Left Termination of the query pattern sublist_in_2(g, a) 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 QDPSizeChangeProof (⇔)
                ↳15 YES
            ↳16 PiDP
                ↳17 UsableRulesProof (⇔)
                ↳18 PiDP
                ↳19 PiDPToQDPProof (⇐)
                ↳20 QDP
                ↳21 NonTerminationProof (⇔)
                ↳22 NO
            ↳23 PiDP
                ↳24 UsableRulesProof (⇔)
                ↳25 PiDP
                ↳26 PiDPToQDPProof (⇐)
                ↳27 QDP
                ↳28 QDPSizeChangeProof (⇔)
                ↳29 YES
    ↳30 PrologToPiTRSProof (⇐)
        ↳31 PiTRS
        ↳32 DependencyPairsProof (⇔)
        ↳33 PiDP
        ↳34 DependencyGraphProof (⇔)
        ↳35 AND
            ↳36 PiDP
                ↳37 UsableRulesProof (⇔)
                ↳38 PiDP
                ↳39 PiDPToQDPProof (⇐)
                ↳40 QDP
                ↳41 QDPSizeChangeProof (⇔)
                ↳42 YES
            ↳43 PiDP
                ↳44 UsableRulesProof (⇔)
                ↳45 PiDP
                ↳46 PiDPToQDPProof (⇐)
                ↳47 QDP
                ↳48 NonTerminationProof (⇔)
                ↳49 NO
            ↳50 PiDP
                ↳51 UsableRulesProof (⇔)
                ↳52 PiDP
                ↳53 PiDPToQDPProof (⇐)
                ↳54 QDP
                ↳55 QDPSizeChangeProof (⇔)
                ↳56 YES

(0) Obligation:

Clauses:

sublist(X, Y) :- ','(append(U, X, V), append(V, W, Y)).
append([], Ys, Ys).
append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs).

Queries:

sublist(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

append7([], T23, T23).
append7(.(T30, T31), X50, .(T30, T33)) :- append7(T31, X50, T33).
append17([], T50, T50).
append17(.(X125, X126), T54, .(X125, X127)) :- append17(X126, T54, X127).
append27([], T72, T72).
append27(.(T79, T82), X189, .(T79, T83)) :- append27(T82, X189, T83).
sublist1(T16, T7) :- append7(T16, X7, T7).
sublist1(T40, T7) :- append17(X88, T40, X89).
sublist1(T40, .(X154, T65)) :- ','(append17(T43, T40, T64), append27(T64, X155, T65)).

Queries:

sublist1(g,a).

(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:
sublist1_in: (b,f)
append7_in: (b,f,f)
append17_in: (f,b,f)
append27_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

sublist1_in_ga(T16, T7) → U4_ga(T16, T7, append7_in_gaa(T16, X7, T7))
append7_in_gaa([], T23, T23) → append7_out_gaa([], T23, T23)
append7_in_gaa(.(T30, T31), X50, .(T30, T33)) → U1_gaa(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
U1_gaa(T30, T31, X50, T33, append7_out_gaa(T31, X50, T33)) → append7_out_gaa(.(T30, T31), X50, .(T30, T33))
U4_ga(T16, T7, append7_out_gaa(T16, X7, T7)) → sublist1_out_ga(T16, T7)
sublist1_in_ga(T40, T7) → U5_ga(T40, T7, append17_in_aga(X88, T40, X89))
append17_in_aga([], T50, T50) → append17_out_aga([], T50, T50)
append17_in_aga(.(X125, X126), T54, .(X125, X127)) → U2_aga(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
U2_aga(X125, X126, T54, X127, append17_out_aga(X126, T54, X127)) → append17_out_aga(.(X125, X126), T54, .(X125, X127))
U5_ga(T40, T7, append17_out_aga(X88, T40, X89)) → sublist1_out_ga(T40, T7)
sublist1_in_ga(T40, .(X154, T65)) → U6_ga(T40, X154, T65, append17_in_aga(T43, T40, T64))
U6_ga(T40, X154, T65, append17_out_aga(T43, T40, T64)) → U7_ga(T40, X154, T65, append27_in_gaa(T64, X155, T65))
append27_in_gaa([], T72, T72) → append27_out_gaa([], T72, T72)
append27_in_gaa(.(T79, T82), X189, .(T79, T83)) → U3_gaa(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
U3_gaa(T79, T82, X189, T83, append27_out_gaa(T82, X189, T83)) → append27_out_gaa(.(T79, T82), X189, .(T79, T83))
U7_ga(T40, X154, T65, append27_out_gaa(T64, X155, T65)) → sublist1_out_ga(T40, .(X154, T65))

The argument filtering Pi contains the following mapping:
sublist1_in_ga(x1, x2)  =  sublist1_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
append7_in_gaa(x1, x2, x3)  =  append7_in_gaa(x1)
[]  =  []
append7_out_gaa(x1, x2, x3)  =  append7_out_gaa
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
append17_in_aga(x1, x2, x3)  =  append17_in_aga(x2)
append17_out_aga(x1, x2, x3)  =  append17_out_aga(x1, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)
append27_in_gaa(x1, x2, x3)  =  append27_in_gaa(x1)
append27_out_gaa(x1, x2, x3)  =  append27_out_gaa
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

sublist1_in_ga(T16, T7) → U4_ga(T16, T7, append7_in_gaa(T16, X7, T7))
append7_in_gaa([], T23, T23) → append7_out_gaa([], T23, T23)
append7_in_gaa(.(T30, T31), X50, .(T30, T33)) → U1_gaa(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
U1_gaa(T30, T31, X50, T33, append7_out_gaa(T31, X50, T33)) → append7_out_gaa(.(T30, T31), X50, .(T30, T33))
U4_ga(T16, T7, append7_out_gaa(T16, X7, T7)) → sublist1_out_ga(T16, T7)
sublist1_in_ga(T40, T7) → U5_ga(T40, T7, append17_in_aga(X88, T40, X89))
append17_in_aga([], T50, T50) → append17_out_aga([], T50, T50)
append17_in_aga(.(X125, X126), T54, .(X125, X127)) → U2_aga(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
U2_aga(X125, X126, T54, X127, append17_out_aga(X126, T54, X127)) → append17_out_aga(.(X125, X126), T54, .(X125, X127))
U5_ga(T40, T7, append17_out_aga(X88, T40, X89)) → sublist1_out_ga(T40, T7)
sublist1_in_ga(T40, .(X154, T65)) → U6_ga(T40, X154, T65, append17_in_aga(T43, T40, T64))
U6_ga(T40, X154, T65, append17_out_aga(T43, T40, T64)) → U7_ga(T40, X154, T65, append27_in_gaa(T64, X155, T65))
append27_in_gaa([], T72, T72) → append27_out_gaa([], T72, T72)
append27_in_gaa(.(T79, T82), X189, .(T79, T83)) → U3_gaa(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
U3_gaa(T79, T82, X189, T83, append27_out_gaa(T82, X189, T83)) → append27_out_gaa(.(T79, T82), X189, .(T79, T83))
U7_ga(T40, X154, T65, append27_out_gaa(T64, X155, T65)) → sublist1_out_ga(T40, .(X154, T65))

