Left Termination of the query pattern append3_in_4(a, a, a, g) w.r.t. the given Prolog program could not be shown:

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
    ↳3 PrologToPiTRSProof (⇐)
        ↳4 PiTRS
        ↳5 DependencyPairsProof (⇔)
        ↳6 PiDP
        ↳7 DependencyGraphProof (⇔)
        ↳8 AND
            ↳9 PiDP
                ↳10 UsableRulesProof (⇔)
                ↳11 PiDP
                ↳12 PiDPToQDPProof (⇐)
                ↳13 QDP
                ↳14 NonTerminationProof (⇔)
                ↳15 NO
            ↳16 PiDP
                ↳17 UsableRulesProof (⇔)
                ↳18 PiDP
                ↳19 PiDPToQDPProof (⇐)
                ↳20 QDP
                ↳21 QDPSizeChangeProof (⇔)
                ↳22 YES
    ↳23 PrologToPiTRSProof (⇐)
        ↳24 PiTRS
        ↳25 DependencyPairsProof (⇔)
        ↳26 PiDP
        ↳27 DependencyGraphProof (⇔)
        ↳28 AND
            ↳29 PiDP
                ↳30 UsableRulesProof (⇔)
                ↳31 PiDP
                ↳32 PiDPToQDPProof (⇐)
                ↳33 QDP
                ↳34 NonTerminationProof (⇔)
                ↳35 NO
            ↳36 PiDP
                ↳37 UsableRulesProof (⇔)
                ↳38 PiDP
                ↳39 PiDPToQDPProof (⇐)
                ↳40 QDP
                ↳41 QDPSizeChangeProof (⇔)
                ↳42 YES

(0) Obligation:

Clauses:

append([], L, L).
append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3).
append3(A, B, C, D) :- ','(append(A, B, E), append(E, C, D)).

Queries:

append3(a,a,a,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

append7([], T29, T29).
append7(.(T38, T42), T43, .(T38, T41)) :- append7(T42, T43, T41).
append19([], T70, T70).
append19(.(T77, T80), T81, .(T77, X85)) :- append19(T80, T81, X85).
append31([], T21, T22, T12) :- append7(T21, T22, T12).
append31(.(T58, T55), T56, T57, T12) :- append19(T55, T56, X56).
append31(.(T62, T55), T56, T63, T12) :- ','(append19(T55, T56, T61), append7(.(T62, T61), T63, T12)).

Queries:

append31(a,a,a,g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
append31_in: (f,f,f,b)
append7_in: (f,f,b)
append19_in: (f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

append31_in_aaag([], T21, T22, T12) → U3_aaag(T21, T22, T12, append7_in_aag(T21, T22, T12))
append7_in_aag([], T29, T29) → append7_out_aag([], T29, T29)
append7_in_aag(.(T38, T42), T43, .(T38, T41)) → U1_aag(T38, T42, T43, T41, append7_in_aag(T42, T43, T41))
U1_aag(T38, T42, T43, T41, append7_out_aag(T42, T43, T41)) → append7_out_aag(.(T38, T42), T43, .(T38, T41))
U3_aaag(T21, T22, T12, append7_out_aag(T21, T22, T12)) → append31_out_aaag([], T21, T22, T12)
append31_in_aaag(.(T58, T55), T56, T57, T12) → U4_aaag(T58, T55, T56, T57, T12, append19_in_aaa(T55, T56, X56))
append19_in_aaa([], T70, T70) → append19_out_aaa([], T70, T70)
append19_in_aaa(.(T77, T80), T81, .(T77, X85)) → U2_aaa(T77, T80, T81, X85, append19_in_aaa(T80, T81, X85))
U2_aaa(T77, T80, T81, X85, append19_out_aaa(T80, T81, X85)) → append19_out_aaa(.(T77, T80), T81, .(T77, X85))
U4_aaag(T58, T55, T56, T57, T12, append19_out_aaa(T55, T56, X56)) → append31_out_aaag(.(T58, T55), T56, T57, T12)
append31_in_aaag(.(T62, T55), T56, T63, T12) → U5_aaag(T62, T55, T56, T63, T12, append19_in_aaa(T55, T56, T61))
U5_aaag(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → U6_aaag(T62, T55, T56, T63, T12, append7_in_aag(.(T62, T61), T63, T12))
U6_aaag(T62, T55, T56, T63, T12, append7_out_aag(.(T62, T61), T63, T12)) → append31_out_aaag(.(T62, T55), T56, T63, T12)

The argument filtering Pi contains the following mapping:
append31_in_aaag(x1, x2, x3, x4)  =  append31_in_aaag(x4)
U3_aaag(x1, x2, x3, x4)  =  U3_aaag(x3, x4)
append7_in_aag(x1, x2, x3)  =  append7_in_aag(x3)
append7_out_aag(x1, x2, x3)  =  append7_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x4, x5)
append31_out_aaag(x1, x2, x3, x4)  =  append31_out_aaag(x1, x4)
U4_aaag(x1, x2, x3, x4, x5, x6)  =  U4_aaag(x5, x6)
append19_in_aaa(x1, x2, x3)  =  append19_in_aaa
append19_out_aaa(x1, x2, x3)  =  append19_out_aaa(x1)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U5_aaag(x1, x2, x3, x4, x5, x6)  =  U5_aaag(x5, x6)
U6_aaag(x1, x2, x3, x4, x5, x6)  =  U6_aaag(x2, x5, x6)
[]  =  []

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

append31_in_aaag([], T21, T22, T12) → U3_aaag(T21, T22, T12, append7_in_aag(T21, T22, T12))
append7_in_aag([], T29, T29) → append7_out_aag([], T29, T29)
append7_in_aag(.(T38, T42), T43, .(T38, T41)) → U1_aag(T38, T42, T43, T41, append7_in_aag(T42, T43, T41))
U1_aag(T38, T42, T43, T41, append7_out_aag(T42, T43, T41)) → append7_out_aag(.(T38, T42), T43, .(T38, T41))
U3_aaag(T21, T22, T12, append7_out_aag(T21, T22, T12)) → append31_out_aaag([], T21, T22, T12)
append31_in_aaag(.(T58, T55), T56, T57, T12) → U4_aaag(T58, T55, T56, T57, T12, append19_in_aaa(T55, T56, X56))
append19_in_aaa([], T70, T70) → append19_out_aaa([], T70, T70)
append19_in_aaa(.(T77, T80), T81, .(T77, X85)) → U2_aaa(T77, T80, T81, X85, append19_in_aaa(T80, T81, X85))
U2_aaa(T77, T80, T81, X85, append19_out_aaa(T80, T81, X85)) → append19_out_aaa(.(T77, T80), T81, .(T77, X85))
U4_aaag(T58, T55, T56, T57, T12, append19_out_aaa(T55, T56, X56)) → append31_out_aaag(.(T58, T55), T56, T57, T12)
append31_in_aaag(.(T62, T55), T56, T63, T12) → U5_aaag(T62, T55, T56, T63, T12, append19_in_aaa(T55, T56, T61))
U5_aaag(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → U6_aaag(T62, T55, T56, T63, T12, append7_in_aag(.(T62, T61), T63, T12))
U6_aaag(T62, T55, T56, T63, T12, append7_out_aag(.(T62, T61), T63, T12)) → append31_out_aaag(.(T62, T55), T56, T63, T12)

