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

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇒, 89 ms)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇒, 21 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 4 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 (⇔, 3 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 (⇔, 118 ms)
        ↳39 QDP
        ↳40 DependencyGraphProof (⇔, 0 ms)
        ↳41 TRUE

(0) Obligation:

Clauses:

ap1(nil, X, X).
ap1(cons(H, X), Y, cons(H, Z)) :- ap1(X, Y, Z).
ap2(nil, X, X).
ap2(cons(H, X), Y, cons(H, Z)) :- ap2(X, Y, Z).
perm(nil, nil).
perm(Xs, cons(X, Ys)) :- ','(ap1(X1s, cons(X, X2s), Xs), ','(ap2(X1s, X2s, Zs), perm(Zs, Ys))).

Query: perm(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

ap1C(cons(X1, X2), X3, X4, cons(X1, X5)) :- ap1C(X2, X3, X4, X5).
ap2E(cons(X1, X2), X3, cons(X1, X4)) :- ap2E(X2, X3, X4).
permA(cons(X1, X2), cons(X1, X3)) :- ','(ap2cB(X2, X4), permA(X4, X3)).
permA(cons(X1, X2), cons(X3, X4)) :- ap1C(X5, X3, X6, X2).
permA(cons(X1, X2), cons(X3, X4)) :- ','(ap1cC(X5, X3, X6, X2), ap2E(X5, X6, X7)).
permA(cons(X1, X2), cons(X3, X4)) :- ','(ap1cC(X5, X3, X6, X2), ','(ap2cD(X1, X5, X6, X7), permA(X7, X4))).

Clauses:

permcA(nil, nil).
permcA(cons(X1, X2), cons(X1, X3)) :- ','(ap2cB(X2, X4), permcA(X4, X3)).
permcA(cons(X1, X2), cons(X3, X4)) :- ','(ap1cC(X5, X3, X6, X2), ','(ap2cD(X1, X5, X6, X7), permcA(X7, X4))).
ap1cC(nil, X1, X2, cons(X1, X2)).
ap1cC(cons(X1, X2), X3, X4, cons(X1, X5)) :- ap1cC(X2, X3, X4, X5).
ap2cE(nil, X1, X1).
ap2cE(cons(X1, X2), X3, cons(X1, X4)) :- ap2cE(X2, X3, X4).
ap2cB(X1, X1).
ap2cD(X1, X2, X3, cons(X1, X4)) :- ap2cE(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)
ap1C_in: (f,f,f,b)
ap1cC_in: (f,f,f,b)
ap2E_in: (b,b,f)
ap2cD_in: (b,b,b,f)
ap2cE_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, ap2cB_in_ga(X2, X4))
U3_GA(X1, X2, X3, ap2cB_out_ga(X2, X4)) → U4_GA(X1, X2, X3, permA_in_ga(X4, X3))
U3_GA(X1, X2, X3, ap2cB_out_ga(X2, X4)) → PERMA_IN_GA(X4, X3)
PERMA_IN_GA(cons(X1, X2), cons(X3, X4)) → U5_GA(X1, X2, X3, X4, ap1C_in_aaag(X5, X3, X6, X2))
PERMA_IN_GA(cons(X1, X2), cons(X3, X4)) → AP1C_IN_AAAG(X5, X3, X6, X2)
AP1C_IN_AAAG(cons(X1, X2), X3, X4, cons(X1, X5)) → U1_AAAG(X1, X2, X3, X4, X5, ap1C_in_aaag(X2, X3, X4, X5))
AP1C_IN_AAAG(cons(X1, X2), X3, X4, cons(X1, X5)) → AP1C_IN_AAAG(X2, X3, X4, X5)
PERMA_IN_GA(cons(X1, X2), cons(X3, X4)) → U6_GA(X1, X2, X3, X4, ap1cC_in_aaag(X5, X3, X6, X2))
U6_GA(X1, X2, X3, X4, ap1cC_out_aaag(X5, X3, X6, X2)) → U7_GA(X1, X2, X3, X4, ap2E_in_gga(X5, X6, X7))
U6_GA(X1, X2, X3, X4, ap1cC_out_aaag(X5, X3, X6, X2)) → AP2E_IN_GGA(X5, X6, X7)
AP2E_IN_GGA(cons(X1, X2), X3, cons(X1, X4)) → U2_GGA(X1, X2, X3, X4, ap2E_in_gga(X2, X3, X4))
AP2E_IN_GGA(cons(X1, X2), X3, cons(X1, X4)) → AP2E_IN_GGA(X2, X3, X4)
U6_GA(X1, X2, X3, X4, ap1cC_out_aaag(X5, X3, X6, X2)) → U8_GA(X1, X2, X3, X4, ap2cD_in_ggga(X1, X5, X6, X7))
U8_GA(X1, X2, X3, X4, ap2cD_out_ggga(X1, X5, X6, X7)) → U9_GA(X1, X2, X3, X4, permA_in_ga(X7, X4))
U8_GA(X1, X2, X3, X4, ap2cD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7, X4)

