Left Termination of the query pattern member_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:

member(X, .(X, X1)).
member(X, .(X2, Xs)) :- member(X, Xs).

Queries:

member(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

member1(T5, .(T5, T6)).
member1(T22, .(T11, .(T22, T23))).
member1(T30, .(T11, .(T31, T33))) :- member1(T30, T33).
member1(T53, .(T42, .(T53, T54))).
member1(T61, .(T42, .(T62, T64))) :- member1(T61, T64).

Queries:

member1(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:
member1_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

member1_in_ga(T5, .(T5, T6)) → member1_out_ga(T5, .(T5, T6))
member1_in_ga(T22, .(T11, .(T22, T23))) → member1_out_ga(T22, .(T11, .(T22, T23)))
member1_in_ga(T30, .(T11, .(T31, T33))) → U1_ga(T30, T11, T31, T33, member1_in_ga(T30, T33))
member1_in_ga(T61, .(T42, .(T62, T64))) → U2_ga(T61, T42, T62, T64, member1_in_ga(T61, T64))
U2_ga(T61, T42, T62, T64, member1_out_ga(T61, T64)) → member1_out_ga(T61, .(T42, .(T62, T64)))
U1_ga(T30, T11, T31, T33, member1_out_ga(T30, T33)) → member1_out_ga(T30, .(T11, .(T31, T33)))

The argument filtering Pi contains the following mapping:
member1_in_ga(x1, x2)  =  member1_in_ga(x1)
member1_out_ga(x1, x2)  =  member1_out_ga
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

member1_in_ga(T5, .(T5, T6)) → member1_out_ga(T5, .(T5, T6))
member1_in_ga(T22, .(T11, .(T22, T23))) → member1_out_ga(T22, .(T11, .(T22, T23)))
member1_in_ga(T30, .(T11, .(T31, T33))) → U1_ga(T30, T11, T31, T33, member1_in_ga(T30, T33))
member1_in_ga(T61, .(T42, .(T62, T64))) → U2_ga(T61, T42, T62, T64, member1_in_ga(T61, T64))
U2_ga(T61, T42, T62, T64, member1_out_ga(T61, T64)) → member1_out_ga(T61, .(T42, .(T62, T64)))
U1_ga(T30, T11, T31, T33, member1_out_ga(T30, T33)) → member1_out_ga(T30, .(T11, .(T31, T33)))

The argument filtering Pi contains the following mapping:
member1_in_ga(x1, x2)  =  member1_in_ga(x1)
member1_out_ga(x1, x2)  =  member1_out_ga
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)

(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:

MEMBER1_IN_GA(T30, .(T11, .(T31, T33))) → U1_GA(T30, T11, T31, T33, member1_in_ga(T30, T33))
MEMBER1_IN_GA(T30, .(T11, .(T31, T33))) → MEMBER1_IN_GA(T30, T33)
MEMBER1_IN_GA(T61, .(T42, .(T62, T64))) → U2_GA(T61, T42, T62, T64, member1_in_ga(T61, T64))

The TRS R consists of the following rules:

member1_in_ga(T5, .(T5, T6)) → member1_out_ga(T5, .(T5, T6))
member1_in_ga(T22, .(T11, .(T22, T23))) → member1_out_ga(T22, .(T11, .(T22, T23)))
member1_in_ga(T30, .(T11, .(T31, T33))) → U1_ga(T30, T11, T31, T33, member1_in_ga(T30, T33))
member1_in_ga(T61, .(T42, .(T62, T64))) → U2_ga(T61, T42, T62, T64, member1_in_ga(T61, T64))
U2_ga(T61, T42, T62, T64, member1_out_ga(T61, T64)) → member1_out_ga(T61, .(T42, .(T62, T64)))
U1_ga(T30, T11, T31, T33, member1_out_ga(T30, T33)) → member1_out_ga(T30, .(T11, .(T31, T33)))

The argument filtering Pi contains the following mapping:
member1_in_ga(x1, x2)  =  member1_in_ga(x1)
member1_out_ga(x1, x2)  =  member1_out_ga
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
MEMBER1_IN_GA(x1, x2)  =  MEMBER1_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x5)

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

(6) Obligation:

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

MEMBER1_IN_GA(T30, .(T11, .(T31, T33))) → U1_GA(T30, T11, T31, T33, member1_in_ga(T30, T33))
MEMBER1_IN_GA(T30, .(T11, .(T31, T33))) → MEMBER1_IN_GA(T30, T33)
MEMBER1_IN_GA(T61, .(T42, .(T62, T64))) → U2_GA(T61, T42, T62, T64, member1_in_ga(T61, T64))

