Left Termination of the query pattern suffix_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 PiDP
        ↳9 UsableRulesProof (⇔)
        ↳10 PiDP
        ↳11 PiDPToQDPProof (⇐)
        ↳12 QDP
        ↳13 NonTerminationProof (⇔)
        ↳14 NO
    ↳15 PrologToPiTRSProof (⇐)
        ↳16 PiTRS
        ↳17 DependencyPairsProof (⇔)
        ↳18 PiDP
        ↳19 DependencyGraphProof (⇔)
        ↳20 PiDP
        ↳21 UsableRulesProof (⇔)
        ↳22 PiDP
        ↳23 PiDPToQDPProof (⇐)
        ↳24 QDP
        ↳25 NonTerminationProof (⇔)
        ↳26 NO

(0) Obligation:

Clauses:

suffix(Xs, Ys) :- app(X1, Xs, Ys).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Queries:

suffix(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

app3([], T12, T12).
app3(.(X29, X30), T17, .(X29, T19)) :- app3(X30, T17, T19).
suffix1(T5, T7) :- app3(X6, T5, T7).

Queries:

suffix1(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:
suffix1_in: (b,f)
app3_in: (f,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

suffix1_in_ga(T5, T7) → U2_ga(T5, T7, app3_in_aga(X6, T5, T7))
app3_in_aga([], T12, T12) → app3_out_aga([], T12, T12)
app3_in_aga(.(X29, X30), T17, .(X29, T19)) → U1_aga(X29, X30, T17, T19, app3_in_aga(X30, T17, T19))
U1_aga(X29, X30, T17, T19, app3_out_aga(X30, T17, T19)) → app3_out_aga(.(X29, X30), T17, .(X29, T19))
U2_ga(T5, T7, app3_out_aga(X6, T5, T7)) → suffix1_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
suffix1_in_ga(x1, x2)  =  suffix1_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
app3_in_aga(x1, x2, x3)  =  app3_in_aga(x2)
app3_out_aga(x1, x2, x3)  =  app3_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
.(x1, x2)  =  .(x2)
suffix1_out_ga(x1, x2)  =  suffix1_out_ga(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

suffix1_in_ga(T5, T7) → U2_ga(T5, T7, app3_in_aga(X6, T5, T7))
app3_in_aga([], T12, T12) → app3_out_aga([], T12, T12)
app3_in_aga(.(X29, X30), T17, .(X29, T19)) → U1_aga(X29, X30, T17, T19, app3_in_aga(X30, T17, T19))
U1_aga(X29, X30, T17, T19, app3_out_aga(X30, T17, T19)) → app3_out_aga(.(X29, X30), T17, .(X29, T19))
U2_ga(T5, T7, app3_out_aga(X6, T5, T7)) → suffix1_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
suffix1_in_ga(x1, x2)  =  suffix1_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
app3_in_aga(x1, x2, x3)  =  app3_in_aga(x2)
app3_out_aga(x1, x2, x3)  =  app3_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
.(x1, x2)  =  .(x2)
suffix1_out_ga(x1, x2)  =  suffix1_out_ga(x2)

(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:

SUFFIX1_IN_GA(T5, T7) → U2_GA(T5, T7, app3_in_aga(X6, T5, T7))
SUFFIX1_IN_GA(T5, T7) → APP3_IN_AGA(X6, T5, T7)
APP3_IN_AGA(.(X29, X30), T17, .(X29, T19)) → U1_AGA(X29, X30, T17, T19, app3_in_aga(X30, T17, T19))
APP3_IN_AGA(.(X29, X30), T17, .(X29, T19)) → APP3_IN_AGA(X30, T17, T19)