The TRS R consists of the following rules:

ap2cB_in_ga(X1, X1) → ap2cB_out_ga(X1, X1)
ap1cC_in_aaag(nil, X1, X2, cons(X1, X2)) → ap1cC_out_aaag(nil, X1, X2, cons(X1, X2))
ap1cC_in_aaag(cons(X1, X2), X3, X4, cons(X1, X5)) → U16_aaag(X1, X2, X3, X4, X5, ap1cC_in_aaag(X2, X3, X4, X5))
U16_aaag(X1, X2, X3, X4, X5, ap1cC_out_aaag(X2, X3, X4, X5)) → ap1cC_out_aaag(cons(X1, X2), X3, X4, cons(X1, X5))
ap2cD_in_ggga(X1, X2, X3, cons(X1, X4)) → U18_ggga(X1, X2, X3, X4, ap2cE_in_gga(X2, X3, X4))
ap2cE_in_gga(nil, X1, X1) → ap2cE_out_gga(nil, X1, X1)
ap2cE_in_gga(cons(X1, X2), X3, cons(X1, X4)) → U17_gga(X1, X2, X3, X4, ap2cE_in_gga(X2, X3, X4))
U17_gga(X1, X2, X3, X4, ap2cE_out_gga(X2, X3, X4)) → ap2cE_out_gga(cons(X1, X2), X3, cons(X1, X4))
U18_ggga(X1, X2, X3, X4, ap2cE_out_gga(X2, X3, X4)) → ap2cD_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)
ap2cB_in_ga(x1, x2)  =  ap2cB_in_ga(x1)
ap2cB_out_ga(x1, x2)  =  ap2cB_out_ga(x1, x2)
ap1C_in_aaag(x1, x2, x3, x4)  =  ap1C_in_aaag(x4)
ap1cC_in_aaag(x1, x2, x3, x4)  =  ap1cC_in_aaag(x4)
ap1cC_out_aaag(x1, x2, x3, x4)  =  ap1cC_out_aaag(x1, x2, x3, x4)
U16_aaag(x1, x2, x3, x4, x5, x6)  =  U16_aaag(x1, x5, x6)
ap2E_in_gga(x1, x2, x3)  =  ap2E_in_gga(x1, x2)
ap2cD_in_ggga(x1, x2, x3, x4)  =  ap2cD_in_ggga(x1, x2, x3)
U18_ggga(x1, x2, x3, x4, x5)  =  U18_ggga(x1, x2, x3, x5)
ap2cE_in_gga(x1, x2, x3)  =  ap2cE_in_gga(x1, x2)
nil  =  nil
ap2cE_out_gga(x1, x2, x3)  =  ap2cE_out_gga(x1, x2, x3)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
ap2cD_out_ggga(x1, x2, x3, x4)  =  ap2cD_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)
AP1C_IN_AAAG(x1, x2, x3, x4)  =  AP1C_IN_AAAG(x4)
U1_AAAG(x1, x2, x3, x4, x5, x6)  =  U1_AAAG(x1, x5, 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)
AP2E_IN_GGA(x1, x2, x3)  =  AP2E_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, ap2cB_in_ga(X2, X4))
U3_GA(X1, X2, X3, ap2cB_out_ga(X2, X4)) → U4_GA(X1, X2, X3, permA_in_ga(X4, X3))
U3_GA(X1, X2, X3, ap2cB_out_ga(X2, X4)) → PERMA_IN_GA(X4, X3)
PERMA_IN_GA(cons(X1, X2), cons(X3, X4)) → U5_GA(X1, X2, X3, X4, ap1C_in_aaag(X5, X3, X6, X2))
PERMA_IN_GA(cons(X1, X2), cons(X3, X4)) → AP1C_IN_AAAG(X5, X3, X6, X2)
AP1C_IN_AAAG(cons(X1, X2), X3, X4, cons(X1, X5)) → U1_AAAG(X1, X2, X3, X4, X5, ap1C_in_aaag(X2, X3, X4, X5))
AP1C_IN_AAAG(cons(X1, X2), X3, X4, cons(X1, X5)) → AP1C_IN_AAAG(X2, X3, X4, X5)
PERMA_IN_GA(cons(X1, X2), cons(X3, X4)) → U6_GA(X1, X2, X3, X4, ap1cC_in_aaag(X5, X3, X6, X2))
U6_GA(X1, X2, X3, X4, ap1cC_out_aaag(X5, X3, X6, X2)) → U7_GA(X1, X2, X3, X4, ap2E_in_gga(X5, X6, X7))
U6_GA(X1, X2, X3, X4, ap1cC_out_aaag(X5, X3, X6, X2)) → AP2E_IN_GGA(X5, X6, X7)
AP2E_IN_GGA(cons(X1, X2), X3, cons(X1, X4)) → U2_GGA(X1, X2, X3, X4, ap2E_in_gga(X2, X3, X4))
AP2E_IN_GGA(cons(X1, X2), X3, cons(X1, X4)) → AP2E_IN_GGA(X2, X3, X4)
U6_GA(X1, X2, X3, X4, ap1cC_out_aaag(X5, X3, X6, X2)) → U8_GA(X1, X2, X3, X4, ap2cD_in_ggga(X1, X5, X6, X7))
U8_GA(X1, X2, X3, X4, ap2cD_out_ggga(X1, X5, X6, X7)) → U9_GA(X1, X2, X3, X4, permA_in_ga(X7, X4))
U8_GA(X1, X2, X3, X4, ap2cD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7, X4)