The TRS R consists of the following rules:

member1_in_ga(T5, .(T5, T6)) → member1_out_ga(T5, .(T5, T6))
member1_in_ga(T22, .(T11, .(T22, T23))) → member1_out_ga(T22, .(T11, .(T22, T23)))
member1_in_ga(T30, .(T11, .(T31, T33))) → U1_ga(T30, T11, T31, T33, member1_in_ga(T30, T33))
member1_in_ga(T61, .(T42, .(T62, T64))) → U2_ga(T61, T42, T62, T64, member1_in_ga(T61, T64))
U2_ga(T61, T42, T62, T64, member1_out_ga(T61, T64)) → member1_out_ga(T61, .(T42, .(T62, T64)))
U1_ga(T30, T11, T31, T33, member1_out_ga(T30, T33)) → member1_out_ga(T30, .(T11, .(T31, T33)))

The argument filtering Pi contains the following mapping:
member1_in_ga(x1, x2)  =  member1_in_ga(x1)
member1_out_ga(x1, x2)  =  member1_out_ga
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
MEMBER1_IN_GA(x1, x2)  =  MEMBER1_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(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 2 less nodes.

(8) Obligation:

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

MEMBER1_IN_GA(T30, .(T11, .(T31, T33))) → MEMBER1_IN_GA(T30, T33)

The TRS R consists of the following rules:

member1_in_ga(T5, .(T5, T6)) → member1_out_ga(T5, .(T5, T6))
member1_in_ga(T22, .(T11, .(T22, T23))) → member1_out_ga(T22, .(T11, .(T22, T23)))
member1_in_ga(T30, .(T11, .(T31, T33))) → U1_ga(T30, T11, T31, T33, member1_in_ga(T30, T33))
member1_in_ga(T61, .(T42, .(T62, T64))) → U2_ga(T61, T42, T62, T64, member1_in_ga(T61, T64))
U2_ga(T61, T42, T62, T64, member1_out_ga(T61, T64)) → member1_out_ga(T61, .(T42, .(T62, T64)))
U1_ga(T30, T11, T31, T33, member1_out_ga(T30, T33)) → member1_out_ga(T30, .(T11, .(T31, T33)))

The argument filtering Pi contains the following mapping:
member1_in_ga(x1, x2)  =  member1_in_ga(x1)
member1_out_ga(x1, x2)  =  member1_out_ga
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
MEMBER1_IN_GA(x1, x2)  =  MEMBER1_IN_GA(x1)

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:

MEMBER1_IN_GA(T30, .(T11, .(T31, T33))) → MEMBER1_IN_GA(T30, T33)

R is empty.
The argument filtering Pi contains the following mapping:
MEMBER1_IN_GA(x1, x2)  =  MEMBER1_IN_GA(x1)

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:

MEMBER1_IN_GA(T30) → MEMBER1_IN_GA(T30)

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 = MEMBER1_IN_GA(T30) evaluates to t =MEMBER1_IN_GA(T30)

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 MEMBER1_IN_GA(T30) to MEMBER1_IN_GA(T30).



(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:
member1_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

member1_in_ga(T5, .(T5, T6)) → member1_out_ga(T5, .(T5, T6))
member1_in_ga(T22, .(T11, .(T22, T23))) → member1_out_ga(T22, .(T11, .(T22, T23)))
member1_in_ga(T30, .(T11, .(T31, T33))) → U1_ga(T30, T11, T31, T33, member1_in_ga(T30, T33))
member1_in_ga(T61, .(T42, .(T62, T64))) → U2_ga(T61, T42, T62, T64, member1_in_ga(T61, T64))
U2_ga(T61, T42, T62, T64, member1_out_ga(T61, T64)) → member1_out_ga(T61, .(T42, .(T62, T64)))
U1_ga(T30, T11, T31, T33, member1_out_ga(T30, T33)) → member1_out_ga(T30, .(T11, .(T31, T33)))

The argument filtering Pi contains the following mapping:
member1_in_ga(x1, x2)  =  member1_in_ga(x1)
member1_out_ga(x1, x2)  =  member1_out_ga(x1)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(16) Obligation:

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

member1_in_ga(T5, .(T5, T6)) → member1_out_ga(T5, .(T5, T6))
member1_in_ga(T22, .(T11, .(T22, T23))) → member1_out_ga(T22, .(T11, .(T22, T23)))
member1_in_ga(T30, .(T11, .(T31, T33))) → U1_ga(T30, T11, T31, T33, member1_in_ga(T30, T33))
member1_in_ga(T61, .(T42, .(T62, T64))) → U2_ga(T61, T42, T62, T64, member1_in_ga(T61, T64))
U2_ga(T61, T42, T62, T64, member1_out_ga(T61, T64)) → member1_out_ga(T61, .(T42, .(T62, T64)))
U1_ga(T30, T11, T31, T33, member1_out_ga(T30, T33)) → member1_out_ga(T30, .(T11, .(T31, T33)))

The argument filtering Pi contains the following mapping:
member1_in_ga(x1, x2)  =  member1_in_ga(x1)
member1_out_ga(x1, x2)  =  member1_out_ga(x1)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x5)

(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:

MEMBER1_IN_GA(T30, .(T11, .(T31, T33))) → U1_GA(T30, T11, T31, T33, member1_in_ga(T30, T33))
MEMBER1_IN_GA(T30, .(T11, .(T31, T33))) → MEMBER1_IN_GA(T30, T33)
MEMBER1_IN_GA(T61, .(T42, .(T62, T64))) → U2_GA(T61, T42, T62, T64, member1_in_ga(T61, T64))

The TRS R consists of the following rules:

member1_in_ga(T5, .(T5, T6)) → member1_out_ga(T5, .(T5, T6))
member1_in_ga(T22, .(T11, .(T22, T23))) → member1_out_ga(T22, .(T11, .(T22, T23)))
member1_in_ga(T30, .(T11, .(T31, T33))) → U1_ga(T30, T11, T31, T33, member1_in_ga(T30, T33))
member1_in_ga(T61, .(T42, .(T62, T64))) → U2_ga(T61, T42, T62, T64, member1_in_ga(T61, T64))
U2_ga(T61, T42, T62, T64, member1_out_ga(T61, T64)) → member1_out_ga(T61, .(T42, .(T62, T64)))
U1_ga(T30, T11, T31, T33, member1_out_ga(T30, T33)) → member1_out_ga(T30, .(T11, .(T31, T33)))

The argument filtering Pi contains the following mapping:
member1_in_ga(x1, x2)  =  member1_in_ga(x1)
member1_out_ga(x1, x2)  =  member1_out_ga(x1)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x5)
MEMBER1_IN_GA(x1, x2)  =  MEMBER1_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x5)

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

(18) Obligation:

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

MEMBER1_IN_GA(T30, .(T11, .(T31, T33))) → U1_GA(T30, T11, T31, T33, member1_in_ga(T30, T33))
MEMBER1_IN_GA(T30, .(T11, .(T31, T33))) → MEMBER1_IN_GA(T30, T33)
MEMBER1_IN_GA(T61, .(T42, .(T62, T64))) → U2_GA(T61, T42, T62, T64, member1_in_ga(T61, T64))

The TRS R consists of the following rules:

member1_in_ga(T5, .(T5, T6)) → member1_out_ga(T5, .(T5, T6))
member1_in_ga(T22, .(T11, .(T22, T23))) → member1_out_ga(T22, .(T11, .(T22, T23)))
member1_in_ga(T30, .(T11, .(T31, T33))) → U1_ga(T30, T11, T31, T33, member1_in_ga(T30, T33))
member1_in_ga(T61, .(T42, .(T62, T64))) → U2_ga(T61, T42, T62, T64, member1_in_ga(T61, T64))
U2_ga(T61, T42, T62, T64, member1_out_ga(T61, T64)) → member1_out_ga(T61, .(T42, .(T62, T64)))
U1_ga(T30, T11, T31, T33, member1_out_ga(T30, T33)) → member1_out_ga(T30, .(T11, .(T31, T33)))

The argument filtering Pi contains the following mapping:
member1_in_ga(x1, x2)  =  member1_in_ga(x1)
member1_out_ga(x1, x2)  =  member1_out_ga(x1)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x5)
MEMBER1_IN_GA(x1, x2)  =  MEMBER1_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, 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 2 less nodes.

(20) Obligation:

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

MEMBER1_IN_GA(T30, .(T11, .(T31, T33))) → MEMBER1_IN_GA(T30, T33)

The TRS R consists of the following rules:

member1_in_ga(T5, .(T5, T6)) → member1_out_ga(T5, .(T5, T6))
member1_in_ga(T22, .(T11, .(T22, T23))) → member1_out_ga(T22, .(T11, .(T22, T23)))
member1_in_ga(T30, .(T11, .(T31, T33))) → U1_ga(T30, T11, T31, T33, member1_in_ga(T30, T33))
member1_in_ga(T61, .(T42, .(T62, T64))) → U2_ga(T61, T42, T62, T64, member1_in_ga(T61, T64))
U2_ga(T61, T42, T62, T64, member1_out_ga(T61, T64)) → member1_out_ga(T61, .(T42, .(T62, T64)))
U1_ga(T30, T11, T31, T33, member1_out_ga(T30, T33)) → member1_out_ga(T30, .(T11, .(T31, T33)))

The argument filtering Pi contains the following mapping:
member1_in_ga(x1, x2)  =  member1_in_ga(x1)
member1_out_ga(x1, x2)  =  member1_out_ga(x1)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x5)
MEMBER1_IN_GA(x1, x2)  =  MEMBER1_IN_GA(x1)

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:

MEMBER1_IN_GA(T30, .(T11, .(T31, T33))) → MEMBER1_IN_GA(T30, T33)

R is empty.
The argument filtering Pi contains the following mapping:
MEMBER1_IN_GA(x1, x2)  =  MEMBER1_IN_GA(x1)

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:

MEMBER1_IN_GA(T30) → MEMBER1_IN_GA(T30)

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 = MEMBER1_IN_GA(T30) evaluates to t =MEMBER1_IN_GA(T30)

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 MEMBER1_IN_GA(T30) to MEMBER1_IN_GA(T30).



(26) NO