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

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇒, 88 ms)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇒, 202 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 QDPSizeChangeProof (⇔, 0 ms)
        ↳25 YES

(0) Obligation:

Clauses:

in_order(void, []).
in_order(tree(X, Left, Right), Xs) :- ','(in_order(Left, Ls), ','(in_order(Right, Rs), app(Ls, .(X, Rs), Xs))).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Query: in_order(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

in_orderA(tree(X1, X2, X3), X4) :- in_orderA(X2, X5).
in_orderA(tree(X1, X2, X3), X4) :- ','(in_ordercA(X2, X5), in_orderA(X3, X6)).
in_orderA(tree(X1, X2, X3), X4) :- ','(in_ordercA(X2, X5), ','(in_ordercA(X3, X6), appB(X5, X1, X6, X4))).
appB(.(X1, X2), X3, X4, .(X1, X5)) :- appB(X2, X3, X4, X5).
appC(.(X1, X2), X3, X4, .(X1, X5)) :- appC(X2, X3, X4, X5).
in_orderD(tree(X1, void, X2), X3) :- in_orderA(X2, X4).
in_orderD(tree(X1, tree(X2, X3, X4), X5), X6) :- in_orderA(X3, X7).
in_orderD(tree(X1, tree(X2, X3, X4), X5), X6) :- ','(in_ordercA(X3, X7), in_orderA(X4, X8)).
in_orderD(tree(X1, tree(X2, X3, X4), X5), X6) :- ','(in_ordercA(X3, X7), ','(in_ordercA(X4, X8), appB(X7, X2, X8, X9))).
in_orderD(tree(X1, tree(X2, X3, X4), X5), X6) :- ','(in_ordercA(X3, X7), ','(in_ordercA(X4, X8), ','(appcB(X7, X2, X8, X9), in_orderA(X5, X10)))).
in_orderD(tree(X1, tree(X2, X3, X4), X5), X6) :- ','(in_ordercA(X3, X7), ','(in_ordercA(X4, X8), ','(appcB(X7, X2, X8, X9), ','(in_ordercA(X5, X10), appC(X9, X1, X10, X6))))).

Clauses:

in_ordercA(void, []).
in_ordercA(tree(X1, X2, X3), X4) :- ','(in_ordercA(X2, X5), ','(in_ordercA(X3, X6), appcB(X5, X1, X6, X4))).
appcB([], X1, X2, .(X1, X2)).
appcB(.(X1, X2), X3, X4, .(X1, X5)) :- appcB(X2, X3, X4, X5).
appcC([], X1, X2, .(X1, X2)).
appcC(.(X1, X2), X3, X4, .(X1, X5)) :- appcC(X2, X3, X4, X5).

Afs:

in_orderD(x1, x2)  =  in_orderD(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:
in_orderD_in: (b,f)
in_orderA_in: (b,f)
in_ordercA_in: (b,f)
appcB_in: (b,b,b,f)
appB_in: (b,b,b,f)
appC_in: (b,b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