The argument filtering Pi contains the following mapping:
append31_in_aaag(x1, x2, x3, x4)  =  append31_in_aaag(x4)
U3_aaag(x1, x2, x3, x4)  =  U3_aaag(x3, x4)
append7_in_aag(x1, x2, x3)  =  append7_in_aag(x3)
append7_out_aag(x1, x2, x3)  =  append7_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x4, x5)
append31_out_aaag(x1, x2, x3, x4)  =  append31_out_aaag(x1, x4)
U4_aaag(x1, x2, x3, x4, x5, x6)  =  U4_aaag(x5, x6)
append19_in_aaa(x1, x2, x3)  =  append19_in_aaa
append19_out_aaa(x1, x2, x3)  =  append19_out_aaa(x1)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U5_aaag(x1, x2, x3, x4, x5, x6)  =  U5_aaag(x5, x6)
U6_aaag(x1, x2, x3, x4, x5, x6)  =  U6_aaag(x2, x5, x6)
[]  =  []

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

APPEND31_IN_AAAG([], T21, T22, T12) → U3_AAAG(T21, T22, T12, append7_in_aag(T21, T22, T12))
APPEND31_IN_AAAG([], T21, T22, T12) → APPEND7_IN_AAG(T21, T22, T12)
APPEND7_IN_AAG(.(T38, T42), T43, .(T38, T41)) → U1_AAG(T38, T42, T43, T41, append7_in_aag(T42, T43, T41))
APPEND7_IN_AAG(.(T38, T42), T43, .(T38, T41)) → APPEND7_IN_AAG(T42, T43, T41)
APPEND31_IN_AAAG(.(T58, T55), T56, T57, T12) → U4_AAAG(T58, T55, T56, T57, T12, append19_in_aaa(T55, T56, X56))
APPEND31_IN_AAAG(.(T58, T55), T56, T57, T12) → APPEND19_IN_AAA(T55, T56, X56)
APPEND19_IN_AAA(.(T77, T80), T81, .(T77, X85)) → U2_AAA(T77, T80, T81, X85, append19_in_aaa(T80, T81, X85))
APPEND19_IN_AAA(.(T77, T80), T81, .(T77, X85)) → APPEND19_IN_AAA(T80, T81, X85)
APPEND31_IN_AAAG(.(T62, T55), T56, T63, T12) → U5_AAAG(T62, T55, T56, T63, T12, append19_in_aaa(T55, T56, T61))
U5_AAAG(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → U6_AAAG(T62, T55, T56, T63, T12, append7_in_aag(.(T62, T61), T63, T12))
U5_AAAG(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → APPEND7_IN_AAG(.(T62, T61), T63, T12)

The TRS R consists of the following rules:

append31_in_aaag([], T21, T22, T12) → U3_aaag(T21, T22, T12, append7_in_aag(T21, T22, T12))
append7_in_aag([], T29, T29) → append7_out_aag([], T29, T29)
append7_in_aag(.(T38, T42), T43, .(T38, T41)) → U1_aag(T38, T42, T43, T41, append7_in_aag(T42, T43, T41))
U1_aag(T38, T42, T43, T41, append7_out_aag(T42, T43, T41)) → append7_out_aag(.(T38, T42), T43, .(T38, T41))
U3_aaag(T21, T22, T12, append7_out_aag(T21, T22, T12)) → append31_out_aaag([], T21, T22, T12)
append31_in_aaag(.(T58, T55), T56, T57, T12) → U4_aaag(T58, T55, T56, T57, T12, append19_in_aaa(T55, T56, X56))
append19_in_aaa([], T70, T70) → append19_out_aaa([], T70, T70)
append19_in_aaa(.(T77, T80), T81, .(T77, X85)) → U2_aaa(T77, T80, T81, X85, append19_in_aaa(T80, T81, X85))
U2_aaa(T77, T80, T81, X85, append19_out_aaa(T80, T81, X85)) → append19_out_aaa(.(T77, T80), T81, .(T77, X85))
U4_aaag(T58, T55, T56, T57, T12, append19_out_aaa(T55, T56, X56)) → append31_out_aaag(.(T58, T55), T56, T57, T12)
append31_in_aaag(.(T62, T55), T56, T63, T12) → U5_aaag(T62, T55, T56, T63, T12, append19_in_aaa(T55, T56, T61))
U5_aaag(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → U6_aaag(T62, T55, T56, T63, T12, append7_in_aag(.(T62, T61), T63, T12))
U6_aaag(T62, T55, T56, T63, T12, append7_out_aag(.(T62, T61), T63, T12)) → append31_out_aaag(.(T62, T55), T56, T63, T12)

