Left Termination of the query pattern sublist_in_2(g, g) w.r.t. the given Prolog program could not be shown:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 TRUE
      ↳14 PiDP
        ↳15 UsableRulesProof (⇔)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇐)
        ↳18 QDP
        ↳19 NonTerminationProof (⇔)
        ↳20 FALSE
  ↳21 PrologToPiTRSProof (⇐)
    ↳22 PiTRS
    ↳23 DependencyPairsProof (⇔)
    ↳24 PiDP
    ↳25 DependencyGraphProof (⇔)
    ↳26 AND
      ↳27 PiDP
        ↳28 UsableRulesProof (⇔)
        ↳29 PiDP
        ↳30 PiDPToQDPProof (⇐)
        ↳31 QDP
        ↳32 QDPSizeChangeProof (⇔)
        ↳33 TRUE
      ↳34 PiDP
        ↳35 UsableRulesProof (⇔)
        ↳36 PiDP
        ↳37 PiDPToQDPProof (⇐)
        ↳38 QDP
        ↳39 NonTerminationProof (⇔)
        ↳40 FALSE

(0) Obligation:

Clauses:

append1([], Ys, Ys).
append1(.(X, Xs), Ys, .(X, Zs)) :- append1(Xs, Ys, Zs).
append2([], Ys, Ys).
append2(.(X, Xs), Ys, .(X, Zs)) :- append2(Xs, Ys, Zs).
sublist(X, Y) :- ','(append1(U, X, V), append2(V, W, Y)).

Queries:

sublist(g,g).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
sublist_in: (b,b)
append1_in: (f,b,f)
append2_in: (b,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

sublist_in_gg(X, Y) → U3_gg(X, Y, append1_in_aga(U, X, V))
append1_in_aga([], Ys, Ys) → append1_out_aga([], Ys, Ys)
append1_in_aga(.(X, Xs), Ys, .(X, Zs)) → U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) → append1_out_aga(.(X, Xs), Ys, .(X, Zs))
U3_gg(X, Y, append1_out_aga(U, X, V)) → U4_gg(X, Y, append2_in_gag(V, W, Y))
append2_in_gag([], Ys, Ys) → append2_out_gag([], Ys, Ys)
append2_in_gag(.(X, Xs), Ys, .(X, Zs)) → U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) → append2_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_gg(X, Y, append2_out_gag(V, W, Y)) → sublist_out_gg(X, Y)

The argument filtering Pi contains the following mapping:
sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x2, x3)
append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
.(x1, x2)  =  .(x2)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
[]  =  []
append2_out_gag(x1, x2, x3)  =  append2_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
sublist_out_gg(x1, x2)  =  sublist_out_gg

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

sublist_in_gg(X, Y) → U3_gg(X, Y, append1_in_aga(U, X, V))
append1_in_aga([], Ys, Ys) → append1_out_aga([], Ys, Ys)
append1_in_aga(.(X, Xs), Ys, .(X, Zs)) → U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) → append1_out_aga(.(X, Xs), Ys, .(X, Zs))
U3_gg(X, Y, append1_out_aga(U, X, V)) → U4_gg(X, Y, append2_in_gag(V, W, Y))
append2_in_gag([], Ys, Ys) → append2_out_gag([], Ys, Ys)
append2_in_gag(.(X, Xs), Ys, .(X, Zs)) → U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) → append2_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_gg(X, Y, append2_out_gag(V, W, Y)) → sublist_out_gg(X, Y)

The argument filtering Pi contains the following mapping:
sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x2, x3)
append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
.(x1, x2)  =  .(x2)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
[]  =  []
append2_out_gag(x1, x2, x3)  =  append2_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
sublist_out_gg(x1, x2)  =  sublist_out_gg

(3) 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:

