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 PrologToPiTRSProof (⇐)
        ↳24 PiTRS
        ↳25 DependencyPairsProof (⇔)
        ↳26 PiDP
        ↳27 DependencyGraphProof (⇔)
        ↳28 AND
            ↳29 PiDP
                ↳30 UsableRulesProof (⇔)
                ↳31 PiDP
                ↳32 PiDPToQDPProof (⇐)
                ↳33 QDP
                ↳34 QDPSizeChangeProof (⇔)
                ↳35 YES
            ↳36 PiDP
                ↳37 UsableRulesProof (⇔)
                ↳38 PiDP
                ↳39 PiDPToQDPProof (⇐)
                ↳40 QDP
                ↳41 NonTerminationProof (⇔)
                ↳42 NO

(0) Obligation:

Clauses:

sublist(Xs, Ys) :- ','(app(X1, Zs, Ys), app(Xs, X2, Zs)).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Queries:

sublist(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

app7([], T26, T26).
app7(.(T33, T34), X57, .(T33, T36)) :- app7(T34, X57, T36).
p3([], T19, T19, T5, X9) :- app7(T5, X9, T19).
p3(.(X84, X85), X86, .(X84, T42), T5, X9) :- p3(X85, X86, T42, T5, X9).
sublist1(T5, T7) :- p3(X7, X8, T7, T5, X9).

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)
p3_in: (f,f,f,b,f)
app7_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(T5, T7) → U4_ga(T5, T7, p3_in_aaaga(X7, X8, T7, T5, X9))
p3_in_aaaga([], T19, T19, T5, X9) → U2_aaaga(T19, T5, X9, app7_in_gaa(T5, X9, T19))
app7_in_gaa([], T26, T26) → app7_out_gaa([], T26, T26)
app7_in_gaa(.(T33, T34), X57, .(T33, T36)) → U1_gaa(T33, T34, X57, T36, app7_in_gaa(T34, X57, T36))
U1_gaa(T33, T34, X57, T36, app7_out_gaa(T34, X57, T36)) → app7_out_gaa(.(T33, T34), X57, .(T33, T36))
U2_aaaga(T19, T5, X9, app7_out_gaa(T5, X9, T19)) → p3_out_aaaga([], T19, T19, T5, X9)
p3_in_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9) → U3_aaaga(X84, X85, X86, T42, T5, X9, p3_in_aaaga(X85, X86, T42, T5, X9))
U3_aaaga(X84, X85, X86, T42, T5, X9, p3_out_aaaga(X85, X86, T42, T5, X9)) → p3_out_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9)
U4_ga(T5, T7, p3_out_aaaga(X7, X8, T7, T5, X9)) → sublist1_out_ga(T5, T7)

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)
p3_in_aaaga(x1, x2, x3, x4, x5)  =  p3_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4)  =  U2_aaaga(x4)
app7_in_gaa(x1, x2, x3)  =  app7_in_gaa(x1)
[]  =  []
app7_out_gaa(x1, x2, x3)  =  app7_out_gaa
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
p3_out_aaaga(x1, x2, x3, x4, x5)  =  p3_out_aaaga(x1)
U3_aaaga(x1, x2, x3, x4, x5, x6, x7)  =  U3_aaaga(x7)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga

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(T5, T7) → U4_ga(T5, T7, p3_in_aaaga(X7, X8, T7, T5, X9))
p3_in_aaaga([], T19, T19, T5, X9) → U2_aaaga(T19, T5, X9, app7_in_gaa(T5, X9, T19))
app7_in_gaa([], T26, T26) → app7_out_gaa([], T26, T26)
app7_in_gaa(.(T33, T34), X57, .(T33, T36)) → U1_gaa(T33, T34, X57, T36, app7_in_gaa(T34, X57, T36))
U1_gaa(T33, T34, X57, T36, app7_out_gaa(T34, X57, T36)) → app7_out_gaa(.(T33, T34), X57, .(T33, T36))
U2_aaaga(T19, T5, X9, app7_out_gaa(T5, X9, T19)) → p3_out_aaaga([], T19, T19, T5, X9)
p3_in_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9) → U3_aaaga(X84, X85, X86, T42, T5, X9, p3_in_aaaga(X85, X86, T42, T5, X9))
U3_aaaga(X84, X85, X86, T42, T5, X9, p3_out_aaaga(X85, X86, T42, T5, X9)) → p3_out_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9)
U4_ga(T5, T7, p3_out_aaaga(X7, X8, T7, T5, X9)) → sublist1_out_ga(T5, T7)