The argument filtering Pi contains the following mapping:
sublist1_in_ga(x1, x2)  =  sublist1_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
append7_in_gaa(x1, x2, x3)  =  append7_in_gaa(x1)
[]  =  []
append7_out_gaa(x1, x2, x3)  =  append7_out_gaa
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
append17_in_aga(x1, x2, x3)  =  append17_in_aga(x2)
append17_out_aga(x1, x2, x3)  =  append17_out_aga(x1, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)
append27_in_gaa(x1, x2, x3)  =  append27_in_gaa(x1)
append27_out_gaa(x1, x2, x3)  =  append27_out_gaa
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)

(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:

SUBLIST1_IN_GA(T16, T7) → U4_GA(T16, T7, append7_in_gaa(T16, X7, T7))
SUBLIST1_IN_GA(T16, T7) → APPEND7_IN_GAA(T16, X7, T7)
APPEND7_IN_GAA(.(T30, T31), X50, .(T30, T33)) → U1_GAA(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
APPEND7_IN_GAA(.(T30, T31), X50, .(T30, T33)) → APPEND7_IN_GAA(T31, X50, T33)
SUBLIST1_IN_GA(T40, T7) → U5_GA(T40, T7, append17_in_aga(X88, T40, X89))
SUBLIST1_IN_GA(T40, T7) → APPEND17_IN_AGA(X88, T40, X89)
APPEND17_IN_AGA(.(X125, X126), T54, .(X125, X127)) → U2_AGA(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
APPEND17_IN_AGA(.(X125, X126), T54, .(X125, X127)) → APPEND17_IN_AGA(X126, T54, X127)
SUBLIST1_IN_GA(T40, .(X154, T65)) → U6_GA(T40, X154, T65, append17_in_aga(T43, T40, T64))
SUBLIST1_IN_GA(T40, .(X154, T65)) → APPEND17_IN_AGA(T43, T40, T64)
U6_GA(T40, X154, T65, append17_out_aga(T43, T40, T64)) → U7_GA(T40, X154, T65, append27_in_gaa(T64, X155, T65))
U6_GA(T40, X154, T65, append17_out_aga(T43, T40, T64)) → APPEND27_IN_GAA(T64, X155, T65)
APPEND27_IN_GAA(.(T79, T82), X189, .(T79, T83)) → U3_GAA(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
APPEND27_IN_GAA(.(T79, T82), X189, .(T79, T83)) → APPEND27_IN_GAA(T82, X189, T83)

The TRS R consists of the following rules:

sublist1_in_ga(T16, T7) → U4_ga(T16, T7, append7_in_gaa(T16, X7, T7))
append7_in_gaa([], T23, T23) → append7_out_gaa([], T23, T23)
append7_in_gaa(.(T30, T31), X50, .(T30, T33)) → U1_gaa(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
U1_gaa(T30, T31, X50, T33, append7_out_gaa(T31, X50, T33)) → append7_out_gaa(.(T30, T31), X50, .(T30, T33))
U4_ga(T16, T7, append7_out_gaa(T16, X7, T7)) → sublist1_out_ga(T16, T7)
sublist1_in_ga(T40, T7) → U5_ga(T40, T7, append17_in_aga(X88, T40, X89))
append17_in_aga([], T50, T50) → append17_out_aga([], T50, T50)
append17_in_aga(.(X125, X126), T54, .(X125, X127)) → U2_aga(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
U2_aga(X125, X126, T54, X127, append17_out_aga(X126, T54, X127)) → append17_out_aga(.(X125, X126), T54, .(X125, X127))
U5_ga(T40, T7, append17_out_aga(X88, T40, X89)) → sublist1_out_ga(T40, T7)
sublist1_in_ga(T40, .(X154, T65)) → U6_ga(T40, X154, T65, append17_in_aga(T43, T40, T64))
U6_ga(T40, X154, T65, append17_out_aga(T43, T40, T64)) → U7_ga(T40, X154, T65, append27_in_gaa(T64, X155, T65))
append27_in_gaa([], T72, T72) → append27_out_gaa([], T72, T72)
append27_in_gaa(.(T79, T82), X189, .(T79, T83)) → U3_gaa(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
U3_gaa(T79, T82, X189, T83, append27_out_gaa(T82, X189, T83)) → append27_out_gaa(.(T79, T82), X189, .(T79, T83))
U7_ga(T40, X154, T65, append27_out_gaa(T64, X155, T65)) → sublist1_out_ga(T40, .(X154, T65))

The argument filtering Pi contains the following mapping:
sublist1_in_ga(x1, x2)  =  sublist1_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
append7_in_gaa(x1, x2, x3)  =  append7_in_gaa(x1)
[]  =  []
append7_out_gaa(x1, x2, x3)  =  append7_out_gaa
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
append17_in_aga(x1, x2, x3)  =  append17_in_aga(x2)
append17_out_aga(x1, x2, x3)  =  append17_out_aga(x1, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)
append27_in_gaa(x1, x2, x3)  =  append27_in_gaa(x1)
append27_out_gaa(x1, x2, x3)  =  append27_out_gaa
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
SUBLIST1_IN_GA(x1, x2)  =  SUBLIST1_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
APPEND7_IN_GAA(x1, x2, x3)  =  APPEND7_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x5)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
APPEND17_IN_AGA(x1, x2, x3)  =  APPEND17_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x5)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x4)
U7_GA(x1, x2, x3, x4)  =  U7_GA(x4)
APPEND27_IN_GAA(x1, x2, x3)  =  APPEND27_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)

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

(6) Obligation:

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

SUBLIST1_IN_GA(T16, T7) → U4_GA(T16, T7, append7_in_gaa(T16, X7, T7))
SUBLIST1_IN_GA(T16, T7) → APPEND7_IN_GAA(T16, X7, T7)
APPEND7_IN_GAA(.(T30, T31), X50, .(T30, T33)) → U1_GAA(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
APPEND7_IN_GAA(.(T30, T31), X50, .(T30, T33)) → APPEND7_IN_GAA(T31, X50, T33)
SUBLIST1_IN_GA(T40, T7) → U5_GA(T40, T7, append17_in_aga(X88, T40, X89))
SUBLIST1_IN_GA(T40, T7) → APPEND17_IN_AGA(X88, T40, X89)
APPEND17_IN_AGA(.(X125, X126), T54, .(X125, X127)) → U2_AGA(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
APPEND17_IN_AGA(.(X125, X126), T54, .(X125, X127)) → APPEND17_IN_AGA(X126, T54, X127)
SUBLIST1_IN_GA(T40, .(X154, T65)) → U6_GA(T40, X154, T65, append17_in_aga(T43, T40, T64))
SUBLIST1_IN_GA(T40, .(X154, T65)) → APPEND17_IN_AGA(T43, T40, T64)
U6_GA(T40, X154, T65, append17_out_aga(T43, T40, T64)) → U7_GA(T40, X154, T65, append27_in_gaa(T64, X155, T65))
U6_GA(T40, X154, T65, append17_out_aga(T43, T40, T64)) → APPEND27_IN_GAA(T64, X155, T65)
APPEND27_IN_GAA(.(T79, T82), X189, .(T79, T83)) → U3_GAA(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
APPEND27_IN_GAA(.(T79, T82), X189, .(T79, T83)) → APPEND27_IN_GAA(T82, X189, T83)

The TRS R consists of the following rules:

sublist1_in_ga(T16, T7) → U4_ga(T16, T7, append7_in_gaa(T16, X7, T7))
append7_in_gaa([], T23, T23) → append7_out_gaa([], T23, T23)
append7_in_gaa(.(T30, T31), X50, .(T30, T33)) → U1_gaa(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
U1_gaa(T30, T31, X50, T33, append7_out_gaa(T31, X50, T33)) → append7_out_gaa(.(T30, T31), X50, .(T30, T33))
U4_ga(T16, T7, append7_out_gaa(T16, X7, T7)) → sublist1_out_ga(T16, T7)
sublist1_in_ga(T40, T7) → U5_ga(T40, T7, append17_in_aga(X88, T40, X89))
append17_in_aga([], T50, T50) → append17_out_aga([], T50, T50)
append17_in_aga(.(X125, X126), T54, .(X125, X127)) → U2_aga(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
U2_aga(X125, X126, T54, X127, append17_out_aga(X126, T54, X127)) → append17_out_aga(.(X125, X126), T54, .(X125, X127))
U5_ga(T40, T7, append17_out_aga(X88, T40, X89)) → sublist1_out_ga(T40, T7)
sublist1_in_ga(T40, .(X154, T65)) → U6_ga(T40, X154, T65, append17_in_aga(T43, T40, T64))
U6_ga(T40, X154, T65, append17_out_aga(T43, T40, T64)) → U7_ga(T40, X154, T65, append27_in_gaa(T64, X155, T65))
append27_in_gaa([], T72, T72) → append27_out_gaa([], T72, T72)
append27_in_gaa(.(T79, T82), X189, .(T79, T83)) → U3_gaa(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
U3_gaa(T79, T82, X189, T83, append27_out_gaa(T82, X189, T83)) → append27_out_gaa(.(T79, T82), X189, .(T79, T83))
U7_ga(T40, X154, T65, append27_out_gaa(T64, X155, T65)) → sublist1_out_ga(T40, .(X154, T65))

The argument filtering Pi contains the following mapping:
sublist1_in_ga(x1, x2)  =  sublist1_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
append7_in_gaa(x1, x2, x3)  =  append7_in_gaa(x1)
[]  =  []
append7_out_gaa(x1, x2, x3)  =  append7_out_gaa
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
append17_in_aga(x1, x2, x3)  =  append17_in_aga(x2)
append17_out_aga(x1, x2, x3)  =  append17_out_aga(x1, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)
append27_in_gaa(x1, x2, x3)  =  append27_in_gaa(x1)
append27_out_gaa(x1, x2, x3)  =  append27_out_gaa
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
SUBLIST1_IN_GA(x1, x2)  =  SUBLIST1_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
APPEND7_IN_GAA(x1, x2, x3)  =  APPEND7_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x5)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
APPEND17_IN_AGA(x1, x2, x3)  =  APPEND17_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x5)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x4)
U7_GA(x1, x2, x3, x4)  =  U7_GA(x4)
APPEND27_IN_GAA(x1, x2, x3)  =  APPEND27_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 11 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APPEND27_IN_GAA(.(T79, T82), X189, .(T79, T83)) → APPEND27_IN_GAA(T82, X189, T83)

The TRS R consists of the following rules:

sublist1_in_ga(T16, T7) → U4_ga(T16, T7, append7_in_gaa(T16, X7, T7))
append7_in_gaa([], T23, T23) → append7_out_gaa([], T23, T23)
append7_in_gaa(.(T30, T31), X50, .(T30, T33)) → U1_gaa(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
U1_gaa(T30, T31, X50, T33, append7_out_gaa(T31, X50, T33)) → append7_out_gaa(.(T30, T31), X50, .(T30, T33))
U4_ga(T16, T7, append7_out_gaa(T16, X7, T7)) → sublist1_out_ga(T16, T7)
sublist1_in_ga(T40, T7) → U5_ga(T40, T7, append17_in_aga(X88, T40, X89))
append17_in_aga([], T50, T50) → append17_out_aga([], T50, T50)
append17_in_aga(.(X125, X126), T54, .(X125, X127)) → U2_aga(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
U2_aga(X125, X126, T54, X127, append17_out_aga(X126, T54, X127)) → append17_out_aga(.(X125, X126), T54, .(X125, X127))
U5_ga(T40, T7, append17_out_aga(X88, T40, X89)) → sublist1_out_ga(T40, T7)
sublist1_in_ga(T40, .(X154, T65)) → U6_ga(T40, X154, T65, append17_in_aga(T43, T40, T64))
U6_ga(T40, X154, T65, append17_out_aga(T43, T40, T64)) → U7_ga(T40, X154, T65, append27_in_gaa(T64, X155, T65))
append27_in_gaa([], T72, T72) → append27_out_gaa([], T72, T72)
append27_in_gaa(.(T79, T82), X189, .(T79, T83)) → U3_gaa(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
U3_gaa(T79, T82, X189, T83, append27_out_gaa(T82, X189, T83)) → append27_out_gaa(.(T79, T82), X189, .(T79, T83))
U7_ga(T40, X154, T65, append27_out_gaa(T64, X155, T65)) → sublist1_out_ga(T40, .(X154, T65))

