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

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇒, 86 ms)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇒, 197 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 (⇒, 10 ms)
        ↳23 QDP
        ↳24 QDPSizeChangeProof (⇔, 0 ms)
        ↳25 YES

(0) Obligation:

Clauses:

inorder(nil, []).
inorder(tree(L, V, R), I) :- ','(inorder(L, LI), ','(inorder(R, RI), append(LI, .(V, RI), I))).
append([], X, X).
append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs).

Query: inorder(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

inorderA(tree(X1, X2, X3), X4) :- inorderA(X1, X5).
inorderA(tree(X1, X2, X3), X4) :- ','(inordercA(X1, X5), inorderA(X3, X6)).
inorderA(tree(X1, X2, X3), X4) :- ','(inordercA(X1, X5), ','(inordercA(X3, X6), appendB(X5, X2, X6, X4))).
appendB(.(X1, X2), X3, X4, .(X1, X5)) :- appendB(X2, X3, X4, X5).
appendC(.(X1, X2), X3, X4, .(X1, X5)) :- appendC(X2, X3, X4, X5).
inorderD(tree(nil, X1, X2), X3) :- inorderA(X2, X4).
inorderD(tree(tree(X1, X2, X3), X4, X5), X6) :- inorderA(X1, X7).
inorderD(tree(tree(X1, X2, X3), X4, X5), X6) :- ','(inordercA(X1, X7), inorderA(X3, X8)).
inorderD(tree(tree(X1, X2, X3), X4, X5), X6) :- ','(inordercA(X1, X7), ','(inordercA(X3, X8), appendB(X7, X2, X8, X9))).
inorderD(tree(tree(X1, X2, X3), X4, X5), X6) :- ','(inordercA(X1, X7), ','(inordercA(X3, X8), ','(appendcB(X7, X2, X8, X9), inorderA(X5, X10)))).
inorderD(tree(tree(X1, X2, X3), X4, X5), X6) :- ','(inordercA(X1, X7), ','(inordercA(X3, X8), ','(appendcB(X7, X2, X8, X9), ','(inordercA(X5, X10), appendC(X9, X4, X10, X6))))).

Clauses:

inordercA(nil, []).
inordercA(tree(X1, X2, X3), X4) :- ','(inordercA(X1, X5), ','(inordercA(X3, X6), appendcB(X5, X2, X6, X4))).
appendcB([], X1, X2, .(X1, X2)).
appendcB(.(X1, X2), X3, X4, .(X1, X5)) :- appendcB(X2, X3, X4, X5).
appendcC([], X1, X2, .(X1, X2)).
appendcC(.(X1, X2), X3, X4, .(X1, X5)) :- appendcC(X2, X3, X4, X5).

Afs:

inorderD(x1, x2)  =  inorderD(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:
inorderD_in: (b,f)
inorderA_in: (b,f)
inordercA_in: (b,f)
appendcB_in: (b,b,b,f)
appendB_in: (b,b,b,f)
appendC_in: (b,b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

