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 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 PiDP
    ↳7 UsableRulesProof (⇔)
    ↳8 PiDP
    ↳9 PiDPToQDPProof (⇐)
    ↳10 QDP
    ↳11 NonTerminationProof (⇔)
    ↳12 FALSE
  ↳13 PrologToPiTRSProof (⇐)
    ↳14 PiTRS
    ↳15 DependencyPairsProof (⇔)
    ↳16 PiDP
    ↳17 DependencyGraphProof (⇔)
    ↳18 PiDP
    ↳19 UsableRulesProof (⇔)
    ↳20 PiDP
    ↳21 PiDPToQDPProof (⇐)
    ↳22 QDP
    ↳23 NonTerminationProof (⇔)
    ↳24 FALSE

(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) PrologToPiTRSProof (SOUND transformation)

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

suffix_in_ga(Xs, Ys) → U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) → suffix_out_ga(Xs, Ys)

The argument filtering Pi contains the following mapping:
suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
.(x1, x2)  =  .(x2)
suffix_out_ga(x1, x2)  =  suffix_out_ga(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

suffix_in_ga(Xs, Ys) → U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) → suffix_out_ga(Xs, Ys)

The argument filtering Pi contains the following mapping:
suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
.(x1, x2)  =  .(x2)
suffix_out_ga(x1, x2)  =  suffix_out_ga(x1, x2)

(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:

SUFFIX_IN_GA(Xs, Ys) → U1_GA(Xs, Ys, app_in_aga(X1, Xs, Ys))
SUFFIX_IN_GA(Xs, Ys) → APP_IN_AGA(X1, Xs, Ys)
APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → U2_AGA(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

suffix_in_ga(Xs, Ys) → U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) → suffix_out_ga(Xs, Ys)

The argument filtering Pi contains the following mapping:
suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
.(x1, x2)  =  .(x2)
suffix_out_ga(x1, x2)  =  suffix_out_ga(x1, x2)
SUFFIX_IN_GA(x1, x2)  =  SUFFIX_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)

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

(4) Obligation:

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

SUFFIX_IN_GA(Xs, Ys) → U1_GA(Xs, Ys, app_in_aga(X1, Xs, Ys))
SUFFIX_IN_GA(Xs, Ys) → APP_IN_AGA(X1, Xs, Ys)
APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → U2_AGA(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

suffix_in_ga(Xs, Ys) → U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) → suffix_out_ga(Xs, Ys)

The argument filtering Pi contains the following mapping:
suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
.(x1, x2)  =  .(x2)
suffix_out_ga(x1, x2)  =  suffix_out_ga(x1, x2)
SUFFIX_IN_GA(x1, x2)  =  SUFFIX_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

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

The TRS R consists of the following rules:

suffix_in_ga(Xs, Ys) → U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) → suffix_out_ga(Xs, Ys)

The argument filtering Pi contains the following mapping:
suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
.(x1, x2)  =  .(x2)
suffix_out_ga(x1, x2)  =  suffix_out_ga(x1, x2)
APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

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

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

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

APP_IN_AGA(Ys) → APP_IN_AGA(Ys)

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

(11) 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 = APP_IN_AGA(Ys) evaluates to t =APP_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 APP_IN_AGA(Ys) to APP_IN_AGA(Ys).



(12) FALSE

(13) PrologToPiTRSProof (SOUND transformation)

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

suffix_in_ga(Xs, Ys) → U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) → suffix_out_ga(Xs, Ys)

The argument filtering Pi contains the following mapping:
suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
.(x1, x2)  =  .(x2)
suffix_out_ga(x1, x2)  =  suffix_out_ga(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(14) Obligation:

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

suffix_in_ga(Xs, Ys) → U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) → suffix_out_ga(Xs, Ys)

The argument filtering Pi contains the following mapping:
suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
.(x1, x2)  =  .(x2)
suffix_out_ga(x1, x2)  =  suffix_out_ga(x2)

(15) 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:

SUFFIX_IN_GA(Xs, Ys) → U1_GA(Xs, Ys, app_in_aga(X1, Xs, Ys))
SUFFIX_IN_GA(Xs, Ys) → APP_IN_AGA(X1, Xs, Ys)
APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → U2_AGA(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

suffix_in_ga(Xs, Ys) → U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) → suffix_out_ga(Xs, Ys)

The argument filtering Pi contains the following mapping:
suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
.(x1, x2)  =  .(x2)
suffix_out_ga(x1, x2)  =  suffix_out_ga(x2)
SUFFIX_IN_GA(x1, x2)  =  SUFFIX_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x5)

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

(16) Obligation:

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

SUFFIX_IN_GA(Xs, Ys) → U1_GA(Xs, Ys, app_in_aga(X1, Xs, Ys))
SUFFIX_IN_GA(Xs, Ys) → APP_IN_AGA(X1, Xs, Ys)
APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → U2_AGA(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

suffix_in_ga(Xs, Ys) → U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) → suffix_out_ga(Xs, Ys)

The argument filtering Pi contains the following mapping:
suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
.(x1, x2)  =  .(x2)
suffix_out_ga(x1, x2)  =  suffix_out_ga(x2)
SUFFIX_IN_GA(x1, x2)  =  SUFFIX_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x5)

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

(17) DependencyGraphProof (EQUIVALENT transformation)

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

(18) Obligation:

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

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

The TRS R consists of the following rules:

suffix_in_ga(Xs, Ys) → U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
app_in_aga([], X, X) → app_out_aga([], X, X)
app_in_aga(.(X, Xs), Ys, .(X, Zs)) → U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) → app_out_aga(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) → suffix_out_ga(Xs, Ys)

The argument filtering Pi contains the following mapping:
suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
.(x1, x2)  =  .(x2)
suffix_out_ga(x1, x2)  =  suffix_out_ga(x2)
APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)

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

(19) UsableRulesProof (EQUIVALENT transformation)

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

(20) Obligation:

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

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

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

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

(21) PiDPToQDPProof (SOUND transformation)

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

(22) Obligation:

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

APP_IN_AGA(Ys) → APP_IN_AGA(Ys)

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

(23) 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 = APP_IN_AGA(Ys) evaluates to t =APP_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 APP_IN_AGA(Ys) to APP_IN_AGA(Ys).



(24) FALSE