The argument filtering Pi contains the following mapping:
sublist1_in_ga(x1, x2)  =  sublist1_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
append7_in_gaa(x1, x2, x3)  =  append7_in_gaa(x1)
[]  =  []
append7_out_gaa(x1, x2, x3)  =  append7_out_gaa
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
append17_in_aga(x1, x2, x3)  =  append17_in_aga(x2)
append17_out_aga(x1, x2, x3)  =  append17_out_aga(x1, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)
append27_in_gaa(x1, x2, x3)  =  append27_in_gaa(x1)
append27_out_gaa(x1, x2, x3)  =  append27_out_gaa
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
APPEND27_IN_GAA(x1, x2, x3)  =  APPEND27_IN_GAA(x1)

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:

APPEND27_IN_GAA(.(T79, T82), X189, .(T79, T83)) → APPEND27_IN_GAA(T82, X189, T83)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND27_IN_GAA(x1, x2, x3)  =  APPEND27_IN_GAA(x1)

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:

APPEND27_IN_GAA(.(T82)) → APPEND27_IN_GAA(T82)

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

(14) 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:

  • APPEND27_IN_GAA(.(T82)) → APPEND27_IN_GAA(T82)
    The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

APPEND17_IN_AGA(.(X125, X126), T54, .(X125, X127)) → APPEND17_IN_AGA(X126, T54, X127)

The TRS R consists of the following rules:

sublist1_in_ga(T16, T7) → U4_ga(T16, T7, append7_in_gaa(T16, X7, T7))
append7_in_gaa([], T23, T23) → append7_out_gaa([], T23, T23)
append7_in_gaa(.(T30, T31), X50, .(T30, T33)) → U1_gaa(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
U1_gaa(T30, T31, X50, T33, append7_out_gaa(T31, X50, T33)) → append7_out_gaa(.(T30, T31), X50, .(T30, T33))
U4_ga(T16, T7, append7_out_gaa(T16, X7, T7)) → sublist1_out_ga(T16, T7)
sublist1_in_ga(T40, T7) → U5_ga(T40, T7, append17_in_aga(X88, T40, X89))
append17_in_aga([], T50, T50) → append17_out_aga([], T50, T50)
append17_in_aga(.(X125, X126), T54, .(X125, X127)) → U2_aga(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
U2_aga(X125, X126, T54, X127, append17_out_aga(X126, T54, X127)) → append17_out_aga(.(X125, X126), T54, .(X125, X127))
U5_ga(T40, T7, append17_out_aga(X88, T40, X89)) → sublist1_out_ga(T40, T7)
sublist1_in_ga(T40, .(X154, T65)) → U6_ga(T40, X154, T65, append17_in_aga(T43, T40, T64))
U6_ga(T40, X154, T65, append17_out_aga(T43, T40, T64)) → U7_ga(T40, X154, T65, append27_in_gaa(T64, X155, T65))
append27_in_gaa([], T72, T72) → append27_out_gaa([], T72, T72)
append27_in_gaa(.(T79, T82), X189, .(T79, T83)) → U3_gaa(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
U3_gaa(T79, T82, X189, T83, append27_out_gaa(T82, X189, T83)) → append27_out_gaa(.(T79, T82), X189, .(T79, T83))
U7_ga(T40, X154, T65, append27_out_gaa(T64, X155, T65)) → sublist1_out_ga(T40, .(X154, T65))

The argument filtering Pi contains the following mapping:
sublist1_in_ga(x1, x2)  =  sublist1_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
append7_in_gaa(x1, x2, x3)  =  append7_in_gaa(x1)
[]  =  []
append7_out_gaa(x1, x2, x3)  =  append7_out_gaa
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
append17_in_aga(x1, x2, x3)  =  append17_in_aga(x2)
append17_out_aga(x1, x2, x3)  =  append17_out_aga(x1, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)
append27_in_gaa(x1, x2, x3)  =  append27_in_gaa(x1)
append27_out_gaa(x1, x2, x3)  =  append27_out_gaa
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
APPEND17_IN_AGA(x1, x2, x3)  =  APPEND17_IN_AGA(x2)

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:

APPEND17_IN_AGA(.(X125, X126), T54, .(X125, X127)) → APPEND17_IN_AGA(X126, T54, X127)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND17_IN_AGA(x1, x2, x3)  =  APPEND17_IN_AGA(x2)

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:

APPEND17_IN_AGA(T54) → APPEND17_IN_AGA(T54)

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

(21) 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 = APPEND17_IN_AGA(T54) evaluates to t =APPEND17_IN_AGA(T54)

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND17_IN_AGA(T54) to APPEND17_IN_AGA(T54).



(22) NO

(23) Obligation:

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

APPEND7_IN_GAA(.(T30, T31), X50, .(T30, T33)) → APPEND7_IN_GAA(T31, X50, T33)

The TRS R consists of the following rules:

sublist1_in_ga(T16, T7) → U4_ga(T16, T7, append7_in_gaa(T16, X7, T7))
append7_in_gaa([], T23, T23) → append7_out_gaa([], T23, T23)
append7_in_gaa(.(T30, T31), X50, .(T30, T33)) → U1_gaa(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
U1_gaa(T30, T31, X50, T33, append7_out_gaa(T31, X50, T33)) → append7_out_gaa(.(T30, T31), X50, .(T30, T33))
U4_ga(T16, T7, append7_out_gaa(T16, X7, T7)) → sublist1_out_ga(T16, T7)
sublist1_in_ga(T40, T7) → U5_ga(T40, T7, append17_in_aga(X88, T40, X89))
append17_in_aga([], T50, T50) → append17_out_aga([], T50, T50)
append17_in_aga(.(X125, X126), T54, .(X125, X127)) → U2_aga(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
U2_aga(X125, X126, T54, X127, append17_out_aga(X126, T54, X127)) → append17_out_aga(.(X125, X126), T54, .(X125, X127))
U5_ga(T40, T7, append17_out_aga(X88, T40, X89)) → sublist1_out_ga(T40, T7)
sublist1_in_ga(T40, .(X154, T65)) → U6_ga(T40, X154, T65, append17_in_aga(T43, T40, T64))
U6_ga(T40, X154, T65, append17_out_aga(T43, T40, T64)) → U7_ga(T40, X154, T65, append27_in_gaa(T64, X155, T65))
append27_in_gaa([], T72, T72) → append27_out_gaa([], T72, T72)
append27_in_gaa(.(T79, T82), X189, .(T79, T83)) → U3_gaa(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
U3_gaa(T79, T82, X189, T83, append27_out_gaa(T82, X189, T83)) → append27_out_gaa(.(T79, T82), X189, .(T79, T83))
U7_ga(T40, X154, T65, append27_out_gaa(T64, X155, T65)) → sublist1_out_ga(T40, .(X154, T65))