The argument filtering Pi contains the following mapping:
append31_in_aaag(x1, x2, x3, x4)  =  append31_in_aaag(x4)
U3_aaag(x1, x2, x3, x4)  =  U3_aaag(x3, x4)
append7_in_aag(x1, x2, x3)  =  append7_in_aag(x3)
append7_out_aag(x1, x2, x3)  =  append7_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x4, x5)
append31_out_aaag(x1, x2, x3, x4)  =  append31_out_aaag(x1, x4)
U4_aaag(x1, x2, x3, x4, x5, x6)  =  U4_aaag(x5, x6)
append19_in_aaa(x1, x2, x3)  =  append19_in_aaa
append19_out_aaa(x1, x2, x3)  =  append19_out_aaa(x1)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U5_aaag(x1, x2, x3, x4, x5, x6)  =  U5_aaag(x5, x6)
U6_aaag(x1, x2, x3, x4, x5, x6)  =  U6_aaag(x2, x5, x6)
[]  =  []
APPEND31_IN_AAAG(x1, x2, x3, x4)  =  APPEND31_IN_AAAG(x4)
U3_AAAG(x1, x2, x3, x4)  =  U3_AAAG(x3, x4)
APPEND7_IN_AAG(x1, x2, x3)  =  APPEND7_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x4, x5)
U4_AAAG(x1, x2, x3, x4, x5, x6)  =  U4_AAAG(x5, x6)
APPEND19_IN_AAA(x1, x2, x3)  =  APPEND19_IN_AAA
U2_AAA(x1, x2, x3, x4, x5)  =  U2_AAA(x5)
U5_AAAG(x1, x2, x3, x4, x5, x6)  =  U5_AAAG(x5, x6)
U6_AAAG(x1, x2, x3, x4, x5, x6)  =  U6_AAAG(x2, x5, x6)

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

(6) Obligation:

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

APPEND31_IN_AAAG([], T21, T22, T12) → U3_AAAG(T21, T22, T12, append7_in_aag(T21, T22, T12))
APPEND31_IN_AAAG([], T21, T22, T12) → APPEND7_IN_AAG(T21, T22, T12)
APPEND7_IN_AAG(.(T38, T42), T43, .(T38, T41)) → U1_AAG(T38, T42, T43, T41, append7_in_aag(T42, T43, T41))
APPEND7_IN_AAG(.(T38, T42), T43, .(T38, T41)) → APPEND7_IN_AAG(T42, T43, T41)
APPEND31_IN_AAAG(.(T58, T55), T56, T57, T12) → U4_AAAG(T58, T55, T56, T57, T12, append19_in_aaa(T55, T56, X56))
APPEND31_IN_AAAG(.(T58, T55), T56, T57, T12) → APPEND19_IN_AAA(T55, T56, X56)
APPEND19_IN_AAA(.(T77, T80), T81, .(T77, X85)) → U2_AAA(T77, T80, T81, X85, append19_in_aaa(T80, T81, X85))
APPEND19_IN_AAA(.(T77, T80), T81, .(T77, X85)) → APPEND19_IN_AAA(T80, T81, X85)
APPEND31_IN_AAAG(.(T62, T55), T56, T63, T12) → U5_AAAG(T62, T55, T56, T63, T12, append19_in_aaa(T55, T56, T61))
U5_AAAG(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → U6_AAAG(T62, T55, T56, T63, T12, append7_in_aag(.(T62, T61), T63, T12))
U5_AAAG(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → APPEND7_IN_AAG(.(T62, T61), T63, T12)

The TRS R consists of the following rules:

append31_in_aaag([], T21, T22, T12) → U3_aaag(T21, T22, T12, append7_in_aag(T21, T22, T12))
append7_in_aag([], T29, T29) → append7_out_aag([], T29, T29)
append7_in_aag(.(T38, T42), T43, .(T38, T41)) → U1_aag(T38, T42, T43, T41, append7_in_aag(T42, T43, T41))
U1_aag(T38, T42, T43, T41, append7_out_aag(T42, T43, T41)) → append7_out_aag(.(T38, T42), T43, .(T38, T41))
U3_aaag(T21, T22, T12, append7_out_aag(T21, T22, T12)) → append31_out_aaag([], T21, T22, T12)
append31_in_aaag(.(T58, T55), T56, T57, T12) → U4_aaag(T58, T55, T56, T57, T12, append19_in_aaa(T55, T56, X56))
append19_in_aaa([], T70, T70) → append19_out_aaa([], T70, T70)
append19_in_aaa(.(T77, T80), T81, .(T77, X85)) → U2_aaa(T77, T80, T81, X85, append19_in_aaa(T80, T81, X85))
U2_aaa(T77, T80, T81, X85, append19_out_aaa(T80, T81, X85)) → append19_out_aaa(.(T77, T80), T81, .(T77, X85))
U4_aaag(T58, T55, T56, T57, T12, append19_out_aaa(T55, T56, X56)) → append31_out_aaag(.(T58, T55), T56, T57, T12)
append31_in_aaag(.(T62, T55), T56, T63, T12) → U5_aaag(T62, T55, T56, T63, T12, append19_in_aaa(T55, T56, T61))
U5_aaag(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → U6_aaag(T62, T55, T56, T63, T12, append7_in_aag(.(T62, T61), T63, T12))
U6_aaag(T62, T55, T56, T63, T12, append7_out_aag(.(T62, T61), T63, T12)) → append31_out_aaag(.(T62, T55), T56, T63, T12)

The argument filtering Pi contains the following mapping:
append31_in_aaag(x1, x2, x3, x4)  =  append31_in_aaag(x4)
U3_aaag(x1, x2, x3, x4)  =  U3_aaag(x3, x4)
append7_in_aag(x1, x2, x3)  =  append7_in_aag(x3)
append7_out_aag(x1, x2, x3)  =  append7_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x4, x5)
append31_out_aaag(x1, x2, x3, x4)  =  append31_out_aaag(x1, x4)
U4_aaag(x1, x2, x3, x4, x5, x6)  =  U4_aaag(x5, x6)
append19_in_aaa(x1, x2, x3)  =  append19_in_aaa
append19_out_aaa(x1, x2, x3)  =  append19_out_aaa(x1)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U5_aaag(x1, x2, x3, x4, x5, x6)  =  U5_aaag(x5, x6)
U6_aaag(x1, x2, x3, x4, x5, x6)  =  U6_aaag(x2, x5, x6)
[]  =  []
APPEND31_IN_AAAG(x1, x2, x3, x4)  =  APPEND31_IN_AAAG(x4)
U3_AAAG(x1, x2, x3, x4)  =  U3_AAAG(x3, x4)
APPEND7_IN_AAG(x1, x2, x3)  =  APPEND7_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x4, x5)
U4_AAAG(x1, x2, x3, x4, x5, x6)  =  U4_AAAG(x5, x6)
APPEND19_IN_AAA(x1, x2, x3)  =  APPEND19_IN_AAA
U2_AAA(x1, x2, x3, x4, x5)  =  U2_AAA(x5)
U5_AAAG(x1, x2, x3, x4, x5, x6)  =  U5_AAAG(x5, x6)
U6_AAAG(x1, x2, x3, x4, x5, x6)  =  U6_AAAG(x2, x5, x6)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 9 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APPEND19_IN_AAA(.(T77, T80), T81, .(T77, X85)) → APPEND19_IN_AAA(T80, T81, X85)

