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

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇒, 86 ms)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇒, 115 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 (⇔, 12 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 (⇔, 111 ms)
        ↳39 QDP
        ↳40 DependencyGraphProof (⇔, 0 ms)
        ↳41 TRUE

(0) Obligation:

Clauses:

append(nil, XS, XS).
append(cons(X, XS1), XS2, cons(X, YS)) :- append(XS1, XS2, YS).
split(XS, nil, XS).
split(cons(X, XS), cons(X, YS1), YS2) :- split(XS, YS1, YS2).
perm(nil, nil).
perm(XS, cons(Y, YS)) :- ','(split(XS, YS1, cons(Y, YS2)), ','(append(YS1, YS2, ZS), perm(ZS, YS))).

Query: perm(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

splitC(cons(X1, X2), cons(X1, X3), X4, X5) :- splitC(X2, X3, X4, X5).
appendE(cons(X1, X2), X3, cons(X1, X4)) :- appendE(X2, X3, X4).
permA(cons(X1, X2), cons(X1, X3)) :- ','(appendcB(X2, X4), permA(X4, X3)).
permA(cons(X1, X2), cons(X3, X4)) :- splitC(X2, X5, X3, X6).
permA(cons(X1, X2), cons(X3, X4)) :- ','(splitcC(X2, X5, X3, X6), appendE(X5, X6, X7)).
permA(cons(X1, X2), cons(X3, X4)) :- ','(splitcC(X2, X5, X3, X6), ','(appendcD(X1, X5, X6, X7), permA(X7, X4))).

Clauses:

permcA(nil, nil).
permcA(cons(X1, X2), cons(X1, X3)) :- ','(appendcB(X2, X4), permcA(X4, X3)).
permcA(cons(X1, X2), cons(X3, X4)) :- ','(splitcC(X2, X5, X3, X6), ','(appendcD(X1, X5, X6, X7), permcA(X7, X4))).
splitcC(cons(X1, X2), nil, X1, X2).
splitcC(cons(X1, X2), cons(X1, X3), X4, X5) :- splitcC(X2, X3, X4, X5).
appendcE(nil, X1, X1).
appendcE(cons(X1, X2), X3, cons(X1, X4)) :- appendcE(X2, X3, X4).
appendcB(X1, X1).
appendcD(X1, X2, X3, cons(X1, X4)) :- appendcE(X2, X3, X4).

Afs:

permA(x1, x2)  =  permA(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:
permA_in: (b,f)
splitC_in: (b,f,f,f)
splitcC_in: (b,f,f,f)
appendE_in: (b,b,f)
appendcD_in: (b,b,b,f)
appendcE_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

PERMA_IN_GA(cons(X1, X2), cons(X1, X3)) → U3_GA(X1, X2, X3, appendcB_in_ga(X2, X4))
U3_GA(X1, X2, X3, appendcB_out_ga(X2, X4)) → U4_GA(X1, X2, X3, permA_in_ga(X4, X3))
U3_GA(X1, X2, X3, appendcB_out_ga(X2, X4)) → PERMA_IN_GA(X4, X3)
PERMA_IN_GA(cons(X1, X2), cons(X3, X4)) → U5_GA(X1, X2, X3, X4, splitC_in_gaaa(X2, X5, X3, X6))
PERMA_IN_GA(cons(X1, X2), cons(X3, X4)) → SPLITC_IN_GAAA(X2, X5, X3, X6)
SPLITC_IN_GAAA(cons(X1, X2), cons(X1, X3), X4, X5) → U1_GAAA(X1, X2, X3, X4, X5, splitC_in_gaaa(X2, X3, X4, X5))
SPLITC_IN_GAAA(cons(X1, X2), cons(X1, X3), X4, X5) → SPLITC_IN_GAAA(X2, X3, X4, X5)
PERMA_IN_GA(cons(X1, X2), cons(X3, X4)) → U6_GA(X1, X2, X3, X4, splitcC_in_gaaa(X2, X5, X3, X6))
U6_GA(X1, X2, X3, X4, splitcC_out_gaaa(X2, X5, X3, X6)) → U7_GA(X1, X2, X3, X4, appendE_in_gga(X5, X6, X7))
U6_GA(X1, X2, X3, X4, splitcC_out_gaaa(X2, X5, X3, X6)) → APPENDE_IN_GGA(X5, X6, X7)
APPENDE_IN_GGA(cons(X1, X2), X3, cons(X1, X4)) → U2_GGA(X1, X2, X3, X4, appendE_in_gga(X2, X3, X4))
APPENDE_IN_GGA(cons(X1, X2), X3, cons(X1, X4)) → APPENDE_IN_GGA(X2, X3, X4)
U6_GA(X1, X2, X3, X4, splitcC_out_gaaa(X2, X5, X3, X6)) → U8_GA(X1, X2, X3, X4, appendcD_in_ggga(X1, X5, X6, X7))
U8_GA(X1, X2, X3, X4, appendcD_out_ggga(X1, X5, X6, X7)) → U9_GA(X1, X2, X3, X4, permA_in_ga(X7, X4))
U8_GA(X1, X2, X3, X4, appendcD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7, X4)

