NO Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Zantema_04/z066-split.srs

Termination w.r.t. Q of the given QTRS could be disproven:

0 QTRS
↳1 QTRSRRRProof (⇔, 84 ms)
↳2 QTRS
↳3 QTRSRRRProof (⇔, 84 ms)
↳4 QTRS
↳5 DependencyPairsProof (⇔, 64 ms)
↳6 QDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 AND
    ↳9 QDP
        ↳10 UsableRulesProof (⇔, 0 ms)
        ↳11 QDP
        ↳12 MRRProof (⇔, 17 ms)
        ↳13 QDP
        ↳14 DependencyGraphProof (⇔, 0 ms)
        ↳15 AND
            ↳16 QDP
                ↳17 UsableRulesProof (⇔, 0 ms)
                ↳18 QDP
                ↳19 QDPSizeChangeProof (⇔, 0 ms)
                ↳20 YES
            ↳21 QDP
                ↳22 QDPOrderProof (⇔, 6 ms)
                ↳23 QDP
                ↳24 PisEmptyProof (⇔, 0 ms)
                ↳25 YES
            ↳26 QDP
                ↳27 QDPOrderProof (⇔, 4 ms)
                ↳28 QDP
                ↳29 DependencyGraphProof (⇔, 0 ms)
                ↳30 TRUE
    ↳31 QDP
        ↳32 UsableRulesProof (⇔, 0 ms)
        ↳33 QDP
        ↳34 QDPSizeChangeProof (⇔, 0 ms)
        ↳35 YES
    ↳36 QDP
        ↳37 UsableRulesProof (⇔, 0 ms)
        ↳38 QDP
        ↳39 QDPSizeChangeProof (⇔, 0 ms)
        ↳40 YES
    ↳41 QDP
        ↳42 UsableRulesProof (⇔, 0 ms)
        ↳43 QDP
        ↳44 QDPSizeChangeProof (⇔, 0 ms)
        ↳45 YES
    ↳46 QDP
        ↳47 UsableRulesProof (⇔, 0 ms)
        ↳48 QDP
        ↳49 QDPSizeChangeProof (⇔, 0 ms)
        ↳50 YES
    ↳51 QDP
        ↳52 UsableRulesProof (⇔, 0 ms)
        ↳53 QDP
        ↳54 QDPSizeChangeProof (⇔, 0 ms)
        ↳55 YES
    ↳56 QDP
        ↳57 UsableRulesProof (⇔, 0 ms)
        ↳58 QDP
        ↳59 QDPSizeChangeProof (⇔, 0 ms)
        ↳60 YES
    ↳61 QDP
        ↳62 UsableRulesProof (⇔, 0 ms)
        ↳63 QDP
        ↳64 QDPSizeChangeProof (⇔, 0 ms)
        ↳65 YES
    ↳66 QDP
        ↳67 UsableRulesProof (⇔, 0 ms)
        ↳68 QDP
        ↳69 QDPSizeChangeProof (⇔, 0 ms)
        ↳70 YES
    ↳71 QDP
        ↳72 UsableRulesProof (⇔, 0 ms)
        ↳73 QDP
        ↳74 QDPSizeChangeProof (⇔, 0 ms)
        ↳75 YES
    ↳76 QDP
        ↳77 UsableRulesProof (⇔, 0 ms)
        ↳78 QDP
        ↳79 QDPSizeChangeProof (⇔, 0 ms)
        ↳80 YES
    ↳81 QDP
        ↳82 UsableRulesProof (⇔, 0 ms)
        ↳83 QDP
        ↳84 QDPSizeChangeProof (⇔, 0 ms)
        ↳85 YES
    ↳86 QDP
        ↳87 UsableRulesProof (⇔, 0 ms)
        ↳88 QDP
        ↳89 QDPSizeChangeProof (⇔, 0 ms)
        ↳90 YES
    ↳91 QDP
        ↳92 UsableRulesProof (⇔, 0 ms)
        ↳93 QDP
        ↳94 QDPSizeChangeProof (⇔, 0 ms)
        ↳95 YES
    ↳96 QDP
        ↳97 UsableRulesProof (⇔, 0 ms)
        ↳98 QDP
        ↳99 QDPSizeChangeProof (⇔, 0 ms)
        ↳100 YES
    ↳101 QDP
        ↳102 UsableRulesProof (⇔, 0 ms)
        ↳103 QDP
        ↳104 QDPSizeChangeProof (⇔, 0 ms)
        ↳105 YES
    ↳106 QDP
        ↳107 UsableRulesProof (⇔, 0 ms)
        ↳108 QDP
        ↳109 QDPSizeChangeProof (⇔, 0 ms)
        ↳110 YES
    ↳111 QDP
        ↳112 UsableRulesProof (⇔, 0 ms)
        ↳113 QDP
        ↳114 QDPSizeChangeProof (⇔, 0 ms)
        ↳115 YES
    ↳116 QDP
        ↳117 UsableRulesProof (⇔, 0 ms)
        ↳118 QDP
        ↳119 QDPSizeChangeProof (⇔, 0 ms)
        ↳120 YES
    ↳121 QDP
        ↳122 UsableRulesProof (⇔, 10 ms)
        ↳123 QDP
        ↳124 NonTerminationLoopProof (⇒, 196 ms)
        ↳125 NO

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Begin(A(x)) → Wait(Right13(x))
Begin(a(x)) → Wait(Right14(x))
Begin(B(x)) → Wait(Right15(x))
Begin(b(x)) → Wait(Right16(x))
Begin(C(x)) → Wait(Right17(x))
Begin(c(x)) → Wait(Right18(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right13(a(End(x))) → Left(End(x))
Right14(A(End(x))) → Left(End(x))
Right15(b(End(x))) → Left(End(x))
Right16(B(End(x))) → Left(End(x))
Right17(c(End(x))) → Left(End(x))
Right18(C(End(x))) → Left(End(x))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))
a(A(x)) → x
A(a(x)) → x
b(B(x)) → x
B(b(x)) → x
c(C(x)) → x
C(c(x)) → x

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(A(x1)) = 1 + x1   
POL(AA(x1)) = 1 + x1   
POL(AB(x1)) = x1   
POL(AC(x1)) = x1   
POL(Aa(x1)) = 1 + x1   
POL(Ab(x1)) = x1   
POL(Ac(x1)) = x1   
POL(B(x1)) = x1   
POL(Begin(x1)) = x1   
POL(C(x1)) = x1   
POL(End(x1)) = x1   
POL(Left(x1)) = x1   
POL(Right1(x1)) = x1   
POL(Right10(x1)) = x1   
POL(Right11(x1)) = 1 + x1   
POL(Right12(x1)) = 1 + x1   
POL(Right13(x1)) = 1 + x1   
POL(Right14(x1)) = x1   
POL(Right15(x1)) = x1   
POL(Right16(x1)) = x1   
POL(Right17(x1)) = x1   
POL(Right18(x1)) = x1   
POL(Right2(x1)) = x1   
POL(Right3(x1)) = 1 + x1   
POL(Right4(x1)) = 1 + x1   
POL(Right5(x1)) = 1 + x1   
POL(Right6(x1)) = x1   
POL(Right7(x1)) = 1 + x1   
POL(Right8(x1)) = x1   
POL(Right9(x1)) = x1   
POL(Wait(x1)) = x1   
POL(a(x1)) = 1 + x1   
POL(b(x1)) = x1   
POL(c(x1)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

Begin(a(x)) → Wait(Right14(x))
Right13(a(End(x))) → Left(End(x))
Right14(A(End(x))) → Left(End(x))
a(A(x)) → x
A(a(x)) → x


(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Begin(A(x)) → Wait(Right13(x))
Begin(B(x)) → Wait(Right15(x))
Begin(b(x)) → Wait(Right16(x))
Begin(C(x)) → Wait(Right17(x))
Begin(c(x)) → Wait(Right18(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right15(b(End(x))) → Left(End(x))
Right16(B(End(x))) → Left(End(x))
Right17(c(End(x))) → Left(End(x))
Right18(C(End(x))) → Left(End(x))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))
b(B(x)) → x
B(b(x)) → x
c(C(x)) → x
C(c(x)) → x

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(A(x1)) = 1 + x1   
POL(AA(x1)) = 1 + x1   
POL(AB(x1)) = 1 + x1   
POL(AC(x1)) = 1 + x1   
POL(Aa(x1)) = x1   
POL(Ab(x1)) = 1 + x1   
POL(Ac(x1)) = 1 + x1   
POL(B(x1)) = 1 + x1   
POL(Begin(x1)) = x1   
POL(C(x1)) = 1 + x1   
POL(End(x1)) = x1   
POL(Left(x1)) = x1   
POL(Right1(x1)) = 2 + x1   
POL(Right10(x1)) = 1 + x1   
POL(Right11(x1)) = 1 + x1   
POL(Right12(x1)) = x1   
POL(Right13(x1)) = x1   
POL(Right14(x1)) = x1   
POL(Right15(x1)) = x1   
POL(Right16(x1)) = x1   
POL(Right17(x1)) = x1   
POL(Right18(x1)) = x1   
POL(Right2(x1)) = 1 + x1   
POL(Right3(x1)) = 2 + x1   
POL(Right4(x1)) = 1 + x1   
POL(Right5(x1)) = 1 + x1   
POL(Right6(x1)) = 1 + x1   
POL(Right7(x1)) = 2 + x1   
POL(Right8(x1)) = 1 + x1   
POL(Right9(x1)) = 2 + x1   
POL(Wait(x1)) = x1   
POL(a(x1)) = x1   
POL(b(x1)) = 1 + x1   
POL(c(x1)) = 1 + x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

Begin(A(x)) → Wait(Right13(x))
Begin(B(x)) → Wait(Right15(x))
Begin(b(x)) → Wait(Right16(x))
Begin(C(x)) → Wait(Right17(x))
Begin(c(x)) → Wait(Right18(x))
Right15(b(End(x))) → Left(End(x))
Right16(B(End(x))) → Left(End(x))
Right17(c(End(x))) → Left(End(x))
Right18(C(End(x))) → Left(End(x))
b(B(x)) → x
B(b(x)) → x
c(C(x)) → x
C(c(x)) → x


(4) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

Q is empty.

