Left Termination of the query pattern suffix_in_2(g, a) w.r.t. the given Prolog program could not be shown:



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof

Clauses:

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

Queries:

suffix(g,a).

We use the technique of [30]. 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(X, 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(X, 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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
  ↳ PrologToPiTRSProof

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

suffix_in_ga(Xs, Ys) → U1_ga(Xs, Ys, app_in_aga(X, 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(X, 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)


Using Dependency Pairs [1,30] 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(X, Xs, Ys))
SUFFIX_IN_GA(Xs, Ys) → APP_IN_AGA(X, 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(X, 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(X, 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)
SUFFIX_IN_GA(x1, x2)  =  SUFFIX_IN_GA(x1)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x5)
U1_GA(x1, x2, x3)  =  U1_GA(x3)

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
  ↳ PrologToPiTRSProof

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

SUFFIX_IN_GA(Xs, Ys) → U1_GA(Xs, Ys, app_in_aga(X, Xs, Ys))
SUFFIX_IN_GA(Xs, Ys) → APP_IN_AGA(X, 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(X, 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(X, 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)
SUFFIX_IN_GA(x1, x2)  =  SUFFIX_IN_GA(x1)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x5)
U1_GA(x1, x2, x3)  =  U1_GA(x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 3 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
  ↳ PrologToPiTRSProof

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(X, 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(X, 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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
  ↳ PrologToPiTRSProof

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ NonTerminationProof
  ↳ PrologToPiTRSProof

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.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

APP_IN_AGA(Ys) → APP_IN_AGA(Ys)

The TRS R consists of the following rules:none


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:



Rewriting sequence

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




We use the technique of [30]. 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(X, 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(X, 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



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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

suffix_in_ga(Xs, Ys) → U1_ga(Xs, Ys, app_in_aga(X, 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(X, 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)


Using Dependency Pairs [1,30] 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(X, Xs, Ys))
SUFFIX_IN_GA(Xs, Ys) → APP_IN_AGA(X, 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(X, 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(X, 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)
SUFFIX_IN_GA(x1, x2)  =  SUFFIX_IN_GA(x1)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)

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

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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

SUFFIX_IN_GA(Xs, Ys) → U1_GA(Xs, Ys, app_in_aga(X, Xs, Ys))
SUFFIX_IN_GA(Xs, Ys) → APP_IN_AGA(X, 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(X, 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(X, 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)
SUFFIX_IN_GA(x1, x2)  =  SUFFIX_IN_GA(x1)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 3 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof

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(X, 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(X, 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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ NonTerminationProof

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.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

APP_IN_AGA(Ys) → APP_IN_AGA(Ys)

The TRS R consists of the following rules:none


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:



Rewriting sequence

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