The TRS R consists of the following rules:

append31_in_aaag([], T21, T22, T12) → U3_aaag(T21, T22, T12, append7_in_aag(T21, T22, T12))
append7_in_aag([], T29, T29) → append7_out_aag([], T29, T29)
append7_in_aag(.(T38, T42), T43, .(T38, T41)) → U1_aag(T38, T42, T43, T41, append7_in_aag(T42, T43, T41))
U1_aag(T38, T42, T43, T41, append7_out_aag(T42, T43, T41)) → append7_out_aag(.(T38, T42), T43, .(T38, T41))
U3_aaag(T21, T22, T12, append7_out_aag(T21, T22, T12)) → append31_out_aaag([], T21, T22, T12)
append31_in_aaag(.(T58, T55), T56, T57, T12) → U4_aaag(T58, T55, T56, T57, T12, append19_in_aaa(T55, T56, X56))
append19_in_aaa([], T70, T70) → append19_out_aaa([], T70, T70)
append19_in_aaa(.(T77, T80), T81, .(T77, X85)) → U2_aaa(T77, T80, T81, X85, append19_in_aaa(T80, T81, X85))
U2_aaa(T77, T80, T81, X85, append19_out_aaa(T80, T81, X85)) → append19_out_aaa(.(T77, T80), T81, .(T77, X85))
U4_aaag(T58, T55, T56, T57, T12, append19_out_aaa(T55, T56, X56)) → append31_out_aaag(.(T58, T55), T56, T57, T12)
append31_in_aaag(.(T62, T55), T56, T63, T12) → U5_aaag(T62, T55, T56, T63, T12, append19_in_aaa(T55, T56, T61))
U5_aaag(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → U6_aaag(T62, T55, T56, T63, T12, append7_in_aag(.(T62, T61), T63, T12))
U6_aaag(T62, T55, T56, T63, T12, append7_out_aag(.(T62, T61), T63, T12)) → append31_out_aaag(.(T62, T55), T56, T63, T12)

The argument filtering Pi contains the following mapping:
append31_in_aaag(x1, x2, x3, x4)  =  append31_in_aaag(x4)
U3_aaag(x1, x2, x3, x4)  =  U3_aaag(x3, x4)
append7_in_aag(x1, x2, x3)  =  append7_in_aag(x3)
append7_out_aag(x1, x2, x3)  =  append7_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x4, x5)
append31_out_aaag(x1, x2, x3, x4)  =  append31_out_aaag(x1, x4)
U4_aaag(x1, x2, x3, x4, x5, x6)  =  U4_aaag(x5, x6)
append19_in_aaa(x1, x2, x3)  =  append19_in_aaa
append19_out_aaa(x1, x2, x3)  =  append19_out_aaa(x1)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U5_aaag(x1, x2, x3, x4, x5, x6)  =  U5_aaag(x5, x6)
U6_aaag(x1, x2, x3, x4, x5, x6)  =  U6_aaag(x2, x5, x6)
[]  =  []
APPEND19_IN_AAA(x1, x2, x3)  =  APPEND19_IN_AAA

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

(10) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(11) Obligation:

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

APPEND19_IN_AAA(.(T77, T80), T81, .(T77, X85)) → APPEND19_IN_AAA(T80, T81, X85)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND19_IN_AAA(x1, x2, x3)  =  APPEND19_IN_AAA

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

(12) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(13) Obligation:

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

APPEND19_IN_AAA → APPEND19_IN_AAA

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

(14) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = APPEND19_IN_AAA evaluates to t =APPEND19_IN_AAA

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND19_IN_AAA to APPEND19_IN_AAA.



(15) NO

(16) Obligation:

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

APPEND7_IN_AAG(.(T38, T42), T43, .(T38, T41)) → APPEND7_IN_AAG(T42, T43, T41)

The TRS R consists of the following rules:

append31_in_aaag([], T21, T22, T12) → U3_aaag(T21, T22, T12, append7_in_aag(T21, T22, T12))
append7_in_aag([], T29, T29) → append7_out_aag([], T29, T29)
append7_in_aag(.(T38, T42), T43, .(T38, T41)) → U1_aag(T38, T42, T43, T41, append7_in_aag(T42, T43, T41))
U1_aag(T38, T42, T43, T41, append7_out_aag(T42, T43, T41)) → append7_out_aag(.(T38, T42), T43, .(T38, T41))
U3_aaag(T21, T22, T12, append7_out_aag(T21, T22, T12)) → append31_out_aaag([], T21, T22, T12)
append31_in_aaag(.(T58, T55), T56, T57, T12) → U4_aaag(T58, T55, T56, T57, T12, append19_in_aaa(T55, T56, X56))
append19_in_aaa([], T70, T70) → append19_out_aaa([], T70, T70)
append19_in_aaa(.(T77, T80), T81, .(T77, X85)) → U2_aaa(T77, T80, T81, X85, append19_in_aaa(T80, T81, X85))
U2_aaa(T77, T80, T81, X85, append19_out_aaa(T80, T81, X85)) → append19_out_aaa(.(T77, T80), T81, .(T77, X85))
U4_aaag(T58, T55, T56, T57, T12, append19_out_aaa(T55, T56, X56)) → append31_out_aaag(.(T58, T55), T56, T57, T12)
append31_in_aaag(.(T62, T55), T56, T63, T12) → U5_aaag(T62, T55, T56, T63, T12, append19_in_aaa(T55, T56, T61))
U5_aaag(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → U6_aaag(T62, T55, T56, T63, T12, append7_in_aag(.(T62, T61), T63, T12))
U6_aaag(T62, T55, T56, T63, T12, append7_out_aag(.(T62, T61), T63, T12)) → append31_out_aaag(.(T62, T55), T56, T63, T12)