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)
p3_in_aaaga(x1, x2, x3, x4, x5)  =  p3_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4)  =  U2_aaaga(x4)
app7_in_gaa(x1, x2, x3)  =  app7_in_gaa(x1)
[]  =  []
app7_out_gaa(x1, x2, x3)  =  app7_out_gaa
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
p3_out_aaaga(x1, x2, x3, x4, x5)  =  p3_out_aaaga(x1)
U3_aaaga(x1, x2, x3, x4, x5, x6, x7)  =  U3_aaaga(x7)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga

(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(T5, T7) → U4_GA(T5, T7, p3_in_aaaga(X7, X8, T7, T5, X9))
SUBLIST1_IN_GA(T5, T7) → P3_IN_AAAGA(X7, X8, T7, T5, X9)
P3_IN_AAAGA([], T19, T19, T5, X9) → U2_AAAGA(T19, T5, X9, app7_in_gaa(T5, X9, T19))
P3_IN_AAAGA([], T19, T19, T5, X9) → APP7_IN_GAA(T5, X9, T19)
APP7_IN_GAA(.(T33, T34), X57, .(T33, T36)) → U1_GAA(T33, T34, X57, T36, app7_in_gaa(T34, X57, T36))
APP7_IN_GAA(.(T33, T34), X57, .(T33, T36)) → APP7_IN_GAA(T34, X57, T36)
P3_IN_AAAGA(.(X84, X85), X86, .(X84, T42), T5, X9) → U3_AAAGA(X84, X85, X86, T42, T5, X9, p3_in_aaaga(X85, X86, T42, T5, X9))
P3_IN_AAAGA(.(X84, X85), X86, .(X84, T42), T5, X9) → P3_IN_AAAGA(X85, X86, T42, T5, X9)

The TRS R consists of the following rules:

sublist1_in_ga(T5, T7) → U4_ga(T5, T7, p3_in_aaaga(X7, X8, T7, T5, X9))
p3_in_aaaga([], T19, T19, T5, X9) → U2_aaaga(T19, T5, X9, app7_in_gaa(T5, X9, T19))
app7_in_gaa([], T26, T26) → app7_out_gaa([], T26, T26)
app7_in_gaa(.(T33, T34), X57, .(T33, T36)) → U1_gaa(T33, T34, X57, T36, app7_in_gaa(T34, X57, T36))
U1_gaa(T33, T34, X57, T36, app7_out_gaa(T34, X57, T36)) → app7_out_gaa(.(T33, T34), X57, .(T33, T36))
U2_aaaga(T19, T5, X9, app7_out_gaa(T5, X9, T19)) → p3_out_aaaga([], T19, T19, T5, X9)
p3_in_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9) → U3_aaaga(X84, X85, X86, T42, T5, X9, p3_in_aaaga(X85, X86, T42, T5, X9))
U3_aaaga(X84, X85, X86, T42, T5, X9, p3_out_aaaga(X85, X86, T42, T5, X9)) → p3_out_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9)
U4_ga(T5, T7, p3_out_aaaga(X7, X8, T7, T5, X9)) → sublist1_out_ga(T5, T7)

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)
p3_in_aaaga(x1, x2, x3, x4, x5)  =  p3_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4)  =  U2_aaaga(x4)
app7_in_gaa(x1, x2, x3)  =  app7_in_gaa(x1)
[]  =  []
app7_out_gaa(x1, x2, x3)  =  app7_out_gaa
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
p3_out_aaaga(x1, x2, x3, x4, x5)  =  p3_out_aaaga(x1)
U3_aaaga(x1, x2, x3, x4, x5, x6, x7)  =  U3_aaaga(x7)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga
SUBLIST1_IN_GA(x1, x2)  =  SUBLIST1_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
P3_IN_AAAGA(x1, x2, x3, x4, x5)  =  P3_IN_AAAGA(x4)
U2_AAAGA(x1, x2, x3, x4)  =  U2_AAAGA(x4)
APP7_IN_GAA(x1, x2, x3)  =  APP7_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x5)
U3_AAAGA(x1, x2, x3, x4, x5, x6, x7)  =  U3_AAAGA(x7)