The argument filtering Pi contains the following mapping:
sublist1_in_ga(x1, x2)  =  sublist1_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
append7_in_gaa(x1, x2, x3)  =  append7_in_gaa(x1)
[]  =  []
append7_out_gaa(x1, x2, x3)  =  append7_out_gaa
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
append17_in_aga(x1, x2, x3)  =  append17_in_aga(x2)
append17_out_aga(x1, x2, x3)  =  append17_out_aga(x1, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)
append27_in_gaa(x1, x2, x3)  =  append27_in_gaa(x1)
append27_out_gaa(x1, x2, x3)  =  append27_out_gaa
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
APPEND7_IN_GAA(x1, x2, x3)  =  APPEND7_IN_GAA(x1)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

APPEND7_IN_GAA(.(T30, T31), X50, .(T30, T33)) → APPEND7_IN_GAA(T31, X50, T33)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND7_IN_GAA(x1, x2, x3)  =  APPEND7_IN_GAA(x1)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

APPEND7_IN_GAA(.(T31)) → APPEND7_IN_GAA(T31)

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

(28) 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_GAA(.(T31)) → APPEND7_IN_GAA(T31)
    The graph contains the following edges 1 > 1

(29) YES

(30) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
sublist1_in: (b,f)
append7_in: (b,f,f)
append17_in: (f,b,f)
append27_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

sublist1_in_ga(T16, T7) → U4_ga(T16, T7, append7_in_gaa(T16, X7, T7))
append7_in_gaa([], T23, T23) → append7_out_gaa([], T23, T23)
append7_in_gaa(.(T30, T31), X50, .(T30, T33)) → U1_gaa(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
U1_gaa(T30, T31, X50, T33, append7_out_gaa(T31, X50, T33)) → append7_out_gaa(.(T30, T31), X50, .(T30, T33))
U4_ga(T16, T7, append7_out_gaa(T16, X7, T7)) → sublist1_out_ga(T16, T7)
sublist1_in_ga(T40, T7) → U5_ga(T40, T7, append17_in_aga(X88, T40, X89))
append17_in_aga([], T50, T50) → append17_out_aga([], T50, T50)
append17_in_aga(.(X125, X126), T54, .(X125, X127)) → U2_aga(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
U2_aga(X125, X126, T54, X127, append17_out_aga(X126, T54, X127)) → append17_out_aga(.(X125, X126), T54, .(X125, X127))
U5_ga(T40, T7, append17_out_aga(X88, T40, X89)) → sublist1_out_ga(T40, T7)
sublist1_in_ga(T40, .(X154, T65)) → U6_ga(T40, X154, T65, append17_in_aga(T43, T40, T64))
U6_ga(T40, X154, T65, append17_out_aga(T43, T40, T64)) → U7_ga(T40, X154, T65, append27_in_gaa(T64, X155, T65))
append27_in_gaa([], T72, T72) → append27_out_gaa([], T72, T72)
append27_in_gaa(.(T79, T82), X189, .(T79, T83)) → U3_gaa(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
U3_gaa(T79, T82, X189, T83, append27_out_gaa(T82, X189, T83)) → append27_out_gaa(.(T79, T82), X189, .(T79, T83))
U7_ga(T40, X154, T65, append27_out_gaa(T64, X155, T65)) → sublist1_out_ga(T40, .(X154, T65))

The argument filtering Pi contains the following mapping:
sublist1_in_ga(x1, x2)  =  sublist1_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
append7_in_gaa(x1, x2, x3)  =  append7_in_gaa(x1)
[]  =  []
append7_out_gaa(x1, x2, x3)  =  append7_out_gaa(x1)
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
append17_in_aga(x1, x2, x3)  =  append17_in_aga(x2)
append17_out_aga(x1, x2, x3)  =  append17_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x1, x4)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
append27_in_gaa(x1, x2, x3)  =  append27_in_gaa(x1)
append27_out_gaa(x1, x2, x3)  =  append27_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(31) Obligation:

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

sublist1_in_ga(T16, T7) → U4_ga(T16, T7, append7_in_gaa(T16, X7, T7))
append7_in_gaa([], T23, T23) → append7_out_gaa([], T23, T23)
append7_in_gaa(.(T30, T31), X50, .(T30, T33)) → U1_gaa(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
U1_gaa(T30, T31, X50, T33, append7_out_gaa(T31, X50, T33)) → append7_out_gaa(.(T30, T31), X50, .(T30, T33))
U4_ga(T16, T7, append7_out_gaa(T16, X7, T7)) → sublist1_out_ga(T16, T7)
sublist1_in_ga(T40, T7) → U5_ga(T40, T7, append17_in_aga(X88, T40, X89))
append17_in_aga([], T50, T50) → append17_out_aga([], T50, T50)
append17_in_aga(.(X125, X126), T54, .(X125, X127)) → U2_aga(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
U2_aga(X125, X126, T54, X127, append17_out_aga(X126, T54, X127)) → append17_out_aga(.(X125, X126), T54, .(X125, X127))
U5_ga(T40, T7, append17_out_aga(X88, T40, X89)) → sublist1_out_ga(T40, T7)
sublist1_in_ga(T40, .(X154, T65)) → U6_ga(T40, X154, T65, append17_in_aga(T43, T40, T64))
U6_ga(T40, X154, T65, append17_out_aga(T43, T40, T64)) → U7_ga(T40, X154, T65, append27_in_gaa(T64, X155, T65))
append27_in_gaa([], T72, T72) → append27_out_gaa([], T72, T72)
append27_in_gaa(.(T79, T82), X189, .(T79, T83)) → U3_gaa(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
U3_gaa(T79, T82, X189, T83, append27_out_gaa(T82, X189, T83)) → append27_out_gaa(.(T79, T82), X189, .(T79, T83))
U7_ga(T40, X154, T65, append27_out_gaa(T64, X155, T65)) → sublist1_out_ga(T40, .(X154, T65))

The argument filtering Pi contains the following mapping:
sublist1_in_ga(x1, x2)  =  sublist1_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
append7_in_gaa(x1, x2, x3)  =  append7_in_gaa(x1)
[]  =  []
append7_out_gaa(x1, x2, x3)  =  append7_out_gaa(x1)
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
append17_in_aga(x1, x2, x3)  =  append17_in_aga(x2)
append17_out_aga(x1, x2, x3)  =  append17_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x1, x4)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
append27_in_gaa(x1, x2, x3)  =  append27_in_gaa(x1)
append27_out_gaa(x1, x2, x3)  =  append27_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)

(32) 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:

SUBLIST1_IN_GA(T16, T7) → U4_GA(T16, T7, append7_in_gaa(T16, X7, T7))
SUBLIST1_IN_GA(T16, T7) → APPEND7_IN_GAA(T16, X7, T7)
APPEND7_IN_GAA(.(T30, T31), X50, .(T30, T33)) → U1_GAA(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
APPEND7_IN_GAA(.(T30, T31), X50, .(T30, T33)) → APPEND7_IN_GAA(T31, X50, T33)
SUBLIST1_IN_GA(T40, T7) → U5_GA(T40, T7, append17_in_aga(X88, T40, X89))
SUBLIST1_IN_GA(T40, T7) → APPEND17_IN_AGA(X88, T40, X89)
APPEND17_IN_AGA(.(X125, X126), T54, .(X125, X127)) → U2_AGA(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
APPEND17_IN_AGA(.(X125, X126), T54, .(X125, X127)) → APPEND17_IN_AGA(X126, T54, X127)
SUBLIST1_IN_GA(T40, .(X154, T65)) → U6_GA(T40, X154, T65, append17_in_aga(T43, T40, T64))
SUBLIST1_IN_GA(T40, .(X154, T65)) → APPEND17_IN_AGA(T43, T40, T64)
U6_GA(T40, X154, T65, append17_out_aga(T43, T40, T64)) → U7_GA(T40, X154, T65, append27_in_gaa(T64, X155, T65))
U6_GA(T40, X154, T65, append17_out_aga(T43, T40, T64)) → APPEND27_IN_GAA(T64, X155, T65)
APPEND27_IN_GAA(.(T79, T82), X189, .(T79, T83)) → U3_GAA(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
APPEND27_IN_GAA(.(T79, T82), X189, .(T79, T83)) → APPEND27_IN_GAA(T82, X189, T83)

The TRS R consists of the following rules:

sublist1_in_ga(T16, T7) → U4_ga(T16, T7, append7_in_gaa(T16, X7, T7))
append7_in_gaa([], T23, T23) → append7_out_gaa([], T23, T23)
append7_in_gaa(.(T30, T31), X50, .(T30, T33)) → U1_gaa(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
U1_gaa(T30, T31, X50, T33, append7_out_gaa(T31, X50, T33)) → append7_out_gaa(.(T30, T31), X50, .(T30, T33))
U4_ga(T16, T7, append7_out_gaa(T16, X7, T7)) → sublist1_out_ga(T16, T7)
sublist1_in_ga(T40, T7) → U5_ga(T40, T7, append17_in_aga(X88, T40, X89))
append17_in_aga([], T50, T50) → append17_out_aga([], T50, T50)
append17_in_aga(.(X125, X126), T54, .(X125, X127)) → U2_aga(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
U2_aga(X125, X126, T54, X127, append17_out_aga(X126, T54, X127)) → append17_out_aga(.(X125, X126), T54, .(X125, X127))
U5_ga(T40, T7, append17_out_aga(X88, T40, X89)) → sublist1_out_ga(T40, T7)
sublist1_in_ga(T40, .(X154, T65)) → U6_ga(T40, X154, T65, append17_in_aga(T43, T40, T64))
U6_ga(T40, X154, T65, append17_out_aga(T43, T40, T64)) → U7_ga(T40, X154, T65, append27_in_gaa(T64, X155, T65))
append27_in_gaa([], T72, T72) → append27_out_gaa([], T72, T72)
append27_in_gaa(.(T79, T82), X189, .(T79, T83)) → U3_gaa(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
U3_gaa(T79, T82, X189, T83, append27_out_gaa(T82, X189, T83)) → append27_out_gaa(.(T79, T82), X189, .(T79, T83))
U7_ga(T40, X154, T65, append27_out_gaa(T64, X155, T65)) → sublist1_out_ga(T40, .(X154, T65))

The argument filtering Pi contains the following mapping:
sublist1_in_ga(x1, x2)  =  sublist1_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
append7_in_gaa(x1, x2, x3)  =  append7_in_gaa(x1)
[]  =  []
append7_out_gaa(x1, x2, x3)  =  append7_out_gaa(x1)
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
append17_in_aga(x1, x2, x3)  =  append17_in_aga(x2)
append17_out_aga(x1, x2, x3)  =  append17_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x1, x4)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
append27_in_gaa(x1, x2, x3)  =  append27_in_gaa(x1)
append27_out_gaa(x1, x2, x3)  =  append27_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
SUBLIST1_IN_GA(x1, x2)  =  SUBLIST1_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
APPEND7_IN_GAA(x1, x2, x3)  =  APPEND7_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x2, x5)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
APPEND17_IN_AGA(x1, x2, x3)  =  APPEND17_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x1, x4)
U7_GA(x1, x2, x3, x4)  =  U7_GA(x1, x4)
APPEND27_IN_GAA(x1, x2, x3)  =  APPEND27_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x2, x5)

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

(33) Obligation:

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

SUBLIST1_IN_GA(T16, T7) → U4_GA(T16, T7, append7_in_gaa(T16, X7, T7))
SUBLIST1_IN_GA(T16, T7) → APPEND7_IN_GAA(T16, X7, T7)
APPEND7_IN_GAA(.(T30, T31), X50, .(T30, T33)) → U1_GAA(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
APPEND7_IN_GAA(.(T30, T31), X50, .(T30, T33)) → APPEND7_IN_GAA(T31, X50, T33)
SUBLIST1_IN_GA(T40, T7) → U5_GA(T40, T7, append17_in_aga(X88, T40, X89))
SUBLIST1_IN_GA(T40, T7) → APPEND17_IN_AGA(X88, T40, X89)
APPEND17_IN_AGA(.(X125, X126), T54, .(X125, X127)) → U2_AGA(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
APPEND17_IN_AGA(.(X125, X126), T54, .(X125, X127)) → APPEND17_IN_AGA(X126, T54, X127)
SUBLIST1_IN_GA(T40, .(X154, T65)) → U6_GA(T40, X154, T65, append17_in_aga(T43, T40, T64))
SUBLIST1_IN_GA(T40, .(X154, T65)) → APPEND17_IN_AGA(T43, T40, T64)
U6_GA(T40, X154, T65, append17_out_aga(T43, T40, T64)) → U7_GA(T40, X154, T65, append27_in_gaa(T64, X155, T65))
U6_GA(T40, X154, T65, append17_out_aga(T43, T40, T64)) → APPEND27_IN_GAA(T64, X155, T65)
APPEND27_IN_GAA(.(T79, T82), X189, .(T79, T83)) → U3_GAA(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
APPEND27_IN_GAA(.(T79, T82), X189, .(T79, T83)) → APPEND27_IN_GAA(T82, X189, T83)