The argument filtering Pi contains the following mapping:
append31_in_aaag(x1, x2, x3, x4)  =  append31_in_aaag(x4)
U3_aaag(x1, x2, x3, x4)  =  U3_aaag(x3, x4)
append7_in_aag(x1, x2, x3)  =  append7_in_aag(x3)
append7_out_aag(x1, x2, x3)  =  append7_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x4, x5)
append31_out_aaag(x1, x2, x3, x4)  =  append31_out_aaag(x1, x4)
U4_aaag(x1, x2, x3, x4, x5, x6)  =  U4_aaag(x5, x6)
append19_in_aaa(x1, x2, x3)  =  append19_in_aaa
append19_out_aaa(x1, x2, x3)  =  append19_out_aaa(x1)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U5_aaag(x1, x2, x3, x4, x5, x6)  =  U5_aaag(x5, x6)
U6_aaag(x1, x2, x3, x4, x5, x6)  =  U6_aaag(x2, x5, x6)
[]  =  []
APPEND7_IN_AAG(x1, x2, x3)  =  APPEND7_IN_AAG(x3)

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

(17) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(18) Obligation:

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

APPEND7_IN_AAG(.(T38, T42), T43, .(T38, T41)) → APPEND7_IN_AAG(T42, T43, T41)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND7_IN_AAG(x1, x2, x3)  =  APPEND7_IN_AAG(x3)

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

(19) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(20) Obligation:

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

APPEND7_IN_AAG(.(T41)) → APPEND7_IN_AAG(T41)

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

(21) 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:

  • APPEND7_IN_AAG(.(T41)) → APPEND7_IN_AAG(T41)
    The graph contains the following edges 1 > 1

(22) YES

(23) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
append31_in: (f,f,f,b)
append7_in: (f,f,b)
append19_in: (f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

append31_in_aaag([], T21, T22, T12) → U3_aaag(T21, T22, T12, append7_in_aag(T21, T22, T12))
append7_in_aag([], T29, T29) → append7_out_aag([], T29, T29)
append7_in_aag(.(T38, T42), T43, .(T38, T41)) → U1_aag(T38, T42, T43, T41, append7_in_aag(T42, T43, T41))
U1_aag(T38, T42, T43, T41, append7_out_aag(T42, T43, T41)) → append7_out_aag(.(T38, T42), T43, .(T38, T41))
U3_aaag(T21, T22, T12, append7_out_aag(T21, T22, T12)) → append31_out_aaag([], T21, T22, T12)
append31_in_aaag(.(T58, T55), T56, T57, T12) → U4_aaag(T58, T55, T56, T57, T12, append19_in_aaa(T55, T56, X56))
append19_in_aaa([], T70, T70) → append19_out_aaa([], T70, T70)
append19_in_aaa(.(T77, T80), T81, .(T77, X85)) → U2_aaa(T77, T80, T81, X85, append19_in_aaa(T80, T81, X85))
U2_aaa(T77, T80, T81, X85, append19_out_aaa(T80, T81, X85)) → append19_out_aaa(.(T77, T80), T81, .(T77, X85))
U4_aaag(T58, T55, T56, T57, T12, append19_out_aaa(T55, T56, X56)) → append31_out_aaag(.(T58, T55), T56, T57, T12)
append31_in_aaag(.(T62, T55), T56, T63, T12) → U5_aaag(T62, T55, T56, T63, T12, append19_in_aaa(T55, T56, T61))
U5_aaag(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → U6_aaag(T62, T55, T56, T63, T12, append7_in_aag(.(T62, T61), T63, T12))
U6_aaag(T62, T55, T56, T63, T12, append7_out_aag(.(T62, T61), T63, T12)) → append31_out_aaag(.(T62, T55), T56, T63, T12)

The argument filtering Pi contains the following mapping:
append31_in_aaag(x1, x2, x3, x4)  =  append31_in_aaag(x4)
U3_aaag(x1, x2, x3, x4)  =  U3_aaag(x4)
append7_in_aag(x1, x2, x3)  =  append7_in_aag(x3)
append7_out_aag(x1, x2, x3)  =  append7_out_aag(x1, x2)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
append31_out_aaag(x1, x2, x3, x4)  =  append31_out_aaag(x1)
U4_aaag(x1, x2, x3, x4, x5, x6)  =  U4_aaag(x6)
append19_in_aaa(x1, x2, x3)  =  append19_in_aaa
append19_out_aaa(x1, x2, x3)  =  append19_out_aaa(x1)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U5_aaag(x1, x2, x3, x4, x5, x6)  =  U5_aaag(x5, x6)
U6_aaag(x1, x2, x3, x4, x5, x6)  =  U6_aaag(x2, x6)
[]  =  []

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(24) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

append31_in_aaag([], T21, T22, T12) → U3_aaag(T21, T22, T12, append7_in_aag(T21, T22, T12))
append7_in_aag([], T29, T29) → append7_out_aag([], T29, T29)
append7_in_aag(.(T38, T42), T43, .(T38, T41)) → U1_aag(T38, T42, T43, T41, append7_in_aag(T42, T43, T41))
U1_aag(T38, T42, T43, T41, append7_out_aag(T42, T43, T41)) → append7_out_aag(.(T38, T42), T43, .(T38, T41))
U3_aaag(T21, T22, T12, append7_out_aag(T21, T22, T12)) → append31_out_aaag([], T21, T22, T12)
append31_in_aaag(.(T58, T55), T56, T57, T12) → U4_aaag(T58, T55, T56, T57, T12, append19_in_aaa(T55, T56, X56))
append19_in_aaa([], T70, T70) → append19_out_aaa([], T70, T70)
append19_in_aaa(.(T77, T80), T81, .(T77, X85)) → U2_aaa(T77, T80, T81, X85, append19_in_aaa(T80, T81, X85))
U2_aaa(T77, T80, T81, X85, append19_out_aaa(T80, T81, X85)) → append19_out_aaa(.(T77, T80), T81, .(T77, X85))
U4_aaag(T58, T55, T56, T57, T12, append19_out_aaa(T55, T56, X56)) → append31_out_aaag(.(T58, T55), T56, T57, T12)
append31_in_aaag(.(T62, T55), T56, T63, T12) → U5_aaag(T62, T55, T56, T63, T12, append19_in_aaa(T55, T56, T61))
U5_aaag(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → U6_aaag(T62, T55, T56, T63, T12, append7_in_aag(.(T62, T61), T63, T12))
U6_aaag(T62, T55, T56, T63, T12, append7_out_aag(.(T62, T61), T63, T12)) → append31_out_aaag(.(T62, T55), T56, T63, T12)