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(T5, T7) → U4_GA(T5, T7, p3_in_aaaga(X7, X8, T7, T5, X9))
SUBLIST1_IN_GA(T5, T7) → P3_IN_AAAGA(X7, X8, T7, T5, X9)
P3_IN_AAAGA([], T19, T19, T5, X9) → U2_AAAGA(T19, T5, X9, app7_in_gaa(T5, X9, T19))
P3_IN_AAAGA([], T19, T19, T5, X9) → APP7_IN_GAA(T5, X9, T19)
APP7_IN_GAA(.(T33, T34), X57, .(T33, T36)) → U1_GAA(T33, T34, X57, T36, app7_in_gaa(T34, X57, T36))
APP7_IN_GAA(.(T33, T34), X57, .(T33, T36)) → APP7_IN_GAA(T34, X57, T36)
P3_IN_AAAGA(.(X84, X85), X86, .(X84, T42), T5, X9) → U3_AAAGA(X84, X85, X86, T42, T5, X9, p3_in_aaaga(X85, X86, T42, T5, X9))
P3_IN_AAAGA(.(X84, X85), X86, .(X84, T42), T5, X9) → P3_IN_AAAGA(X85, X86, T42, T5, X9)

The TRS R consists of the following rules:

sublist1_in_ga(T5, T7) → U4_ga(T5, T7, p3_in_aaaga(X7, X8, T7, T5, X9))
p3_in_aaaga([], T19, T19, T5, X9) → U2_aaaga(T19, T5, X9, app7_in_gaa(T5, X9, T19))
app7_in_gaa([], T26, T26) → app7_out_gaa([], T26, T26)
app7_in_gaa(.(T33, T34), X57, .(T33, T36)) → U1_gaa(T33, T34, X57, T36, app7_in_gaa(T34, X57, T36))
U1_gaa(T33, T34, X57, T36, app7_out_gaa(T34, X57, T36)) → app7_out_gaa(.(T33, T34), X57, .(T33, T36))
U2_aaaga(T19, T5, X9, app7_out_gaa(T5, X9, T19)) → p3_out_aaaga([], T19, T19, T5, X9)
p3_in_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9) → U3_aaaga(X84, X85, X86, T42, T5, X9, p3_in_aaaga(X85, X86, T42, T5, X9))
U3_aaaga(X84, X85, X86, T42, T5, X9, p3_out_aaaga(X85, X86, T42, T5, X9)) → p3_out_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9)
U4_ga(T5, T7, p3_out_aaaga(X7, X8, T7, T5, X9)) → sublist1_out_ga(T5, T7)

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)
p3_in_aaaga(x1, x2, x3, x4, x5)  =  p3_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4)  =  U2_aaaga(x4)
app7_in_gaa(x1, x2, x3)  =  app7_in_gaa(x1)
[]  =  []
app7_out_gaa(x1, x2, x3)  =  app7_out_gaa
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
p3_out_aaaga(x1, x2, x3, x4, x5)  =  p3_out_aaaga(x1)
U3_aaaga(x1, x2, x3, x4, x5, x6, x7)  =  U3_aaaga(x7)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga
SUBLIST1_IN_GA(x1, x2)  =  SUBLIST1_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
P3_IN_AAAGA(x1, x2, x3, x4, x5)  =  P3_IN_AAAGA(x4)
U2_AAAGA(x1, x2, x3, x4)  =  U2_AAAGA(x4)
APP7_IN_GAA(x1, x2, x3)  =  APP7_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x5)
U3_AAAGA(x1, x2, x3, x4, x5, x6, x7)  =  U3_AAAGA(x7)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Complex Obligation (AND)

(9) Obligation:

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

APP7_IN_GAA(.(T33, T34), X57, .(T33, T36)) → APP7_IN_GAA(T34, X57, T36)