The TRS R consists of the following rules:

sublist1_in_ga(T16, T7) → U4_ga(T16, T7, append7_in_gaa(T16, X7, T7))
append7_in_gaa([], T23, T23) → append7_out_gaa([], T23, T23)
append7_in_gaa(.(T30, T31), X50, .(T30, T33)) → U1_gaa(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
U1_gaa(T30, T31, X50, T33, append7_out_gaa(T31, X50, T33)) → append7_out_gaa(.(T30, T31), X50, .(T30, T33))
U4_ga(T16, T7, append7_out_gaa(T16, X7, T7)) → sublist1_out_ga(T16, T7)
sublist1_in_ga(T40, T7) → U5_ga(T40, T7, append17_in_aga(X88, T40, X89))
append17_in_aga([], T50, T50) → append17_out_aga([], T50, T50)
append17_in_aga(.(X125, X126), T54, .(X125, X127)) → U2_aga(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
U2_aga(X125, X126, T54, X127, append17_out_aga(X126, T54, X127)) → append17_out_aga(.(X125, X126), T54, .(X125, X127))
U5_ga(T40, T7, append17_out_aga(X88, T40, X89)) → sublist1_out_ga(T40, T7)
sublist1_in_ga(T40, .(X154, T65)) → U6_ga(T40, X154, T65, append17_in_aga(T43, T40, T64))
U6_ga(T40, X154, T65, append17_out_aga(T43, T40, T64)) → U7_ga(T40, X154, T65, append27_in_gaa(T64, X155, T65))
append27_in_gaa([], T72, T72) → append27_out_gaa([], T72, T72)
append27_in_gaa(.(T79, T82), X189, .(T79, T83)) → U3_gaa(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
U3_gaa(T79, T82, X189, T83, append27_out_gaa(T82, X189, T83)) → append27_out_gaa(.(T79, T82), X189, .(T79, T83))
U7_ga(T40, X154, T65, append27_out_gaa(T64, X155, T65)) → sublist1_out_ga(T40, .(X154, T65))

The argument filtering Pi contains the following mapping:
sublist1_in_ga(x1, x2)  =  sublist1_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
append7_in_gaa(x1, x2, x3)  =  append7_in_gaa(x1)
[]  =  []
append7_out_gaa(x1, x2, x3)  =  append7_out_gaa(x1)
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
append17_in_aga(x1, x2, x3)  =  append17_in_aga(x2)
append17_out_aga(x1, x2, x3)  =  append17_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x1, x4)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
append27_in_gaa(x1, x2, x3)  =  append27_in_gaa(x1)
append27_out_gaa(x1, x2, x3)  =  append27_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
SUBLIST1_IN_GA(x1, x2)  =  SUBLIST1_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
APPEND7_IN_GAA(x1, x2, x3)  =  APPEND7_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x2, x5)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
APPEND17_IN_AGA(x1, x2, x3)  =  APPEND17_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x1, x4)
U7_GA(x1, x2, x3, x4)  =  U7_GA(x1, x4)
APPEND27_IN_GAA(x1, x2, x3)  =  APPEND27_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x2, x5)

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

(34) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 11 less nodes.

(35) Complex Obligation (AND)

(36) Obligation:

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

APPEND27_IN_GAA(.(T79, T82), X189, .(T79, T83)) → APPEND27_IN_GAA(T82, X189, T83)

The TRS R consists of the following rules:

sublist1_in_ga(T16, T7) → U4_ga(T16, T7, append7_in_gaa(T16, X7, T7))
append7_in_gaa([], T23, T23) → append7_out_gaa([], T23, T23)
append7_in_gaa(.(T30, T31), X50, .(T30, T33)) → U1_gaa(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
U1_gaa(T30, T31, X50, T33, append7_out_gaa(T31, X50, T33)) → append7_out_gaa(.(T30, T31), X50, .(T30, T33))
U4_ga(T16, T7, append7_out_gaa(T16, X7, T7)) → sublist1_out_ga(T16, T7)
sublist1_in_ga(T40, T7) → U5_ga(T40, T7, append17_in_aga(X88, T40, X89))
append17_in_aga([], T50, T50) → append17_out_aga([], T50, T50)
append17_in_aga(.(X125, X126), T54, .(X125, X127)) → U2_aga(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
U2_aga(X125, X126, T54, X127, append17_out_aga(X126, T54, X127)) → append17_out_aga(.(X125, X126), T54, .(X125, X127))
U5_ga(T40, T7, append17_out_aga(X88, T40, X89)) → sublist1_out_ga(T40, T7)
sublist1_in_ga(T40, .(X154, T65)) → U6_ga(T40, X154, T65, append17_in_aga(T43, T40, T64))
U6_ga(T40, X154, T65, append17_out_aga(T43, T40, T64)) → U7_ga(T40, X154, T65, append27_in_gaa(T64, X155, T65))
append27_in_gaa([], T72, T72) → append27_out_gaa([], T72, T72)
append27_in_gaa(.(T79, T82), X189, .(T79, T83)) → U3_gaa(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
U3_gaa(T79, T82, X189, T83, append27_out_gaa(T82, X189, T83)) → append27_out_gaa(.(T79, T82), X189, .(T79, T83))
U7_ga(T40, X154, T65, append27_out_gaa(T64, X155, T65)) → sublist1_out_ga(T40, .(X154, T65))

The argument filtering Pi contains the following mapping:
sublist1_in_ga(x1, x2)  =  sublist1_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
append7_in_gaa(x1, x2, x3)  =  append7_in_gaa(x1)
[]  =  []
append7_out_gaa(x1, x2, x3)  =  append7_out_gaa(x1)
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
append17_in_aga(x1, x2, x3)  =  append17_in_aga(x2)
append17_out_aga(x1, x2, x3)  =  append17_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x1, x4)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
append27_in_gaa(x1, x2, x3)  =  append27_in_gaa(x1)
append27_out_gaa(x1, x2, x3)  =  append27_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
APPEND27_IN_GAA(x1, x2, x3)  =  APPEND27_IN_GAA(x1)

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:

APPEND27_IN_GAA(.(T79, T82), X189, .(T79, T83)) → APPEND27_IN_GAA(T82, X189, T83)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND27_IN_GAA(x1, x2, x3)  =  APPEND27_IN_GAA(x1)

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:

APPEND27_IN_GAA(.(T82)) → APPEND27_IN_GAA(T82)

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:

  • APPEND27_IN_GAA(.(T82)) → APPEND27_IN_GAA(T82)
    The graph contains the following edges 1 > 1

(42) YES

(43) Obligation:

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

APPEND17_IN_AGA(.(X125, X126), T54, .(X125, X127)) → APPEND17_IN_AGA(X126, T54, X127)