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(6) Obligation:

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

BEGIN(b(c(x))) → WAIT(Right1(x))
BEGIN(b(c(x))) → RIGHT1(x)
BEGIN(c(x)) → WAIT(Right2(x))
BEGIN(c(x)) → RIGHT2(x)
BEGIN(B(A(x))) → WAIT(Right3(x))
BEGIN(B(A(x))) → RIGHT3(x)
BEGIN(A(x)) → WAIT(Right4(x))
BEGIN(A(x)) → RIGHT4(x)
BEGIN(a(C(x))) → WAIT(Right5(x))
BEGIN(a(C(x))) → RIGHT5(x)
BEGIN(C(x)) → WAIT(Right6(x))
BEGIN(C(x)) → RIGHT6(x)
BEGIN(A(B(x))) → WAIT(Right7(x))
BEGIN(A(B(x))) → RIGHT7(x)
BEGIN(B(x)) → WAIT(Right8(x))
BEGIN(B(x)) → RIGHT8(x)
BEGIN(c(b(x))) → WAIT(Right9(x))
BEGIN(c(b(x))) → RIGHT9(x)
BEGIN(b(x)) → WAIT(Right10(x))
BEGIN(b(x)) → RIGHT10(x)
BEGIN(C(a(x))) → WAIT(Right11(x))
BEGIN(C(a(x))) → RIGHT11(x)
BEGIN(a(x)) → WAIT(Right12(x))
BEGIN(a(x)) → RIGHT12(x)
RIGHT1(a(End(x))) → C1(b(a(End(x))))
RIGHT1(a(End(x))) → B1(a(End(x)))
RIGHT2(a(b(End(x)))) → C1(b(a(End(x))))
RIGHT2(a(b(End(x)))) → B1(a(End(x)))
RIGHT2(a(b(End(x)))) → A1(End(x))
RIGHT3(C(End(x))) → A2(B(C(End(x))))
RIGHT3(C(End(x))) → B2(C(End(x)))
RIGHT4(C(B(End(x)))) → A2(B(C(End(x))))
RIGHT4(C(B(End(x)))) → B2(C(End(x)))
RIGHT4(C(B(End(x)))) → C2(End(x))
RIGHT5(b(End(x))) → C2(a(b(End(x))))
RIGHT5(b(End(x))) → A1(b(End(x)))
RIGHT6(b(a(End(x)))) → C2(a(b(End(x))))
RIGHT6(b(a(End(x)))) → A1(b(End(x)))
RIGHT6(b(a(End(x)))) → B1(End(x))
RIGHT7(c(End(x))) → B2(A(c(End(x))))
RIGHT7(c(End(x))) → A2(c(End(x)))
RIGHT8(c(A(End(x)))) → B2(A(c(End(x))))
RIGHT8(c(A(End(x)))) → A2(c(End(x)))
RIGHT8(c(A(End(x)))) → C1(End(x))
RIGHT9(A(End(x))) → B1(c(A(End(x))))
RIGHT9(A(End(x))) → C1(A(End(x)))
RIGHT10(A(c(End(x)))) → B1(c(A(End(x))))
RIGHT10(A(c(End(x)))) → C1(A(End(x)))
RIGHT10(A(c(End(x)))) → A2(End(x))
RIGHT11(B(End(x))) → A1(C(B(End(x))))
RIGHT11(B(End(x))) → C2(B(End(x)))
RIGHT12(B(C(End(x)))) → A1(C(B(End(x))))
RIGHT12(B(C(End(x)))) → C2(B(End(x)))
RIGHT12(B(C(End(x)))) → B2(End(x))
RIGHT1(a(x)) → AA1(Right1(x))
RIGHT1(a(x)) → RIGHT1(x)
RIGHT2(a(x)) → AA1(Right2(x))
RIGHT2(a(x)) → RIGHT2(x)
RIGHT3(a(x)) → AA1(Right3(x))
RIGHT3(a(x)) → RIGHT3(x)
RIGHT4(a(x)) → AA1(Right4(x))
RIGHT4(a(x)) → RIGHT4(x)
RIGHT5(a(x)) → AA1(Right5(x))
RIGHT5(a(x)) → RIGHT5(x)
RIGHT6(a(x)) → AA1(Right6(x))
RIGHT6(a(x)) → RIGHT6(x)
RIGHT7(a(x)) → AA1(Right7(x))
RIGHT7(a(x)) → RIGHT7(x)
RIGHT8(a(x)) → AA1(Right8(x))
RIGHT8(a(x)) → RIGHT8(x)
RIGHT9(a(x)) → AA1(Right9(x))
RIGHT9(a(x)) → RIGHT9(x)
RIGHT10(a(x)) → AA1(Right10(x))
RIGHT10(a(x)) → RIGHT10(x)
RIGHT11(a(x)) → AA1(Right11(x))
RIGHT11(a(x)) → RIGHT11(x)
RIGHT12(a(x)) → AA1(Right12(x))
RIGHT12(a(x)) → RIGHT12(x)
RIGHT13(a(x)) → AA1(Right13(x))
RIGHT13(a(x)) → RIGHT13(x)
RIGHT14(a(x)) → AA1(Right14(x))
RIGHT14(a(x)) → RIGHT14(x)
RIGHT15(a(x)) → AA1(Right15(x))
RIGHT15(a(x)) → RIGHT15(x)
RIGHT16(a(x)) → AA1(Right16(x))
RIGHT16(a(x)) → RIGHT16(x)
RIGHT17(a(x)) → AA1(Right17(x))
RIGHT17(a(x)) → RIGHT17(x)
RIGHT18(a(x)) → AA1(Right18(x))
RIGHT18(a(x)) → RIGHT18(x)
RIGHT1(b(x)) → AB1(Right1(x))
RIGHT1(b(x)) → RIGHT1(x)
RIGHT2(b(x)) → AB1(Right2(x))
RIGHT2(b(x)) → RIGHT2(x)
RIGHT3(b(x)) → AB1(Right3(x))
RIGHT3(b(x)) → RIGHT3(x)
RIGHT4(b(x)) → AB1(Right4(x))
RIGHT4(b(x)) → RIGHT4(x)
RIGHT5(b(x)) → AB1(Right5(x))
RIGHT5(b(x)) → RIGHT5(x)
RIGHT6(b(x)) → AB1(Right6(x))
RIGHT6(b(x)) → RIGHT6(x)
RIGHT7(b(x)) → AB1(Right7(x))
RIGHT7(b(x)) → RIGHT7(x)
RIGHT8(b(x)) → AB1(Right8(x))
RIGHT8(b(x)) → RIGHT8(x)
RIGHT9(b(x)) → AB1(Right9(x))
RIGHT9(b(x)) → RIGHT9(x)
RIGHT10(b(x)) → AB1(Right10(x))
RIGHT10(b(x)) → RIGHT10(x)
RIGHT11(b(x)) → AB1(Right11(x))
RIGHT11(b(x)) → RIGHT11(x)
RIGHT12(b(x)) → AB1(Right12(x))
RIGHT12(b(x)) → RIGHT12(x)
RIGHT13(b(x)) → AB1(Right13(x))
RIGHT13(b(x)) → RIGHT13(x)
RIGHT14(b(x)) → AB1(Right14(x))
RIGHT14(b(x)) → RIGHT14(x)
RIGHT15(b(x)) → AB1(Right15(x))
RIGHT15(b(x)) → RIGHT15(x)
RIGHT16(b(x)) → AB1(Right16(x))
RIGHT16(b(x)) → RIGHT16(x)
RIGHT17(b(x)) → AB1(Right17(x))
RIGHT17(b(x)) → RIGHT17(x)
RIGHT18(b(x)) → AB1(Right18(x))
RIGHT18(b(x)) → RIGHT18(x)
RIGHT1(c(x)) → AC1(Right1(x))
RIGHT1(c(x)) → RIGHT1(x)
RIGHT2(c(x)) → AC1(Right2(x))
RIGHT2(c(x)) → RIGHT2(x)
RIGHT3(c(x)) → AC1(Right3(x))
RIGHT3(c(x)) → RIGHT3(x)
RIGHT4(c(x)) → AC1(Right4(x))
RIGHT4(c(x)) → RIGHT4(x)
RIGHT5(c(x)) → AC1(Right5(x))
RIGHT5(c(x)) → RIGHT5(x)
RIGHT6(c(x)) → AC1(Right6(x))
RIGHT6(c(x)) → RIGHT6(x)
RIGHT7(c(x)) → AC1(Right7(x))
RIGHT7(c(x)) → RIGHT7(x)
RIGHT8(c(x)) → AC1(Right8(x))
RIGHT8(c(x)) → RIGHT8(x)
RIGHT9(c(x)) → AC1(Right9(x))
RIGHT9(c(x)) → RIGHT9(x)
RIGHT10(c(x)) → AC1(Right10(x))
RIGHT10(c(x)) → RIGHT10(x)
RIGHT11(c(x)) → AC1(Right11(x))
RIGHT11(c(x)) → RIGHT11(x)
RIGHT12(c(x)) → AC1(Right12(x))
RIGHT12(c(x)) → RIGHT12(x)