The TRS R consists of the following rules:

sublist1_in_ga(T5, T7) → U4_ga(T5, T7, p3_in_aaaga(X7, X8, T7, T5, X9))
p3_in_aaaga([], T19, T19, T5, X9) → U2_aaaga(T19, T5, X9, app7_in_gaa(T5, X9, T19))
app7_in_gaa([], T26, T26) → app7_out_gaa([], T26, T26)
app7_in_gaa(.(T33, T34), X57, .(T33, T36)) → U1_gaa(T33, T34, X57, T36, app7_in_gaa(T34, X57, T36))
U1_gaa(T33, T34, X57, T36, app7_out_gaa(T34, X57, T36)) → app7_out_gaa(.(T33, T34), X57, .(T33, T36))
U2_aaaga(T19, T5, X9, app7_out_gaa(T5, X9, T19)) → p3_out_aaaga([], T19, T19, T5, X9)
p3_in_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9) → U3_aaaga(X84, X85, X86, T42, T5, X9, p3_in_aaaga(X85, X86, T42, T5, X9))
U3_aaaga(X84, X85, X86, T42, T5, X9, p3_out_aaaga(X85, X86, T42, T5, X9)) → p3_out_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9)
U4_ga(T5, T7, p3_out_aaaga(X7, X8, T7, T5, X9)) → sublist1_out_ga(T5, T7)

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)
p3_in_aaaga(x1, x2, x3, x4, x5)  =  p3_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4)  =  U2_aaaga(x4)
app7_in_gaa(x1, x2, x3)  =  app7_in_gaa(x1)
[]  =  []
app7_out_gaa(x1, x2, x3)  =  app7_out_gaa
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
p3_out_aaaga(x1, x2, x3, x4, x5)  =  p3_out_aaaga(x1)
U3_aaaga(x1, x2, x3, x4, x5, x6, x7)  =  U3_aaaga(x7)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga
APP7_IN_GAA(x1, x2, x3)  =  APP7_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:

APP7_IN_GAA(.(T33, T34), X57, .(T33, T36)) → APP7_IN_GAA(T34, X57, T36)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP7_IN_GAA(x1, x2, x3)  =  APP7_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:

APP7_IN_GAA(.(T34)) → APP7_IN_GAA(T34)

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:

  • APP7_IN_GAA(.(T34)) → APP7_IN_GAA(T34)
    The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

P3_IN_AAAGA(.(X84, X85), X86, .(X84, T42), T5, X9) → P3_IN_AAAGA(X85, X86, T42, T5, X9)

The TRS R consists of the following rules:

sublist1_in_ga(T5, T7) → U4_ga(T5, T7, p3_in_aaaga(X7, X8, T7, T5, X9))
p3_in_aaaga([], T19, T19, T5, X9) → U2_aaaga(T19, T5, X9, app7_in_gaa(T5, X9, T19))
app7_in_gaa([], T26, T26) → app7_out_gaa([], T26, T26)
app7_in_gaa(.(T33, T34), X57, .(T33, T36)) → U1_gaa(T33, T34, X57, T36, app7_in_gaa(T34, X57, T36))
U1_gaa(T33, T34, X57, T36, app7_out_gaa(T34, X57, T36)) → app7_out_gaa(.(T33, T34), X57, .(T33, T36))
U2_aaaga(T19, T5, X9, app7_out_gaa(T5, X9, T19)) → p3_out_aaaga([], T19, T19, T5, X9)
p3_in_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9) → U3_aaaga(X84, X85, X86, T42, T5, X9, p3_in_aaaga(X85, X86, T42, T5, X9))
U3_aaaga(X84, X85, X86, T42, T5, X9, p3_out_aaaga(X85, X86, T42, T5, X9)) → p3_out_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9)
U4_ga(T5, T7, p3_out_aaaga(X7, X8, T7, T5, X9)) → sublist1_out_ga(T5, T7)

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)
p3_in_aaaga(x1, x2, x3, x4, x5)  =  p3_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4)  =  U2_aaaga(x4)
app7_in_gaa(x1, x2, x3)  =  app7_in_gaa(x1)
[]  =  []
app7_out_gaa(x1, x2, x3)  =  app7_out_gaa
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
p3_out_aaaga(x1, x2, x3, x4, x5)  =  p3_out_aaaga(x1)
U3_aaaga(x1, x2, x3, x4, x5, x6, x7)  =  U3_aaaga(x7)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga
P3_IN_AAAGA(x1, x2, x3, x4, x5)  =  P3_IN_AAAGA(x4)

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:

P3_IN_AAAGA(.(X84, X85), X86, .(X84, T42), T5, X9) → P3_IN_AAAGA(X85, X86, T42, T5, X9)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
P3_IN_AAAGA(x1, x2, x3, x4, x5)  =  P3_IN_AAAGA(x4)

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:

P3_IN_AAAGA(T5) → P3_IN_AAAGA(T5)

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 = P3_IN_AAAGA(T5) evaluates to t =P3_IN_AAAGA(T5)

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 P3_IN_AAAGA(T5) to P3_IN_AAAGA(T5).