The argument filtering Pi contains the following mapping:
append31_in_aaag(x1, x2, x3, x4)  =  append31_in_aaag(x4)
U3_aaag(x1, x2, x3, x4)  =  U3_aaag(x4)
append7_in_aag(x1, x2, x3)  =  append7_in_aag(x3)
append7_out_aag(x1, x2, x3)  =  append7_out_aag(x1, x2)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
append31_out_aaag(x1, x2, x3, x4)  =  append31_out_aaag(x1)
U4_aaag(x1, x2, x3, x4, x5, x6)  =  U4_aaag(x6)
append19_in_aaa(x1, x2, x3)  =  append19_in_aaa
append19_out_aaa(x1, x2, x3)  =  append19_out_aaa(x1)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U5_aaag(x1, x2, x3, x4, x5, x6)  =  U5_aaag(x5, x6)
U6_aaag(x1, x2, x3, x4, x5, x6)  =  U6_aaag(x2, x6)
[]  =  []

(25) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

APPEND31_IN_AAAG([], T21, T22, T12) → U3_AAAG(T21, T22, T12, append7_in_aag(T21, T22, T12))
APPEND31_IN_AAAG([], T21, T22, T12) → APPEND7_IN_AAG(T21, T22, T12)
APPEND7_IN_AAG(.(T38, T42), T43, .(T38, T41)) → U1_AAG(T38, T42, T43, T41, append7_in_aag(T42, T43, T41))
APPEND7_IN_AAG(.(T38, T42), T43, .(T38, T41)) → APPEND7_IN_AAG(T42, T43, T41)
APPEND31_IN_AAAG(.(T58, T55), T56, T57, T12) → U4_AAAG(T58, T55, T56, T57, T12, append19_in_aaa(T55, T56, X56))
APPEND31_IN_AAAG(.(T58, T55), T56, T57, T12) → APPEND19_IN_AAA(T55, T56, X56)
APPEND19_IN_AAA(.(T77, T80), T81, .(T77, X85)) → U2_AAA(T77, T80, T81, X85, append19_in_aaa(T80, T81, X85))
APPEND19_IN_AAA(.(T77, T80), T81, .(T77, X85)) → APPEND19_IN_AAA(T80, T81, X85)
APPEND31_IN_AAAG(.(T62, T55), T56, T63, T12) → U5_AAAG(T62, T55, T56, T63, T12, append19_in_aaa(T55, T56, T61))
U5_AAAG(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → U6_AAAG(T62, T55, T56, T63, T12, append7_in_aag(.(T62, T61), T63, T12))
U5_AAAG(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → APPEND7_IN_AAG(.(T62, T61), T63, T12)

The TRS R consists of the following rules:

append31_in_aaag([], T21, T22, T12) → U3_aaag(T21, T22, T12, append7_in_aag(T21, T22, T12))
append7_in_aag([], T29, T29) → append7_out_aag([], T29, T29)
append7_in_aag(.(T38, T42), T43, .(T38, T41)) → U1_aag(T38, T42, T43, T41, append7_in_aag(T42, T43, T41))
U1_aag(T38, T42, T43, T41, append7_out_aag(T42, T43, T41)) → append7_out_aag(.(T38, T42), T43, .(T38, T41))
U3_aaag(T21, T22, T12, append7_out_aag(T21, T22, T12)) → append31_out_aaag([], T21, T22, T12)
append31_in_aaag(.(T58, T55), T56, T57, T12) → U4_aaag(T58, T55, T56, T57, T12, append19_in_aaa(T55, T56, X56))
append19_in_aaa([], T70, T70) → append19_out_aaa([], T70, T70)
append19_in_aaa(.(T77, T80), T81, .(T77, X85)) → U2_aaa(T77, T80, T81, X85, append19_in_aaa(T80, T81, X85))
U2_aaa(T77, T80, T81, X85, append19_out_aaa(T80, T81, X85)) → append19_out_aaa(.(T77, T80), T81, .(T77, X85))
U4_aaag(T58, T55, T56, T57, T12, append19_out_aaa(T55, T56, X56)) → append31_out_aaag(.(T58, T55), T56, T57, T12)
append31_in_aaag(.(T62, T55), T56, T63, T12) → U5_aaag(T62, T55, T56, T63, T12, append19_in_aaa(T55, T56, T61))
U5_aaag(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → U6_aaag(T62, T55, T56, T63, T12, append7_in_aag(.(T62, T61), T63, T12))
U6_aaag(T62, T55, T56, T63, T12, append7_out_aag(.(T62, T61), T63, T12)) → append31_out_aaag(.(T62, T55), T56, T63, T12)

The argument filtering Pi contains the following mapping:
append31_in_aaag(x1, x2, x3, x4)  =  append31_in_aaag(x4)
U3_aaag(x1, x2, x3, x4)  =  U3_aaag(x4)
append7_in_aag(x1, x2, x3)  =  append7_in_aag(x3)
append7_out_aag(x1, x2, x3)  =  append7_out_aag(x1, x2)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
append31_out_aaag(x1, x2, x3, x4)  =  append31_out_aaag(x1)
U4_aaag(x1, x2, x3, x4, x5, x6)  =  U4_aaag(x6)
append19_in_aaa(x1, x2, x3)  =  append19_in_aaa
append19_out_aaa(x1, x2, x3)  =  append19_out_aaa(x1)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U5_aaag(x1, x2, x3, x4, x5, x6)  =  U5_aaag(x5, x6)
U6_aaag(x1, x2, x3, x4, x5, x6)  =  U6_aaag(x2, x6)
[]  =  []
APPEND31_IN_AAAG(x1, x2, x3, x4)  =  APPEND31_IN_AAAG(x4)
U3_AAAG(x1, x2, x3, x4)  =  U3_AAAG(x4)
APPEND7_IN_AAG(x1, x2, x3)  =  APPEND7_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x5)
U4_AAAG(x1, x2, x3, x4, x5, x6)  =  U4_AAAG(x6)
APPEND19_IN_AAA(x1, x2, x3)  =  APPEND19_IN_AAA
U2_AAA(x1, x2, x3, x4, x5)  =  U2_AAA(x5)
U5_AAAG(x1, x2, x3, x4, x5, x6)  =  U5_AAAG(x5, x6)
U6_AAAG(x1, x2, x3, x4, x5, x6)  =  U6_AAAG(x2, x6)

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

(26) Obligation:

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

APPEND31_IN_AAAG([], T21, T22, T12) → U3_AAAG(T21, T22, T12, append7_in_aag(T21, T22, T12))
APPEND31_IN_AAAG([], T21, T22, T12) → APPEND7_IN_AAG(T21, T22, T12)
APPEND7_IN_AAG(.(T38, T42), T43, .(T38, T41)) → U1_AAG(T38, T42, T43, T41, append7_in_aag(T42, T43, T41))
APPEND7_IN_AAG(.(T38, T42), T43, .(T38, T41)) → APPEND7_IN_AAG(T42, T43, T41)
APPEND31_IN_AAAG(.(T58, T55), T56, T57, T12) → U4_AAAG(T58, T55, T56, T57, T12, append19_in_aaa(T55, T56, X56))
APPEND31_IN_AAAG(.(T58, T55), T56, T57, T12) → APPEND19_IN_AAA(T55, T56, X56)
APPEND19_IN_AAA(.(T77, T80), T81, .(T77, X85)) → U2_AAA(T77, T80, T81, X85, append19_in_aaa(T80, T81, X85))
APPEND19_IN_AAA(.(T77, T80), T81, .(T77, X85)) → APPEND19_IN_AAA(T80, T81, X85)
APPEND31_IN_AAAG(.(T62, T55), T56, T63, T12) → U5_AAAG(T62, T55, T56, T63, T12, append19_in_aaa(T55, T56, T61))
U5_AAAG(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → U6_AAAG(T62, T55, T56, T63, T12, append7_in_aag(.(T62, T61), T63, T12))
U5_AAAG(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → APPEND7_IN_AAG(.(T62, T61), T63, T12)

The TRS R consists of the following rules:

append31_in_aaag([], T21, T22, T12) → U3_aaag(T21, T22, T12, append7_in_aag(T21, T22, T12))
append7_in_aag([], T29, T29) → append7_out_aag([], T29, T29)
append7_in_aag(.(T38, T42), T43, .(T38, T41)) → U1_aag(T38, T42, T43, T41, append7_in_aag(T42, T43, T41))
U1_aag(T38, T42, T43, T41, append7_out_aag(T42, T43, T41)) → append7_out_aag(.(T38, T42), T43, .(T38, T41))
U3_aaag(T21, T22, T12, append7_out_aag(T21, T22, T12)) → append31_out_aaag([], T21, T22, T12)
append31_in_aaag(.(T58, T55), T56, T57, T12) → U4_aaag(T58, T55, T56, T57, T12, append19_in_aaa(T55, T56, X56))
append19_in_aaa([], T70, T70) → append19_out_aaa([], T70, T70)
append19_in_aaa(.(T77, T80), T81, .(T77, X85)) → U2_aaa(T77, T80, T81, X85, append19_in_aaa(T80, T81, X85))
U2_aaa(T77, T80, T81, X85, append19_out_aaa(T80, T81, X85)) → append19_out_aaa(.(T77, T80), T81, .(T77, X85))
U4_aaag(T58, T55, T56, T57, T12, append19_out_aaa(T55, T56, X56)) → append31_out_aaag(.(T58, T55), T56, T57, T12)
append31_in_aaag(.(T62, T55), T56, T63, T12) → U5_aaag(T62, T55, T56, T63, T12, append19_in_aaa(T55, T56, T61))
U5_aaag(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → U6_aaag(T62, T55, T56, T63, T12, append7_in_aag(.(T62, T61), T63, T12))
U6_aaag(T62, T55, T56, T63, T12, append7_out_aag(.(T62, T61), T63, T12)) → append31_out_aaag(.(T62, T55), T56, T63, T12)

The argument filtering Pi contains the following mapping:
append31_in_aaag(x1, x2, x3, x4)  =  append31_in_aaag(x4)
U3_aaag(x1, x2, x3, x4)  =  U3_aaag(x4)
append7_in_aag(x1, x2, x3)  =  append7_in_aag(x3)
append7_out_aag(x1, x2, x3)  =  append7_out_aag(x1, x2)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
append31_out_aaag(x1, x2, x3, x4)  =  append31_out_aaag(x1)
U4_aaag(x1, x2, x3, x4, x5, x6)  =  U4_aaag(x6)
append19_in_aaa(x1, x2, x3)  =  append19_in_aaa
append19_out_aaa(x1, x2, x3)  =  append19_out_aaa(x1)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U5_aaag(x1, x2, x3, x4, x5, x6)  =  U5_aaag(x5, x6)
U6_aaag(x1, x2, x3, x4, x5, x6)  =  U6_aaag(x2, x6)
[]  =  []
APPEND31_IN_AAAG(x1, x2, x3, x4)  =  APPEND31_IN_AAAG(x4)
U3_AAAG(x1, x2, x3, x4)  =  U3_AAAG(x4)
APPEND7_IN_AAG(x1, x2, x3)  =  APPEND7_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x5)
U4_AAAG(x1, x2, x3, x4, x5, x6)  =  U4_AAAG(x6)
APPEND19_IN_AAA(x1, x2, x3)  =  APPEND19_IN_AAA
U2_AAA(x1, x2, x3, x4, x5)  =  U2_AAA(x5)
U5_AAAG(x1, x2, x3, x4, x5, x6)  =  U5_AAAG(x5, x6)
U6_AAAG(x1, x2, x3, x4, x5, x6)  =  U6_AAAG(x2, x6)

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

(27) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 9 less nodes.

(28) Complex Obligation (AND)

(29) Obligation:

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

APPEND19_IN_AAA(.(T77, T80), T81, .(T77, X85)) → APPEND19_IN_AAA(T80, T81, X85)

The TRS R consists of the following rules:

append31_in_aaag([], T21, T22, T12) → U3_aaag(T21, T22, T12, append7_in_aag(T21, T22, T12))
append7_in_aag([], T29, T29) → append7_out_aag([], T29, T29)
append7_in_aag(.(T38, T42), T43, .(T38, T41)) → U1_aag(T38, T42, T43, T41, append7_in_aag(T42, T43, T41))
U1_aag(T38, T42, T43, T41, append7_out_aag(T42, T43, T41)) → append7_out_aag(.(T38, T42), T43, .(T38, T41))
U3_aaag(T21, T22, T12, append7_out_aag(T21, T22, T12)) → append31_out_aaag([], T21, T22, T12)
append31_in_aaag(.(T58, T55), T56, T57, T12) → U4_aaag(T58, T55, T56, T57, T12, append19_in_aaa(T55, T56, X56))
append19_in_aaa([], T70, T70) → append19_out_aaa([], T70, T70)
append19_in_aaa(.(T77, T80), T81, .(T77, X85)) → U2_aaa(T77, T80, T81, X85, append19_in_aaa(T80, T81, X85))
U2_aaa(T77, T80, T81, X85, append19_out_aaa(T80, T81, X85)) → append19_out_aaa(.(T77, T80), T81, .(T77, X85))
U4_aaag(T58, T55, T56, T57, T12, append19_out_aaa(T55, T56, X56)) → append31_out_aaag(.(T58, T55), T56, T57, T12)
append31_in_aaag(.(T62, T55), T56, T63, T12) → U5_aaag(T62, T55, T56, T63, T12, append19_in_aaa(T55, T56, T61))
U5_aaag(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → U6_aaag(T62, T55, T56, T63, T12, append7_in_aag(.(T62, T61), T63, T12))
U6_aaag(T62, T55, T56, T63, T12, append7_out_aag(.(T62, T61), T63, T12)) → append31_out_aaag(.(T62, T55), T56, T63, T12)