INORDERD_IN_GA(tree(nil, X1, X2), X3) → U8_GA(X1, X2, X3, inorderA_in_ga(X2, X4))
INORDERD_IN_GA(tree(nil, X1, X2), X3) → INORDERA_IN_GA(X2, X4)
INORDERA_IN_GA(tree(X1, X2, X3), X4) → U1_GA(X1, X2, X3, X4, inorderA_in_ga(X1, X5))
INORDERA_IN_GA(tree(X1, X2, X3), X4) → INORDERA_IN_GA(X1, X5)
INORDERA_IN_GA(tree(X1, X2, X3), X4) → U2_GA(X1, X2, X3, X4, inordercA_in_ga(X1, X5))
U2_GA(X1, X2, X3, X4, inordercA_out_ga(X1, X5)) → U3_GA(X1, X2, X3, X4, inorderA_in_ga(X3, X6))
U2_GA(X1, X2, X3, X4, inordercA_out_ga(X1, X5)) → INORDERA_IN_GA(X3, X6)
U2_GA(X1, X2, X3, X4, inordercA_out_ga(X1, X5)) → U4_GA(X1, X2, X3, X4, X5, inordercA_in_ga(X3, X6))
U4_GA(X1, X2, X3, X4, X5, inordercA_out_ga(X3, X6)) → U5_GA(X1, X2, X3, X4, appendB_in_ggga(X5, X2, X6, X4))
U4_GA(X1, X2, X3, X4, X5, inordercA_out_ga(X3, X6)) → APPENDB_IN_GGGA(X5, X2, X6, X4)
APPENDB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → U6_GGGA(X1, X2, X3, X4, X5, appendB_in_ggga(X2, X3, X4, X5))
APPENDB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPENDB_IN_GGGA(X2, X3, X4, X5)
INORDERD_IN_GA(tree(tree(X1, X2, X3), X4, X5), X6) → U9_GA(X1, X2, X3, X4, X5, X6, inorderA_in_ga(X1, X7))
INORDERD_IN_GA(tree(tree(X1, X2, X3), X4, X5), X6) → INORDERA_IN_GA(X1, X7)
INORDERD_IN_GA(tree(tree(X1, X2, X3), X4, X5), X6) → U10_GA(X1, X2, X3, X4, X5, X6, inordercA_in_ga(X1, X7))
U10_GA(X1, X2, X3, X4, X5, X6, inordercA_out_ga(X1, X7)) → U11_GA(X1, X2, X3, X4, X5, X6, inorderA_in_ga(X3, X8))
U10_GA(X1, X2, X3, X4, X5, X6, inordercA_out_ga(X1, X7)) → INORDERA_IN_GA(X3, X8)
U10_GA(X1, X2, X3, X4, X5, X6, inordercA_out_ga(X1, X7)) → U12_GA(X1, X2, X3, X4, X5, X6, X7, inordercA_in_ga(X3, X8))
U12_GA(X1, X2, X3, X4, X5, X6, X7, inordercA_out_ga(X3, X8)) → U13_GA(X1, X2, X3, X4, X5, X6, appendB_in_ggga(X7, X2, X8, X9))
U12_GA(X1, X2, X3, X4, X5, X6, X7, inordercA_out_ga(X3, X8)) → APPENDB_IN_GGGA(X7, X2, X8, X9)
U12_GA(X1, X2, X3, X4, X5, X6, X7, inordercA_out_ga(X3, X8)) → U14_GA(X1, X2, X3, X4, X5, X6, appendcB_in_ggga(X7, X2, X8, X9))
U14_GA(X1, X2, X3, X4, X5, X6, appendcB_out_ggga(X7, X2, X8, X9)) → U15_GA(X1, X2, X3, X4, X5, X6, inorderA_in_ga(X5, X10))
U14_GA(X1, X2, X3, X4, X5, X6, appendcB_out_ggga(X7, X2, X8, X9)) → INORDERA_IN_GA(X5, X10)
U14_GA(X1, X2, X3, X4, X5, X6, appendcB_out_ggga(X7, X2, X8, X9)) → U16_GA(X1, X2, X3, X4, X5, X6, X9, inordercA_in_ga(X5, X10))
U16_GA(X1, X2, X3, X4, X5, X6, X9, inordercA_out_ga(X5, X10)) → U17_GA(X1, X2, X3, X4, X5, X6, appendC_in_ggga(X9, X4, X10, X6))
U16_GA(X1, X2, X3, X4, X5, X6, X9, inordercA_out_ga(X5, X10)) → APPENDC_IN_GGGA(X9, X4, X10, X6)
APPENDC_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → U7_GGGA(X1, X2, X3, X4, X5, appendC_in_ggga(X2, X3, X4, X5))
APPENDC_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPENDC_IN_GGGA(X2, X3, X4, X5)

The TRS R consists of the following rules:

inordercA_in_ga(nil, []) → inordercA_out_ga(nil, [])
inordercA_in_ga(tree(X1, X2, X3), X4) → U19_ga(X1, X2, X3, X4, inordercA_in_ga(X1, X5))
U19_ga(X1, X2, X3, X4, inordercA_out_ga(X1, X5)) → U20_ga(X1, X2, X3, X4, X5, inordercA_in_ga(X3, X6))
U20_ga(X1, X2, X3, X4, X5, inordercA_out_ga(X3, X6)) → U21_ga(X1, X2, X3, X4, appendcB_in_ggga(X5, X2, X6, X4))
appendcB_in_ggga([], X1, X2, .(X1, X2)) → appendcB_out_ggga([], X1, X2, .(X1, X2))
appendcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U22_ggga(X1, X2, X3, X4, X5, appendcB_in_ggga(X2, X3, X4, X5))
U22_ggga(X1, X2, X3, X4, X5, appendcB_out_ggga(X2, X3, X4, X5)) → appendcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U21_ga(X1, X2, X3, X4, appendcB_out_ggga(X5, X2, X6, X4)) → inordercA_out_ga(tree(X1, X2, X3), X4)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
nil  =  nil
inorderA_in_ga(x1, x2)  =  inorderA_in_ga(x1)
inordercA_in_ga(x1, x2)  =  inordercA_in_ga(x1)
inordercA_out_ga(x1, x2)  =  inordercA_out_ga(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5, x6)  =  U20_ga(x1, x2, x3, x5, x6)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
appendcB_in_ggga(x1, x2, x3, x4)  =  appendcB_in_ggga(x1, x2, x3)
[]  =  []
appendcB_out_ggga(x1, x2, x3, x4)  =  appendcB_out_ggga(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
U22_ggga(x1, x2, x3, x4, x5, x6)  =  U22_ggga(x1, x2, x3, x4, x6)
appendB_in_ggga(x1, x2, x3, x4)  =  appendB_in_ggga(x1, x2, x3)
appendC_in_ggga(x1, x2, x3, x4)  =  appendC_in_ggga(x1, x2, x3)
INORDERD_IN_GA(x1, x2)  =  INORDERD_IN_GA(x1)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x2, x4)
INORDERA_IN_GA(x1, x2)  =  INORDERA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x3, x5)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x2, x3, x5, x6)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x3, x5)
APPENDB_IN_GGGA(x1, x2, x3, x4)  =  APPENDB_IN_GGGA(x1, x2, x3)
U6_GGGA(x1, x2, x3, x4, x5, x6)  =  U6_GGGA(x1, x2, x3, x4, x6)
U9_GA(x1, x2, x3, x4, x5, x6, x7)  =  U9_GA(x1, x2, x3, x4, x5, x7)
U10_GA(x1, x2, x3, x4, x5, x6, x7)  =  U10_GA(x1, x2, x3, x4, x5, x7)
U11_GA(x1, x2, x3, x4, x5, x6, x7)  =  U11_GA(x1, x2, x3, x4, x5, x7)
U12_GA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_GA(x1, x2, x3, x4, x5, x7, x8)
U13_GA(x1, x2, x3, x4, x5, x6, x7)  =  U13_GA(x1, x2, x3, x4, x5, x7)
U14_GA(x1, x2, x3, x4, x5, x6, x7)  =  U14_GA(x1, x2, x3, x4, x5, x7)
U15_GA(x1, x2, x3, x4, x5, x6, x7)  =  U15_GA(x1, x2, x3, x4, x5, x7)
U16_GA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_GA(x1, x2, x3, x4, x5, x7, x8)
U17_GA(x1, x2, x3, x4, x5, x6, x7)  =  U17_GA(x1, x2, x3, x4, x5, x7)
APPENDC_IN_GGGA(x1, x2, x3, x4)  =  APPENDC_IN_GGGA(x1, x2, x3)
U7_GGGA(x1, x2, x3, x4, x5, x6)  =  U7_GGGA(x1, x2, x3, x4, x6)

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:

INORDERD_IN_GA(tree(nil, X1, X2), X3) → U8_GA(X1, X2, X3, inorderA_in_ga(X2, X4))
INORDERD_IN_GA(tree(nil, X1, X2), X3) → INORDERA_IN_GA(X2, X4)
INORDERA_IN_GA(tree(X1, X2, X3), X4) → U1_GA(X1, X2, X3, X4, inorderA_in_ga(X1, X5))
INORDERA_IN_GA(tree(X1, X2, X3), X4) → INORDERA_IN_GA(X1, X5)
INORDERA_IN_GA(tree(X1, X2, X3), X4) → U2_GA(X1, X2, X3, X4, inordercA_in_ga(X1, X5))
U2_GA(X1, X2, X3, X4, inordercA_out_ga(X1, X5)) → U3_GA(X1, X2, X3, X4, inorderA_in_ga(X3, X6))
U2_GA(X1, X2, X3, X4, inordercA_out_ga(X1, X5)) → INORDERA_IN_GA(X3, X6)
U2_GA(X1, X2, X3, X4, inordercA_out_ga(X1, X5)) → U4_GA(X1, X2, X3, X4, X5, inordercA_in_ga(X3, X6))
U4_GA(X1, X2, X3, X4, X5, inordercA_out_ga(X3, X6)) → U5_GA(X1, X2, X3, X4, appendB_in_ggga(X5, X2, X6, X4))
U4_GA(X1, X2, X3, X4, X5, inordercA_out_ga(X3, X6)) → APPENDB_IN_GGGA(X5, X2, X6, X4)
APPENDB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → U6_GGGA(X1, X2, X3, X4, X5, appendB_in_ggga(X2, X3, X4, X5))
APPENDB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPENDB_IN_GGGA(X2, X3, X4, X5)
INORDERD_IN_GA(tree(tree(X1, X2, X3), X4, X5), X6) → U9_GA(X1, X2, X3, X4, X5, X6, inorderA_in_ga(X1, X7))
INORDERD_IN_GA(tree(tree(X1, X2, X3), X4, X5), X6) → INORDERA_IN_GA(X1, X7)
INORDERD_IN_GA(tree(tree(X1, X2, X3), X4, X5), X6) → U10_GA(X1, X2, X3, X4, X5, X6, inordercA_in_ga(X1, X7))
U10_GA(X1, X2, X3, X4, X5, X6, inordercA_out_ga(X1, X7)) → U11_GA(X1, X2, X3, X4, X5, X6, inorderA_in_ga(X3, X8))
U10_GA(X1, X2, X3, X4, X5, X6, inordercA_out_ga(X1, X7)) → INORDERA_IN_GA(X3, X8)
U10_GA(X1, X2, X3, X4, X5, X6, inordercA_out_ga(X1, X7)) → U12_GA(X1, X2, X3, X4, X5, X6, X7, inordercA_in_ga(X3, X8))
U12_GA(X1, X2, X3, X4, X5, X6, X7, inordercA_out_ga(X3, X8)) → U13_GA(X1, X2, X3, X4, X5, X6, appendB_in_ggga(X7, X2, X8, X9))
U12_GA(X1, X2, X3, X4, X5, X6, X7, inordercA_out_ga(X3, X8)) → APPENDB_IN_GGGA(X7, X2, X8, X9)
U12_GA(X1, X2, X3, X4, X5, X6, X7, inordercA_out_ga(X3, X8)) → U14_GA(X1, X2, X3, X4, X5, X6, appendcB_in_ggga(X7, X2, X8, X9))
U14_GA(X1, X2, X3, X4, X5, X6, appendcB_out_ggga(X7, X2, X8, X9)) → U15_GA(X1, X2, X3, X4, X5, X6, inorderA_in_ga(X5, X10))
U14_GA(X1, X2, X3, X4, X5, X6, appendcB_out_ggga(X7, X2, X8, X9)) → INORDERA_IN_GA(X5, X10)
U14_GA(X1, X2, X3, X4, X5, X6, appendcB_out_ggga(X7, X2, X8, X9)) → U16_GA(X1, X2, X3, X4, X5, X6, X9, inordercA_in_ga(X5, X10))
U16_GA(X1, X2, X3, X4, X5, X6, X9, inordercA_out_ga(X5, X10)) → U17_GA(X1, X2, X3, X4, X5, X6, appendC_in_ggga(X9, X4, X10, X6))
U16_GA(X1, X2, X3, X4, X5, X6, X9, inordercA_out_ga(X5, X10)) → APPENDC_IN_GGGA(X9, X4, X10, X6)
APPENDC_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → U7_GGGA(X1, X2, X3, X4, X5, appendC_in_ggga(X2, X3, X4, X5))
APPENDC_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPENDC_IN_GGGA(X2, X3, X4, X5)

The TRS R consists of the following rules:

inordercA_in_ga(nil, []) → inordercA_out_ga(nil, [])
inordercA_in_ga(tree(X1, X2, X3), X4) → U19_ga(X1, X2, X3, X4, inordercA_in_ga(X1, X5))
U19_ga(X1, X2, X3, X4, inordercA_out_ga(X1, X5)) → U20_ga(X1, X2, X3, X4, X5, inordercA_in_ga(X3, X6))
U20_ga(X1, X2, X3, X4, X5, inordercA_out_ga(X3, X6)) → U21_ga(X1, X2, X3, X4, appendcB_in_ggga(X5, X2, X6, X4))
appendcB_in_ggga([], X1, X2, .(X1, X2)) → appendcB_out_ggga([], X1, X2, .(X1, X2))
appendcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U22_ggga(X1, X2, X3, X4, X5, appendcB_in_ggga(X2, X3, X4, X5))
U22_ggga(X1, X2, X3, X4, X5, appendcB_out_ggga(X2, X3, X4, X5)) → appendcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U21_ga(X1, X2, X3, X4, appendcB_out_ggga(X5, X2, X6, X4)) → inordercA_out_ga(tree(X1, X2, X3), X4)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
nil  =  nil
inorderA_in_ga(x1, x2)  =  inorderA_in_ga(x1)
inordercA_in_ga(x1, x2)  =  inordercA_in_ga(x1)
inordercA_out_ga(x1, x2)  =  inordercA_out_ga(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5, x6)  =  U20_ga(x1, x2, x3, x5, x6)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
appendcB_in_ggga(x1, x2, x3, x4)  =  appendcB_in_ggga(x1, x2, x3)
[]  =  []
appendcB_out_ggga(x1, x2, x3, x4)  =  appendcB_out_ggga(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
U22_ggga(x1, x2, x3, x4, x5, x6)  =  U22_ggga(x1, x2, x3, x4, x6)
appendB_in_ggga(x1, x2, x3, x4)  =  appendB_in_ggga(x1, x2, x3)
appendC_in_ggga(x1, x2, x3, x4)  =  appendC_in_ggga(x1, x2, x3)
INORDERD_IN_GA(x1, x2)  =  INORDERD_IN_GA(x1)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x2, x4)
INORDERA_IN_GA(x1, x2)  =  INORDERA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x3, x5)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x2, x3, x5, x6)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x3, x5)
APPENDB_IN_GGGA(x1, x2, x3, x4)  =  APPENDB_IN_GGGA(x1, x2, x3)
U6_GGGA(x1, x2, x3, x4, x5, x6)  =  U6_GGGA(x1, x2, x3, x4, x6)
U9_GA(x1, x2, x3, x4, x5, x6, x7)  =  U9_GA(x1, x2, x3, x4, x5, x7)
U10_GA(x1, x2, x3, x4, x5, x6, x7)  =  U10_GA(x1, x2, x3, x4, x5, x7)
U11_GA(x1, x2, x3, x4, x5, x6, x7)  =  U11_GA(x1, x2, x3, x4, x5, x7)
U12_GA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_GA(x1, x2, x3, x4, x5, x7, x8)
U13_GA(x1, x2, x3, x4, x5, x6, x7)  =  U13_GA(x1, x2, x3, x4, x5, x7)
U14_GA(x1, x2, x3, x4, x5, x6, x7)  =  U14_GA(x1, x2, x3, x4, x5, x7)
U15_GA(x1, x2, x3, x4, x5, x6, x7)  =  U15_GA(x1, x2, x3, x4, x5, x7)
U16_GA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U16_GA(x1, x2, x3, x4, x5, x7, x8)
U17_GA(x1, x2, x3, x4, x5, x6, x7)  =  U17_GA(x1, x2, x3, x4, x5, x7)
APPENDC_IN_GGGA(x1, x2, x3, x4)  =  APPENDC_IN_GGGA(x1, x2, x3)
U7_GGGA(x1, x2, x3, x4, x5, x6)  =  U7_GGGA(x1, x2, x3, x4, x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 23 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

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

The TRS R consists of the following rules:

inordercA_in_ga(nil, []) → inordercA_out_ga(nil, [])
inordercA_in_ga(tree(X1, X2, X3), X4) → U19_ga(X1, X2, X3, X4, inordercA_in_ga(X1, X5))
U19_ga(X1, X2, X3, X4, inordercA_out_ga(X1, X5)) → U20_ga(X1, X2, X3, X4, X5, inordercA_in_ga(X3, X6))
U20_ga(X1, X2, X3, X4, X5, inordercA_out_ga(X3, X6)) → U21_ga(X1, X2, X3, X4, appendcB_in_ggga(X5, X2, X6, X4))
appendcB_in_ggga([], X1, X2, .(X1, X2)) → appendcB_out_ggga([], X1, X2, .(X1, X2))
appendcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U22_ggga(X1, X2, X3, X4, X5, appendcB_in_ggga(X2, X3, X4, X5))
U22_ggga(X1, X2, X3, X4, X5, appendcB_out_ggga(X2, X3, X4, X5)) → appendcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U21_ga(X1, X2, X3, X4, appendcB_out_ggga(X5, X2, X6, X4)) → inordercA_out_ga(tree(X1, X2, X3), X4)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
nil  =  nil
inordercA_in_ga(x1, x2)  =  inordercA_in_ga(x1)
inordercA_out_ga(x1, x2)  =  inordercA_out_ga(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5, x6)  =  U20_ga(x1, x2, x3, x5, x6)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
appendcB_in_ggga(x1, x2, x3, x4)  =  appendcB_in_ggga(x1, x2, x3)
[]  =  []
appendcB_out_ggga(x1, x2, x3, x4)  =  appendcB_out_ggga(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
U22_ggga(x1, x2, x3, x4, x5, x6)  =  U22_ggga(x1, x2, x3, x4, x6)
APPENDC_IN_GGGA(x1, x2, x3, x4)  =  APPENDC_IN_GGGA(x1, x2, x3)

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:

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

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

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:

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

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:

  • APPENDC_IN_GGGA(.(X1, X2), X3, X4) → APPENDC_IN_GGGA(X2, X3, X4)
    The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

(13) YES

(14) Obligation:

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

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

The TRS R consists of the following rules:

inordercA_in_ga(nil, []) → inordercA_out_ga(nil, [])
inordercA_in_ga(tree(X1, X2, X3), X4) → U19_ga(X1, X2, X3, X4, inordercA_in_ga(X1, X5))
U19_ga(X1, X2, X3, X4, inordercA_out_ga(X1, X5)) → U20_ga(X1, X2, X3, X4, X5, inordercA_in_ga(X3, X6))
U20_ga(X1, X2, X3, X4, X5, inordercA_out_ga(X3, X6)) → U21_ga(X1, X2, X3, X4, appendcB_in_ggga(X5, X2, X6, X4))
appendcB_in_ggga([], X1, X2, .(X1, X2)) → appendcB_out_ggga([], X1, X2, .(X1, X2))
appendcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U22_ggga(X1, X2, X3, X4, X5, appendcB_in_ggga(X2, X3, X4, X5))
U22_ggga(X1, X2, X3, X4, X5, appendcB_out_ggga(X2, X3, X4, X5)) → appendcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U21_ga(X1, X2, X3, X4, appendcB_out_ggga(X5, X2, X6, X4)) → inordercA_out_ga(tree(X1, X2, X3), X4)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
nil  =  nil
inordercA_in_ga(x1, x2)  =  inordercA_in_ga(x1)
inordercA_out_ga(x1, x2)  =  inordercA_out_ga(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5, x6)  =  U20_ga(x1, x2, x3, x5, x6)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
appendcB_in_ggga(x1, x2, x3, x4)  =  appendcB_in_ggga(x1, x2, x3)
[]  =  []
appendcB_out_ggga(x1, x2, x3, x4)  =  appendcB_out_ggga(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
U22_ggga(x1, x2, x3, x4, x5, x6)  =  U22_ggga(x1, x2, x3, x4, x6)
APPENDB_IN_GGGA(x1, x2, x3, x4)  =  APPENDB_IN_GGGA(x1, x2, x3)

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:

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

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

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:

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

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:

  • APPENDB_IN_GGGA(.(X1, X2), X3, X4) → APPENDB_IN_GGGA(X2, X3, X4)
    The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

(20) YES

(21) Obligation:

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

INORDERA_IN_GA(tree(X1, X2, X3), X4) → U2_GA(X1, X2, X3, X4, inordercA_in_ga(X1, X5))
U2_GA(X1, X2, X3, X4, inordercA_out_ga(X1, X5)) → INORDERA_IN_GA(X3, X6)
INORDERA_IN_GA(tree(X1, X2, X3), X4) → INORDERA_IN_GA(X1, X5)