(22) NO

(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:
sublist1_in: (b,f)
p3_in: (f,f,f,b,f)
app7_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(T5, T7) → U4_ga(T5, T7, p3_in_aaaga(X7, X8, T7, T5, X9))
p3_in_aaaga([], T19, T19, T5, X9) → U2_aaaga(T19, T5, X9, app7_in_gaa(T5, X9, T19))
app7_in_gaa([], T26, T26) → app7_out_gaa([], T26, T26)
app7_in_gaa(.(T33, T34), X57, .(T33, T36)) → U1_gaa(T33, T34, X57, T36, app7_in_gaa(T34, X57, T36))
U1_gaa(T33, T34, X57, T36, app7_out_gaa(T34, X57, T36)) → app7_out_gaa(.(T33, T34), X57, .(T33, T36))
U2_aaaga(T19, T5, X9, app7_out_gaa(T5, X9, T19)) → p3_out_aaaga([], T19, T19, T5, X9)
p3_in_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9) → U3_aaaga(X84, X85, X86, T42, T5, X9, p3_in_aaaga(X85, X86, T42, T5, X9))
U3_aaaga(X84, X85, X86, T42, T5, X9, p3_out_aaaga(X85, X86, T42, T5, X9)) → p3_out_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9)
U4_ga(T5, T7, p3_out_aaaga(X7, X8, T7, T5, X9)) → sublist1_out_ga(T5, T7)

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)
p3_in_aaaga(x1, x2, x3, x4, x5)  =  p3_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4)  =  U2_aaaga(x2, x4)
app7_in_gaa(x1, x2, x3)  =  app7_in_gaa(x1)
[]  =  []
app7_out_gaa(x1, x2, x3)  =  app7_out_gaa(x1)
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
p3_out_aaaga(x1, x2, x3, x4, x5)  =  p3_out_aaaga(x1, x4)
U3_aaaga(x1, x2, x3, x4, x5, x6, x7)  =  U3_aaaga(x5, x7)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(24) Obligation:

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

sublist1_in_ga(T5, T7) → U4_ga(T5, T7, p3_in_aaaga(X7, X8, T7, T5, X9))
p3_in_aaaga([], T19, T19, T5, X9) → U2_aaaga(T19, T5, X9, app7_in_gaa(T5, X9, T19))
app7_in_gaa([], T26, T26) → app7_out_gaa([], T26, T26)
app7_in_gaa(.(T33, T34), X57, .(T33, T36)) → U1_gaa(T33, T34, X57, T36, app7_in_gaa(T34, X57, T36))
U1_gaa(T33, T34, X57, T36, app7_out_gaa(T34, X57, T36)) → app7_out_gaa(.(T33, T34), X57, .(T33, T36))
U2_aaaga(T19, T5, X9, app7_out_gaa(T5, X9, T19)) → p3_out_aaaga([], T19, T19, T5, X9)
p3_in_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9) → U3_aaaga(X84, X85, X86, T42, T5, X9, p3_in_aaaga(X85, X86, T42, T5, X9))
U3_aaaga(X84, X85, X86, T42, T5, X9, p3_out_aaaga(X85, X86, T42, T5, X9)) → p3_out_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9)
U4_ga(T5, T7, p3_out_aaaga(X7, X8, T7, T5, X9)) → sublist1_out_ga(T5, T7)

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)
p3_in_aaaga(x1, x2, x3, x4, x5)  =  p3_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4)  =  U2_aaaga(x2, x4)
app7_in_gaa(x1, x2, x3)  =  app7_in_gaa(x1)
[]  =  []
app7_out_gaa(x1, x2, x3)  =  app7_out_gaa(x1)
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
p3_out_aaaga(x1, x2, x3, x4, x5)  =  p3_out_aaaga(x1, x4)
U3_aaaga(x1, x2, x3, x4, x5, x6, x7)  =  U3_aaaga(x5, x7)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga(x1)

(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:

SUBLIST1_IN_GA(T5, T7) → U4_GA(T5, T7, p3_in_aaaga(X7, X8, T7, T5, X9))
SUBLIST1_IN_GA(T5, T7) → P3_IN_AAAGA(X7, X8, T7, T5, X9)
P3_IN_AAAGA([], T19, T19, T5, X9) → U2_AAAGA(T19, T5, X9, app7_in_gaa(T5, X9, T19))
P3_IN_AAAGA([], T19, T19, T5, X9) → APP7_IN_GAA(T5, X9, T19)
APP7_IN_GAA(.(T33, T34), X57, .(T33, T36)) → U1_GAA(T33, T34, X57, T36, app7_in_gaa(T34, X57, T36))
APP7_IN_GAA(.(T33, T34), X57, .(T33, T36)) → APP7_IN_GAA(T34, X57, T36)
P3_IN_AAAGA(.(X84, X85), X86, .(X84, T42), T5, X9) → U3_AAAGA(X84, X85, X86, T42, T5, X9, p3_in_aaaga(X85, X86, T42, T5, X9))
P3_IN_AAAGA(.(X84, X85), X86, .(X84, T42), T5, X9) → P3_IN_AAAGA(X85, X86, T42, T5, X9)

The TRS R consists of the following rules:

sublist1_in_ga(T5, T7) → U4_ga(T5, T7, p3_in_aaaga(X7, X8, T7, T5, X9))
p3_in_aaaga([], T19, T19, T5, X9) → U2_aaaga(T19, T5, X9, app7_in_gaa(T5, X9, T19))
app7_in_gaa([], T26, T26) → app7_out_gaa([], T26, T26)
app7_in_gaa(.(T33, T34), X57, .(T33, T36)) → U1_gaa(T33, T34, X57, T36, app7_in_gaa(T34, X57, T36))
U1_gaa(T33, T34, X57, T36, app7_out_gaa(T34, X57, T36)) → app7_out_gaa(.(T33, T34), X57, .(T33, T36))
U2_aaaga(T19, T5, X9, app7_out_gaa(T5, X9, T19)) → p3_out_aaaga([], T19, T19, T5, X9)
p3_in_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9) → U3_aaaga(X84, X85, X86, T42, T5, X9, p3_in_aaaga(X85, X86, T42, T5, X9))
U3_aaaga(X84, X85, X86, T42, T5, X9, p3_out_aaaga(X85, X86, T42, T5, X9)) → p3_out_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9)
U4_ga(T5, T7, p3_out_aaaga(X7, X8, T7, T5, X9)) → sublist1_out_ga(T5, T7)

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)
p3_in_aaaga(x1, x2, x3, x4, x5)  =  p3_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4)  =  U2_aaaga(x2, x4)
app7_in_gaa(x1, x2, x3)  =  app7_in_gaa(x1)
[]  =  []
app7_out_gaa(x1, x2, x3)  =  app7_out_gaa(x1)
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
p3_out_aaaga(x1, x2, x3, x4, x5)  =  p3_out_aaaga(x1, x4)
U3_aaaga(x1, x2, x3, x4, x5, x6, x7)  =  U3_aaaga(x5, x7)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga(x1)
SUBLIST1_IN_GA(x1, x2)  =  SUBLIST1_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
P3_IN_AAAGA(x1, x2, x3, x4, x5)  =  P3_IN_AAAGA(x4)
U2_AAAGA(x1, x2, x3, x4)  =  U2_AAAGA(x2, x4)
APP7_IN_GAA(x1, x2, x3)  =  APP7_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x2, x5)
U3_AAAGA(x1, x2, x3, x4, x5, x6, x7)  =  U3_AAAGA(x5, x7)

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

(26) Obligation:

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

SUBLIST1_IN_GA(T5, T7) → U4_GA(T5, T7, p3_in_aaaga(X7, X8, T7, T5, X9))
SUBLIST1_IN_GA(T5, T7) → P3_IN_AAAGA(X7, X8, T7, T5, X9)
P3_IN_AAAGA([], T19, T19, T5, X9) → U2_AAAGA(T19, T5, X9, app7_in_gaa(T5, X9, T19))
P3_IN_AAAGA([], T19, T19, T5, X9) → APP7_IN_GAA(T5, X9, T19)
APP7_IN_GAA(.(T33, T34), X57, .(T33, T36)) → U1_GAA(T33, T34, X57, T36, app7_in_gaa(T34, X57, T36))
APP7_IN_GAA(.(T33, T34), X57, .(T33, T36)) → APP7_IN_GAA(T34, X57, T36)
P3_IN_AAAGA(.(X84, X85), X86, .(X84, T42), T5, X9) → U3_AAAGA(X84, X85, X86, T42, T5, X9, p3_in_aaaga(X85, X86, T42, T5, X9))
P3_IN_AAAGA(.(X84, X85), X86, .(X84, T42), T5, X9) → P3_IN_AAAGA(X85, X86, T42, T5, X9)