The TRS R consists of the following rules:

suffix1_in_ga(T5, T7) → U2_ga(T5, T7, app3_in_aga(X6, T5, T7))
app3_in_aga([], T12, T12) → app3_out_aga([], T12, T12)
app3_in_aga(.(X29, X30), T17, .(X29, T19)) → U1_aga(X29, X30, T17, T19, app3_in_aga(X30, T17, T19))
U1_aga(X29, X30, T17, T19, app3_out_aga(X30, T17, T19)) → app3_out_aga(.(X29, X30), T17, .(X29, T19))
U2_ga(T5, T7, app3_out_aga(X6, T5, T7)) → suffix1_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
suffix1_in_ga(x1, x2)  =  suffix1_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
app3_in_aga(x1, x2, x3)  =  app3_in_aga(x2)
app3_out_aga(x1, x2, x3)  =  app3_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
.(x1, x2)  =  .(x2)
suffix1_out_ga(x1, x2)  =  suffix1_out_ga(x2)
SUFFIX1_IN_GA(x1, x2)  =  SUFFIX1_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x3)
APP3_IN_AGA(x1, x2, x3)  =  APP3_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x5)

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

(6) Obligation:

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

SUFFIX1_IN_GA(T5, T7) → U2_GA(T5, T7, app3_in_aga(X6, T5, T7))
SUFFIX1_IN_GA(T5, T7) → APP3_IN_AGA(X6, T5, T7)
APP3_IN_AGA(.(X29, X30), T17, .(X29, T19)) → U1_AGA(X29, X30, T17, T19, app3_in_aga(X30, T17, T19))
APP3_IN_AGA(.(X29, X30), T17, .(X29, T19)) → APP3_IN_AGA(X30, T17, T19)

The TRS R consists of the following rules:

suffix1_in_ga(T5, T7) → U2_ga(T5, T7, app3_in_aga(X6, T5, T7))
app3_in_aga([], T12, T12) → app3_out_aga([], T12, T12)
app3_in_aga(.(X29, X30), T17, .(X29, T19)) → U1_aga(X29, X30, T17, T19, app3_in_aga(X30, T17, T19))
U1_aga(X29, X30, T17, T19, app3_out_aga(X30, T17, T19)) → app3_out_aga(.(X29, X30), T17, .(X29, T19))
U2_ga(T5, T7, app3_out_aga(X6, T5, T7)) → suffix1_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
suffix1_in_ga(x1, x2)  =  suffix1_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
app3_in_aga(x1, x2, x3)  =  app3_in_aga(x2)
app3_out_aga(x1, x2, x3)  =  app3_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
.(x1, x2)  =  .(x2)
suffix1_out_ga(x1, x2)  =  suffix1_out_ga(x2)
SUFFIX1_IN_GA(x1, x2)  =  SUFFIX1_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x3)
APP3_IN_AGA(x1, x2, x3)  =  APP3_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(8) Obligation:

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

APP3_IN_AGA(.(X29, X30), T17, .(X29, T19)) → APP3_IN_AGA(X30, T17, T19)

The TRS R consists of the following rules:

suffix1_in_ga(T5, T7) → U2_ga(T5, T7, app3_in_aga(X6, T5, T7))
app3_in_aga([], T12, T12) → app3_out_aga([], T12, T12)
app3_in_aga(.(X29, X30), T17, .(X29, T19)) → U1_aga(X29, X30, T17, T19, app3_in_aga(X30, T17, T19))
U1_aga(X29, X30, T17, T19, app3_out_aga(X30, T17, T19)) → app3_out_aga(.(X29, X30), T17, .(X29, T19))
U2_ga(T5, T7, app3_out_aga(X6, T5, T7)) → suffix1_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
suffix1_in_ga(x1, x2)  =  suffix1_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
app3_in_aga(x1, x2, x3)  =  app3_in_aga(x2)
app3_out_aga(x1, x2, x3)  =  app3_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
.(x1, x2)  =  .(x2)
suffix1_out_ga(x1, x2)  =  suffix1_out_ga(x2)
APP3_IN_AGA(x1, x2, x3)  =  APP3_IN_AGA(x2)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

APP3_IN_AGA(.(X29, X30), T17, .(X29, T19)) → APP3_IN_AGA(X30, T17, T19)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP3_IN_AGA(x1, x2, x3)  =  APP3_IN_AGA(x2)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

APP3_IN_AGA(T17) → APP3_IN_AGA(T17)

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

(13) 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 = APP3_IN_AGA(T17) evaluates to t =APP3_IN_AGA(T17)

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 APP3_IN_AGA(T17) to APP3_IN_AGA(T17).