The TRS R consists of the following rules:

appendcB_in_ga(X1, X1) → appendcB_out_ga(X1, X1)
splitcC_in_gaaa(cons(X1, X2), nil, X1, X2) → splitcC_out_gaaa(cons(X1, X2), nil, X1, X2)
splitcC_in_gaaa(cons(X1, X2), cons(X1, X3), X4, X5) → U16_gaaa(X1, X2, X3, X4, X5, splitcC_in_gaaa(X2, X3, X4, X5))
U16_gaaa(X1, X2, X3, X4, X5, splitcC_out_gaaa(X2, X3, X4, X5)) → splitcC_out_gaaa(cons(X1, X2), cons(X1, X3), X4, X5)
appendcD_in_ggga(X1, X2, X3, cons(X1, X4)) → U18_ggga(X1, X2, X3, X4, appendcE_in_gga(X2, X3, X4))
appendcE_in_gga(nil, X1, X1) → appendcE_out_gga(nil, X1, X1)
appendcE_in_gga(cons(X1, X2), X3, cons(X1, X4)) → U17_gga(X1, X2, X3, X4, appendcE_in_gga(X2, X3, X4))
U17_gga(X1, X2, X3, X4, appendcE_out_gga(X2, X3, X4)) → appendcE_out_gga(cons(X1, X2), X3, cons(X1, X4))
U18_ggga(X1, X2, X3, X4, appendcE_out_gga(X2, X3, X4)) → appendcD_out_ggga(X1, X2, X3, cons(X1, X4))

The argument filtering Pi contains the following mapping:
permA_in_ga(x1, x2)  =  permA_in_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
appendcB_in_ga(x1, x2)  =  appendcB_in_ga(x1)
appendcB_out_ga(x1, x2)  =  appendcB_out_ga(x1, x2)
splitC_in_gaaa(x1, x2, x3, x4)  =  splitC_in_gaaa(x1)
splitcC_in_gaaa(x1, x2, x3, x4)  =  splitcC_in_gaaa(x1)
splitcC_out_gaaa(x1, x2, x3, x4)  =  splitcC_out_gaaa(x1, x2, x3, x4)
U16_gaaa(x1, x2, x3, x4, x5, x6)  =  U16_gaaa(x1, x2, x6)
appendE_in_gga(x1, x2, x3)  =  appendE_in_gga(x1, x2)
appendcD_in_ggga(x1, x2, x3, x4)  =  appendcD_in_ggga(x1, x2, x3)
U18_ggga(x1, x2, x3, x4, x5)  =  U18_ggga(x1, x2, x3, x5)
appendcE_in_gga(x1, x2, x3)  =  appendcE_in_gga(x1, x2)
nil  =  nil
appendcE_out_gga(x1, x2, x3)  =  appendcE_out_gga(x1, x2, x3)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
appendcD_out_ggga(x1, x2, x3, x4)  =  appendcD_out_ggga(x1, x2, x3, x4)
PERMA_IN_GA(x1, x2)  =  PERMA_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x2, x4)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x2, x4)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x5)
SPLITC_IN_GAAA(x1, x2, x3, x4)  =  SPLITC_IN_GAAA(x1)
U1_GAAA(x1, x2, x3, x4, x5, x6)  =  U1_GAAA(x1, x2, x6)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x5)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x5)
APPENDE_IN_GGA(x1, x2, x3)  =  APPENDE_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x5)

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:

PERMA_IN_GA(cons(X1, X2), cons(X1, X3)) → U3_GA(X1, X2, X3, appendcB_in_ga(X2, X4))
U3_GA(X1, X2, X3, appendcB_out_ga(X2, X4)) → U4_GA(X1, X2, X3, permA_in_ga(X4, X3))
U3_GA(X1, X2, X3, appendcB_out_ga(X2, X4)) → PERMA_IN_GA(X4, X3)
PERMA_IN_GA(cons(X1, X2), cons(X3, X4)) → U5_GA(X1, X2, X3, X4, splitC_in_gaaa(X2, X5, X3, X6))
PERMA_IN_GA(cons(X1, X2), cons(X3, X4)) → SPLITC_IN_GAAA(X2, X5, X3, X6)
SPLITC_IN_GAAA(cons(X1, X2), cons(X1, X3), X4, X5) → U1_GAAA(X1, X2, X3, X4, X5, splitC_in_gaaa(X2, X3, X4, X5))
SPLITC_IN_GAAA(cons(X1, X2), cons(X1, X3), X4, X5) → SPLITC_IN_GAAA(X2, X3, X4, X5)
PERMA_IN_GA(cons(X1, X2), cons(X3, X4)) → U6_GA(X1, X2, X3, X4, splitcC_in_gaaa(X2, X5, X3, X6))
U6_GA(X1, X2, X3, X4, splitcC_out_gaaa(X2, X5, X3, X6)) → U7_GA(X1, X2, X3, X4, appendE_in_gga(X5, X6, X7))
U6_GA(X1, X2, X3, X4, splitcC_out_gaaa(X2, X5, X3, X6)) → APPENDE_IN_GGA(X5, X6, X7)
APPENDE_IN_GGA(cons(X1, X2), X3, cons(X1, X4)) → U2_GGA(X1, X2, X3, X4, appendE_in_gga(X2, X3, X4))
APPENDE_IN_GGA(cons(X1, X2), X3, cons(X1, X4)) → APPENDE_IN_GGA(X2, X3, X4)
U6_GA(X1, X2, X3, X4, splitcC_out_gaaa(X2, X5, X3, X6)) → U8_GA(X1, X2, X3, X4, appendcD_in_ggga(X1, X5, X6, X7))
U8_GA(X1, X2, X3, X4, appendcD_out_ggga(X1, X5, X6, X7)) → U9_GA(X1, X2, X3, X4, permA_in_ga(X7, X4))
U8_GA(X1, X2, X3, X4, appendcD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7, X4)