IN_ORDERD_IN_GA(tree(X1, void, X2), X3) → U8_GA(X1, X2, X3, in_orderA_in_ga(X2, X4))
IN_ORDERD_IN_GA(tree(X1, void, X2), X3) → IN_ORDERA_IN_GA(X2, X4)
IN_ORDERA_IN_GA(tree(X1, X2, X3), X4) → U1_GA(X1, X2, X3, X4, in_orderA_in_ga(X2, X5))
IN_ORDERA_IN_GA(tree(X1, X2, X3), X4) → IN_ORDERA_IN_GA(X2, X5)
IN_ORDERA_IN_GA(tree(X1, X2, X3), X4) → U2_GA(X1, X2, X3, X4, in_ordercA_in_ga(X2, X5))
U2_GA(X1, X2, X3, X4, in_ordercA_out_ga(X2, X5)) → U3_GA(X1, X2, X3, X4, in_orderA_in_ga(X3, X6))
U2_GA(X1, X2, X3, X4, in_ordercA_out_ga(X2, X5)) → IN_ORDERA_IN_GA(X3, X6)
U2_GA(X1, X2, X3, X4, in_ordercA_out_ga(X2, X5)) → U4_GA(X1, X2, X3, X4, X5, in_ordercA_in_ga(X3, X6))
U4_GA(X1, X2, X3, X4, X5, in_ordercA_out_ga(X3, X6)) → U5_GA(X1, X2, X3, X4, appB_in_ggga(X5, X1, X6, X4))
U4_GA(X1, X2, X3, X4, X5, in_ordercA_out_ga(X3, X6)) → APPB_IN_GGGA(X5, X1, X6, X4)
APPB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → U6_GGGA(X1, X2, X3, X4, X5, appB_in_ggga(X2, X3, X4, X5))
APPB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPB_IN_GGGA(X2, X3, X4, X5)
IN_ORDERD_IN_GA(tree(X1, tree(X2, X3, X4), X5), X6) → U9_GA(X1, X2, X3, X4, X5, X6, in_orderA_in_ga(X3, X7))
IN_ORDERD_IN_GA(tree(X1, tree(X2, X3, X4), X5), X6) → IN_ORDERA_IN_GA(X3, X7)
IN_ORDERD_IN_GA(tree(X1, tree(X2, X3, X4), X5), X6) → U10_GA(X1, X2, X3, X4, X5, X6, in_ordercA_in_ga(X3, X7))
U10_GA(X1, X2, X3, X4, X5, X6, in_ordercA_out_ga(X3, X7)) → U11_GA(X1, X2, X3, X4, X5, X6, in_orderA_in_ga(X4, X8))
U10_GA(X1, X2, X3, X4, X5, X6, in_ordercA_out_ga(X3, X7)) → IN_ORDERA_IN_GA(X4, X8)
U10_GA(X1, X2, X3, X4, X5, X6, in_ordercA_out_ga(X3, X7)) → U12_GA(X1, X2, X3, X4, X5, X6, X7, in_ordercA_in_ga(X4, X8))
U12_GA(X1, X2, X3, X4, X5, X6, X7, in_ordercA_out_ga(X4, X8)) → U13_GA(X1, X2, X3, X4, X5, X6, appB_in_ggga(X7, X2, X8, X9))
U12_GA(X1, X2, X3, X4, X5, X6, X7, in_ordercA_out_ga(X4, X8)) → APPB_IN_GGGA(X7, X2, X8, X9)
U12_GA(X1, X2, X3, X4, X5, X6, X7, in_ordercA_out_ga(X4, X8)) → U14_GA(X1, X2, X3, X4, X5, X6, appcB_in_ggga(X7, X2, X8, X9))
U14_GA(X1, X2, X3, X4, X5, X6, appcB_out_ggga(X7, X2, X8, X9)) → U15_GA(X1, X2, X3, X4, X5, X6, in_orderA_in_ga(X5, X10))
U14_GA(X1, X2, X3, X4, X5, X6, appcB_out_ggga(X7, X2, X8, X9)) → IN_ORDERA_IN_GA(X5, X10)
U14_GA(X1, X2, X3, X4, X5, X6, appcB_out_ggga(X7, X2, X8, X9)) → U16_GA(X1, X2, X3, X4, X5, X6, X9, in_ordercA_in_ga(X5, X10))
U16_GA(X1, X2, X3, X4, X5, X6, X9, in_ordercA_out_ga(X5, X10)) → U17_GA(X1, X2, X3, X4, X5, X6, appC_in_ggga(X9, X1, X10, X6))
U16_GA(X1, X2, X3, X4, X5, X6, X9, in_ordercA_out_ga(X5, X10)) → APPC_IN_GGGA(X9, X1, X10, X6)
APPC_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → U7_GGGA(X1, X2, X3, X4, X5, appC_in_ggga(X2, X3, X4, X5))
APPC_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPC_IN_GGGA(X2, X3, X4, X5)