RIGHT13(c(x)) → AC1(Right13(x))
RIGHT13(c(x)) → RIGHT13(x)
RIGHT14(c(x)) → AC1(Right14(x))
RIGHT14(c(x)) → RIGHT14(x)
RIGHT15(c(x)) → AC1(Right15(x))
RIGHT15(c(x)) → RIGHT15(x)
RIGHT16(c(x)) → AC1(Right16(x))
RIGHT16(c(x)) → RIGHT16(x)
RIGHT17(c(x)) → AC1(Right17(x))
RIGHT17(c(x)) → RIGHT17(x)
RIGHT18(c(x)) → AC1(Right18(x))
RIGHT18(c(x)) → RIGHT18(x)
RIGHT1(C(x)) → AC2(Right1(x))
RIGHT1(C(x)) → RIGHT1(x)
RIGHT2(C(x)) → AC2(Right2(x))
RIGHT2(C(x)) → RIGHT2(x)
RIGHT3(C(x)) → AC2(Right3(x))
RIGHT3(C(x)) → RIGHT3(x)
RIGHT4(C(x)) → AC2(Right4(x))
RIGHT4(C(x)) → RIGHT4(x)
RIGHT5(C(x)) → AC2(Right5(x))
RIGHT5(C(x)) → RIGHT5(x)
RIGHT6(C(x)) → AC2(Right6(x))
RIGHT6(C(x)) → RIGHT6(x)
RIGHT7(C(x)) → AC2(Right7(x))
RIGHT7(C(x)) → RIGHT7(x)
RIGHT8(C(x)) → AC2(Right8(x))
RIGHT8(C(x)) → RIGHT8(x)
RIGHT9(C(x)) → AC2(Right9(x))
RIGHT9(C(x)) → RIGHT9(x)
RIGHT10(C(x)) → AC2(Right10(x))
RIGHT10(C(x)) → RIGHT10(x)
RIGHT11(C(x)) → AC2(Right11(x))
RIGHT11(C(x)) → RIGHT11(x)
RIGHT12(C(x)) → AC2(Right12(x))
RIGHT12(C(x)) → RIGHT12(x)
RIGHT13(C(x)) → AC2(Right13(x))
RIGHT13(C(x)) → RIGHT13(x)
RIGHT14(C(x)) → AC2(Right14(x))
RIGHT14(C(x)) → RIGHT14(x)
RIGHT15(C(x)) → AC2(Right15(x))
RIGHT15(C(x)) → RIGHT15(x)
RIGHT16(C(x)) → AC2(Right16(x))
RIGHT16(C(x)) → RIGHT16(x)
RIGHT17(C(x)) → AC2(Right17(x))
RIGHT17(C(x)) → RIGHT17(x)
RIGHT18(C(x)) → AC2(Right18(x))
RIGHT18(C(x)) → RIGHT18(x)
RIGHT1(B(x)) → AB2(Right1(x))
RIGHT1(B(x)) → RIGHT1(x)
RIGHT2(B(x)) → AB2(Right2(x))
RIGHT2(B(x)) → RIGHT2(x)
RIGHT3(B(x)) → AB2(Right3(x))
RIGHT3(B(x)) → RIGHT3(x)
RIGHT4(B(x)) → AB2(Right4(x))
RIGHT4(B(x)) → RIGHT4(x)
RIGHT5(B(x)) → AB2(Right5(x))
RIGHT5(B(x)) → RIGHT5(x)
RIGHT6(B(x)) → AB2(Right6(x))
RIGHT6(B(x)) → RIGHT6(x)
RIGHT7(B(x)) → AB2(Right7(x))
RIGHT7(B(x)) → RIGHT7(x)
RIGHT8(B(x)) → AB2(Right8(x))
RIGHT8(B(x)) → RIGHT8(x)
RIGHT9(B(x)) → AB2(Right9(x))
RIGHT9(B(x)) → RIGHT9(x)
RIGHT10(B(x)) → AB2(Right10(x))
RIGHT10(B(x)) → RIGHT10(x)
RIGHT11(B(x)) → AB2(Right11(x))
RIGHT11(B(x)) → RIGHT11(x)
RIGHT12(B(x)) → AB2(Right12(x))
RIGHT12(B(x)) → RIGHT12(x)
RIGHT13(B(x)) → AB2(Right13(x))
RIGHT13(B(x)) → RIGHT13(x)
RIGHT14(B(x)) → AB2(Right14(x))
RIGHT14(B(x)) → RIGHT14(x)
RIGHT15(B(x)) → AB2(Right15(x))
RIGHT15(B(x)) → RIGHT15(x)
RIGHT16(B(x)) → AB2(Right16(x))
RIGHT16(B(x)) → RIGHT16(x)
RIGHT17(B(x)) → AB2(Right17(x))
RIGHT17(B(x)) → RIGHT17(x)
RIGHT18(B(x)) → AB2(Right18(x))
RIGHT18(B(x)) → RIGHT18(x)
RIGHT1(A(x)) → AA2(Right1(x))
RIGHT1(A(x)) → RIGHT1(x)
RIGHT2(A(x)) → AA2(Right2(x))
RIGHT2(A(x)) → RIGHT2(x)
RIGHT3(A(x)) → AA2(Right3(x))
RIGHT3(A(x)) → RIGHT3(x)
RIGHT4(A(x)) → AA2(Right4(x))
RIGHT4(A(x)) → RIGHT4(x)
RIGHT5(A(x)) → AA2(Right5(x))
RIGHT5(A(x)) → RIGHT5(x)
RIGHT6(A(x)) → AA2(Right6(x))
RIGHT6(A(x)) → RIGHT6(x)
RIGHT7(A(x)) → AA2(Right7(x))
RIGHT7(A(x)) → RIGHT7(x)
RIGHT8(A(x)) → AA2(Right8(x))
RIGHT8(A(x)) → RIGHT8(x)
RIGHT9(A(x)) → AA2(Right9(x))
RIGHT9(A(x)) → RIGHT9(x)
RIGHT10(A(x)) → AA2(Right10(x))
RIGHT10(A(x)) → RIGHT10(x)
RIGHT11(A(x)) → AA2(Right11(x))
RIGHT11(A(x)) → RIGHT11(x)
RIGHT12(A(x)) → AA2(Right12(x))
RIGHT12(A(x)) → RIGHT12(x)
RIGHT13(A(x)) → AA2(Right13(x))
RIGHT13(A(x)) → RIGHT13(x)
RIGHT14(A(x)) → AA2(Right14(x))
RIGHT14(A(x)) → RIGHT14(x)
RIGHT15(A(x)) → AA2(Right15(x))
RIGHT15(A(x)) → RIGHT15(x)
RIGHT16(A(x)) → AA2(Right16(x))
RIGHT16(A(x)) → RIGHT16(x)
RIGHT17(A(x)) → AA2(Right17(x))
RIGHT17(A(x)) → RIGHT17(x)
RIGHT18(A(x)) → AA2(Right18(x))
RIGHT18(A(x)) → RIGHT18(x)
AA1(Left(x)) → A1(x)
AB1(Left(x)) → B1(x)
AC1(Left(x)) → C1(x)
AC2(Left(x)) → C2(x)
AB2(Left(x)) → B2(x)
AA2(Left(x)) → A2(x)
WAIT(Left(x)) → BEGIN(x)
A1(b(c(x))) → C1(b(a(x)))
A1(b(c(x))) → B1(a(x))
A1(b(c(x))) → A1(x)
C2(B(A(x))) → A2(B(C(x)))
C2(B(A(x))) → B2(C(x))
C2(B(A(x))) → C2(x)
B1(a(C(x))) → C2(a(b(x)))
B1(a(C(x))) → A1(b(x))
B1(a(C(x))) → B1(x)
C1(A(B(x))) → B2(A(c(x)))
C1(A(B(x))) → A2(c(x))
C1(A(B(x))) → C1(x)
A2(c(b(x))) → B1(c(A(x)))
A2(c(b(x))) → C1(A(x))
A2(c(b(x))) → A2(x)
B2(C(a(x))) → A1(C(B(x)))
B2(C(a(x))) → C2(B(x))
B2(C(a(x))) → B2(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 20 SCCs with 156 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

C1(A(B(x))) → B2(A(c(x)))
B2(C(a(x))) → A1(C(B(x)))
A1(b(c(x))) → C1(b(a(x)))
C1(A(B(x))) → A2(c(x))
A2(c(b(x))) → B1(c(A(x)))
B1(a(C(x))) → C2(a(b(x)))
C2(B(A(x))) → A2(B(C(x)))
A2(c(b(x))) → C1(A(x))
C1(A(B(x))) → C1(x)
A2(c(b(x))) → A2(x)
C2(B(A(x))) → B2(C(x))
B2(C(a(x))) → C2(B(x))
C2(B(A(x))) → C2(x)
B2(C(a(x))) → B2(x)
B1(a(C(x))) → A1(b(x))
A1(b(c(x))) → B1(a(x))
B1(a(C(x))) → B1(x)
A1(b(c(x))) → A1(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(10) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(11) Obligation:

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

C1(A(B(x))) → B2(A(c(x)))
B2(C(a(x))) → A1(C(B(x)))
A1(b(c(x))) → C1(b(a(x)))
C1(A(B(x))) → A2(c(x))
A2(c(b(x))) → B1(c(A(x)))
B1(a(C(x))) → C2(a(b(x)))
C2(B(A(x))) → A2(B(C(x)))
A2(c(b(x))) → C1(A(x))
C1(A(B(x))) → C1(x)
A2(c(b(x))) → A2(x)
C2(B(A(x))) → B2(C(x))
B2(C(a(x))) → C2(B(x))
C2(B(A(x))) → C2(x)
B2(C(a(x))) → B2(x)
B1(a(C(x))) → A1(b(x))
A1(b(c(x))) → B1(a(x))
B1(a(C(x))) → B1(x)
A1(b(c(x))) → A1(x)

The TRS R consists of the following rules:

c(A(B(x))) → B(A(c(x)))
B(C(a(x))) → a(C(B(x)))
a(b(c(x))) → c(b(a(x)))
A(c(b(x))) → b(c(A(x)))
b(a(C(x))) → C(a(b(x)))
C(B(A(x))) → A(B(C(x)))

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

(12) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

A2(c(b(x))) → B1(c(A(x)))
A2(c(b(x))) → C1(A(x))
A2(c(b(x))) → A2(x)
A1(b(c(x))) → A1(x)

Strictly oriented rules of the TRS R:

A(c(b(x))) → b(c(A(x)))

Used ordering: Polynomial interpretation [POLO]:

POL(A(x1)) = 2·x1   
POL(A1(x1)) = x1   
POL(A2(x1)) = 2·x1   
POL(B(x1)) = 2·x1   
POL(B1(x1)) = 2 + 2·x1   
POL(B2(x1)) = 2·x1   
POL(C(x1)) = x1   
POL(C1(x1)) = x1   
POL(C2(x1)) = x1   
POL(a(x1)) = x1   
POL(b(x1)) = 2 + 2·x1   
POL(c(x1)) = x1   

(13) Obligation:

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

C1(A(B(x))) → B2(A(c(x)))
B2(C(a(x))) → A1(C(B(x)))
A1(b(c(x))) → C1(b(a(x)))
C1(A(B(x))) → A2(c(x))
B1(a(C(x))) → C2(a(b(x)))
C2(B(A(x))) → A2(B(C(x)))
C1(A(B(x))) → C1(x)
C2(B(A(x))) → B2(C(x))
B2(C(a(x))) → C2(B(x))
C2(B(A(x))) → C2(x)
B2(C(a(x))) → B2(x)
B1(a(C(x))) → A1(b(x))
A1(b(c(x))) → B1(a(x))
B1(a(C(x))) → B1(x)

The TRS R consists of the following rules:

c(A(B(x))) → B(A(c(x)))
B(C(a(x))) → a(C(B(x)))
a(b(c(x))) → c(b(a(x)))
b(a(C(x))) → C(a(b(x)))
C(B(A(x))) → A(B(C(x)))

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

(14) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 6 less nodes.

(15) Complex Obligation (AND)

(16) Obligation:

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

C1(A(B(x))) → C1(x)

The TRS R consists of the following rules:

c(A(B(x))) → B(A(c(x)))
B(C(a(x))) → a(C(B(x)))
a(b(c(x))) → c(b(a(x)))
b(a(C(x))) → C(a(b(x)))
C(B(A(x))) → A(B(C(x)))

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

(17) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(18) Obligation:

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

C1(A(B(x))) → C1(x)

R is empty.
Q is empty.
We have to consider all minimal (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:

  • C1(A(B(x))) → C1(x)
    The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

B2(C(a(x))) → C2(B(x))
C2(B(A(x))) → B2(C(x))
B2(C(a(x))) → B2(x)
C2(B(A(x))) → C2(x)

The TRS R consists of the following rules:

c(A(B(x))) → B(A(c(x)))
B(C(a(x))) → a(C(B(x)))
a(b(c(x))) → c(b(a(x)))
b(a(C(x))) → C(a(b(x)))
C(B(A(x))) → A(B(C(x)))

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

(22) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


B2(C(a(x))) → C2(B(x))
C2(B(A(x))) → B2(C(x))
B2(C(a(x))) → B2(x)
C2(B(A(x))) → C2(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(A(x1)) = 1 + x1   
POL(B(x1)) = x1   
POL(B2(x1)) = x1   
POL(C(x1)) = 1 + x1   
POL(C2(x1)) = 1 + x1   
POL(a(x1)) = 1 + x1   
POL(b(x1)) = x1   
POL(c(x1)) = 1 + x1   

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

B(C(a(x))) → a(C(B(x)))
C(B(A(x))) → A(B(C(x)))
a(b(c(x))) → c(b(a(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))

(23) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

c(A(B(x))) → B(A(c(x)))
B(C(a(x))) → a(C(B(x)))
a(b(c(x))) → c(b(a(x)))
b(a(C(x))) → C(a(b(x)))
C(B(A(x))) → A(B(C(x)))

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

(24) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(25) YES

(26) Obligation:

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

B1(a(C(x))) → A1(b(x))
A1(b(c(x))) → B1(a(x))
B1(a(C(x))) → B1(x)

The TRS R consists of the following rules:

c(A(B(x))) → B(A(c(x)))
B(C(a(x))) → a(C(B(x)))
a(b(c(x))) → c(b(a(x)))
b(a(C(x))) → C(a(b(x)))
C(B(A(x))) → A(B(C(x)))

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

(27) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


B1(a(C(x))) → A1(b(x))
B1(a(C(x))) → B1(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(A(x1)) = 0   
POL(A1(x1)) = x1   
POL(B(x1)) = 0   
POL(B1(x1)) = x1   
POL(C(x1)) = 1 + x1   
POL(a(x1)) = x1   
POL(b(x1)) = x1   
POL(c(x1)) = x1   

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

b(a(C(x))) → C(a(b(x)))
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
c(A(B(x))) → B(A(c(x)))

(28) Obligation:

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

A1(b(c(x))) → B1(a(x))

The TRS R consists of the following rules:

c(A(B(x))) → B(A(c(x)))
B(C(a(x))) → a(C(B(x)))
a(b(c(x))) → c(b(a(x)))
b(a(C(x))) → C(a(b(x)))
C(B(A(x))) → A(B(C(x)))

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

(29) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(30) TRUE

(31) Obligation:

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

RIGHT18(b(x)) → RIGHT18(x)
RIGHT18(a(x)) → RIGHT18(x)
RIGHT18(c(x)) → RIGHT18(x)
RIGHT18(C(x)) → RIGHT18(x)
RIGHT18(B(x)) → RIGHT18(x)
RIGHT18(A(x)) → RIGHT18(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(32) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(33) Obligation:

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

RIGHT18(b(x)) → RIGHT18(x)
RIGHT18(a(x)) → RIGHT18(x)
RIGHT18(c(x)) → RIGHT18(x)
RIGHT18(C(x)) → RIGHT18(x)
RIGHT18(B(x)) → RIGHT18(x)
RIGHT18(A(x)) → RIGHT18(x)

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

(34) 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:

  • RIGHT18(b(x)) → RIGHT18(x)
    The graph contains the following edges 1 > 1

  • RIGHT18(a(x)) → RIGHT18(x)
    The graph contains the following edges 1 > 1

  • RIGHT18(c(x)) → RIGHT18(x)
    The graph contains the following edges 1 > 1

  • RIGHT18(C(x)) → RIGHT18(x)
    The graph contains the following edges 1 > 1

  • RIGHT18(B(x)) → RIGHT18(x)
    The graph contains the following edges 1 > 1

  • RIGHT18(A(x)) → RIGHT18(x)
    The graph contains the following edges 1 > 1

(35) YES

(36) Obligation:

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

RIGHT17(b(x)) → RIGHT17(x)
RIGHT17(a(x)) → RIGHT17(x)
RIGHT17(c(x)) → RIGHT17(x)
RIGHT17(C(x)) → RIGHT17(x)
RIGHT17(B(x)) → RIGHT17(x)
RIGHT17(A(x)) → RIGHT17(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(37) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(38) Obligation:

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

RIGHT17(b(x)) → RIGHT17(x)
RIGHT17(a(x)) → RIGHT17(x)
RIGHT17(c(x)) → RIGHT17(x)
RIGHT17(C(x)) → RIGHT17(x)
RIGHT17(B(x)) → RIGHT17(x)
RIGHT17(A(x)) → RIGHT17(x)

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

(39) 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:

  • RIGHT17(b(x)) → RIGHT17(x)
    The graph contains the following edges 1 > 1

  • RIGHT17(a(x)) → RIGHT17(x)
    The graph contains the following edges 1 > 1

  • RIGHT17(c(x)) → RIGHT17(x)
    The graph contains the following edges 1 > 1

  • RIGHT17(C(x)) → RIGHT17(x)
    The graph contains the following edges 1 > 1

  • RIGHT17(B(x)) → RIGHT17(x)
    The graph contains the following edges 1 > 1

  • RIGHT17(A(x)) → RIGHT17(x)
    The graph contains the following edges 1 > 1

(40) YES

(41) Obligation:

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

RIGHT16(b(x)) → RIGHT16(x)
RIGHT16(a(x)) → RIGHT16(x)
RIGHT16(c(x)) → RIGHT16(x)
RIGHT16(C(x)) → RIGHT16(x)
RIGHT16(B(x)) → RIGHT16(x)
RIGHT16(A(x)) → RIGHT16(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(42) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(43) Obligation:

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

RIGHT16(b(x)) → RIGHT16(x)
RIGHT16(a(x)) → RIGHT16(x)
RIGHT16(c(x)) → RIGHT16(x)
RIGHT16(C(x)) → RIGHT16(x)
RIGHT16(B(x)) → RIGHT16(x)
RIGHT16(A(x)) → RIGHT16(x)

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

(44) 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:

  • RIGHT16(b(x)) → RIGHT16(x)
    The graph contains the following edges 1 > 1

  • RIGHT16(a(x)) → RIGHT16(x)
    The graph contains the following edges 1 > 1

  • RIGHT16(c(x)) → RIGHT16(x)
    The graph contains the following edges 1 > 1

  • RIGHT16(C(x)) → RIGHT16(x)
    The graph contains the following edges 1 > 1

  • RIGHT16(B(x)) → RIGHT16(x)
    The graph contains the following edges 1 > 1

  • RIGHT16(A(x)) → RIGHT16(x)
    The graph contains the following edges 1 > 1

(45) YES

(46) Obligation:

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

RIGHT15(b(x)) → RIGHT15(x)
RIGHT15(a(x)) → RIGHT15(x)
RIGHT15(c(x)) → RIGHT15(x)
RIGHT15(C(x)) → RIGHT15(x)
RIGHT15(B(x)) → RIGHT15(x)
RIGHT15(A(x)) → RIGHT15(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(47) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(48) Obligation:

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

RIGHT15(b(x)) → RIGHT15(x)
RIGHT15(a(x)) → RIGHT15(x)
RIGHT15(c(x)) → RIGHT15(x)
RIGHT15(C(x)) → RIGHT15(x)
RIGHT15(B(x)) → RIGHT15(x)
RIGHT15(A(x)) → RIGHT15(x)

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

(49) 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:

  • RIGHT15(b(x)) → RIGHT15(x)
    The graph contains the following edges 1 > 1

  • RIGHT15(a(x)) → RIGHT15(x)
    The graph contains the following edges 1 > 1

  • RIGHT15(c(x)) → RIGHT15(x)
    The graph contains the following edges 1 > 1

  • RIGHT15(C(x)) → RIGHT15(x)
    The graph contains the following edges 1 > 1

  • RIGHT15(B(x)) → RIGHT15(x)
    The graph contains the following edges 1 > 1

  • RIGHT15(A(x)) → RIGHT15(x)
    The graph contains the following edges 1 > 1

(50) YES

(51) Obligation:

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

RIGHT14(b(x)) → RIGHT14(x)
RIGHT14(a(x)) → RIGHT14(x)
RIGHT14(c(x)) → RIGHT14(x)
RIGHT14(C(x)) → RIGHT14(x)
RIGHT14(B(x)) → RIGHT14(x)
RIGHT14(A(x)) → RIGHT14(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(52) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(53) Obligation:

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

RIGHT14(b(x)) → RIGHT14(x)
RIGHT14(a(x)) → RIGHT14(x)
RIGHT14(c(x)) → RIGHT14(x)
RIGHT14(C(x)) → RIGHT14(x)
RIGHT14(B(x)) → RIGHT14(x)
RIGHT14(A(x)) → RIGHT14(x)

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

(54) 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:

  • RIGHT14(b(x)) → RIGHT14(x)
    The graph contains the following edges 1 > 1

  • RIGHT14(a(x)) → RIGHT14(x)
    The graph contains the following edges 1 > 1

  • RIGHT14(c(x)) → RIGHT14(x)
    The graph contains the following edges 1 > 1

  • RIGHT14(C(x)) → RIGHT14(x)
    The graph contains the following edges 1 > 1

  • RIGHT14(B(x)) → RIGHT14(x)
    The graph contains the following edges 1 > 1

  • RIGHT14(A(x)) → RIGHT14(x)
    The graph contains the following edges 1 > 1

(55) YES

(56) Obligation:

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

RIGHT13(b(x)) → RIGHT13(x)
RIGHT13(a(x)) → RIGHT13(x)
RIGHT13(c(x)) → RIGHT13(x)
RIGHT13(C(x)) → RIGHT13(x)
RIGHT13(B(x)) → RIGHT13(x)
RIGHT13(A(x)) → RIGHT13(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(57) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(58) Obligation:

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

RIGHT13(b(x)) → RIGHT13(x)
RIGHT13(a(x)) → RIGHT13(x)
RIGHT13(c(x)) → RIGHT13(x)
RIGHT13(C(x)) → RIGHT13(x)
RIGHT13(B(x)) → RIGHT13(x)
RIGHT13(A(x)) → RIGHT13(x)

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

(59) 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:

  • RIGHT13(b(x)) → RIGHT13(x)
    The graph contains the following edges 1 > 1

  • RIGHT13(a(x)) → RIGHT13(x)
    The graph contains the following edges 1 > 1

  • RIGHT13(c(x)) → RIGHT13(x)
    The graph contains the following edges 1 > 1

  • RIGHT13(C(x)) → RIGHT13(x)
    The graph contains the following edges 1 > 1

  • RIGHT13(B(x)) → RIGHT13(x)
    The graph contains the following edges 1 > 1

  • RIGHT13(A(x)) → RIGHT13(x)
    The graph contains the following edges 1 > 1

(60) YES

(61) Obligation:

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

RIGHT12(b(x)) → RIGHT12(x)
RIGHT12(a(x)) → RIGHT12(x)
RIGHT12(c(x)) → RIGHT12(x)
RIGHT12(C(x)) → RIGHT12(x)
RIGHT12(B(x)) → RIGHT12(x)
RIGHT12(A(x)) → RIGHT12(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(62) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(63) Obligation:

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

RIGHT12(b(x)) → RIGHT12(x)
RIGHT12(a(x)) → RIGHT12(x)
RIGHT12(c(x)) → RIGHT12(x)
RIGHT12(C(x)) → RIGHT12(x)
RIGHT12(B(x)) → RIGHT12(x)
RIGHT12(A(x)) → RIGHT12(x)

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

(64) 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:

  • RIGHT12(b(x)) → RIGHT12(x)
    The graph contains the following edges 1 > 1

  • RIGHT12(a(x)) → RIGHT12(x)
    The graph contains the following edges 1 > 1

  • RIGHT12(c(x)) → RIGHT12(x)
    The graph contains the following edges 1 > 1

  • RIGHT12(C(x)) → RIGHT12(x)
    The graph contains the following edges 1 > 1

  • RIGHT12(B(x)) → RIGHT12(x)
    The graph contains the following edges 1 > 1

  • RIGHT12(A(x)) → RIGHT12(x)
    The graph contains the following edges 1 > 1

(65) YES

(66) Obligation:

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

RIGHT11(b(x)) → RIGHT11(x)
RIGHT11(a(x)) → RIGHT11(x)
RIGHT11(c(x)) → RIGHT11(x)
RIGHT11(C(x)) → RIGHT11(x)
RIGHT11(B(x)) → RIGHT11(x)
RIGHT11(A(x)) → RIGHT11(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(67) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(68) Obligation:

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

RIGHT11(b(x)) → RIGHT11(x)
RIGHT11(a(x)) → RIGHT11(x)
RIGHT11(c(x)) → RIGHT11(x)
RIGHT11(C(x)) → RIGHT11(x)
RIGHT11(B(x)) → RIGHT11(x)
RIGHT11(A(x)) → RIGHT11(x)

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

(69) 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:

  • RIGHT11(b(x)) → RIGHT11(x)
    The graph contains the following edges 1 > 1

  • RIGHT11(a(x)) → RIGHT11(x)
    The graph contains the following edges 1 > 1

  • RIGHT11(c(x)) → RIGHT11(x)
    The graph contains the following edges 1 > 1

  • RIGHT11(C(x)) → RIGHT11(x)
    The graph contains the following edges 1 > 1

  • RIGHT11(B(x)) → RIGHT11(x)
    The graph contains the following edges 1 > 1

  • RIGHT11(A(x)) → RIGHT11(x)
    The graph contains the following edges 1 > 1

(70) YES

(71) Obligation:

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

RIGHT10(b(x)) → RIGHT10(x)
RIGHT10(a(x)) → RIGHT10(x)
RIGHT10(c(x)) → RIGHT10(x)
RIGHT10(C(x)) → RIGHT10(x)
RIGHT10(B(x)) → RIGHT10(x)
RIGHT10(A(x)) → RIGHT10(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(72) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(73) Obligation:

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

RIGHT10(b(x)) → RIGHT10(x)
RIGHT10(a(x)) → RIGHT10(x)
RIGHT10(c(x)) → RIGHT10(x)
RIGHT10(C(x)) → RIGHT10(x)
RIGHT10(B(x)) → RIGHT10(x)
RIGHT10(A(x)) → RIGHT10(x)