The TRS R consists of the following rules:

appendcB_in_ga(X1, X1) → appendcB_out_ga(X1, X1)
splitcC_in_gaaa(cons(X1, X2), nil, X1, X2) → splitcC_out_gaaa(cons(X1, X2), nil, X1, X2)
splitcC_in_gaaa(cons(X1, X2), cons(X1, X3), X4, X5) → U16_gaaa(X1, X2, X3, X4, X5, splitcC_in_gaaa(X2, X3, X4, X5))
U16_gaaa(X1, X2, X3, X4, X5, splitcC_out_gaaa(X2, X3, X4, X5)) → splitcC_out_gaaa(cons(X1, X2), cons(X1, X3), X4, X5)
appendcD_in_ggga(X1, X2, X3, cons(X1, X4)) → U18_ggga(X1, X2, X3, X4, appendcE_in_gga(X2, X3, X4))
appendcE_in_gga(nil, X1, X1) → appendcE_out_gga(nil, X1, X1)
appendcE_in_gga(cons(X1, X2), X3, cons(X1, X4)) → U17_gga(X1, X2, X3, X4, appendcE_in_gga(X2, X3, X4))
U17_gga(X1, X2, X3, X4, appendcE_out_gga(X2, X3, X4)) → appendcE_out_gga(cons(X1, X2), X3, cons(X1, X4))
U18_ggga(X1, X2, X3, X4, appendcE_out_gga(X2, X3, X4)) → appendcD_out_ggga(X1, X2, X3, cons(X1, X4))

The argument filtering Pi contains the following mapping:
permA_in_ga(x1, x2)  =  permA_in_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
appendcB_in_ga(x1, x2)  =  appendcB_in_ga(x1)
appendcB_out_ga(x1, x2)  =  appendcB_out_ga(x1, x2)
splitC_in_gaaa(x1, x2, x3, x4)  =  splitC_in_gaaa(x1)
splitcC_in_gaaa(x1, x2, x3, x4)  =  splitcC_in_gaaa(x1)
splitcC_out_gaaa(x1, x2, x3, x4)  =  splitcC_out_gaaa(x1, x2, x3, x4)
U16_gaaa(x1, x2, x3, x4, x5, x6)  =  U16_gaaa(x1, x2, x6)
appendE_in_gga(x1, x2, x3)  =  appendE_in_gga(x1, x2)
appendcD_in_ggga(x1, x2, x3, x4)  =  appendcD_in_ggga(x1, x2, x3)
U18_ggga(x1, x2, x3, x4, x5)  =  U18_ggga(x1, x2, x3, x5)
appendcE_in_gga(x1, x2, x3)  =  appendcE_in_gga(x1, x2)
nil  =  nil
appendcE_out_gga(x1, x2, x3)  =  appendcE_out_gga(x1, x2, x3)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
appendcD_out_ggga(x1, x2, x3, x4)  =  appendcD_out_ggga(x1, x2, x3, x4)
PERMA_IN_GA(x1, x2)  =  PERMA_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x2, x4)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x2, x4)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x5)
SPLITC_IN_GAAA(x1, x2, x3, x4)  =  SPLITC_IN_GAAA(x1)
U1_GAAA(x1, x2, x3, x4, x5, x6)  =  U1_GAAA(x1, x2, x6)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x5)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x5)
APPENDE_IN_GGA(x1, x2, x3)  =  APPENDE_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x5)

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:

APPENDE_IN_GGA(cons(X1, X2), X3, cons(X1, X4)) → APPENDE_IN_GGA(X2, X3, X4)

The TRS R consists of the following rules:

appendcB_in_ga(X1, X1) → appendcB_out_ga(X1, X1)
splitcC_in_gaaa(cons(X1, X2), nil, X1, X2) → splitcC_out_gaaa(cons(X1, X2), nil, X1, X2)
splitcC_in_gaaa(cons(X1, X2), cons(X1, X3), X4, X5) → U16_gaaa(X1, X2, X3, X4, X5, splitcC_in_gaaa(X2, X3, X4, X5))
U16_gaaa(X1, X2, X3, X4, X5, splitcC_out_gaaa(X2, X3, X4, X5)) → splitcC_out_gaaa(cons(X1, X2), cons(X1, X3), X4, X5)
appendcD_in_ggga(X1, X2, X3, cons(X1, X4)) → U18_ggga(X1, X2, X3, X4, appendcE_in_gga(X2, X3, X4))
appendcE_in_gga(nil, X1, X1) → appendcE_out_gga(nil, X1, X1)
appendcE_in_gga(cons(X1, X2), X3, cons(X1, X4)) → U17_gga(X1, X2, X3, X4, appendcE_in_gga(X2, X3, X4))
U17_gga(X1, X2, X3, X4, appendcE_out_gga(X2, X3, X4)) → appendcE_out_gga(cons(X1, X2), X3, cons(X1, X4))
U18_ggga(X1, X2, X3, X4, appendcE_out_gga(X2, X3, X4)) → appendcD_out_ggga(X1, X2, X3, cons(X1, X4))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
appendcB_in_ga(x1, x2)  =  appendcB_in_ga(x1)
appendcB_out_ga(x1, x2)  =  appendcB_out_ga(x1, x2)
splitcC_in_gaaa(x1, x2, x3, x4)  =  splitcC_in_gaaa(x1)
splitcC_out_gaaa(x1, x2, x3, x4)  =  splitcC_out_gaaa(x1, x2, x3, x4)
U16_gaaa(x1, x2, x3, x4, x5, x6)  =  U16_gaaa(x1, x2, x6)
appendcD_in_ggga(x1, x2, x3, x4)  =  appendcD_in_ggga(x1, x2, x3)
U18_ggga(x1, x2, x3, x4, x5)  =  U18_ggga(x1, x2, x3, x5)
appendcE_in_gga(x1, x2, x3)  =  appendcE_in_gga(x1, x2)
nil  =  nil
appendcE_out_gga(x1, x2, x3)  =  appendcE_out_gga(x1, x2, x3)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
appendcD_out_ggga(x1, x2, x3, x4)  =  appendcD_out_ggga(x1, x2, x3, x4)
APPENDE_IN_GGA(x1, x2, x3)  =  APPENDE_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:

APPENDE_IN_GGA(cons(X1, X2), X3, cons(X1, X4)) → APPENDE_IN_GGA(X2, X3, X4)

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

APPENDE_IN_GGA(cons(X1, X2), X3) → APPENDE_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:

  • APPENDE_IN_GGA(cons(X1, X2), X3) → APPENDE_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:

SPLITC_IN_GAAA(cons(X1, X2), cons(X1, X3), X4, X5) → SPLITC_IN_GAAA(X2, X3, X4, X5)

The TRS R consists of the following rules:

appendcB_in_ga(X1, X1) → appendcB_out_ga(X1, X1)
splitcC_in_gaaa(cons(X1, X2), nil, X1, X2) → splitcC_out_gaaa(cons(X1, X2), nil, X1, X2)
splitcC_in_gaaa(cons(X1, X2), cons(X1, X3), X4, X5) → U16_gaaa(X1, X2, X3, X4, X5, splitcC_in_gaaa(X2, X3, X4, X5))
U16_gaaa(X1, X2, X3, X4, X5, splitcC_out_gaaa(X2, X3, X4, X5)) → splitcC_out_gaaa(cons(X1, X2), cons(X1, X3), X4, X5)
appendcD_in_ggga(X1, X2, X3, cons(X1, X4)) → U18_ggga(X1, X2, X3, X4, appendcE_in_gga(X2, X3, X4))
appendcE_in_gga(nil, X1, X1) → appendcE_out_gga(nil, X1, X1)
appendcE_in_gga(cons(X1, X2), X3, cons(X1, X4)) → U17_gga(X1, X2, X3, X4, appendcE_in_gga(X2, X3, X4))
U17_gga(X1, X2, X3, X4, appendcE_out_gga(X2, X3, X4)) → appendcE_out_gga(cons(X1, X2), X3, cons(X1, X4))
U18_ggga(X1, X2, X3, X4, appendcE_out_gga(X2, X3, X4)) → appendcD_out_ggga(X1, X2, X3, cons(X1, X4))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
appendcB_in_ga(x1, x2)  =  appendcB_in_ga(x1)
appendcB_out_ga(x1, x2)  =  appendcB_out_ga(x1, x2)
splitcC_in_gaaa(x1, x2, x3, x4)  =  splitcC_in_gaaa(x1)
splitcC_out_gaaa(x1, x2, x3, x4)  =  splitcC_out_gaaa(x1, x2, x3, x4)
U16_gaaa(x1, x2, x3, x4, x5, x6)  =  U16_gaaa(x1, x2, x6)
appendcD_in_ggga(x1, x2, x3, x4)  =  appendcD_in_ggga(x1, x2, x3)
U18_ggga(x1, x2, x3, x4, x5)  =  U18_ggga(x1, x2, x3, x5)
appendcE_in_gga(x1, x2, x3)  =  appendcE_in_gga(x1, x2)
nil  =  nil
appendcE_out_gga(x1, x2, x3)  =  appendcE_out_gga(x1, x2, x3)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
appendcD_out_ggga(x1, x2, x3, x4)  =  appendcD_out_ggga(x1, x2, x3, x4)
SPLITC_IN_GAAA(x1, x2, x3, x4)  =  SPLITC_IN_GAAA(x1)

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:

SPLITC_IN_GAAA(cons(X1, X2), cons(X1, X3), X4, X5) → SPLITC_IN_GAAA(X2, X3, X4, X5)

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

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:

SPLITC_IN_GAAA(cons(X1, X2)) → SPLITC_IN_GAAA(X2)

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:

  • SPLITC_IN_GAAA(cons(X1, X2)) → SPLITC_IN_GAAA(X2)
    The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