SUBLIST_IN_GG(X, Y) → U3_GG(X, Y, append1_in_aga(U, X, V))
SUBLIST_IN_GG(X, Y) → APPEND1_IN_AGA(U, X, V)
APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → U1_AGA(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_AGA(Xs, Ys, Zs)
U3_GG(X, Y, append1_out_aga(U, X, V)) → U4_GG(X, Y, append2_in_gag(V, W, Y))
U3_GG(X, Y, append1_out_aga(U, X, V)) → APPEND2_IN_GAG(V, W, Y)
APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → U2_GAG(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_in_gg(X, Y) → U3_gg(X, Y, append1_in_aga(U, X, V))
append1_in_aga([], Ys, Ys) → append1_out_aga([], Ys, Ys)
append1_in_aga(.(X, Xs), Ys, .(X, Zs)) → U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) → append1_out_aga(.(X, Xs), Ys, .(X, Zs))
U3_gg(X, Y, append1_out_aga(U, X, V)) → U4_gg(X, Y, append2_in_gag(V, W, Y))
append2_in_gag([], Ys, Ys) → append2_out_gag([], Ys, Ys)
append2_in_gag(.(X, Xs), Ys, .(X, Zs)) → U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) → append2_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_gg(X, Y, append2_out_gag(V, W, Y)) → sublist_out_gg(X, Y)

The argument filtering Pi contains the following mapping:
sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x2, x3)
append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
.(x1, x2)  =  .(x2)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
[]  =  []
append2_out_gag(x1, x2, x3)  =  append2_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
sublist_out_gg(x1, x2)  =  sublist_out_gg
SUBLIST_IN_GG(x1, x2)  =  SUBLIST_IN_GG(x1, x2)
U3_GG(x1, x2, x3)  =  U3_GG(x2, x3)
APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x5)
U4_GG(x1, x2, x3)  =  U4_GG(x3)
APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x5)

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

(4) Obligation:

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

SUBLIST_IN_GG(X, Y) → U3_GG(X, Y, append1_in_aga(U, X, V))
SUBLIST_IN_GG(X, Y) → APPEND1_IN_AGA(U, X, V)
APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → U1_AGA(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_AGA(Xs, Ys, Zs)
U3_GG(X, Y, append1_out_aga(U, X, V)) → U4_GG(X, Y, append2_in_gag(V, W, Y))
U3_GG(X, Y, append1_out_aga(U, X, V)) → APPEND2_IN_GAG(V, W, Y)
APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → U2_GAG(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_in_gg(X, Y) → U3_gg(X, Y, append1_in_aga(U, X, V))
append1_in_aga([], Ys, Ys) → append1_out_aga([], Ys, Ys)
append1_in_aga(.(X, Xs), Ys, .(X, Zs)) → U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) → append1_out_aga(.(X, Xs), Ys, .(X, Zs))
U3_gg(X, Y, append1_out_aga(U, X, V)) → U4_gg(X, Y, append2_in_gag(V, W, Y))
append2_in_gag([], Ys, Ys) → append2_out_gag([], Ys, Ys)
append2_in_gag(.(X, Xs), Ys, .(X, Zs)) → U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) → append2_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_gg(X, Y, append2_out_gag(V, W, Y)) → sublist_out_gg(X, Y)

The argument filtering Pi contains the following mapping:
sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x2, x3)
append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
.(x1, x2)  =  .(x2)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
[]  =  []
append2_out_gag(x1, x2, x3)  =  append2_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
sublist_out_gg(x1, x2)  =  sublist_out_gg
SUBLIST_IN_GG(x1, x2)  =  SUBLIST_IN_GG(x1, x2)
U3_GG(x1, x2, x3)  =  U3_GG(x2, x3)
APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x5)
U4_GG(x1, x2, x3)  =  U4_GG(x3)
APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_in_gg(X, Y) → U3_gg(X, Y, append1_in_aga(U, X, V))
append1_in_aga([], Ys, Ys) → append1_out_aga([], Ys, Ys)
append1_in_aga(.(X, Xs), Ys, .(X, Zs)) → U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) → append1_out_aga(.(X, Xs), Ys, .(X, Zs))
U3_gg(X, Y, append1_out_aga(U, X, V)) → U4_gg(X, Y, append2_in_gag(V, W, Y))
append2_in_gag([], Ys, Ys) → append2_out_gag([], Ys, Ys)
append2_in_gag(.(X, Xs), Ys, .(X, Zs)) → U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) → append2_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_gg(X, Y, append2_out_gag(V, W, Y)) → sublist_out_gg(X, Y)