The TRS R consists of the following rules:

ap2cB_in_ga(X1, X1) → ap2cB_out_ga(X1, X1)
ap1cC_in_aaag(nil, X1, X2, cons(X1, X2)) → ap1cC_out_aaag(nil, X1, X2, cons(X1, X2))
ap1cC_in_aaag(cons(X1, X2), X3, X4, cons(X1, X5)) → U16_aaag(X1, X2, X3, X4, X5, ap1cC_in_aaag(X2, X3, X4, X5))
U16_aaag(X1, X2, X3, X4, X5, ap1cC_out_aaag(X2, X3, X4, X5)) → ap1cC_out_aaag(cons(X1, X2), X3, X4, cons(X1, X5))
ap2cD_in_ggga(X1, X2, X3, cons(X1, X4)) → U18_ggga(X1, X2, X3, X4, ap2cE_in_gga(X2, X3, X4))
ap2cE_in_gga(nil, X1, X1) → ap2cE_out_gga(nil, X1, X1)
ap2cE_in_gga(cons(X1, X2), X3, cons(X1, X4)) → U17_gga(X1, X2, X3, X4, ap2cE_in_gga(X2, X3, X4))
U17_gga(X1, X2, X3, X4, ap2cE_out_gga(X2, X3, X4)) → ap2cE_out_gga(cons(X1, X2), X3, cons(X1, X4))
U18_ggga(X1, X2, X3, X4, ap2cE_out_gga(X2, X3, X4)) → ap2cD_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)
ap2cB_in_ga(x1, x2)  =  ap2cB_in_ga(x1)
ap2cB_out_ga(x1, x2)  =  ap2cB_out_ga(x1, x2)
ap1C_in_aaag(x1, x2, x3, x4)  =  ap1C_in_aaag(x4)
ap1cC_in_aaag(x1, x2, x3, x4)  =  ap1cC_in_aaag(x4)
ap1cC_out_aaag(x1, x2, x3, x4)  =  ap1cC_out_aaag(x1, x2, x3, x4)
U16_aaag(x1, x2, x3, x4, x5, x6)  =  U16_aaag(x1, x5, x6)
ap2E_in_gga(x1, x2, x3)  =  ap2E_in_gga(x1, x2)
ap2cD_in_ggga(x1, x2, x3, x4)  =  ap2cD_in_ggga(x1, x2, x3)
U18_ggga(x1, x2, x3, x4, x5)  =  U18_ggga(x1, x2, x3, x5)
ap2cE_in_gga(x1, x2, x3)  =  ap2cE_in_gga(x1, x2)
nil  =  nil
ap2cE_out_gga(x1, x2, x3)  =  ap2cE_out_gga(x1, x2, x3)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
ap2cD_out_ggga(x1, x2, x3, x4)  =  ap2cD_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)
AP1C_IN_AAAG(x1, x2, x3, x4)  =  AP1C_IN_AAAG(x4)
U1_AAAG(x1, x2, x3, x4, x5, x6)  =  U1_AAAG(x1, x5, 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)
AP2E_IN_GGA(x1, x2, x3)  =  AP2E_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:

AP2E_IN_GGA(cons(X1, X2), X3, cons(X1, X4)) → AP2E_IN_GGA(X2, X3, X4)

The TRS R consists of the following rules:

ap2cB_in_ga(X1, X1) → ap2cB_out_ga(X1, X1)
ap1cC_in_aaag(nil, X1, X2, cons(X1, X2)) → ap1cC_out_aaag(nil, X1, X2, cons(X1, X2))
ap1cC_in_aaag(cons(X1, X2), X3, X4, cons(X1, X5)) → U16_aaag(X1, X2, X3, X4, X5, ap1cC_in_aaag(X2, X3, X4, X5))
U16_aaag(X1, X2, X3, X4, X5, ap1cC_out_aaag(X2, X3, X4, X5)) → ap1cC_out_aaag(cons(X1, X2), X3, X4, cons(X1, X5))
ap2cD_in_ggga(X1, X2, X3, cons(X1, X4)) → U18_ggga(X1, X2, X3, X4, ap2cE_in_gga(X2, X3, X4))
ap2cE_in_gga(nil, X1, X1) → ap2cE_out_gga(nil, X1, X1)
ap2cE_in_gga(cons(X1, X2), X3, cons(X1, X4)) → U17_gga(X1, X2, X3, X4, ap2cE_in_gga(X2, X3, X4))
U17_gga(X1, X2, X3, X4, ap2cE_out_gga(X2, X3, X4)) → ap2cE_out_gga(cons(X1, X2), X3, cons(X1, X4))
U18_ggga(X1, X2, X3, X4, ap2cE_out_gga(X2, X3, X4)) → ap2cD_out_ggga(X1, X2, X3, cons(X1, X4))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
ap2cB_in_ga(x1, x2)  =  ap2cB_in_ga(x1)
ap2cB_out_ga(x1, x2)  =  ap2cB_out_ga(x1, x2)
ap1cC_in_aaag(x1, x2, x3, x4)  =  ap1cC_in_aaag(x4)
ap1cC_out_aaag(x1, x2, x3, x4)  =  ap1cC_out_aaag(x1, x2, x3, x4)
U16_aaag(x1, x2, x3, x4, x5, x6)  =  U16_aaag(x1, x5, x6)
ap2cD_in_ggga(x1, x2, x3, x4)  =  ap2cD_in_ggga(x1, x2, x3)
U18_ggga(x1, x2, x3, x4, x5)  =  U18_ggga(x1, x2, x3, x5)
ap2cE_in_gga(x1, x2, x3)  =  ap2cE_in_gga(x1, x2)
nil  =  nil
ap2cE_out_gga(x1, x2, x3)  =  ap2cE_out_gga(x1, x2, x3)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
ap2cD_out_ggga(x1, x2, x3, x4)  =  ap2cD_out_ggga(x1, x2, x3, x4)
AP2E_IN_GGA(x1, x2, x3)  =  AP2E_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:

AP2E_IN_GGA(cons(X1, X2), X3, cons(X1, X4)) → AP2E_IN_GGA(X2, X3, X4)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
AP2E_IN_GGA(x1, x2, x3)  =  AP2E_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:

AP2E_IN_GGA(cons(X1, X2), X3) → AP2E_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:

  • AP2E_IN_GGA(cons(X1, X2), X3) → AP2E_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:

AP1C_IN_AAAG(cons(X1, X2), X3, X4, cons(X1, X5)) → AP1C_IN_AAAG(X2, X3, X4, X5)

The TRS R consists of the following rules:

ap2cB_in_ga(X1, X1) → ap2cB_out_ga(X1, X1)
ap1cC_in_aaag(nil, X1, X2, cons(X1, X2)) → ap1cC_out_aaag(nil, X1, X2, cons(X1, X2))
ap1cC_in_aaag(cons(X1, X2), X3, X4, cons(X1, X5)) → U16_aaag(X1, X2, X3, X4, X5, ap1cC_in_aaag(X2, X3, X4, X5))
U16_aaag(X1, X2, X3, X4, X5, ap1cC_out_aaag(X2, X3, X4, X5)) → ap1cC_out_aaag(cons(X1, X2), X3, X4, cons(X1, X5))
ap2cD_in_ggga(X1, X2, X3, cons(X1, X4)) → U18_ggga(X1, X2, X3, X4, ap2cE_in_gga(X2, X3, X4))
ap2cE_in_gga(nil, X1, X1) → ap2cE_out_gga(nil, X1, X1)
ap2cE_in_gga(cons(X1, X2), X3, cons(X1, X4)) → U17_gga(X1, X2, X3, X4, ap2cE_in_gga(X2, X3, X4))
U17_gga(X1, X2, X3, X4, ap2cE_out_gga(X2, X3, X4)) → ap2cE_out_gga(cons(X1, X2), X3, cons(X1, X4))
U18_ggga(X1, X2, X3, X4, ap2cE_out_gga(X2, X3, X4)) → ap2cD_out_ggga(X1, X2, X3, cons(X1, X4))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
ap2cB_in_ga(x1, x2)  =  ap2cB_in_ga(x1)
ap2cB_out_ga(x1, x2)  =  ap2cB_out_ga(x1, x2)
ap1cC_in_aaag(x1, x2, x3, x4)  =  ap1cC_in_aaag(x4)
ap1cC_out_aaag(x1, x2, x3, x4)  =  ap1cC_out_aaag(x1, x2, x3, x4)
U16_aaag(x1, x2, x3, x4, x5, x6)  =  U16_aaag(x1, x5, x6)
ap2cD_in_ggga(x1, x2, x3, x4)  =  ap2cD_in_ggga(x1, x2, x3)
U18_ggga(x1, x2, x3, x4, x5)  =  U18_ggga(x1, x2, x3, x5)
ap2cE_in_gga(x1, x2, x3)  =  ap2cE_in_gga(x1, x2)
nil  =  nil
ap2cE_out_gga(x1, x2, x3)  =  ap2cE_out_gga(x1, x2, x3)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
ap2cD_out_ggga(x1, x2, x3, x4)  =  ap2cD_out_ggga(x1, x2, x3, x4)
AP1C_IN_AAAG(x1, x2, x3, x4)  =  AP1C_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:

AP1C_IN_AAAG(cons(X1, X2), X3, X4, cons(X1, X5)) → AP1C_IN_AAAG(X2, X3, X4, X5)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
AP1C_IN_AAAG(x1, x2, x3, x4)  =  AP1C_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:

AP1C_IN_AAAG(cons(X1, X5)) → AP1C_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:

  • AP1C_IN_AAAG(cons(X1, X5)) → AP1C_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:

U3_GA(X1, X2, X3, ap2cB_out_ga(X2, X4)) → PERMA_IN_GA(X4, X3)
PERMA_IN_GA(cons(X1, X2), cons(X1, X3)) → U3_GA(X1, X2, X3, ap2cB_in_ga(X2, X4))
PERMA_IN_GA(cons(X1, X2), cons(X3, X4)) → U6_GA(X1, X2, X3, X4, ap1cC_in_aaag(X5, X3, X6, X2))
U6_GA(X1, X2, X3, X4, ap1cC_out_aaag(X5, X3, X6, X2)) → U8_GA(X1, X2, X3, X4, ap2cD_in_ggga(X1, X5, X6, X7))
U8_GA(X1, X2, X3, X4, ap2cD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7, X4)

The TRS R consists of the following rules:

ap2cB_in_ga(X1, X1) → ap2cB_out_ga(X1, X1)
ap1cC_in_aaag(nil, X1, X2, cons(X1, X2)) → ap1cC_out_aaag(nil, X1, X2, cons(X1, X2))
ap1cC_in_aaag(cons(X1, X2), X3, X4, cons(X1, X5)) → U16_aaag(X1, X2, X3, X4, X5, ap1cC_in_aaag(X2, X3, X4, X5))
U16_aaag(X1, X2, X3, X4, X5, ap1cC_out_aaag(X2, X3, X4, X5)) → ap1cC_out_aaag(cons(X1, X2), X3, X4, cons(X1, X5))
ap2cD_in_ggga(X1, X2, X3, cons(X1, X4)) → U18_ggga(X1, X2, X3, X4, ap2cE_in_gga(X2, X3, X4))
ap2cE_in_gga(nil, X1, X1) → ap2cE_out_gga(nil, X1, X1)
ap2cE_in_gga(cons(X1, X2), X3, cons(X1, X4)) → U17_gga(X1, X2, X3, X4, ap2cE_in_gga(X2, X3, X4))
U17_gga(X1, X2, X3, X4, ap2cE_out_gga(X2, X3, X4)) → ap2cE_out_gga(cons(X1, X2), X3, cons(X1, X4))
U18_ggga(X1, X2, X3, X4, ap2cE_out_gga(X2, X3, X4)) → ap2cD_out_ggga(X1, X2, X3, cons(X1, X4))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
ap2cB_in_ga(x1, x2)  =  ap2cB_in_ga(x1)
ap2cB_out_ga(x1, x2)  =  ap2cB_out_ga(x1, x2)
ap1cC_in_aaag(x1, x2, x3, x4)  =  ap1cC_in_aaag(x4)
ap1cC_out_aaag(x1, x2, x3, x4)  =  ap1cC_out_aaag(x1, x2, x3, x4)
U16_aaag(x1, x2, x3, x4, x5, x6)  =  U16_aaag(x1, x5, x6)
ap2cD_in_ggga(x1, x2, x3, x4)  =  ap2cD_in_ggga(x1, x2, x3)
U18_ggga(x1, x2, x3, x4, x5)  =  U18_ggga(x1, x2, x3, x5)
ap2cE_in_gga(x1, x2, x3)  =  ap2cE_in_gga(x1, x2)
nil  =  nil
ap2cE_out_gga(x1, x2, x3)  =  ap2cE_out_gga(x1, x2, x3)
U17_gga(x1, x2, x3, x4, x5)  =  U17_gga(x1, x2, x3, x5)
ap2cD_out_ggga(x1, x2, x3, x4)  =  ap2cD_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, ap2cB_out_ga(X2, X4)) → PERMA_IN_GA(X4)
PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, ap2cB_in_ga(X2))
PERMA_IN_GA(cons(X1, X2)) → U6_GA(X1, X2, ap1cC_in_aaag(X2))
U6_GA(X1, X2, ap1cC_out_aaag(X5, X3, X6, X2)) → U8_GA(X1, X2, ap2cD_in_ggga(X1, X5, X6))
U8_GA(X1, X2, ap2cD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7)

The TRS R consists of the following rules:

ap2cB_in_ga(X1) → ap2cB_out_ga(X1, X1)
ap1cC_in_aaag(cons(X1, X2)) → ap1cC_out_aaag(nil, X1, X2, cons(X1, X2))
ap1cC_in_aaag(cons(X1, X5)) → U16_aaag(X1, X5, ap1cC_in_aaag(X5))
U16_aaag(X1, X5, ap1cC_out_aaag(X2, X3, X4, X5)) → ap1cC_out_aaag(cons(X1, X2), X3, X4, cons(X1, X5))
ap2cD_in_ggga(X1, X2, X3) → U18_ggga(X1, X2, X3, ap2cE_in_gga(X2, X3))
ap2cE_in_gga(nil, X1) → ap2cE_out_gga(nil, X1, X1)
ap2cE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, ap2cE_in_gga(X2, X3))
U17_gga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cE_out_gga(cons(X1, X2), X3, cons(X1, X4))
U18_ggga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cD_out_ggga(X1, X2, X3, cons(X1, X4))

The set Q consists of the following terms:

ap2cB_in_ga(x0)
ap1cC_in_aaag(x0)
U16_aaag(x0, x1, x2)
ap2cD_in_ggga(x0, x1, x2)
ap2cE_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, ap2cB_in_ga(X2)) at position [2] we obtained the following new rules [LPAR04]:

PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, ap2cB_out_ga(X2, X2))

(25) Obligation:

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

U3_GA(X1, X2, ap2cB_out_ga(X2, X4)) → PERMA_IN_GA(X4)
PERMA_IN_GA(cons(X1, X2)) → U6_GA(X1, X2, ap1cC_in_aaag(X2))
U6_GA(X1, X2, ap1cC_out_aaag(X5, X3, X6, X2)) → U8_GA(X1, X2, ap2cD_in_ggga(X1, X5, X6))
U8_GA(X1, X2, ap2cD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7)
PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, ap2cB_out_ga(X2, X2))

The TRS R consists of the following rules:

ap2cB_in_ga(X1) → ap2cB_out_ga(X1, X1)
ap1cC_in_aaag(cons(X1, X2)) → ap1cC_out_aaag(nil, X1, X2, cons(X1, X2))
ap1cC_in_aaag(cons(X1, X5)) → U16_aaag(X1, X5, ap1cC_in_aaag(X5))
U16_aaag(X1, X5, ap1cC_out_aaag(X2, X3, X4, X5)) → ap1cC_out_aaag(cons(X1, X2), X3, X4, cons(X1, X5))
ap2cD_in_ggga(X1, X2, X3) → U18_ggga(X1, X2, X3, ap2cE_in_gga(X2, X3))
ap2cE_in_gga(nil, X1) → ap2cE_out_gga(nil, X1, X1)
ap2cE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, ap2cE_in_gga(X2, X3))
U17_gga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cE_out_gga(cons(X1, X2), X3, cons(X1, X4))
U18_ggga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cD_out_ggga(X1, X2, X3, cons(X1, X4))

The set Q consists of the following terms:

ap2cB_in_ga(x0)
ap1cC_in_aaag(x0)
U16_aaag(x0, x1, x2)
ap2cD_in_ggga(x0, x1, x2)
ap2cE_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, ap2cB_out_ga(X2, X4)) → PERMA_IN_GA(X4)
PERMA_IN_GA(cons(X1, X2)) → U6_GA(X1, X2, ap1cC_in_aaag(X2))
U6_GA(X1, X2, ap1cC_out_aaag(X5, X3, X6, X2)) → U8_GA(X1, X2, ap2cD_in_ggga(X1, X5, X6))
U8_GA(X1, X2, ap2cD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7)
PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, ap2cB_out_ga(X2, X2))

The TRS R consists of the following rules:

ap2cD_in_ggga(X1, X2, X3) → U18_ggga(X1, X2, X3, ap2cE_in_gga(X2, X3))
ap2cE_in_gga(nil, X1) → ap2cE_out_gga(nil, X1, X1)
ap2cE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, ap2cE_in_gga(X2, X3))
U18_ggga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cD_out_ggga(X1, X2, X3, cons(X1, X4))
U17_gga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cE_out_gga(cons(X1, X2), X3, cons(X1, X4))
ap1cC_in_aaag(cons(X1, X2)) → ap1cC_out_aaag(nil, X1, X2, cons(X1, X2))
ap1cC_in_aaag(cons(X1, X5)) → U16_aaag(X1, X5, ap1cC_in_aaag(X5))
U16_aaag(X1, X5, ap1cC_out_aaag(X2, X3, X4, X5)) → ap1cC_out_aaag(cons(X1, X2), X3, X4, cons(X1, X5))

The set Q consists of the following terms:

ap2cB_in_ga(x0)
ap1cC_in_aaag(x0)
U16_aaag(x0, x1, x2)
ap2cD_in_ggga(x0, x1, x2)
ap2cE_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].

ap2cB_in_ga(x0)

(29) Obligation:

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

U3_GA(X1, X2, ap2cB_out_ga(X2, X4)) → PERMA_IN_GA(X4)
PERMA_IN_GA(cons(X1, X2)) → U6_GA(X1, X2, ap1cC_in_aaag(X2))
U6_GA(X1, X2, ap1cC_out_aaag(X5, X3, X6, X2)) → U8_GA(X1, X2, ap2cD_in_ggga(X1, X5, X6))
U8_GA(X1, X2, ap2cD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7)
PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, ap2cB_out_ga(X2, X2))

The TRS R consists of the following rules:

ap2cD_in_ggga(X1, X2, X3) → U18_ggga(X1, X2, X3, ap2cE_in_gga(X2, X3))
ap2cE_in_gga(nil, X1) → ap2cE_out_gga(nil, X1, X1)
ap2cE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, ap2cE_in_gga(X2, X3))
U18_ggga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cD_out_ggga(X1, X2, X3, cons(X1, X4))
U17_gga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cE_out_gga(cons(X1, X2), X3, cons(X1, X4))
ap1cC_in_aaag(cons(X1, X2)) → ap1cC_out_aaag(nil, X1, X2, cons(X1, X2))
ap1cC_in_aaag(cons(X1, X5)) → U16_aaag(X1, X5, ap1cC_in_aaag(X5))
U16_aaag(X1, X5, ap1cC_out_aaag(X2, X3, X4, X5)) → ap1cC_out_aaag(cons(X1, X2), X3, X4, cons(X1, X5))