(14) NO

(15) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
suffix1_in: (b,f)
app3_in: (f,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

suffix1_in_ga(T5, T7) → U2_ga(T5, T7, app3_in_aga(X6, T5, T7))
app3_in_aga([], T12, T12) → app3_out_aga([], T12, T12)
app3_in_aga(.(X29, X30), T17, .(X29, T19)) → U1_aga(X29, X30, T17, T19, app3_in_aga(X30, T17, T19))
U1_aga(X29, X30, T17, T19, app3_out_aga(X30, T17, T19)) → app3_out_aga(.(X29, X30), T17, .(X29, T19))
U2_ga(T5, T7, app3_out_aga(X6, T5, T7)) → suffix1_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
suffix1_in_ga(x1, x2)  =  suffix1_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
app3_in_aga(x1, x2, x3)  =  app3_in_aga(x2)
app3_out_aga(x1, x2, x3)  =  app3_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
.(x1, x2)  =  .(x2)
suffix1_out_ga(x1, x2)  =  suffix1_out_ga(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(16) Obligation:

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

suffix1_in_ga(T5, T7) → U2_ga(T5, T7, app3_in_aga(X6, T5, T7))
app3_in_aga([], T12, T12) → app3_out_aga([], T12, T12)
app3_in_aga(.(X29, X30), T17, .(X29, T19)) → U1_aga(X29, X30, T17, T19, app3_in_aga(X30, T17, T19))
U1_aga(X29, X30, T17, T19, app3_out_aga(X30, T17, T19)) → app3_out_aga(.(X29, X30), T17, .(X29, T19))
U2_ga(T5, T7, app3_out_aga(X6, T5, T7)) → suffix1_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
suffix1_in_ga(x1, x2)  =  suffix1_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
app3_in_aga(x1, x2, x3)  =  app3_in_aga(x2)
app3_out_aga(x1, x2, x3)  =  app3_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
.(x1, x2)  =  .(x2)
suffix1_out_ga(x1, x2)  =  suffix1_out_ga(x1, x2)

(17) 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:

SUFFIX1_IN_GA(T5, T7) → U2_GA(T5, T7, app3_in_aga(X6, T5, T7))
SUFFIX1_IN_GA(T5, T7) → APP3_IN_AGA(X6, T5, T7)
APP3_IN_AGA(.(X29, X30), T17, .(X29, T19)) → U1_AGA(X29, X30, T17, T19, app3_in_aga(X30, T17, T19))
APP3_IN_AGA(.(X29, X30), T17, .(X29, T19)) → APP3_IN_AGA(X30, T17, T19)

The TRS R consists of the following rules:

suffix1_in_ga(T5, T7) → U2_ga(T5, T7, app3_in_aga(X6, T5, T7))
app3_in_aga([], T12, T12) → app3_out_aga([], T12, T12)
app3_in_aga(.(X29, X30), T17, .(X29, T19)) → U1_aga(X29, X30, T17, T19, app3_in_aga(X30, T17, T19))
U1_aga(X29, X30, T17, T19, app3_out_aga(X30, T17, T19)) → app3_out_aga(.(X29, X30), T17, .(X29, T19))
U2_ga(T5, T7, app3_out_aga(X6, T5, T7)) → suffix1_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
suffix1_in_ga(x1, x2)  =  suffix1_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
app3_in_aga(x1, x2, x3)  =  app3_in_aga(x2)
app3_out_aga(x1, x2, x3)  =  app3_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
.(x1, x2)  =  .(x2)
suffix1_out_ga(x1, x2)  =  suffix1_out_ga(x1, x2)
SUFFIX1_IN_GA(x1, x2)  =  SUFFIX1_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
APP3_IN_AGA(x1, x2, x3)  =  APP3_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, x5)