The argument filtering Pi contains the following mapping:
append31_in_aaag(x1, x2, x3, x4)  =  append31_in_aaag(x4)
U3_aaag(x1, x2, x3, x4)  =  U3_aaag(x4)
append7_in_aag(x1, x2, x3)  =  append7_in_aag(x3)
append7_out_aag(x1, x2, x3)  =  append7_out_aag(x1, x2)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
append31_out_aaag(x1, x2, x3, x4)  =  append31_out_aaag(x1)
U4_aaag(x1, x2, x3, x4, x5, x6)  =  U4_aaag(x6)
append19_in_aaa(x1, x2, x3)  =  append19_in_aaa
append19_out_aaa(x1, x2, x3)  =  append19_out_aaa(x1)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U5_aaag(x1, x2, x3, x4, x5, x6)  =  U5_aaag(x5, x6)
U6_aaag(x1, x2, x3, x4, x5, x6)  =  U6_aaag(x2, x6)
[]  =  []
APPEND19_IN_AAA(x1, x2, x3)  =  APPEND19_IN_AAA

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

(30) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(31) Obligation:

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

APPEND19_IN_AAA(.(T77, T80), T81, .(T77, X85)) → APPEND19_IN_AAA(T80, T81, X85)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND19_IN_AAA(x1, x2, x3)  =  APPEND19_IN_AAA

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

(32) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(33) Obligation:

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

APPEND19_IN_AAA → APPEND19_IN_AAA

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

(34) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = APPEND19_IN_AAA evaluates to t =APPEND19_IN_AAA

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND19_IN_AAA to APPEND19_IN_AAA.



(35) NO

(36) Obligation:

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

APPEND7_IN_AAG(.(T38, T42), T43, .(T38, T41)) → APPEND7_IN_AAG(T42, T43, T41)

The TRS R consists of the following rules:

append31_in_aaag([], T21, T22, T12) → U3_aaag(T21, T22, T12, append7_in_aag(T21, T22, T12))
append7_in_aag([], T29, T29) → append7_out_aag([], T29, T29)
append7_in_aag(.(T38, T42), T43, .(T38, T41)) → U1_aag(T38, T42, T43, T41, append7_in_aag(T42, T43, T41))
U1_aag(T38, T42, T43, T41, append7_out_aag(T42, T43, T41)) → append7_out_aag(.(T38, T42), T43, .(T38, T41))
U3_aaag(T21, T22, T12, append7_out_aag(T21, T22, T12)) → append31_out_aaag([], T21, T22, T12)
append31_in_aaag(.(T58, T55), T56, T57, T12) → U4_aaag(T58, T55, T56, T57, T12, append19_in_aaa(T55, T56, X56))
append19_in_aaa([], T70, T70) → append19_out_aaa([], T70, T70)
append19_in_aaa(.(T77, T80), T81, .(T77, X85)) → U2_aaa(T77, T80, T81, X85, append19_in_aaa(T80, T81, X85))
U2_aaa(T77, T80, T81, X85, append19_out_aaa(T80, T81, X85)) → append19_out_aaa(.(T77, T80), T81, .(T77, X85))
U4_aaag(T58, T55, T56, T57, T12, append19_out_aaa(T55, T56, X56)) → append31_out_aaag(.(T58, T55), T56, T57, T12)
append31_in_aaag(.(T62, T55), T56, T63, T12) → U5_aaag(T62, T55, T56, T63, T12, append19_in_aaa(T55, T56, T61))
U5_aaag(T62, T55, T56, T63, T12, append19_out_aaa(T55, T56, T61)) → U6_aaag(T62, T55, T56, T63, T12, append7_in_aag(.(T62, T61), T63, T12))
U6_aaag(T62, T55, T56, T63, T12, append7_out_aag(.(T62, T61), T63, T12)) → append31_out_aaag(.(T62, T55), T56, T63, T12)

The argument filtering Pi contains the following mapping:
append31_in_aaag(x1, x2, x3, x4)  =  append31_in_aaag(x4)
U3_aaag(x1, x2, x3, x4)  =  U3_aaag(x4)
append7_in_aag(x1, x2, x3)  =  append7_in_aag(x3)
append7_out_aag(x1, x2, x3)  =  append7_out_aag(x1, x2)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
append31_out_aaag(x1, x2, x3, x4)  =  append31_out_aaag(x1)
U4_aaag(x1, x2, x3, x4, x5, x6)  =  U4_aaag(x6)
append19_in_aaa(x1, x2, x3)  =  append19_in_aaa
append19_out_aaa(x1, x2, x3)  =  append19_out_aaa(x1)
U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
U5_aaag(x1, x2, x3, x4, x5, x6)  =  U5_aaag(x5, x6)
U6_aaag(x1, x2, x3, x4, x5, x6)  =  U6_aaag(x2, x6)
[]  =  []
APPEND7_IN_AAG(x1, x2, x3)  =  APPEND7_IN_AAG(x3)

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

(37) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(38) Obligation:

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

APPEND7_IN_AAG(.(T38, T42), T43, .(T38, T41)) → APPEND7_IN_AAG(T42, T43, T41)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND7_IN_AAG(x1, x2, x3)  =  APPEND7_IN_AAG(x3)

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

(39) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(40) Obligation:

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

APPEND7_IN_AAG(.(T41)) → APPEND7_IN_AAG(T41)

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

(41) 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:

  • APPEND7_IN_AAG(.(T41)) → APPEND7_IN_AAG(T41)
    The graph contains the following edges 1 > 1

(42) YES