The set Q consists of the following terms:

ap1cC_in_aaag(x0)
U16_aaag(x0, x1, x2)
ap2cD_in_ggga(x0, x1, x2)
ap2cE_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, ap1cC_out_aaag(X5, X3, X6, X2)) → U8_GA(X1, X2, ap2cD_in_ggga(X1, X5, X6)) at position [2] we obtained the following new rules [LPAR04]:

U6_GA(X1, X2, ap1cC_out_aaag(X5, X3, X6, X2)) → U8_GA(X1, X2, U18_ggga(X1, X5, X6, ap2cE_in_gga(X5, X6)))

(31) Obligation:

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

U3_GA(X1, X2, ap2cB_out_ga(X2, X4)) → PERMA_IN_GA(X4)
PERMA_IN_GA(cons(X1, X2)) → U6_GA(X1, X2, ap1cC_in_aaag(X2))
U8_GA(X1, X2, ap2cD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7)
PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, ap2cB_out_ga(X2, X2))
U6_GA(X1, X2, ap1cC_out_aaag(X5, X3, X6, X2)) → U8_GA(X1, X2, U18_ggga(X1, X5, X6, ap2cE_in_gga(X5, X6)))

The TRS R consists of the following rules:

ap2cD_in_ggga(X1, X2, X3) → U18_ggga(X1, X2, X3, ap2cE_in_gga(X2, X3))
ap2cE_in_gga(nil, X1) → ap2cE_out_gga(nil, X1, X1)
ap2cE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, ap2cE_in_gga(X2, X3))
U18_ggga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cD_out_ggga(X1, X2, X3, cons(X1, X4))
U17_gga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cE_out_gga(cons(X1, X2), X3, cons(X1, X4))
ap1cC_in_aaag(cons(X1, X2)) → ap1cC_out_aaag(nil, X1, X2, cons(X1, X2))
ap1cC_in_aaag(cons(X1, X5)) → U16_aaag(X1, X5, ap1cC_in_aaag(X5))
U16_aaag(X1, X5, ap1cC_out_aaag(X2, X3, X4, X5)) → ap1cC_out_aaag(cons(X1, X2), X3, X4, cons(X1, X5))

The set Q consists of the following terms:

ap1cC_in_aaag(x0)
U16_aaag(x0, x1, x2)
ap2cD_in_ggga(x0, x1, x2)
ap2cE_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, ap2cB_out_ga(X2, X4)) → PERMA_IN_GA(X4)
PERMA_IN_GA(cons(X1, X2)) → U6_GA(X1, X2, ap1cC_in_aaag(X2))
U8_GA(X1, X2, ap2cD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7)
PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, ap2cB_out_ga(X2, X2))
U6_GA(X1, X2, ap1cC_out_aaag(X5, X3, X6, X2)) → U8_GA(X1, X2, U18_ggga(X1, X5, X6, ap2cE_in_gga(X5, X6)))

The TRS R consists of the following rules:

ap2cE_in_gga(nil, X1) → ap2cE_out_gga(nil, X1, X1)
ap2cE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, ap2cE_in_gga(X2, X3))
U18_ggga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cD_out_ggga(X1, X2, X3, cons(X1, X4))
U17_gga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cE_out_gga(cons(X1, X2), X3, cons(X1, X4))
ap1cC_in_aaag(cons(X1, X2)) → ap1cC_out_aaag(nil, X1, X2, cons(X1, X2))
ap1cC_in_aaag(cons(X1, X5)) → U16_aaag(X1, X5, ap1cC_in_aaag(X5))
U16_aaag(X1, X5, ap1cC_out_aaag(X2, X3, X4, X5)) → ap1cC_out_aaag(cons(X1, X2), X3, X4, cons(X1, X5))

The set Q consists of the following terms:

ap1cC_in_aaag(x0)
U16_aaag(x0, x1, x2)
ap2cD_in_ggga(x0, x1, x2)
ap2cE_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].

ap2cD_in_ggga(x0, x1, x2)

(35) Obligation:

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

U3_GA(X1, X2, ap2cB_out_ga(X2, X4)) → PERMA_IN_GA(X4)
PERMA_IN_GA(cons(X1, X2)) → U6_GA(X1, X2, ap1cC_in_aaag(X2))
U8_GA(X1, X2, ap2cD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7)
PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, ap2cB_out_ga(X2, X2))
U6_GA(X1, X2, ap1cC_out_aaag(X5, X3, X6, X2)) → U8_GA(X1, X2, U18_ggga(X1, X5, X6, ap2cE_in_gga(X5, X6)))

The TRS R consists of the following rules:

ap2cE_in_gga(nil, X1) → ap2cE_out_gga(nil, X1, X1)
ap2cE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, ap2cE_in_gga(X2, X3))
U18_ggga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cD_out_ggga(X1, X2, X3, cons(X1, X4))
U17_gga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cE_out_gga(cons(X1, X2), X3, cons(X1, X4))
ap1cC_in_aaag(cons(X1, X2)) → ap1cC_out_aaag(nil, X1, X2, cons(X1, X2))
ap1cC_in_aaag(cons(X1, X5)) → U16_aaag(X1, X5, ap1cC_in_aaag(X5))
U16_aaag(X1, X5, ap1cC_out_aaag(X2, X3, X4, X5)) → ap1cC_out_aaag(cons(X1, X2), X3, X4, cons(X1, X5))