The argument filtering Pi contains the following mapping:
sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x2, x3)
append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
.(x1, x2)  =  .(x2)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
[]  =  []
append2_out_gag(x1, x2, x3)  =  append2_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
sublist_out_gg(x1, x2)  =  sublist_out_gg
APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_GAG(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

APPEND2_IN_GAG(.(Xs), .(Zs)) → APPEND2_IN_GAG(Xs, Zs)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • APPEND2_IN_GAG(.(Xs), .(Zs)) → APPEND2_IN_GAG(Xs, Zs)
    The graph contains the following edges 1 > 1, 2 > 2

(13) TRUE

(14) Obligation:

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

APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_AGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_in_gg(X, Y) → U3_gg(X, Y, append1_in_aga(U, X, V))
append1_in_aga([], Ys, Ys) → append1_out_aga([], Ys, Ys)
append1_in_aga(.(X, Xs), Ys, .(X, Zs)) → U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) → append1_out_aga(.(X, Xs), Ys, .(X, Zs))
U3_gg(X, Y, append1_out_aga(U, X, V)) → U4_gg(X, Y, append2_in_gag(V, W, Y))
append2_in_gag([], Ys, Ys) → append2_out_gag([], Ys, Ys)
append2_in_gag(.(X, Xs), Ys, .(X, Zs)) → U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) → append2_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_gg(X, Y, append2_out_gag(V, W, Y)) → sublist_out_gg(X, Y)

The argument filtering Pi contains the following mapping:
sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x2, x3)
append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
.(x1, x2)  =  .(x2)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
[]  =  []
append2_out_gag(x1, x2, x3)  =  append2_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
sublist_out_gg(x1, x2)  =  sublist_out_gg
APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_AGA(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

APPEND1_IN_AGA(Ys) → APPEND1_IN_AGA(Ys)

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

(19) 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 = APPEND1_IN_AGA(Ys) evaluates to t =APPEND1_IN_AGA(Ys)

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 APPEND1_IN_AGA(Ys) to APPEND1_IN_AGA(Ys).



(20) FALSE

(21) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
sublist_in: (b,b)
append1_in: (f,b,f)
append2_in: (b,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

sublist_in_gg(X, Y) → U3_gg(X, Y, append1_in_aga(U, X, V))
append1_in_aga([], Ys, Ys) → append1_out_aga([], Ys, Ys)
append1_in_aga(.(X, Xs), Ys, .(X, Zs)) → U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) → append1_out_aga(.(X, Xs), Ys, .(X, Zs))
U3_gg(X, Y, append1_out_aga(U, X, V)) → U4_gg(X, Y, append2_in_gag(V, W, Y))
append2_in_gag([], Ys, Ys) → append2_out_gag([], Ys, Ys)
append2_in_gag(.(X, Xs), Ys, .(X, Zs)) → U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) → append2_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_gg(X, Y, append2_out_gag(V, W, Y)) → sublist_out_gg(X, Y)

The argument filtering Pi contains the following mapping:
sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x1, x2, x3)
append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
.(x1, x2)  =  .(x2)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
[]  =  []
append2_out_gag(x1, x2, x3)  =  append2_out_gag(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
sublist_out_gg(x1, x2)  =  sublist_out_gg(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(22) Obligation:

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

sublist_in_gg(X, Y) → U3_gg(X, Y, append1_in_aga(U, X, V))
append1_in_aga([], Ys, Ys) → append1_out_aga([], Ys, Ys)
append1_in_aga(.(X, Xs), Ys, .(X, Zs)) → U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) → append1_out_aga(.(X, Xs), Ys, .(X, Zs))
U3_gg(X, Y, append1_out_aga(U, X, V)) → U4_gg(X, Y, append2_in_gag(V, W, Y))
append2_in_gag([], Ys, Ys) → append2_out_gag([], Ys, Ys)
append2_in_gag(.(X, Xs), Ys, .(X, Zs)) → U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) → append2_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_gg(X, Y, append2_out_gag(V, W, Y)) → sublist_out_gg(X, Y)