The TRS R consists of the following rules:

sublist1_in_ga(T5, T7) → U4_ga(T5, T7, p3_in_aaaga(X7, X8, T7, T5, X9))
p3_in_aaaga([], T19, T19, T5, X9) → U2_aaaga(T19, T5, X9, app7_in_gaa(T5, X9, T19))
app7_in_gaa([], T26, T26) → app7_out_gaa([], T26, T26)
app7_in_gaa(.(T33, T34), X57, .(T33, T36)) → U1_gaa(T33, T34, X57, T36, app7_in_gaa(T34, X57, T36))
U1_gaa(T33, T34, X57, T36, app7_out_gaa(T34, X57, T36)) → app7_out_gaa(.(T33, T34), X57, .(T33, T36))
U2_aaaga(T19, T5, X9, app7_out_gaa(T5, X9, T19)) → p3_out_aaaga([], T19, T19, T5, X9)
p3_in_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9) → U3_aaaga(X84, X85, X86, T42, T5, X9, p3_in_aaaga(X85, X86, T42, T5, X9))
U3_aaaga(X84, X85, X86, T42, T5, X9, p3_out_aaaga(X85, X86, T42, T5, X9)) → p3_out_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9)
U4_ga(T5, T7, p3_out_aaaga(X7, X8, T7, T5, X9)) → sublist1_out_ga(T5, T7)

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)
p3_in_aaaga(x1, x2, x3, x4, x5)  =  p3_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4)  =  U2_aaaga(x2, x4)
app7_in_gaa(x1, x2, x3)  =  app7_in_gaa(x1)
[]  =  []
app7_out_gaa(x1, x2, x3)  =  app7_out_gaa(x1)
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
p3_out_aaaga(x1, x2, x3, x4, x5)  =  p3_out_aaaga(x1, x4)
U3_aaaga(x1, x2, x3, x4, x5, x6, x7)  =  U3_aaaga(x5, x7)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga(x1)
SUBLIST1_IN_GA(x1, x2)  =  SUBLIST1_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
P3_IN_AAAGA(x1, x2, x3, x4, x5)  =  P3_IN_AAAGA(x4)
U2_AAAGA(x1, x2, x3, x4)  =  U2_AAAGA(x2, x4)
APP7_IN_GAA(x1, x2, x3)  =  APP7_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x2, x5)
U3_AAAGA(x1, x2, x3, x4, x5, x6, x7)  =  U3_AAAGA(x5, x7)

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

(27) DependencyGraphProof (EQUIVALENT transformation)

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

(28) Complex Obligation (AND)

(29) Obligation:

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

APP7_IN_GAA(.(T33, T34), X57, .(T33, T36)) → APP7_IN_GAA(T34, X57, T36)

The TRS R consists of the following rules:

sublist1_in_ga(T5, T7) → U4_ga(T5, T7, p3_in_aaaga(X7, X8, T7, T5, X9))
p3_in_aaaga([], T19, T19, T5, X9) → U2_aaaga(T19, T5, X9, app7_in_gaa(T5, X9, T19))
app7_in_gaa([], T26, T26) → app7_out_gaa([], T26, T26)
app7_in_gaa(.(T33, T34), X57, .(T33, T36)) → U1_gaa(T33, T34, X57, T36, app7_in_gaa(T34, X57, T36))
U1_gaa(T33, T34, X57, T36, app7_out_gaa(T34, X57, T36)) → app7_out_gaa(.(T33, T34), X57, .(T33, T36))
U2_aaaga(T19, T5, X9, app7_out_gaa(T5, X9, T19)) → p3_out_aaaga([], T19, T19, T5, X9)
p3_in_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9) → U3_aaaga(X84, X85, X86, T42, T5, X9, p3_in_aaaga(X85, X86, T42, T5, X9))
U3_aaaga(X84, X85, X86, T42, T5, X9, p3_out_aaaga(X85, X86, T42, T5, X9)) → p3_out_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9)
U4_ga(T5, T7, p3_out_aaaga(X7, X8, T7, T5, X9)) → sublist1_out_ga(T5, T7)