U3_GA(X1, X2, X3, appendcB_out_ga(X2, X4)) → PERMA_IN_GA(X4, X3)
PERMA_IN_GA(cons(X1, X2), cons(X1, X3)) → U3_GA(X1, X2, X3, appendcB_in_ga(X2, X4))
PERMA_IN_GA(cons(X1, X2), cons(X3, X4)) → U6_GA(X1, X2, X3, X4, splitcC_in_gaaa(X2, X5, X3, X6))
U6_GA(X1, X2, X3, X4, splitcC_out_gaaa(X2, X5, X3, X6)) → U8_GA(X1, X2, X3, X4, appendcD_in_ggga(X1, X5, X6, X7))
U8_GA(X1, X2, X3, X4, appendcD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7, X4)

The TRS R consists of the following rules:

appendcB_in_ga(X1, X1) → appendcB_out_ga(X1, X1)
splitcC_in_gaaa(cons(X1, X2), nil, X1, X2) → splitcC_out_gaaa(cons(X1, X2), nil, X1, X2)
splitcC_in_gaaa(cons(X1, X2), cons(X1, X3), X4, X5) → U16_gaaa(X1, X2, X3, X4, X5, splitcC_in_gaaa(X2, X3, X4, X5))
U16_gaaa(X1, X2, X3, X4, X5, splitcC_out_gaaa(X2, X3, X4, X5)) → splitcC_out_gaaa(cons(X1, X2), cons(X1, X3), X4, X5)
appendcD_in_ggga(X1, X2, X3, cons(X1, X4)) → U18_ggga(X1, X2, X3, X4, appendcE_in_gga(X2, X3, X4))
appendcE_in_gga(nil, X1, X1) → appendcE_out_gga(nil, X1, X1)
appendcE_in_gga(cons(X1, X2), X3, cons(X1, X4)) → U17_gga(X1, X2, X3, X4, appendcE_in_gga(X2, X3, X4))
U17_gga(X1, X2, X3, X4, appendcE_out_gga(X2, X3, X4)) → appendcE_out_gga(cons(X1, X2), X3, cons(X1, X4))
U18_ggga(X1, X2, X3, X4, appendcE_out_gga(X2, X3, X4)) → appendcD_out_ggga(X1, X2, X3, cons(X1, X4))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
appendcB_in_ga(x1, x2)  =  appendcB_in_ga(x1)
appendcB_out_ga(x1, x2)  =  appendcB_out_ga(x1, x2)
splitcC_in_gaaa(x1, x2, x3, x4)  =  splitcC_in_gaaa(x1)
splitcC_out_gaaa(x1, x2, x3, x4)  =  splitcC_out_gaaa(x1, x2, x3, x4)
U16_gaaa(x1, x2, x3, x4, x5, x6)  =  U16_gaaa(x1, x2, x6)
appendcD_in_ggga(x1, x2, x3, x4)  =  appendcD_in_ggga(x1, x2, x3)
U18_ggga(x1, x2, x3, x4, x5)  =  U18_ggga(x1, x2, x3, x5)
appendcE_in_gga(x1, x2, x3)  =  appendcE_in_gga(x1, x2)
nil  =  nil
appendcE_out_gga(x1, x2, x3)  =  appendcE_out_gga(x1, x2, x3)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
appendcD_out_ggga(x1, x2, x3, x4)  =  appendcD_out_ggga(x1, x2, x3, x4)
PERMA_IN_GA(x1, x2)  =  PERMA_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x2, x4)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x5)

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:

U3_GA(X1, X2, appendcB_out_ga(X2, X4)) → PERMA_IN_GA(X4)
PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, appendcB_in_ga(X2))
PERMA_IN_GA(cons(X1, X2)) → U6_GA(X1, X2, splitcC_in_gaaa(X2))
U6_GA(X1, X2, splitcC_out_gaaa(X2, X5, X3, X6)) → U8_GA(X1, X2, appendcD_in_ggga(X1, X5, X6))
U8_GA(X1, X2, appendcD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7)

The TRS R consists of the following rules:

appendcB_in_ga(X1) → appendcB_out_ga(X1, X1)
splitcC_in_gaaa(cons(X1, X2)) → splitcC_out_gaaa(cons(X1, X2), nil, X1, X2)
splitcC_in_gaaa(cons(X1, X2)) → U16_gaaa(X1, X2, splitcC_in_gaaa(X2))
U16_gaaa(X1, X2, splitcC_out_gaaa(X2, X3, X4, X5)) → splitcC_out_gaaa(cons(X1, X2), cons(X1, X3), X4, X5)
appendcD_in_ggga(X1, X2, X3) → U18_ggga(X1, X2, X3, appendcE_in_gga(X2, X3))
appendcE_in_gga(nil, X1) → appendcE_out_gga(nil, X1, X1)
appendcE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, appendcE_in_gga(X2, X3))
U17_gga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcE_out_gga(cons(X1, X2), X3, cons(X1, X4))
U18_ggga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcD_out_ggga(X1, X2, X3, cons(X1, X4))

The set Q consists of the following terms:

appendcB_in_ga(x0)
splitcC_in_gaaa(x0)
U16_gaaa(x0, x1, x2)
appendcD_in_ggga(x0, x1, x2)
appendcE_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.