The TRS R consists of the following rules:

inordercA_in_ga(nil, []) → inordercA_out_ga(nil, [])
inordercA_in_ga(tree(X1, X2, X3), X4) → U19_ga(X1, X2, X3, X4, inordercA_in_ga(X1, X5))
U19_ga(X1, X2, X3, X4, inordercA_out_ga(X1, X5)) → U20_ga(X1, X2, X3, X4, X5, inordercA_in_ga(X3, X6))
U20_ga(X1, X2, X3, X4, X5, inordercA_out_ga(X3, X6)) → U21_ga(X1, X2, X3, X4, appendcB_in_ggga(X5, X2, X6, X4))
appendcB_in_ggga([], X1, X2, .(X1, X2)) → appendcB_out_ggga([], X1, X2, .(X1, X2))
appendcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U22_ggga(X1, X2, X3, X4, X5, appendcB_in_ggga(X2, X3, X4, X5))
U22_ggga(X1, X2, X3, X4, X5, appendcB_out_ggga(X2, X3, X4, X5)) → appendcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U21_ga(X1, X2, X3, X4, appendcB_out_ggga(X5, X2, X6, X4)) → inordercA_out_ga(tree(X1, X2, X3), X4)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
nil  =  nil
inordercA_in_ga(x1, x2)  =  inordercA_in_ga(x1)
inordercA_out_ga(x1, x2)  =  inordercA_out_ga(x1, x2)
U19_ga(x1, x2, x3, x4, x5)  =  U19_ga(x1, x2, x3, x5)
U20_ga(x1, x2, x3, x4, x5, x6)  =  U20_ga(x1, x2, x3, x5, x6)
U21_ga(x1, x2, x3, x4, x5)  =  U21_ga(x1, x2, x3, x5)
appendcB_in_ggga(x1, x2, x3, x4)  =  appendcB_in_ggga(x1, x2, x3)
[]  =  []
appendcB_out_ggga(x1, x2, x3, x4)  =  appendcB_out_ggga(x1, x2, x3, x4)
.(x1, x2)  =  .(x1, x2)
U22_ggga(x1, x2, x3, x4, x5, x6)  =  U22_ggga(x1, x2, x3, x4, x6)
INORDERA_IN_GA(x1, x2)  =  INORDERA_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, 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:

INORDERA_IN_GA(tree(X1, X2, X3)) → U2_GA(X1, X2, X3, inordercA_in_ga(X1))
U2_GA(X1, X2, X3, inordercA_out_ga(X1, X5)) → INORDERA_IN_GA(X3)
INORDERA_IN_GA(tree(X1, X2, X3)) → INORDERA_IN_GA(X1)

The TRS R consists of the following rules:

inordercA_in_ga(nil) → inordercA_out_ga(nil, [])
inordercA_in_ga(tree(X1, X2, X3)) → U19_ga(X1, X2, X3, inordercA_in_ga(X1))
U19_ga(X1, X2, X3, inordercA_out_ga(X1, X5)) → U20_ga(X1, X2, X3, X5, inordercA_in_ga(X3))
U20_ga(X1, X2, X3, X5, inordercA_out_ga(X3, X6)) → U21_ga(X1, X2, X3, appendcB_in_ggga(X5, X2, X6))
appendcB_in_ggga([], X1, X2) → appendcB_out_ggga([], X1, X2, .(X1, X2))
appendcB_in_ggga(.(X1, X2), X3, X4) → U22_ggga(X1, X2, X3, X4, appendcB_in_ggga(X2, X3, X4))
U22_ggga(X1, X2, X3, X4, appendcB_out_ggga(X2, X3, X4, X5)) → appendcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U21_ga(X1, X2, X3, appendcB_out_ggga(X5, X2, X6, X4)) → inordercA_out_ga(tree(X1, X2, X3), X4)

The set Q consists of the following terms:

inordercA_in_ga(x0)
U19_ga(x0, x1, x2, x3)
U20_ga(x0, x1, x2, x3, x4)
appendcB_in_ggga(x0, x1, x2)
U22_ggga(x0, x1, x2, x3, x4)
U21_ga(x0, x1, x2, x3)

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

(24) 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:

  • U2_GA(X1, X2, X3, inordercA_out_ga(X1, X5)) → INORDERA_IN_GA(X3)
    The graph contains the following edges 3 >= 1

  • INORDERA_IN_GA(tree(X1, X2, X3)) → INORDERA_IN_GA(X1)
    The graph contains the following edges 1 > 1

  • INORDERA_IN_GA(tree(X1, X2, X3)) → U2_GA(X1, X2, X3, inordercA_in_ga(X1))
    The graph contains the following edges 1 > 1, 1 > 2, 1 > 3

(25) YES