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)
p3_in_aaaga(x1, x2, x3, x4, x5)  =  p3_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4)  =  U2_aaaga(x2, x4)
app7_in_gaa(x1, x2, x3)  =  app7_in_gaa(x1)
[]  =  []
app7_out_gaa(x1, x2, x3)  =  app7_out_gaa(x1)
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
p3_out_aaaga(x1, x2, x3, x4, x5)  =  p3_out_aaaga(x1, x4)
U3_aaaga(x1, x2, x3, x4, x5, x6, x7)  =  U3_aaaga(x5, x7)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga(x1)
APP7_IN_GAA(x1, x2, x3)  =  APP7_IN_GAA(x1)

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:

APP7_IN_GAA(.(T33, T34), X57, .(T33, T36)) → APP7_IN_GAA(T34, X57, T36)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP7_IN_GAA(x1, x2, x3)  =  APP7_IN_GAA(x1)

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:

APP7_IN_GAA(.(T34)) → APP7_IN_GAA(T34)

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

(34) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

  • APP7_IN_GAA(.(T34)) → APP7_IN_GAA(T34)
    The graph contains the following edges 1 > 1

(35) YES

(36) Obligation:

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

P3_IN_AAAGA(.(X84, X85), X86, .(X84, T42), T5, X9) → P3_IN_AAAGA(X85, X86, T42, T5, X9)

The TRS R consists of the following rules:

sublist1_in_ga(T5, T7) → U4_ga(T5, T7, p3_in_aaaga(X7, X8, T7, T5, X9))
p3_in_aaaga([], T19, T19, T5, X9) → U2_aaaga(T19, T5, X9, app7_in_gaa(T5, X9, T19))
app7_in_gaa([], T26, T26) → app7_out_gaa([], T26, T26)
app7_in_gaa(.(T33, T34), X57, .(T33, T36)) → U1_gaa(T33, T34, X57, T36, app7_in_gaa(T34, X57, T36))
U1_gaa(T33, T34, X57, T36, app7_out_gaa(T34, X57, T36)) → app7_out_gaa(.(T33, T34), X57, .(T33, T36))
U2_aaaga(T19, T5, X9, app7_out_gaa(T5, X9, T19)) → p3_out_aaaga([], T19, T19, T5, X9)
p3_in_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9) → U3_aaaga(X84, X85, X86, T42, T5, X9, p3_in_aaaga(X85, X86, T42, T5, X9))
U3_aaaga(X84, X85, X86, T42, T5, X9, p3_out_aaaga(X85, X86, T42, T5, X9)) → p3_out_aaaga(.(X84, X85), X86, .(X84, T42), T5, X9)
U4_ga(T5, T7, p3_out_aaaga(X7, X8, T7, T5, X9)) → sublist1_out_ga(T5, T7)

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)
p3_in_aaaga(x1, x2, x3, x4, x5)  =  p3_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4)  =  U2_aaaga(x2, x4)
app7_in_gaa(x1, x2, x3)  =  app7_in_gaa(x1)
[]  =  []
app7_out_gaa(x1, x2, x3)  =  app7_out_gaa(x1)
.(x1, x2)  =  .(x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x2, x5)
p3_out_aaaga(x1, x2, x3, x4, x5)  =  p3_out_aaaga(x1, x4)
U3_aaaga(x1, x2, x3, x4, x5, x6, x7)  =  U3_aaaga(x5, x7)
sublist1_out_ga(x1, x2)  =  sublist1_out_ga(x1)
P3_IN_AAAGA(x1, x2, x3, x4, x5)  =  P3_IN_AAAGA(x4)

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:

P3_IN_AAAGA(.(X84, X85), X86, .(X84, T42), T5, X9) → P3_IN_AAAGA(X85, X86, T42, T5, X9)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
P3_IN_AAAGA(x1, x2, x3, x4, x5)  =  P3_IN_AAAGA(x4)

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:

P3_IN_AAAGA(T5) → P3_IN_AAAGA(T5)

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

(41) 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 = P3_IN_AAAGA(T5) evaluates to t =P3_IN_AAAGA(T5)

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 P3_IN_AAAGA(T5) to P3_IN_AAAGA(T5).



(42) NO