(24) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, appendcB_in_ga(X2)) at position [2] we obtained the following new rules [LPAR04]:

PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, appendcB_out_ga(X2, X2))

(25) Obligation:

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

U3_GA(X1, X2, appendcB_out_ga(X2, X4)) → PERMA_IN_GA(X4)
PERMA_IN_GA(cons(X1, X2)) → U6_GA(X1, X2, splitcC_in_gaaa(X2))
U6_GA(X1, X2, splitcC_out_gaaa(X2, X5, X3, X6)) → U8_GA(X1, X2, appendcD_in_ggga(X1, X5, X6))
U8_GA(X1, X2, appendcD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7)
PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, appendcB_out_ga(X2, X2))

The TRS R consists of the following rules:

appendcB_in_ga(X1) → appendcB_out_ga(X1, X1)
splitcC_in_gaaa(cons(X1, X2)) → splitcC_out_gaaa(cons(X1, X2), nil, X1, X2)
splitcC_in_gaaa(cons(X1, X2)) → U16_gaaa(X1, X2, splitcC_in_gaaa(X2))
U16_gaaa(X1, X2, splitcC_out_gaaa(X2, X3, X4, X5)) → splitcC_out_gaaa(cons(X1, X2), cons(X1, X3), X4, X5)
appendcD_in_ggga(X1, X2, X3) → U18_ggga(X1, X2, X3, appendcE_in_gga(X2, X3))
appendcE_in_gga(nil, X1) → appendcE_out_gga(nil, X1, X1)
appendcE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, appendcE_in_gga(X2, X3))
U17_gga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcE_out_gga(cons(X1, X2), X3, cons(X1, X4))
U18_ggga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcD_out_ggga(X1, X2, X3, cons(X1, X4))

The set Q consists of the following terms:

appendcB_in_ga(x0)
splitcC_in_gaaa(x0)
U16_gaaa(x0, x1, x2)
appendcD_in_ggga(x0, x1, x2)
appendcE_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.

(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:

U3_GA(X1, X2, appendcB_out_ga(X2, X4)) → PERMA_IN_GA(X4)
PERMA_IN_GA(cons(X1, X2)) → U6_GA(X1, X2, splitcC_in_gaaa(X2))
U6_GA(X1, X2, splitcC_out_gaaa(X2, X5, X3, X6)) → U8_GA(X1, X2, appendcD_in_ggga(X1, X5, X6))
U8_GA(X1, X2, appendcD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7)
PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, appendcB_out_ga(X2, X2))

The TRS R consists of the following rules:

appendcD_in_ggga(X1, X2, X3) → U18_ggga(X1, X2, X3, appendcE_in_gga(X2, X3))
appendcE_in_gga(nil, X1) → appendcE_out_gga(nil, X1, X1)
appendcE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, appendcE_in_gga(X2, X3))
U18_ggga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcD_out_ggga(X1, X2, X3, cons(X1, X4))
U17_gga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcE_out_gga(cons(X1, X2), X3, cons(X1, X4))
splitcC_in_gaaa(cons(X1, X2)) → splitcC_out_gaaa(cons(X1, X2), nil, X1, X2)
splitcC_in_gaaa(cons(X1, X2)) → U16_gaaa(X1, X2, splitcC_in_gaaa(X2))
U16_gaaa(X1, X2, splitcC_out_gaaa(X2, X3, X4, X5)) → splitcC_out_gaaa(cons(X1, X2), cons(X1, X3), X4, X5)

The set Q consists of the following terms:

appendcB_in_ga(x0)
splitcC_in_gaaa(x0)
U16_gaaa(x0, x1, x2)
appendcD_in_ggga(x0, x1, x2)
appendcE_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.

(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].

appendcB_in_ga(x0)

(29) Obligation:

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

U3_GA(X1, X2, appendcB_out_ga(X2, X4)) → PERMA_IN_GA(X4)
PERMA_IN_GA(cons(X1, X2)) → U6_GA(X1, X2, splitcC_in_gaaa(X2))
U6_GA(X1, X2, splitcC_out_gaaa(X2, X5, X3, X6)) → U8_GA(X1, X2, appendcD_in_ggga(X1, X5, X6))
U8_GA(X1, X2, appendcD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7)
PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, appendcB_out_ga(X2, X2))

The TRS R consists of the following rules:

appendcD_in_ggga(X1, X2, X3) → U18_ggga(X1, X2, X3, appendcE_in_gga(X2, X3))
appendcE_in_gga(nil, X1) → appendcE_out_gga(nil, X1, X1)
appendcE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, appendcE_in_gga(X2, X3))
U18_ggga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcD_out_ggga(X1, X2, X3, cons(X1, X4))
U17_gga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcE_out_gga(cons(X1, X2), X3, cons(X1, X4))
splitcC_in_gaaa(cons(X1, X2)) → splitcC_out_gaaa(cons(X1, X2), nil, X1, X2)
splitcC_in_gaaa(cons(X1, X2)) → U16_gaaa(X1, X2, splitcC_in_gaaa(X2))
U16_gaaa(X1, X2, splitcC_out_gaaa(X2, X3, X4, X5)) → splitcC_out_gaaa(cons(X1, X2), cons(X1, X3), X4, X5)

The set Q consists of the following terms:

splitcC_in_gaaa(x0)
U16_gaaa(x0, x1, x2)
appendcD_in_ggga(x0, x1, x2)
appendcE_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.