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

(74) 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:

  • RIGHT10(b(x)) → RIGHT10(x)
    The graph contains the following edges 1 > 1

  • RIGHT10(a(x)) → RIGHT10(x)
    The graph contains the following edges 1 > 1

  • RIGHT10(c(x)) → RIGHT10(x)
    The graph contains the following edges 1 > 1

  • RIGHT10(C(x)) → RIGHT10(x)
    The graph contains the following edges 1 > 1

  • RIGHT10(B(x)) → RIGHT10(x)
    The graph contains the following edges 1 > 1

  • RIGHT10(A(x)) → RIGHT10(x)
    The graph contains the following edges 1 > 1

(75) YES

(76) Obligation:

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

RIGHT9(b(x)) → RIGHT9(x)
RIGHT9(a(x)) → RIGHT9(x)
RIGHT9(c(x)) → RIGHT9(x)
RIGHT9(C(x)) → RIGHT9(x)
RIGHT9(B(x)) → RIGHT9(x)
RIGHT9(A(x)) → RIGHT9(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(77) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(78) Obligation:

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

RIGHT9(b(x)) → RIGHT9(x)
RIGHT9(a(x)) → RIGHT9(x)
RIGHT9(c(x)) → RIGHT9(x)
RIGHT9(C(x)) → RIGHT9(x)
RIGHT9(B(x)) → RIGHT9(x)
RIGHT9(A(x)) → RIGHT9(x)

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

(79) 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:

  • RIGHT9(b(x)) → RIGHT9(x)
    The graph contains the following edges 1 > 1

  • RIGHT9(a(x)) → RIGHT9(x)
    The graph contains the following edges 1 > 1

  • RIGHT9(c(x)) → RIGHT9(x)
    The graph contains the following edges 1 > 1

  • RIGHT9(C(x)) → RIGHT9(x)
    The graph contains the following edges 1 > 1

  • RIGHT9(B(x)) → RIGHT9(x)
    The graph contains the following edges 1 > 1

  • RIGHT9(A(x)) → RIGHT9(x)
    The graph contains the following edges 1 > 1

(80) YES

(81) Obligation:

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

RIGHT8(b(x)) → RIGHT8(x)
RIGHT8(a(x)) → RIGHT8(x)
RIGHT8(c(x)) → RIGHT8(x)
RIGHT8(C(x)) → RIGHT8(x)
RIGHT8(B(x)) → RIGHT8(x)
RIGHT8(A(x)) → RIGHT8(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(82) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(83) Obligation:

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

RIGHT8(b(x)) → RIGHT8(x)
RIGHT8(a(x)) → RIGHT8(x)
RIGHT8(c(x)) → RIGHT8(x)
RIGHT8(C(x)) → RIGHT8(x)
RIGHT8(B(x)) → RIGHT8(x)
RIGHT8(A(x)) → RIGHT8(x)

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

(84) 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:

  • RIGHT8(b(x)) → RIGHT8(x)
    The graph contains the following edges 1 > 1

  • RIGHT8(a(x)) → RIGHT8(x)
    The graph contains the following edges 1 > 1

  • RIGHT8(c(x)) → RIGHT8(x)
    The graph contains the following edges 1 > 1

  • RIGHT8(C(x)) → RIGHT8(x)
    The graph contains the following edges 1 > 1

  • RIGHT8(B(x)) → RIGHT8(x)
    The graph contains the following edges 1 > 1

  • RIGHT8(A(x)) → RIGHT8(x)
    The graph contains the following edges 1 > 1

(85) YES

(86) Obligation:

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

RIGHT7(b(x)) → RIGHT7(x)
RIGHT7(a(x)) → RIGHT7(x)
RIGHT7(c(x)) → RIGHT7(x)
RIGHT7(C(x)) → RIGHT7(x)
RIGHT7(B(x)) → RIGHT7(x)
RIGHT7(A(x)) → RIGHT7(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(87) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(88) Obligation:

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

RIGHT7(b(x)) → RIGHT7(x)
RIGHT7(a(x)) → RIGHT7(x)
RIGHT7(c(x)) → RIGHT7(x)
RIGHT7(C(x)) → RIGHT7(x)
RIGHT7(B(x)) → RIGHT7(x)
RIGHT7(A(x)) → RIGHT7(x)

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

(89) 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:

  • RIGHT7(b(x)) → RIGHT7(x)
    The graph contains the following edges 1 > 1

  • RIGHT7(a(x)) → RIGHT7(x)
    The graph contains the following edges 1 > 1

  • RIGHT7(c(x)) → RIGHT7(x)
    The graph contains the following edges 1 > 1

  • RIGHT7(C(x)) → RIGHT7(x)
    The graph contains the following edges 1 > 1

  • RIGHT7(B(x)) → RIGHT7(x)
    The graph contains the following edges 1 > 1

  • RIGHT7(A(x)) → RIGHT7(x)
    The graph contains the following edges 1 > 1

(90) YES

(91) Obligation:

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

RIGHT6(b(x)) → RIGHT6(x)
RIGHT6(a(x)) → RIGHT6(x)
RIGHT6(c(x)) → RIGHT6(x)
RIGHT6(C(x)) → RIGHT6(x)
RIGHT6(B(x)) → RIGHT6(x)
RIGHT6(A(x)) → RIGHT6(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(92) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(93) Obligation:

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

RIGHT6(b(x)) → RIGHT6(x)
RIGHT6(a(x)) → RIGHT6(x)
RIGHT6(c(x)) → RIGHT6(x)
RIGHT6(C(x)) → RIGHT6(x)
RIGHT6(B(x)) → RIGHT6(x)
RIGHT6(A(x)) → RIGHT6(x)

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

(94) 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:

  • RIGHT6(b(x)) → RIGHT6(x)
    The graph contains the following edges 1 > 1

  • RIGHT6(a(x)) → RIGHT6(x)
    The graph contains the following edges 1 > 1

  • RIGHT6(c(x)) → RIGHT6(x)
    The graph contains the following edges 1 > 1

  • RIGHT6(C(x)) → RIGHT6(x)
    The graph contains the following edges 1 > 1

  • RIGHT6(B(x)) → RIGHT6(x)
    The graph contains the following edges 1 > 1

  • RIGHT6(A(x)) → RIGHT6(x)
    The graph contains the following edges 1 > 1

(95) YES

(96) Obligation:

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

RIGHT5(b(x)) → RIGHT5(x)
RIGHT5(a(x)) → RIGHT5(x)
RIGHT5(c(x)) → RIGHT5(x)
RIGHT5(C(x)) → RIGHT5(x)
RIGHT5(B(x)) → RIGHT5(x)
RIGHT5(A(x)) → RIGHT5(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(97) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(98) Obligation:

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

RIGHT5(b(x)) → RIGHT5(x)
RIGHT5(a(x)) → RIGHT5(x)
RIGHT5(c(x)) → RIGHT5(x)
RIGHT5(C(x)) → RIGHT5(x)
RIGHT5(B(x)) → RIGHT5(x)
RIGHT5(A(x)) → RIGHT5(x)

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

(99) 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:

  • RIGHT5(b(x)) → RIGHT5(x)
    The graph contains the following edges 1 > 1

  • RIGHT5(a(x)) → RIGHT5(x)
    The graph contains the following edges 1 > 1

  • RIGHT5(c(x)) → RIGHT5(x)
    The graph contains the following edges 1 > 1

  • RIGHT5(C(x)) → RIGHT5(x)
    The graph contains the following edges 1 > 1

  • RIGHT5(B(x)) → RIGHT5(x)
    The graph contains the following edges 1 > 1

  • RIGHT5(A(x)) → RIGHT5(x)
    The graph contains the following edges 1 > 1

(100) YES

(101) Obligation:

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

RIGHT4(b(x)) → RIGHT4(x)
RIGHT4(a(x)) → RIGHT4(x)
RIGHT4(c(x)) → RIGHT4(x)
RIGHT4(C(x)) → RIGHT4(x)
RIGHT4(B(x)) → RIGHT4(x)
RIGHT4(A(x)) → RIGHT4(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(102) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(103) Obligation:

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

RIGHT4(b(x)) → RIGHT4(x)
RIGHT4(a(x)) → RIGHT4(x)
RIGHT4(c(x)) → RIGHT4(x)
RIGHT4(C(x)) → RIGHT4(x)
RIGHT4(B(x)) → RIGHT4(x)
RIGHT4(A(x)) → RIGHT4(x)

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

(104) 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:

  • RIGHT4(b(x)) → RIGHT4(x)
    The graph contains the following edges 1 > 1

  • RIGHT4(a(x)) → RIGHT4(x)
    The graph contains the following edges 1 > 1

  • RIGHT4(c(x)) → RIGHT4(x)
    The graph contains the following edges 1 > 1

  • RIGHT4(C(x)) → RIGHT4(x)
    The graph contains the following edges 1 > 1

  • RIGHT4(B(x)) → RIGHT4(x)
    The graph contains the following edges 1 > 1

  • RIGHT4(A(x)) → RIGHT4(x)
    The graph contains the following edges 1 > 1

(105) YES

(106) Obligation:

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

RIGHT3(b(x)) → RIGHT3(x)
RIGHT3(a(x)) → RIGHT3(x)
RIGHT3(c(x)) → RIGHT3(x)
RIGHT3(C(x)) → RIGHT3(x)
RIGHT3(B(x)) → RIGHT3(x)
RIGHT3(A(x)) → RIGHT3(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(107) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(108) Obligation:

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

RIGHT3(b(x)) → RIGHT3(x)
RIGHT3(a(x)) → RIGHT3(x)
RIGHT3(c(x)) → RIGHT3(x)
RIGHT3(C(x)) → RIGHT3(x)
RIGHT3(B(x)) → RIGHT3(x)
RIGHT3(A(x)) → RIGHT3(x)

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

(109) 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:

  • RIGHT3(b(x)) → RIGHT3(x)
    The graph contains the following edges 1 > 1

  • RIGHT3(a(x)) → RIGHT3(x)
    The graph contains the following edges 1 > 1

  • RIGHT3(c(x)) → RIGHT3(x)
    The graph contains the following edges 1 > 1

  • RIGHT3(C(x)) → RIGHT3(x)
    The graph contains the following edges 1 > 1

  • RIGHT3(B(x)) → RIGHT3(x)
    The graph contains the following edges 1 > 1

  • RIGHT3(A(x)) → RIGHT3(x)
    The graph contains the following edges 1 > 1

(110) YES

(111) Obligation:

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

RIGHT2(b(x)) → RIGHT2(x)
RIGHT2(a(x)) → RIGHT2(x)
RIGHT2(c(x)) → RIGHT2(x)
RIGHT2(C(x)) → RIGHT2(x)
RIGHT2(B(x)) → RIGHT2(x)
RIGHT2(A(x)) → RIGHT2(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(112) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(113) Obligation:

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

RIGHT2(b(x)) → RIGHT2(x)
RIGHT2(a(x)) → RIGHT2(x)
RIGHT2(c(x)) → RIGHT2(x)
RIGHT2(C(x)) → RIGHT2(x)
RIGHT2(B(x)) → RIGHT2(x)
RIGHT2(A(x)) → RIGHT2(x)

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

(114) 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:

  • RIGHT2(b(x)) → RIGHT2(x)
    The graph contains the following edges 1 > 1

  • RIGHT2(a(x)) → RIGHT2(x)
    The graph contains the following edges 1 > 1

  • RIGHT2(c(x)) → RIGHT2(x)
    The graph contains the following edges 1 > 1

  • RIGHT2(C(x)) → RIGHT2(x)
    The graph contains the following edges 1 > 1

  • RIGHT2(B(x)) → RIGHT2(x)
    The graph contains the following edges 1 > 1

  • RIGHT2(A(x)) → RIGHT2(x)
    The graph contains the following edges 1 > 1

(115) YES

(116) Obligation:

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

RIGHT1(b(x)) → RIGHT1(x)
RIGHT1(a(x)) → RIGHT1(x)
RIGHT1(c(x)) → RIGHT1(x)
RIGHT1(C(x)) → RIGHT1(x)
RIGHT1(B(x)) → RIGHT1(x)
RIGHT1(A(x)) → RIGHT1(x)

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(117) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(118) Obligation:

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

RIGHT1(b(x)) → RIGHT1(x)
RIGHT1(a(x)) → RIGHT1(x)
RIGHT1(c(x)) → RIGHT1(x)
RIGHT1(C(x)) → RIGHT1(x)
RIGHT1(B(x)) → RIGHT1(x)
RIGHT1(A(x)) → RIGHT1(x)

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

(119) 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:

  • RIGHT1(b(x)) → RIGHT1(x)
    The graph contains the following edges 1 > 1

  • RIGHT1(a(x)) → RIGHT1(x)
    The graph contains the following edges 1 > 1

  • RIGHT1(c(x)) → RIGHT1(x)
    The graph contains the following edges 1 > 1

  • RIGHT1(C(x)) → RIGHT1(x)
    The graph contains the following edges 1 > 1

  • RIGHT1(B(x)) → RIGHT1(x)
    The graph contains the following edges 1 > 1

  • RIGHT1(A(x)) → RIGHT1(x)
    The graph contains the following edges 1 > 1

(120) YES

(121) Obligation:

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

WAIT(Left(x)) → BEGIN(x)
BEGIN(b(c(x))) → WAIT(Right1(x))
BEGIN(c(x)) → WAIT(Right2(x))
BEGIN(B(A(x))) → WAIT(Right3(x))
BEGIN(A(x)) → WAIT(Right4(x))
BEGIN(a(C(x))) → WAIT(Right5(x))
BEGIN(C(x)) → WAIT(Right6(x))
BEGIN(A(B(x))) → WAIT(Right7(x))
BEGIN(B(x)) → WAIT(Right8(x))
BEGIN(c(b(x))) → WAIT(Right9(x))
BEGIN(b(x)) → WAIT(Right10(x))
BEGIN(C(a(x))) → WAIT(Right11(x))
BEGIN(a(x)) → WAIT(Right12(x))

The TRS R consists of the following rules:

Begin(b(c(x))) → Wait(Right1(x))
Begin(c(x)) → Wait(Right2(x))
Begin(B(A(x))) → Wait(Right3(x))
Begin(A(x)) → Wait(Right4(x))
Begin(a(C(x))) → Wait(Right5(x))
Begin(C(x)) → Wait(Right6(x))
Begin(A(B(x))) → Wait(Right7(x))
Begin(B(x)) → Wait(Right8(x))
Begin(c(b(x))) → Wait(Right9(x))
Begin(b(x)) → Wait(Right10(x))
Begin(C(a(x))) → Wait(Right11(x))
Begin(a(x)) → Wait(Right12(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right2(a(x)) → Aa(Right2(x))
Right3(a(x)) → Aa(Right3(x))
Right4(a(x)) → Aa(Right4(x))
Right5(a(x)) → Aa(Right5(x))
Right6(a(x)) → Aa(Right6(x))
Right7(a(x)) → Aa(Right7(x))
Right8(a(x)) → Aa(Right8(x))
Right9(a(x)) → Aa(Right9(x))
Right10(a(x)) → Aa(Right10(x))
Right11(a(x)) → Aa(Right11(x))
Right12(a(x)) → Aa(Right12(x))
Right13(a(x)) → Aa(Right13(x))
Right14(a(x)) → Aa(Right14(x))
Right15(a(x)) → Aa(Right15(x))
Right16(a(x)) → Aa(Right16(x))
Right17(a(x)) → Aa(Right17(x))
Right18(a(x)) → Aa(Right18(x))
Right1(b(x)) → Ab(Right1(x))
Right2(b(x)) → Ab(Right2(x))
Right3(b(x)) → Ab(Right3(x))
Right4(b(x)) → Ab(Right4(x))
Right5(b(x)) → Ab(Right5(x))
Right6(b(x)) → Ab(Right6(x))
Right7(b(x)) → Ab(Right7(x))
Right8(b(x)) → Ab(Right8(x))
Right9(b(x)) → Ab(Right9(x))
Right10(b(x)) → Ab(Right10(x))
Right11(b(x)) → Ab(Right11(x))
Right12(b(x)) → Ab(Right12(x))
Right13(b(x)) → Ab(Right13(x))
Right14(b(x)) → Ab(Right14(x))
Right15(b(x)) → Ab(Right15(x))
Right16(b(x)) → Ab(Right16(x))
Right17(b(x)) → Ab(Right17(x))
Right18(b(x)) → Ab(Right18(x))
Right1(c(x)) → Ac(Right1(x))
Right2(c(x)) → Ac(Right2(x))
Right3(c(x)) → Ac(Right3(x))
Right4(c(x)) → Ac(Right4(x))
Right5(c(x)) → Ac(Right5(x))
Right6(c(x)) → Ac(Right6(x))
Right7(c(x)) → Ac(Right7(x))
Right8(c(x)) → Ac(Right8(x))
Right9(c(x)) → Ac(Right9(x))
Right10(c(x)) → Ac(Right10(x))
Right11(c(x)) → Ac(Right11(x))
Right12(c(x)) → Ac(Right12(x))
Right13(c(x)) → Ac(Right13(x))
Right14(c(x)) → Ac(Right14(x))
Right15(c(x)) → Ac(Right15(x))
Right16(c(x)) → Ac(Right16(x))
Right17(c(x)) → Ac(Right17(x))
Right18(c(x)) → Ac(Right18(x))
Right1(C(x)) → AC(Right1(x))
Right2(C(x)) → AC(Right2(x))
Right3(C(x)) → AC(Right3(x))
Right4(C(x)) → AC(Right4(x))
Right5(C(x)) → AC(Right5(x))
Right6(C(x)) → AC(Right6(x))
Right7(C(x)) → AC(Right7(x))
Right8(C(x)) → AC(Right8(x))
Right9(C(x)) → AC(Right9(x))
Right10(C(x)) → AC(Right10(x))
Right11(C(x)) → AC(Right11(x))
Right12(C(x)) → AC(Right12(x))
Right13(C(x)) → AC(Right13(x))
Right14(C(x)) → AC(Right14(x))
Right15(C(x)) → AC(Right15(x))
Right16(C(x)) → AC(Right16(x))
Right17(C(x)) → AC(Right17(x))
Right18(C(x)) → AC(Right18(x))
Right1(B(x)) → AB(Right1(x))
Right2(B(x)) → AB(Right2(x))
Right3(B(x)) → AB(Right3(x))
Right4(B(x)) → AB(Right4(x))
Right5(B(x)) → AB(Right5(x))
Right6(B(x)) → AB(Right6(x))
Right7(B(x)) → AB(Right7(x))
Right8(B(x)) → AB(Right8(x))
Right9(B(x)) → AB(Right9(x))
Right10(B(x)) → AB(Right10(x))
Right11(B(x)) → AB(Right11(x))
Right12(B(x)) → AB(Right12(x))
Right13(B(x)) → AB(Right13(x))
Right14(B(x)) → AB(Right14(x))
Right15(B(x)) → AB(Right15(x))
Right16(B(x)) → AB(Right16(x))
Right17(B(x)) → AB(Right17(x))
Right18(B(x)) → AB(Right18(x))
Right1(A(x)) → AA(Right1(x))
Right2(A(x)) → AA(Right2(x))
Right3(A(x)) → AA(Right3(x))
Right4(A(x)) → AA(Right4(x))
Right5(A(x)) → AA(Right5(x))
Right6(A(x)) → AA(Right6(x))
Right7(A(x)) → AA(Right7(x))
Right8(A(x)) → AA(Right8(x))
Right9(A(x)) → AA(Right9(x))
Right10(A(x)) → AA(Right10(x))
Right11(A(x)) → AA(Right11(x))
Right12(A(x)) → AA(Right12(x))
Right13(A(x)) → AA(Right13(x))
Right14(A(x)) → AA(Right14(x))
Right15(A(x)) → AA(Right15(x))
Right16(A(x)) → AA(Right16(x))
Right17(A(x)) → AA(Right17(x))
Right18(A(x)) → AA(Right18(x))
Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Ac(Left(x)) → Left(c(x))
AC(Left(x)) → Left(C(x))
AB(Left(x)) → Left(B(x))
AA(Left(x)) → Left(A(x))
Wait(Left(x)) → Begin(x)
a(b(c(x))) → c(b(a(x)))
C(B(A(x))) → A(B(C(x)))
b(a(C(x))) → C(a(b(x)))
c(A(B(x))) → B(A(c(x)))
A(c(b(x))) → b(c(A(x)))
B(C(a(x))) → a(C(B(x)))