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

(18) Obligation:

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

SUFFIX1_IN_GA(T5, T7) → U2_GA(T5, T7, app3_in_aga(X6, T5, T7))
SUFFIX1_IN_GA(T5, T7) → APP3_IN_AGA(X6, T5, T7)
APP3_IN_AGA(.(X29, X30), T17, .(X29, T19)) → U1_AGA(X29, X30, T17, T19, app3_in_aga(X30, T17, T19))
APP3_IN_AGA(.(X29, X30), T17, .(X29, T19)) → APP3_IN_AGA(X30, T17, T19)

The TRS R consists of the following rules:

suffix1_in_ga(T5, T7) → U2_ga(T5, T7, app3_in_aga(X6, T5, T7))
app3_in_aga([], T12, T12) → app3_out_aga([], T12, T12)
app3_in_aga(.(X29, X30), T17, .(X29, T19)) → U1_aga(X29, X30, T17, T19, app3_in_aga(X30, T17, T19))
U1_aga(X29, X30, T17, T19, app3_out_aga(X30, T17, T19)) → app3_out_aga(.(X29, X30), T17, .(X29, T19))
U2_ga(T5, T7, app3_out_aga(X6, T5, T7)) → suffix1_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
suffix1_in_ga(x1, x2)  =  suffix1_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
app3_in_aga(x1, x2, x3)  =  app3_in_aga(x2)
app3_out_aga(x1, x2, x3)  =  app3_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
.(x1, x2)  =  .(x2)
suffix1_out_ga(x1, x2)  =  suffix1_out_ga(x1, x2)
SUFFIX1_IN_GA(x1, x2)  =  SUFFIX1_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
APP3_IN_AGA(x1, x2, x3)  =  APP3_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, x5)

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

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(20) Obligation:

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

APP3_IN_AGA(.(X29, X30), T17, .(X29, T19)) → APP3_IN_AGA(X30, T17, T19)

The TRS R consists of the following rules:

suffix1_in_ga(T5, T7) → U2_ga(T5, T7, app3_in_aga(X6, T5, T7))
app3_in_aga([], T12, T12) → app3_out_aga([], T12, T12)
app3_in_aga(.(X29, X30), T17, .(X29, T19)) → U1_aga(X29, X30, T17, T19, app3_in_aga(X30, T17, T19))
U1_aga(X29, X30, T17, T19, app3_out_aga(X30, T17, T19)) → app3_out_aga(.(X29, X30), T17, .(X29, T19))
U2_ga(T5, T7, app3_out_aga(X6, T5, T7)) → suffix1_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
suffix1_in_ga(x1, x2)  =  suffix1_in_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
app3_in_aga(x1, x2, x3)  =  app3_in_aga(x2)
app3_out_aga(x1, x2, x3)  =  app3_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
.(x1, x2)  =  .(x2)
suffix1_out_ga(x1, x2)  =  suffix1_out_ga(x1, x2)
APP3_IN_AGA(x1, x2, x3)  =  APP3_IN_AGA(x2)

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

(21) UsableRulesProof (EQUIVALENT transformation)

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

(22) Obligation:

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

APP3_IN_AGA(.(X29, X30), T17, .(X29, T19)) → APP3_IN_AGA(X30, T17, T19)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP3_IN_AGA(x1, x2, x3)  =  APP3_IN_AGA(x2)

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

(23) PiDPToQDPProof (SOUND transformation)

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

(24) Obligation:

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

APP3_IN_AGA(T17) → APP3_IN_AGA(T17)

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

(25) 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 = APP3_IN_AGA(T17) evaluates to t =APP3_IN_AGA(T17)

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 APP3_IN_AGA(T17) to APP3_IN_AGA(T17).



(26) NO