(30) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U6_GA(X1, X2, splitcC_out_gaaa(X2, X5, X3, X6)) → U8_GA(X1, X2, appendcD_in_ggga(X1, X5, X6)) at position [2] we obtained the following new rules [LPAR04]:

U6_GA(X1, X2, splitcC_out_gaaa(X2, X5, X3, X6)) → U8_GA(X1, X2, U18_ggga(X1, X5, X6, appendcE_in_gga(X5, X6)))

(31) Obligation:

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

U3_GA(X1, X2, appendcB_out_ga(X2, X4)) → PERMA_IN_GA(X4)
PERMA_IN_GA(cons(X1, X2)) → U6_GA(X1, X2, splitcC_in_gaaa(X2))
U8_GA(X1, X2, appendcD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7)
PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, appendcB_out_ga(X2, X2))
U6_GA(X1, X2, splitcC_out_gaaa(X2, X5, X3, X6)) → U8_GA(X1, X2, U18_ggga(X1, X5, X6, appendcE_in_gga(X5, X6)))

The TRS R consists of the following rules:

appendcD_in_ggga(X1, X2, X3) → U18_ggga(X1, X2, X3, appendcE_in_gga(X2, X3))
appendcE_in_gga(nil, X1) → appendcE_out_gga(nil, X1, X1)
appendcE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, appendcE_in_gga(X2, X3))
U18_ggga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcD_out_ggga(X1, X2, X3, cons(X1, X4))
U17_gga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcE_out_gga(cons(X1, X2), X3, cons(X1, X4))
splitcC_in_gaaa(cons(X1, X2)) → splitcC_out_gaaa(cons(X1, X2), nil, X1, X2)
splitcC_in_gaaa(cons(X1, X2)) → U16_gaaa(X1, X2, splitcC_in_gaaa(X2))
U16_gaaa(X1, X2, splitcC_out_gaaa(X2, X3, X4, X5)) → splitcC_out_gaaa(cons(X1, X2), cons(X1, X3), X4, X5)

The set Q consists of the following terms:

splitcC_in_gaaa(x0)
U16_gaaa(x0, x1, x2)
appendcD_in_ggga(x0, x1, x2)
appendcE_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.

(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:

U3_GA(X1, X2, appendcB_out_ga(X2, X4)) → PERMA_IN_GA(X4)
PERMA_IN_GA(cons(X1, X2)) → U6_GA(X1, X2, splitcC_in_gaaa(X2))
U8_GA(X1, X2, appendcD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7)
PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, appendcB_out_ga(X2, X2))
U6_GA(X1, X2, splitcC_out_gaaa(X2, X5, X3, X6)) → U8_GA(X1, X2, U18_ggga(X1, X5, X6, appendcE_in_gga(X5, X6)))

The TRS R consists of the following rules:

appendcE_in_gga(nil, X1) → appendcE_out_gga(nil, X1, X1)
appendcE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, appendcE_in_gga(X2, X3))
U18_ggga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcD_out_ggga(X1, X2, X3, cons(X1, X4))
U17_gga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcE_out_gga(cons(X1, X2), X3, cons(X1, X4))
splitcC_in_gaaa(cons(X1, X2)) → splitcC_out_gaaa(cons(X1, X2), nil, X1, X2)
splitcC_in_gaaa(cons(X1, X2)) → U16_gaaa(X1, X2, splitcC_in_gaaa(X2))
U16_gaaa(X1, X2, splitcC_out_gaaa(X2, X3, X4, X5)) → splitcC_out_gaaa(cons(X1, X2), cons(X1, X3), X4, X5)

The set Q consists of the following terms:

splitcC_in_gaaa(x0)
U16_gaaa(x0, x1, x2)
appendcD_in_ggga(x0, x1, x2)
appendcE_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.

(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].

appendcD_in_ggga(x0, x1, x2)

(35) Obligation:

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

U3_GA(X1, X2, appendcB_out_ga(X2, X4)) → PERMA_IN_GA(X4)
PERMA_IN_GA(cons(X1, X2)) → U6_GA(X1, X2, splitcC_in_gaaa(X2))
U8_GA(X1, X2, appendcD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7)
PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, appendcB_out_ga(X2, X2))
U6_GA(X1, X2, splitcC_out_gaaa(X2, X5, X3, X6)) → U8_GA(X1, X2, U18_ggga(X1, X5, X6, appendcE_in_gga(X5, X6)))

The TRS R consists of the following rules:

appendcE_in_gga(nil, X1) → appendcE_out_gga(nil, X1, X1)
appendcE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, appendcE_in_gga(X2, X3))
U18_ggga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcD_out_ggga(X1, X2, X3, cons(X1, X4))
U17_gga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcE_out_gga(cons(X1, X2), X3, cons(X1, X4))
splitcC_in_gaaa(cons(X1, X2)) → splitcC_out_gaaa(cons(X1, X2), nil, X1, X2)
splitcC_in_gaaa(cons(X1, X2)) → U16_gaaa(X1, X2, splitcC_in_gaaa(X2))
U16_gaaa(X1, X2, splitcC_out_gaaa(X2, X3, X4, X5)) → splitcC_out_gaaa(cons(X1, X2), cons(X1, X3), X4, X5)