The argument filtering Pi contains the following mapping:
sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x1, x2, x3)
append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
.(x1, x2)  =  .(x2)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
[]  =  []
append2_out_gag(x1, x2, x3)  =  append2_out_gag(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
sublist_out_gg(x1, x2)  =  sublist_out_gg(x1, x2)

(23) 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:

SUBLIST_IN_GG(X, Y) → U3_GG(X, Y, append1_in_aga(U, X, V))
SUBLIST_IN_GG(X, Y) → APPEND1_IN_AGA(U, X, V)
APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → U1_AGA(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_AGA(Xs, Ys, Zs)
U3_GG(X, Y, append1_out_aga(U, X, V)) → U4_GG(X, Y, append2_in_gag(V, W, Y))
U3_GG(X, Y, append1_out_aga(U, X, V)) → APPEND2_IN_GAG(V, W, Y)
APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → U2_GAG(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_in_gg(X, Y) → U3_gg(X, Y, append1_in_aga(U, X, V))
append1_in_aga([], Ys, Ys) → append1_out_aga([], Ys, Ys)
append1_in_aga(.(X, Xs), Ys, .(X, Zs)) → U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) → append1_out_aga(.(X, Xs), Ys, .(X, Zs))
U3_gg(X, Y, append1_out_aga(U, X, V)) → U4_gg(X, Y, append2_in_gag(V, W, Y))
append2_in_gag([], Ys, Ys) → append2_out_gag([], Ys, Ys)
append2_in_gag(.(X, Xs), Ys, .(X, Zs)) → U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) → append2_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_gg(X, Y, append2_out_gag(V, W, Y)) → sublist_out_gg(X, Y)

The argument filtering Pi contains the following mapping:
sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x1, x2, x3)
append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
.(x1, x2)  =  .(x2)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
[]  =  []
append2_out_gag(x1, x2, x3)  =  append2_out_gag(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
sublist_out_gg(x1, x2)  =  sublist_out_gg(x1, x2)
SUBLIST_IN_GG(x1, x2)  =  SUBLIST_IN_GG(x1, x2)
U3_GG(x1, x2, x3)  =  U3_GG(x1, x2, x3)
APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, x5)
U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)
APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x2, x4, x5)

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

(24) Obligation:

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

SUBLIST_IN_GG(X, Y) → U3_GG(X, Y, append1_in_aga(U, X, V))
SUBLIST_IN_GG(X, Y) → APPEND1_IN_AGA(U, X, V)
APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → U1_AGA(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_AGA(Xs, Ys, Zs)
U3_GG(X, Y, append1_out_aga(U, X, V)) → U4_GG(X, Y, append2_in_gag(V, W, Y))
U3_GG(X, Y, append1_out_aga(U, X, V)) → APPEND2_IN_GAG(V, W, Y)
APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → U2_GAG(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_in_gg(X, Y) → U3_gg(X, Y, append1_in_aga(U, X, V))
append1_in_aga([], Ys, Ys) → append1_out_aga([], Ys, Ys)
append1_in_aga(.(X, Xs), Ys, .(X, Zs)) → U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) → append1_out_aga(.(X, Xs), Ys, .(X, Zs))
U3_gg(X, Y, append1_out_aga(U, X, V)) → U4_gg(X, Y, append2_in_gag(V, W, Y))
append2_in_gag([], Ys, Ys) → append2_out_gag([], Ys, Ys)
append2_in_gag(.(X, Xs), Ys, .(X, Zs)) → U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) → append2_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_gg(X, Y, append2_out_gag(V, W, Y)) → sublist_out_gg(X, Y)

