

PERMB_IN_GA(.(X1, X2)) → U4_GA(X1, X2, appcA_in_aaag(X2))
U6_GA(X1, X2, appcC_out_ggga(X1, X5, X6, X7)) → PERMB_IN_GA(X7)
U8_GA(X1, X2, appcD_out_ga(X2, X4)) → PERMB_IN_GA(X4)
U4_GA(X1, X2, appcA_out_aaag(X5, X3, X6, X2)) → U6_GA(X1, X2, U18_ggga(X1, X5, X6, appcE_in_gga(X5, X6)))
PERMB_IN_GA(.(X1, X2)) → U8_GA(X1, X2, appcD_out_ga(X2, X2))

The TRS R consists of the following rules:

appcE_in_gga(.(X1, X2), X3) → U17_gga(X1, X2, X3, appcE_in_gga(X2, X3))
appcE_in_gga([], X1) → appcE_out_gga([], X1, X1)
U18_ggga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcC_out_ggga(X1, X2, X3, .(X1, X4))
U17_gga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcE_out_gga(.(X1, X2), X3, .(X1, X4))
appcA_in_aaag(.(X1, X5)) → U11_aaag(X1, X5, appcA_in_aaag(X5))
appcA_in_aaag(.(X1, X2)) → appcA_out_aaag([], X1, X2, .(X1, X2))
U11_aaag(X1, X5, appcA_out_aaag(X2, X3, X4, X5)) → appcA_out_aaag(.(X1, X2), X3, X4, .(X1, X5))

The set Q consists of the following terms:

appcA_in_aaag(x0)
U11_aaag(x0, x1, x2)
appcE_in_gga(x0, x1)
U17_gga(x0, x1, x2, x3)
U18_ggga(x0, x1, x2, x3)

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

(36) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U8_GA(X1, X2, appcD_out_ga(X2, X4)) → PERMB_IN_GA(X4) we obtained the following new rules [LPAR04]:

U8_GA(z0, z1, appcD_out_ga(z1, z1)) → PERMB_IN_GA(z1)

(37) Obligation:

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

PERMB_IN_GA(.(X1, X2)) → U4_GA(X1, X2, appcA_in_aaag(X2))
U6_GA(X1, X2, appcC_out_ggga(X1, X5, X6, X7)) → PERMB_IN_GA(X7)
U4_GA(X1, X2, appcA_out_aaag(X5, X3, X6, X2)) → U6_GA(X1, X2, U18_ggga(X1, X5, X6, appcE_in_gga(X5, X6)))
PERMB_IN_GA(.(X1, X2)) → U8_GA(X1, X2, appcD_out_ga(X2, X2))
U8_GA(z0, z1, appcD_out_ga(z1, z1)) → PERMB_IN_GA(z1)

The TRS R consists of the following rules:

appcE_in_gga(.(X1, X2), X3) → U17_gga(X1, X2, X3, appcE_in_gga(X2, X3))
appcE_in_gga([], X1) → appcE_out_gga([], X1, X1)
U18_ggga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcC_out_ggga(X1, X2, X3, .(X1, X4))
U17_gga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcE_out_gga(.(X1, X2), X3, .(X1, X4))
appcA_in_aaag(.(X1, X5)) → U11_aaag(X1, X5, appcA_in_aaag(X5))
appcA_in_aaag(.(X1, X2)) → appcA_out_aaag([], X1, X2, .(X1, X2))
U11_aaag(X1, X5, appcA_out_aaag(X2, X3, X4, X5)) → appcA_out_aaag(.(X1, X2), X3, X4, .(X1, X5))

The set Q consists of the following terms:

appcA_in_aaag(x0)
U11_aaag(x0, x1, x2)
appcE_in_gga(x0, x1)
U17_gga(x0, x1, x2, x3)
U18_ggga(x0, x1, x2, x3)

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

(38) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


U4_GA(X1, X2, appcA_out_aaag(X5, X3, X6, X2)) → U6_GA(X1, X2, U18_ggga(X1, X5, X6, appcE_in_gga(X5, X6)))
U8_GA(z0, z1, appcD_out_ga(z1, z1)) → PERMB_IN_GA(z1)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x2   
POL(PERMB_IN_GA(x1)) = x1   
POL(U11_aaag(x1, x2, x3)) = 1 + x3   
POL(U17_gga(x1, x2, x3, x4)) = 1 + x4   
POL(U18_ggga(x1, x2, x3, x4)) = x4   
POL(U4_GA(x1, x2, x3)) = 1 + x3   
POL(U6_GA(x1, x2, x3)) = x3   
POL(U8_GA(x1, x2, x3)) = 1 + x2   
POL([]) = 0   
POL(appcA_in_aaag(x1)) = x1   
POL(appcA_out_aaag(x1, x2, x3, x4)) = 1 + x1 + x3   
POL(appcC_out_ggga(x1, x2, x3, x4)) = x4   
POL(appcD_out_ga(x1, x2)) = 0   
POL(appcE_in_gga(x1, x2)) = 1 + x1 + x2   
POL(appcE_out_gga(x1, x2, x3)) = 1 + x3   

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

appcA_in_aaag(.(X1, X5)) → U11_aaag(X1, X5, appcA_in_aaag(X5))
appcA_in_aaag(.(X1, X2)) → appcA_out_aaag([], X1, X2, .(X1, X2))
appcE_in_gga(.(X1, X2), X3) → U17_gga(X1, X2, X3, appcE_in_gga(X2, X3))
appcE_in_gga([], X1) → appcE_out_gga([], X1, X1)
U18_ggga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcC_out_ggga(X1, X2, X3, .(X1, X4))
U17_gga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcE_out_gga(.(X1, X2), X3, .(X1, X4))
U11_aaag(X1, X5, appcA_out_aaag(X2, X3, X4, X5)) → appcA_out_aaag(.(X1, X2), X3, X4, .(X1, X5))

(39) Obligation:

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

PERMB_IN_GA(.(X1, X2)) → U4_GA(X1, X2, appcA_in_aaag(X2))
U6_GA(X1, X2, appcC_out_ggga(X1, X5, X6, X7)) → PERMB_IN_GA(X7)
PERMB_IN_GA(.(X1, X2)) → U8_GA(X1, X2, appcD_out_ga(X2, X2))

The TRS R consists of the following rules:

appcE_in_gga(.(X1, X2), X3) → U17_gga(X1, X2, X3, appcE_in_gga(X2, X3))
appcE_in_gga([], X1) → appcE_out_gga([], X1, X1)
U18_ggga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcC_out_ggga(X1, X2, X3, .(X1, X4))
U17_gga(X1, X2, X3, appcE_out_gga(X2, X3, X4)) → appcE_out_gga(.(X1, X2), X3, .(X1, X4))
appcA_in_aaag(.(X1, X5)) → U11_aaag(X1, X5, appcA_in_aaag(X5))
appcA_in_aaag(.(X1, X2)) → appcA_out_aaag([], X1, X2, .(X1, X2))
U11_aaag(X1, X5, appcA_out_aaag(X2, X3, X4, X5)) → appcA_out_aaag(.(X1, X2), X3, X4, .(X1, X5))

The set Q consists of the following terms:

appcA_in_aaag(x0)
U11_aaag(x0, x1, x2)
appcE_in_gga(x0, x1)
U17_gga(x0, x1, x2, x3)
U18_ggga(x0, x1, x2, x3)

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

(40) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(41) TRUE