The TRS R consists of the following rules:

in_ordercA_in_ga(void, []) → in_ordercA_out_ga(void, [])
in_ordercA_in_ga(tree(X1, X2, X3), X4) → U19_ga(X1, X2, X3, X4, in_ordercA_in_ga(X2, X5))
U19_ga(X1, X2, X3, X4, in_ordercA_out_ga(X2, X5)) → U20_ga(X1, X2, X3, X4, X5, in_ordercA_in_ga(X3, X6))
U20_ga(X1, X2, X3, X4, X5, in_ordercA_out_ga(X3, X6)) → U21_ga(X1, X2, X3, X4, appcB_in_ggga(X5, X1, X6, X4))
appcB_in_ggga([], X1, X2, .(X1, X2)) → appcB_out_ggga([], X1, X2, .(X1, X2))
appcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U22_ggga(X1, X2, X3, X4, X5, appcB_in_ggga(X2, X3, X4, X5))
U22_ggga(X1, X2, X3, X4, X5, appcB_out_ggga(X2, X3, X4, X5)) → appcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U21_ga(X1, X2, X3, X4, appcB_out_ggga(X5, X1, X6, X4)) → in_ordercA_out_ga(tree(X1, X2, X3), X4)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
in_orderA_in_ga(x1, x2)  =  in_orderA_in_ga(x1)
in_ordercA_in_ga(x1, x2)  =  in_ordercA_in_ga(x1)
in_ordercA_out_ga(x1, x2)  =  in_ordercA_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)
appcB_in_ggga(x1, x2, x3, x4)  =  appcB_in_ggga(x1, x2, x3)
[]  =  []
appcB_out_ggga(x1, x2, x3, x4)  =  appcB_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)
appB_in_ggga(x1, x2, x3, x4)  =  appB_in_ggga(x1, x2, x3)
appC_in_ggga(x1, x2, x3, x4)  =  appC_in_ggga(x1, x2, x3)
IN_ORDERD_IN_GA(x1, x2)  =  IN_ORDERD_IN_GA(x1)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x2, x4)
IN_ORDERA_IN_GA(x1, x2)  =  IN_ORDERA_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)
APPB_IN_GGGA(x1, x2, x3, x4)  =  APPB_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)
APPC_IN_GGGA(x1, x2, x3, x4)  =  APPC_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:

IN_ORDERD_IN_GA(tree(X1, void, X2), X3) → U8_GA(X1, X2, X3, in_orderA_in_ga(X2, X4))
IN_ORDERD_IN_GA(tree(X1, void, X2), X3) → IN_ORDERA_IN_GA(X2, X4)
IN_ORDERA_IN_GA(tree(X1, X2, X3), X4) → U1_GA(X1, X2, X3, X4, in_orderA_in_ga(X2, X5))
IN_ORDERA_IN_GA(tree(X1, X2, X3), X4) → IN_ORDERA_IN_GA(X2, X5)
IN_ORDERA_IN_GA(tree(X1, X2, X3), X4) → U2_GA(X1, X2, X3, X4, in_ordercA_in_ga(X2, X5))
U2_GA(X1, X2, X3, X4, in_ordercA_out_ga(X2, X5)) → U3_GA(X1, X2, X3, X4, in_orderA_in_ga(X3, X6))
U2_GA(X1, X2, X3, X4, in_ordercA_out_ga(X2, X5)) → IN_ORDERA_IN_GA(X3, X6)
U2_GA(X1, X2, X3, X4, in_ordercA_out_ga(X2, X5)) → U4_GA(X1, X2, X3, X4, X5, in_ordercA_in_ga(X3, X6))
U4_GA(X1, X2, X3, X4, X5, in_ordercA_out_ga(X3, X6)) → U5_GA(X1, X2, X3, X4, appB_in_ggga(X5, X1, X6, X4))
U4_GA(X1, X2, X3, X4, X5, in_ordercA_out_ga(X3, X6)) → APPB_IN_GGGA(X5, X1, X6, X4)
APPB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → U6_GGGA(X1, X2, X3, X4, X5, appB_in_ggga(X2, X3, X4, X5))
APPB_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPB_IN_GGGA(X2, X3, X4, X5)
IN_ORDERD_IN_GA(tree(X1, tree(X2, X3, X4), X5), X6) → U9_GA(X1, X2, X3, X4, X5, X6, in_orderA_in_ga(X3, X7))
IN_ORDERD_IN_GA(tree(X1, tree(X2, X3, X4), X5), X6) → IN_ORDERA_IN_GA(X3, X7)
IN_ORDERD_IN_GA(tree(X1, tree(X2, X3, X4), X5), X6) → U10_GA(X1, X2, X3, X4, X5, X6, in_ordercA_in_ga(X3, X7))
U10_GA(X1, X2, X3, X4, X5, X6, in_ordercA_out_ga(X3, X7)) → U11_GA(X1, X2, X3, X4, X5, X6, in_orderA_in_ga(X4, X8))
U10_GA(X1, X2, X3, X4, X5, X6, in_ordercA_out_ga(X3, X7)) → IN_ORDERA_IN_GA(X4, X8)
U10_GA(X1, X2, X3, X4, X5, X6, in_ordercA_out_ga(X3, X7)) → U12_GA(X1, X2, X3, X4, X5, X6, X7, in_ordercA_in_ga(X4, X8))
U12_GA(X1, X2, X3, X4, X5, X6, X7, in_ordercA_out_ga(X4, X8)) → U13_GA(X1, X2, X3, X4, X5, X6, appB_in_ggga(X7, X2, X8, X9))
U12_GA(X1, X2, X3, X4, X5, X6, X7, in_ordercA_out_ga(X4, X8)) → APPB_IN_GGGA(X7, X2, X8, X9)
U12_GA(X1, X2, X3, X4, X5, X6, X7, in_ordercA_out_ga(X4, X8)) → U14_GA(X1, X2, X3, X4, X5, X6, appcB_in_ggga(X7, X2, X8, X9))
U14_GA(X1, X2, X3, X4, X5, X6, appcB_out_ggga(X7, X2, X8, X9)) → U15_GA(X1, X2, X3, X4, X5, X6, in_orderA_in_ga(X5, X10))
U14_GA(X1, X2, X3, X4, X5, X6, appcB_out_ggga(X7, X2, X8, X9)) → IN_ORDERA_IN_GA(X5, X10)
U14_GA(X1, X2, X3, X4, X5, X6, appcB_out_ggga(X7, X2, X8, X9)) → U16_GA(X1, X2, X3, X4, X5, X6, X9, in_ordercA_in_ga(X5, X10))
U16_GA(X1, X2, X3, X4, X5, X6, X9, in_ordercA_out_ga(X5, X10)) → U17_GA(X1, X2, X3, X4, X5, X6, appC_in_ggga(X9, X1, X10, X6))
U16_GA(X1, X2, X3, X4, X5, X6, X9, in_ordercA_out_ga(X5, X10)) → APPC_IN_GGGA(X9, X1, X10, X6)
APPC_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → U7_GGGA(X1, X2, X3, X4, X5, appC_in_ggga(X2, X3, X4, X5))
APPC_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) → APPC_IN_GGGA(X2, X3, X4, X5)