The argument filtering Pi contains the following mapping:
sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x1, x2, x3)
append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
.(x1, x2)  =  .(x2)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
[]  =  []
append2_out_gag(x1, x2, x3)  =  append2_out_gag(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
sublist_out_gg(x1, x2)  =  sublist_out_gg(x1, x2)
SUBLIST_IN_GG(x1, x2)  =  SUBLIST_IN_GG(x1, x2)
U3_GG(x1, x2, x3)  =  U3_GG(x1, x2, x3)
APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, x5)
U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)
APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x2, x4, x5)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(26) Complex Obligation (AND)

(27) Obligation:

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

APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_in_gg(X, Y) → U3_gg(X, Y, append1_in_aga(U, X, V))
append1_in_aga([], Ys, Ys) → append1_out_aga([], Ys, Ys)
append1_in_aga(.(X, Xs), Ys, .(X, Zs)) → U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) → append1_out_aga(.(X, Xs), Ys, .(X, Zs))
U3_gg(X, Y, append1_out_aga(U, X, V)) → U4_gg(X, Y, append2_in_gag(V, W, Y))
append2_in_gag([], Ys, Ys) → append2_out_gag([], Ys, Ys)
append2_in_gag(.(X, Xs), Ys, .(X, Zs)) → U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) → append2_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_gg(X, Y, append2_out_gag(V, W, Y)) → sublist_out_gg(X, Y)

The argument filtering Pi contains the following mapping:
sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x1, x2, x3)
append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
.(x1, x2)  =  .(x2)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
[]  =  []
append2_out_gag(x1, x2, x3)  =  append2_out_gag(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
sublist_out_gg(x1, x2)  =  sublist_out_gg(x1, x2)
APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)

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

(28) UsableRulesProof (EQUIVALENT transformation)

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

(29) Obligation:

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

APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_GAG(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)

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

(30) PiDPToQDPProof (SOUND transformation)

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

(31) Obligation:

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

APPEND2_IN_GAG(.(Xs), .(Zs)) → APPEND2_IN_GAG(Xs, Zs)

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

(32) 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:

  • APPEND2_IN_GAG(.(Xs), .(Zs)) → APPEND2_IN_GAG(Xs, Zs)
    The graph contains the following edges 1 > 1, 2 > 2

(33) TRUE

(34) Obligation:

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

APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_AGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_in_gg(X, Y) → U3_gg(X, Y, append1_in_aga(U, X, V))
append1_in_aga([], Ys, Ys) → append1_out_aga([], Ys, Ys)
append1_in_aga(.(X, Xs), Ys, .(X, Zs)) → U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) → append1_out_aga(.(X, Xs), Ys, .(X, Zs))
U3_gg(X, Y, append1_out_aga(U, X, V)) → U4_gg(X, Y, append2_in_gag(V, W, Y))
append2_in_gag([], Ys, Ys) → append2_out_gag([], Ys, Ys)
append2_in_gag(.(X, Xs), Ys, .(X, Zs)) → U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) → append2_out_gag(.(X, Xs), Ys, .(X, Zs))
U4_gg(X, Y, append2_out_gag(V, W, Y)) → sublist_out_gg(X, Y)

The argument filtering Pi contains the following mapping:
sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x1, x2, x3)
append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
.(x1, x2)  =  .(x2)
U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
[]  =  []
append2_out_gag(x1, x2, x3)  =  append2_out_gag(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
sublist_out_gg(x1, x2)  =  sublist_out_gg(x1, x2)
APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)

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

(35) UsableRulesProof (EQUIVALENT transformation)

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

(36) Obligation:

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

APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_AGA(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)

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

(37) PiDPToQDPProof (SOUND transformation)

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

(38) Obligation:

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

APPEND1_IN_AGA(Ys) → APPEND1_IN_AGA(Ys)

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

(39) 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 = APPEND1_IN_AGA(Ys) evaluates to t =APPEND1_IN_AGA(Ys)

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 APPEND1_IN_AGA(Ys) to APPEND1_IN_AGA(Ys).



(40) FALSE