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

(122) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(123) Obligation:

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

WAIT(Left(x)) → BEGIN(x)
BEGIN(b(c(x))) → WAIT(Right1(x))
BEGIN(c(x)) → WAIT(Right2(x))
BEGIN(B(A(x))) → WAIT(Right3(x))
BEGIN(A(x)) → WAIT(Right4(x))
BEGIN(a(C(x))) → WAIT(Right5(x))
BEGIN(C(x)) → WAIT(Right6(x))
BEGIN(A(B(x))) → WAIT(Right7(x))
BEGIN(B(x)) → WAIT(Right8(x))
BEGIN(c(b(x))) → WAIT(Right9(x))
BEGIN(b(x)) → WAIT(Right10(x))
BEGIN(C(a(x))) → WAIT(Right11(x))
BEGIN(a(x)) → WAIT(Right12(x))

The TRS R consists of the following rules:

Right12(B(C(End(x)))) → Left(a(C(B(End(x)))))
Right12(a(x)) → Aa(Right12(x))
Right12(b(x)) → Ab(Right12(x))
Right12(c(x)) → Ac(Right12(x))
Right12(C(x)) → AC(Right12(x))
Right12(B(x)) → AB(Right12(x))
Right12(A(x)) → AA(Right12(x))
AA(Left(x)) → Left(A(x))
A(c(b(x))) → b(c(A(x)))
b(a(C(x))) → C(a(b(x)))
C(B(A(x))) → A(B(C(x)))
c(A(B(x))) → B(A(c(x)))
B(C(a(x))) → a(C(B(x)))
a(b(c(x))) → c(b(a(x)))
AB(Left(x)) → Left(B(x))
AC(Left(x)) → Left(C(x))
Ac(Left(x)) → Left(c(x))
Ab(Left(x)) → Left(b(x))
Aa(Left(x)) → Left(a(x))
Right11(B(End(x))) → Left(a(C(B(End(x)))))
Right11(a(x)) → Aa(Right11(x))
Right11(b(x)) → Ab(Right11(x))
Right11(c(x)) → Ac(Right11(x))
Right11(C(x)) → AC(Right11(x))
Right11(B(x)) → AB(Right11(x))
Right11(A(x)) → AA(Right11(x))
Right10(A(c(End(x)))) → Left(b(c(A(End(x)))))
Right10(a(x)) → Aa(Right10(x))
Right10(b(x)) → Ab(Right10(x))
Right10(c(x)) → Ac(Right10(x))
Right10(C(x)) → AC(Right10(x))
Right10(B(x)) → AB(Right10(x))
Right10(A(x)) → AA(Right10(x))
Right9(A(End(x))) → Left(b(c(A(End(x)))))
Right9(a(x)) → Aa(Right9(x))
Right9(b(x)) → Ab(Right9(x))
Right9(c(x)) → Ac(Right9(x))
Right9(C(x)) → AC(Right9(x))
Right9(B(x)) → AB(Right9(x))
Right9(A(x)) → AA(Right9(x))
Right8(c(A(End(x)))) → Left(B(A(c(End(x)))))
Right8(a(x)) → Aa(Right8(x))
Right8(b(x)) → Ab(Right8(x))
Right8(c(x)) → Ac(Right8(x))
Right8(C(x)) → AC(Right8(x))
Right8(B(x)) → AB(Right8(x))
Right8(A(x)) → AA(Right8(x))
Right7(c(End(x))) → Left(B(A(c(End(x)))))
Right7(a(x)) → Aa(Right7(x))
Right7(b(x)) → Ab(Right7(x))
Right7(c(x)) → Ac(Right7(x))
Right7(C(x)) → AC(Right7(x))
Right7(B(x)) → AB(Right7(x))
Right7(A(x)) → AA(Right7(x))
Right6(b(a(End(x)))) → Left(C(a(b(End(x)))))
Right6(a(x)) → Aa(Right6(x))
Right6(b(x)) → Ab(Right6(x))
Right6(c(x)) → Ac(Right6(x))
Right6(C(x)) → AC(Right6(x))
Right6(B(x)) → AB(Right6(x))
Right6(A(x)) → AA(Right6(x))
Right5(b(End(x))) → Left(C(a(b(End(x)))))
Right5(a(x)) → Aa(Right5(x))
Right5(b(x)) → Ab(Right5(x))
Right5(c(x)) → Ac(Right5(x))
Right5(C(x)) → AC(Right5(x))
Right5(B(x)) → AB(Right5(x))
Right5(A(x)) → AA(Right5(x))
Right4(C(B(End(x)))) → Left(A(B(C(End(x)))))
Right4(a(x)) → Aa(Right4(x))
Right4(b(x)) → Ab(Right4(x))
Right4(c(x)) → Ac(Right4(x))
Right4(C(x)) → AC(Right4(x))
Right4(B(x)) → AB(Right4(x))
Right4(A(x)) → AA(Right4(x))
Right3(C(End(x))) → Left(A(B(C(End(x)))))
Right3(a(x)) → Aa(Right3(x))
Right3(b(x)) → Ab(Right3(x))
Right3(c(x)) → Ac(Right3(x))
Right3(C(x)) → AC(Right3(x))
Right3(B(x)) → AB(Right3(x))
Right3(A(x)) → AA(Right3(x))
Right2(a(b(End(x)))) → Left(c(b(a(End(x)))))
Right2(a(x)) → Aa(Right2(x))
Right2(b(x)) → Ab(Right2(x))
Right2(c(x)) → Ac(Right2(x))
Right2(C(x)) → AC(Right2(x))
Right2(B(x)) → AB(Right2(x))
Right2(A(x)) → AA(Right2(x))
Right1(a(End(x))) → Left(c(b(a(End(x)))))
Right1(a(x)) → Aa(Right1(x))
Right1(b(x)) → Ab(Right1(x))
Right1(c(x)) → Ac(Right1(x))
Right1(C(x)) → AC(Right1(x))
Right1(B(x)) → AB(Right1(x))
Right1(A(x)) → AA(Right1(x))

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

(124) NonTerminationLoopProof (COMPLETE transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = BEGIN(A(B(A(c(End(x)))))) evaluates to t =BEGIN(A(B(A(c(End(x))))))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]



Rewriting sequence

BEGIN(A(B(A(c(End(x)))))) → WAIT(Right7(A(c(End(x)))))
with rule BEGIN(A(B(x'))) → WAIT(Right7(x')) at position [] and matcher [x' / A(c(End(x)))]

WAIT(Right7(A(c(End(x))))) → WAIT(AA(Right7(c(End(x)))))
with rule Right7(A(x')) → AA(Right7(x')) at position [0] and matcher [x' / c(End(x))]

WAIT(AA(Right7(c(End(x))))) → WAIT(AA(Left(B(A(c(End(x)))))))
with rule Right7(c(End(x'))) → Left(B(A(c(End(x'))))) at position [0,0] and matcher [x' / x]

WAIT(AA(Left(B(A(c(End(x))))))) → WAIT(Left(A(B(A(c(End(x)))))))
with rule AA(Left(x')) → Left(A(x')) at position [0] and matcher [x' / B(A(c(End(x))))]

WAIT(Left(A(B(A(c(End(x))))))) → BEGIN(A(B(A(c(End(x))))))
with rule WAIT(Left(x)) → BEGIN(x)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.



(125) NO