The set Q consists of the following terms:

splitcC_in_gaaa(x0)
U16_gaaa(x0, x1, x2)
appendcE_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 U3_GA(X1, X2, appendcB_out_ga(X2, X4)) → PERMA_IN_GA(X4) we obtained the following new rules [LPAR04]:

U3_GA(z0, z1, appendcB_out_ga(z1, z1)) → PERMA_IN_GA(z1)

(37) Obligation:

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

PERMA_IN_GA(cons(X1, X2)) → U6_GA(X1, X2, splitcC_in_gaaa(X2))
U8_GA(X1, X2, appendcD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7)
PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, appendcB_out_ga(X2, X2))
U6_GA(X1, X2, splitcC_out_gaaa(X2, X5, X3, X6)) → U8_GA(X1, X2, U18_ggga(X1, X5, X6, appendcE_in_gga(X5, X6)))
U3_GA(z0, z1, appendcB_out_ga(z1, z1)) → PERMA_IN_GA(z1)

The TRS R consists of the following rules:

appendcE_in_gga(nil, X1) → appendcE_out_gga(nil, X1, X1)
appendcE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, appendcE_in_gga(X2, X3))
U18_ggga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcD_out_ggga(X1, X2, X3, cons(X1, X4))
U17_gga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcE_out_gga(cons(X1, X2), X3, cons(X1, X4))
splitcC_in_gaaa(cons(X1, X2)) → splitcC_out_gaaa(cons(X1, X2), nil, X1, X2)
splitcC_in_gaaa(cons(X1, X2)) → U16_gaaa(X1, X2, splitcC_in_gaaa(X2))
U16_gaaa(X1, X2, splitcC_out_gaaa(X2, X3, X4, X5)) → splitcC_out_gaaa(cons(X1, X2), cons(X1, X3), X4, X5)

The set Q consists of the following terms:

splitcC_in_gaaa(x0)
U16_gaaa(x0, x1, x2)
appendcE_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.


U6_GA(X1, X2, splitcC_out_gaaa(X2, X5, X3, X6)) → U8_GA(X1, X2, U18_ggga(X1, X5, X6, appendcE_in_gga(X5, X6)))
U3_GA(z0, z1, appendcB_out_ga(z1, z1)) → PERMA_IN_GA(z1)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(PERMA_IN_GA(x1)) = x1   
POL(U16_gaaa(x1, x2, x3)) = 1 + x3   
POL(U17_gga(x1, x2, x3, x4)) = 1 + x4   
POL(U18_ggga(x1, x2, x3, x4)) = x4   
POL(U3_GA(x1, x2, x3)) = 1 + x2   
POL(U6_GA(x1, x2, x3)) = 1 + x3   
POL(U8_GA(x1, x2, x3)) = x3   
POL(appendcB_out_ga(x1, x2)) = 0   
POL(appendcD_out_ggga(x1, x2, x3, x4)) = x4   
POL(appendcE_in_gga(x1, x2)) = 1 + x1 + x2   
POL(appendcE_out_gga(x1, x2, x3)) = 1 + x3   
POL(cons(x1, x2)) = 1 + x2   
POL(nil) = 0   
POL(splitcC_in_gaaa(x1)) = x1   
POL(splitcC_out_gaaa(x1, x2, x3, x4)) = 1 + x2 + x4   

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

splitcC_in_gaaa(cons(X1, X2)) → splitcC_out_gaaa(cons(X1, X2), nil, X1, X2)
splitcC_in_gaaa(cons(X1, X2)) → U16_gaaa(X1, X2, splitcC_in_gaaa(X2))
appendcE_in_gga(nil, X1) → appendcE_out_gga(nil, X1, X1)
appendcE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, appendcE_in_gga(X2, X3))
U18_ggga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcD_out_ggga(X1, X2, X3, cons(X1, X4))
U17_gga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcE_out_gga(cons(X1, X2), X3, cons(X1, X4))
U16_gaaa(X1, X2, splitcC_out_gaaa(X2, X3, X4, X5)) → splitcC_out_gaaa(cons(X1, X2), cons(X1, X3), X4, X5)

(39) Obligation:

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

PERMA_IN_GA(cons(X1, X2)) → U6_GA(X1, X2, splitcC_in_gaaa(X2))
U8_GA(X1, X2, appendcD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7)
PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, appendcB_out_ga(X2, X2))

The TRS R consists of the following rules:

appendcE_in_gga(nil, X1) → appendcE_out_gga(nil, X1, X1)
appendcE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, appendcE_in_gga(X2, X3))
U18_ggga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcD_out_ggga(X1, X2, X3, cons(X1, X4))
U17_gga(X1, X2, X3, appendcE_out_gga(X2, X3, X4)) → appendcE_out_gga(cons(X1, X2), X3, cons(X1, X4))
splitcC_in_gaaa(cons(X1, X2)) → splitcC_out_gaaa(cons(X1, X2), nil, X1, X2)
splitcC_in_gaaa(cons(X1, X2)) → U16_gaaa(X1, X2, splitcC_in_gaaa(X2))
U16_gaaa(X1, X2, splitcC_out_gaaa(X2, X3, X4, X5)) → splitcC_out_gaaa(cons(X1, X2), cons(X1, X3), X4, X5)

The set Q consists of the following terms:

splitcC_in_gaaa(x0)
U16_gaaa(x0, x1, x2)
appendcE_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