The set Q consists of the following terms:

ap1cC_in_aaag(x0)
U16_aaag(x0, x1, x2)
ap2cE_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, ap2cB_out_ga(X2, X4)) → PERMA_IN_GA(X4) we obtained the following new rules [LPAR04]:

U3_GA(z0, z1, ap2cB_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, ap1cC_in_aaag(X2))
U8_GA(X1, X2, ap2cD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7)
PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, ap2cB_out_ga(X2, X2))
U6_GA(X1, X2, ap1cC_out_aaag(X5, X3, X6, X2)) → U8_GA(X1, X2, U18_ggga(X1, X5, X6, ap2cE_in_gga(X5, X6)))
U3_GA(z0, z1, ap2cB_out_ga(z1, z1)) → PERMA_IN_GA(z1)

The TRS R consists of the following rules:

ap2cE_in_gga(nil, X1) → ap2cE_out_gga(nil, X1, X1)
ap2cE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, ap2cE_in_gga(X2, X3))
U18_ggga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cD_out_ggga(X1, X2, X3, cons(X1, X4))
U17_gga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cE_out_gga(cons(X1, X2), X3, cons(X1, X4))
ap1cC_in_aaag(cons(X1, X2)) → ap1cC_out_aaag(nil, X1, X2, cons(X1, X2))
ap1cC_in_aaag(cons(X1, X5)) → U16_aaag(X1, X5, ap1cC_in_aaag(X5))
U16_aaag(X1, X5, ap1cC_out_aaag(X2, X3, X4, X5)) → ap1cC_out_aaag(cons(X1, X2), X3, X4, cons(X1, X5))

The set Q consists of the following terms:

ap1cC_in_aaag(x0)
U16_aaag(x0, x1, x2)
ap2cE_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, ap1cC_out_aaag(X5, X3, X6, X2)) → U8_GA(X1, X2, U18_ggga(X1, X5, X6, ap2cE_in_gga(X5, X6)))
U3_GA(z0, z1, ap2cB_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_aaag(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)) = x2 + x3   
POL(U6_GA(x1, x2, x3)) = 1 + x3   
POL(U8_GA(x1, x2, x3)) = x3   
POL(ap1cC_in_aaag(x1)) = x1   
POL(ap1cC_out_aaag(x1, x2, x3, x4)) = 1 + x1 + x3   
POL(ap2cB_out_ga(x1, x2)) = 1   
POL(ap2cD_out_ggga(x1, x2, x3, x4)) = x4   
POL(ap2cE_in_gga(x1, x2)) = 1 + x1 + x2   
POL(ap2cE_out_gga(x1, x2, x3)) = 1 + x3   
POL(cons(x1, x2)) = 1 + x2   
POL(nil) = 0   

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

ap1cC_in_aaag(cons(X1, X2)) → ap1cC_out_aaag(nil, X1, X2, cons(X1, X2))
ap1cC_in_aaag(cons(X1, X5)) → U16_aaag(X1, X5, ap1cC_in_aaag(X5))
ap2cE_in_gga(nil, X1) → ap2cE_out_gga(nil, X1, X1)
ap2cE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, ap2cE_in_gga(X2, X3))
U18_ggga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cD_out_ggga(X1, X2, X3, cons(X1, X4))
U17_gga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cE_out_gga(cons(X1, X2), X3, cons(X1, X4))
U16_aaag(X1, X5, ap1cC_out_aaag(X2, X3, X4, X5)) → ap1cC_out_aaag(cons(X1, X2), X3, X4, cons(X1, X5))

(39) Obligation:

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

PERMA_IN_GA(cons(X1, X2)) → U6_GA(X1, X2, ap1cC_in_aaag(X2))
U8_GA(X1, X2, ap2cD_out_ggga(X1, X5, X6, X7)) → PERMA_IN_GA(X7)
PERMA_IN_GA(cons(X1, X2)) → U3_GA(X1, X2, ap2cB_out_ga(X2, X2))

The TRS R consists of the following rules:

ap2cE_in_gga(nil, X1) → ap2cE_out_gga(nil, X1, X1)
ap2cE_in_gga(cons(X1, X2), X3) → U17_gga(X1, X2, X3, ap2cE_in_gga(X2, X3))
U18_ggga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cD_out_ggga(X1, X2, X3, cons(X1, X4))
U17_gga(X1, X2, X3, ap2cE_out_gga(X2, X3, X4)) → ap2cE_out_gga(cons(X1, X2), X3, cons(X1, X4))
ap1cC_in_aaag(cons(X1, X2)) → ap1cC_out_aaag(nil, X1, X2, cons(X1, X2))
ap1cC_in_aaag(cons(X1, X5)) → U16_aaag(X1, X5, ap1cC_in_aaag(X5))
U16_aaag(X1, X5, ap1cC_out_aaag(X2, X3, X4, X5)) → ap1cC_out_aaag(cons(X1, X2), X3, X4, cons(X1, X5))

The set Q consists of the following terms:

ap1cC_in_aaag(x0)
U16_aaag(x0, x1, x2)
ap2cE_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