The TRS R consists of the following rules:

in_ordercA_in_ga(void, []) → in_ordercA_out_ga(void, [])
in_ordercA_in_ga(tree(X1, X2, X3), X4) → U19_ga(X1, X2, X3, X4, in_ordercA_in_ga(X2, X5))
U19_ga(X1, X2, X3, X4, in_ordercA_out_ga(X2, X5)) → U20_ga(X1, X2, X3, X4, X5, in_ordercA_in_ga(X3, X6))
U20_ga(X1, X2, X3, X4, X5, in_ordercA_out_ga(X3, X6)) → U21_ga(X1, X2, X3, X4, appcB_in_ggga(X5, X1, X6, X4))
appcB_in_ggga([], X1, X2, .(X1, X2)) → appcB_out_ggga([], X1, X2, .(X1, X2))
appcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U22_ggga(X1, X2, X3, X4, X5, appcB_in_ggga(X2, X3, X4, X5))
U22_ggga(X1, X2, X3, X4, X5, appcB_out_ggga(X2, X3, X4, X5)) → appcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U21_ga(X1, X2, X3, X4, appcB_out_ggga(X5, X1, X6, X4)) → in_ordercA_out_ga(tree(X1, X2, X3), X4)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
in_orderA_in_ga(x1, x2)  =  in_orderA_in_ga(x1)
in_ordercA_in_ga(x1, x2)  =  in_ordercA_in_ga(x1)
in_ordercA_out_ga(x1, x2)  =  in_ordercA_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)
appcB_in_ggga(x1, x2, x3, x4)  =  appcB_in_ggga(x1, x2, x3)
[]  =  []
appcB_out_ggga(x1, x2, x3, x4)  =  appcB_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)
appB_in_ggga(x1, x2, x3, x4)  =  appB_in_ggga(x1, x2, x3)
appC_in_ggga(x1, x2, x3, x4)  =  appC_in_ggga(x1, x2, x3)
IN_ORDERD_IN_GA(x1, x2)  =  IN_ORDERD_IN_GA(x1)
U8_GA(x1, x2, x3, x4)  =  U8_GA(x1, x2, x4)
IN_ORDERA_IN_GA(x1, x2)  =  IN_ORDERA_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)
APPB_IN_GGGA(x1, x2, x3, x4)  =  APPB_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)
APPC_IN_GGGA(x1, x2, x3, x4)  =  APPC_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:

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

The TRS R consists of the following rules:

in_ordercA_in_ga(void, []) → in_ordercA_out_ga(void, [])
in_ordercA_in_ga(tree(X1, X2, X3), X4) → U19_ga(X1, X2, X3, X4, in_ordercA_in_ga(X2, X5))
U19_ga(X1, X2, X3, X4, in_ordercA_out_ga(X2, X5)) → U20_ga(X1, X2, X3, X4, X5, in_ordercA_in_ga(X3, X6))
U20_ga(X1, X2, X3, X4, X5, in_ordercA_out_ga(X3, X6)) → U21_ga(X1, X2, X3, X4, appcB_in_ggga(X5, X1, X6, X4))
appcB_in_ggga([], X1, X2, .(X1, X2)) → appcB_out_ggga([], X1, X2, .(X1, X2))
appcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U22_ggga(X1, X2, X3, X4, X5, appcB_in_ggga(X2, X3, X4, X5))
U22_ggga(X1, X2, X3, X4, X5, appcB_out_ggga(X2, X3, X4, X5)) → appcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U21_ga(X1, X2, X3, X4, appcB_out_ggga(X5, X1, X6, X4)) → in_ordercA_out_ga(tree(X1, X2, X3), X4)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
in_ordercA_in_ga(x1, x2)  =  in_ordercA_in_ga(x1)
in_ordercA_out_ga(x1, x2)  =  in_ordercA_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)
appcB_in_ggga(x1, x2, x3, x4)  =  appcB_in_ggga(x1, x2, x3)
[]  =  []
appcB_out_ggga(x1, x2, x3, x4)  =  appcB_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)
APPC_IN_GGGA(x1, x2, x3, x4)  =  APPC_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:

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

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

APPC_IN_GGGA(.(X1, X2), X3, X4) → APPC_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:

  • APPC_IN_GGGA(.(X1, X2), X3, X4) → APPC_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:

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

The TRS R consists of the following rules:

in_ordercA_in_ga(void, []) → in_ordercA_out_ga(void, [])
in_ordercA_in_ga(tree(X1, X2, X3), X4) → U19_ga(X1, X2, X3, X4, in_ordercA_in_ga(X2, X5))
U19_ga(X1, X2, X3, X4, in_ordercA_out_ga(X2, X5)) → U20_ga(X1, X2, X3, X4, X5, in_ordercA_in_ga(X3, X6))
U20_ga(X1, X2, X3, X4, X5, in_ordercA_out_ga(X3, X6)) → U21_ga(X1, X2, X3, X4, appcB_in_ggga(X5, X1, X6, X4))
appcB_in_ggga([], X1, X2, .(X1, X2)) → appcB_out_ggga([], X1, X2, .(X1, X2))
appcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U22_ggga(X1, X2, X3, X4, X5, appcB_in_ggga(X2, X3, X4, X5))
U22_ggga(X1, X2, X3, X4, X5, appcB_out_ggga(X2, X3, X4, X5)) → appcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U21_ga(X1, X2, X3, X4, appcB_out_ggga(X5, X1, X6, X4)) → in_ordercA_out_ga(tree(X1, X2, X3), X4)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
in_ordercA_in_ga(x1, x2)  =  in_ordercA_in_ga(x1)
in_ordercA_out_ga(x1, x2)  =  in_ordercA_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)
appcB_in_ggga(x1, x2, x3, x4)  =  appcB_in_ggga(x1, x2, x3)
[]  =  []
appcB_out_ggga(x1, x2, x3, x4)  =  appcB_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)
APPB_IN_GGGA(x1, x2, x3, x4)  =  APPB_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:

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

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

APPB_IN_GGGA(.(X1, X2), X3, X4) → APPB_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:

  • APPB_IN_GGGA(.(X1, X2), X3, X4) → APPB_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:

IN_ORDERA_IN_GA(tree(X1, X2, X3), X4) → U2_GA(X1, X2, X3, X4, in_ordercA_in_ga(X2, X5))
U2_GA(X1, X2, X3, X4, in_ordercA_out_ga(X2, X5)) → IN_ORDERA_IN_GA(X3, X6)
IN_ORDERA_IN_GA(tree(X1, X2, X3), X4) → IN_ORDERA_IN_GA(X2, X5)

The TRS R consists of the following rules:

in_ordercA_in_ga(void, []) → in_ordercA_out_ga(void, [])
in_ordercA_in_ga(tree(X1, X2, X3), X4) → U19_ga(X1, X2, X3, X4, in_ordercA_in_ga(X2, X5))
U19_ga(X1, X2, X3, X4, in_ordercA_out_ga(X2, X5)) → U20_ga(X1, X2, X3, X4, X5, in_ordercA_in_ga(X3, X6))
U20_ga(X1, X2, X3, X4, X5, in_ordercA_out_ga(X3, X6)) → U21_ga(X1, X2, X3, X4, appcB_in_ggga(X5, X1, X6, X4))
appcB_in_ggga([], X1, X2, .(X1, X2)) → appcB_out_ggga([], X1, X2, .(X1, X2))
appcB_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) → U22_ggga(X1, X2, X3, X4, X5, appcB_in_ggga(X2, X3, X4, X5))
U22_ggga(X1, X2, X3, X4, X5, appcB_out_ggga(X2, X3, X4, X5)) → appcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U21_ga(X1, X2, X3, X4, appcB_out_ggga(X5, X1, X6, X4)) → in_ordercA_out_ga(tree(X1, X2, X3), X4)

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
in_ordercA_in_ga(x1, x2)  =  in_ordercA_in_ga(x1)
in_ordercA_out_ga(x1, x2)  =  in_ordercA_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)
appcB_in_ggga(x1, x2, x3, x4)  =  appcB_in_ggga(x1, x2, x3)
[]  =  []
appcB_out_ggga(x1, x2, x3, x4)  =  appcB_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)
IN_ORDERA_IN_GA(x1, x2)  =  IN_ORDERA_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:

IN_ORDERA_IN_GA(tree(X1, X2, X3)) → U2_GA(X1, X2, X3, in_ordercA_in_ga(X2))
U2_GA(X1, X2, X3, in_ordercA_out_ga(X2, X5)) → IN_ORDERA_IN_GA(X3)
IN_ORDERA_IN_GA(tree(X1, X2, X3)) → IN_ORDERA_IN_GA(X2)

The TRS R consists of the following rules:

in_ordercA_in_ga(void) → in_ordercA_out_ga(void, [])
in_ordercA_in_ga(tree(X1, X2, X3)) → U19_ga(X1, X2, X3, in_ordercA_in_ga(X2))
U19_ga(X1, X2, X3, in_ordercA_out_ga(X2, X5)) → U20_ga(X1, X2, X3, X5, in_ordercA_in_ga(X3))
U20_ga(X1, X2, X3, X5, in_ordercA_out_ga(X3, X6)) → U21_ga(X1, X2, X3, appcB_in_ggga(X5, X1, X6))
appcB_in_ggga([], X1, X2) → appcB_out_ggga([], X1, X2, .(X1, X2))
appcB_in_ggga(.(X1, X2), X3, X4) → U22_ggga(X1, X2, X3, X4, appcB_in_ggga(X2, X3, X4))
U22_ggga(X1, X2, X3, X4, appcB_out_ggga(X2, X3, X4, X5)) → appcB_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
U21_ga(X1, X2, X3, appcB_out_ggga(X5, X1, X6, X4)) → in_ordercA_out_ga(tree(X1, X2, X3), X4)

The set Q consists of the following terms:

in_ordercA_in_ga(x0)
U19_ga(x0, x1, x2, x3)
U20_ga(x0, x1, x2, x3, x4)
appcB_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, in_ordercA_out_ga(X2, X5)) → IN_ORDERA_IN_GA(X3)
    The graph contains the following edges 3 >= 1

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

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

(25) YES