The TRS R consists of the following rules:

sublist1_in_ga(T16, T7) → U4_ga(T16, T7, append7_in_gaa(T16, X7, T7))
append7_in_gaa([], T23, T23) → append7_out_gaa([], T23, T23)
append7_in_gaa(.(T30, T31), X50, .(T30, T33)) → U1_gaa(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
U1_gaa(T30, T31, X50, T33, append7_out_gaa(T31, X50, T33)) → append7_out_gaa(.(T30, T31), X50, .(T30, T33))
U4_ga(T16, T7, append7_out_gaa(T16, X7, T7)) → sublist1_out_ga(T16, T7)
sublist1_in_ga(T40, T7) → U5_ga(T40, T7, append17_in_aga(X88, T40, X89))
append17_in_aga([], T50, T50) → append17_out_aga([], T50, T50)
append17_in_aga(.(X125, X126), T54, .(X125, X127)) → U2_aga(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
U2_aga(X125, X126, T54, X127, append17_out_aga(X126, T54, X127)) → append17_out_aga(.(X125, X126), T54, .(X125, X127))
U5_ga(T40, T7, append17_out_aga(X88, T40, X89)) → sublist1_out_ga(T40, T7)
sublist1_in_ga(T40, .(X154, T65)) → U6_ga(T40, X154, T65, append17_in_aga(T43, T40, T64))
U6_ga(T40, X154, T65, append17_out_aga(T43, T40, T64)) → U7_ga(T40, X154, T65, append27_in_gaa(T64, X155, T65))
append27_in_gaa([], T72, T72) → append27_out_gaa([], T72, T72)
append27_in_gaa(.(T79, T82), X189, .(T79, T83)) → U3_gaa(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
U3_gaa(T79, T82, X189, T83, append27_out_gaa(T82, X189, T83)) → append27_out_gaa(.(T79, T82), X189, .(T79, T83))
U7_ga(T40, X154, T65, append27_out_gaa(T64, X155, T65)) → sublist1_out_ga(T40, .(X154, T65))

The argument filtering Pi contains the following mapping:
sublist1_in_ga(x1, x2)  =  sublist1_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
append7_in_gaa(x1, x2, x3)  =  append7_in_gaa(x1)
[]  =  []
append7_out_gaa(x1, x2, x3)  =  append7_out_gaa(x1)
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
append17_in_aga(x1, x2, x3)  =  append17_in_aga(x2)
append17_out_aga(x1, x2, x3)  =  append17_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x1, x4)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
append27_in_gaa(x1, x2, x3)  =  append27_in_gaa(x1)
append27_out_gaa(x1, x2, x3)  =  append27_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
APPEND17_IN_AGA(x1, x2, x3)  =  APPEND17_IN_AGA(x2)

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

(44) UsableRulesProof (EQUIVALENT transformation)

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

(45) Obligation:

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

APPEND17_IN_AGA(.(X125, X126), T54, .(X125, X127)) → APPEND17_IN_AGA(X126, T54, X127)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND17_IN_AGA(x1, x2, x3)  =  APPEND17_IN_AGA(x2)

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

(46) PiDPToQDPProof (SOUND transformation)

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

(47) Obligation:

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

APPEND17_IN_AGA(T54) → APPEND17_IN_AGA(T54)

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

(48) 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 = APPEND17_IN_AGA(T54) evaluates to t =APPEND17_IN_AGA(T54)

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND17_IN_AGA(T54) to APPEND17_IN_AGA(T54).



(49) NO

(50) Obligation:

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

APPEND7_IN_GAA(.(T30, T31), X50, .(T30, T33)) → APPEND7_IN_GAA(T31, X50, T33)

The TRS R consists of the following rules:

sublist1_in_ga(T16, T7) → U4_ga(T16, T7, append7_in_gaa(T16, X7, T7))
append7_in_gaa([], T23, T23) → append7_out_gaa([], T23, T23)
append7_in_gaa(.(T30, T31), X50, .(T30, T33)) → U1_gaa(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
U1_gaa(T30, T31, X50, T33, append7_out_gaa(T31, X50, T33)) → append7_out_gaa(.(T30, T31), X50, .(T30, T33))
U4_ga(T16, T7, append7_out_gaa(T16, X7, T7)) → sublist1_out_ga(T16, T7)
sublist1_in_ga(T40, T7) → U5_ga(T40, T7, append17_in_aga(X88, T40, X89))
append17_in_aga([], T50, T50) → append17_out_aga([], T50, T50)
append17_in_aga(.(X125, X126), T54, .(X125, X127)) → U2_aga(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
U2_aga(X125, X126, T54, X127, append17_out_aga(X126, T54, X127)) → append17_out_aga(.(X125, X126), T54, .(X125, X127))
U5_ga(T40, T7, append17_out_aga(X88, T40, X89)) → sublist1_out_ga(T40, T7)
sublist1_in_ga(T40, .(X154, T65)) → U6_ga(T40, X154, T65, append17_in_aga(T43, T40, T64))
U6_ga(T40, X154, T65, append17_out_aga(T43, T40, T64)) → U7_ga(T40, X154, T65, append27_in_gaa(T64, X155, T65))
append27_in_gaa([], T72, T72) → append27_out_gaa([], T72, T72)
append27_in_gaa(.(T79, T82), X189, .(T79, T83)) → U3_gaa(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
U3_gaa(T79, T82, X189, T83, append27_out_gaa(T82, X189, T83)) → append27_out_gaa(.(T79, T82), X189, .(T79, T83))
U7_ga(T40, X154, T65, append27_out_gaa(T64, X155, T65)) → sublist1_out_ga(T40, .(X154, T65))

The argument filtering Pi contains the following mapping:
sublist1_in_ga(x1, x2)  =  sublist1_in_ga(x1)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
append7_in_gaa(x1, x2, x3)  =  append7_in_gaa(x1)
[]  =  []
append7_out_gaa(x1, x2, x3)  =  append7_out_gaa(x1)
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
append17_in_aga(x1, x2, x3)  =  append17_in_aga(x2)
append17_out_aga(x1, x2, x3)  =  append17_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x1, x4)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
append27_in_gaa(x1, x2, x3)  =  append27_in_gaa(x1)
append27_out_gaa(x1, x2, x3)  =  append27_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
APPEND7_IN_GAA(x1, x2, x3)  =  APPEND7_IN_GAA(x1)

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

(51) UsableRulesProof (EQUIVALENT transformation)

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

(52) Obligation:

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

APPEND7_IN_GAA(.(T30, T31), X50, .(T30, T33)) → APPEND7_IN_GAA(T31, X50, T33)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND7_IN_GAA(x1, x2, x3)  =  APPEND7_IN_GAA(x1)

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

(53) PiDPToQDPProof (SOUND transformation)

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

(54) Obligation:

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

APPEND7_IN_GAA(.(T31)) → APPEND7_IN_GAA(T31)

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

(55) 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_GAA(.(T31)) → APPEND7_IN_GAA(T31)
    The graph contains the following edges 1 > 1

(56) YES