Left Termination of the query pattern subset_in_2(a, g) w.r.t. the given Prolog program could not be shown:

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇐)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇐)
↳4 PiDP
↳5 DependencyGraphProof (⇔)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 YES
    ↳14 PiDP
        ↳15 PiDPToQDPProof (⇐)
        ↳16 QDP
        ↳17 Narrowing (⇐)
        ↳18 QDP
        ↳19 Instantiation (⇔)
        ↳20 QDP
        ↳21 ForwardInstantiation (⇔)
        ↳22 QDP
        ↳23 NonTerminationProof (⇔)
        ↳24 NO

(0) Obligation:

Clauses:

subset([], X1).
subset(.(X, Xs), Ys) :- ','(member(X, Ys), subset(Xs, Ys)).
member(X, .(X, X2)).
member(X, .(X3, Xs)) :- member(X, Xs).

Queries:

subset(a,g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

member15(T67, .(T65, T66)) :- member15(T67, T66).
p13(T37, T36, T38, T35) :- member15(T37, T36).
p13(T37, T36, T43, T35) :- ','(memberc15(T37, T36), subset1(T43, .(T35, T36))).
subset1(.(T21, T23), .(T21, T22)) :- subset1(T23, .(T21, T22)).
subset1(.(T37, T38), .(T35, T36)) :- p13(T37, T36, T38, T35).
subset1(.(T92, T94), .(T92, T93)) :- subset1(T94, .(T92, T93)).
subset1(.(T108, T109), .(T106, T107)) :- p13(T108, T107, T109, T106).

Clauses:

subsetc1([], T4).
subsetc1(.(T21, T23), .(T21, T22)) :- subsetc1(T23, .(T21, T22)).
subsetc1(.(T37, T38), .(T35, T36)) :- qc13(T37, T36, T38, T35).
subsetc1(.(T92, T94), .(T92, T93)) :- subsetc1(T94, .(T92, T93)).
subsetc1(.(T108, T109), .(T106, T107)) :- qc13(T108, T107, T109, T106).
memberc15(T56, .(T56, T57)).
memberc15(T67, .(T65, T66)) :- memberc15(T67, T66).
qc13(T37, T36, T43, T35) :- ','(memberc15(T37, T36), subsetc1(T43, .(T35, T36))).

Afs:

subset1(x1, x2)  =  subset1(x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
subset1_in: (f,b)
p13_in: (f,b,f,b)
member15_in: (f,b)
memberc15_in: (f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

SUBSET1_IN_AG(.(T21, T23), .(T21, T22)) → U5_AG(T21, T23, T22, subset1_in_ag(T23, .(T21, T22)))
SUBSET1_IN_AG(.(T21, T23), .(T21, T22)) → SUBSET1_IN_AG(T23, .(T21, T22))
SUBSET1_IN_AG(.(T37, T38), .(T35, T36)) → U6_AG(T37, T38, T35, T36, p13_in_agag(T37, T36, T38, T35))
SUBSET1_IN_AG(.(T37, T38), .(T35, T36)) → P13_IN_AGAG(T37, T36, T38, T35)
P13_IN_AGAG(T37, T36, T38, T35) → U2_AGAG(T37, T36, T38, T35, member15_in_ag(T37, T36))
P13_IN_AGAG(T37, T36, T38, T35) → MEMBER15_IN_AG(T37, T36)
MEMBER15_IN_AG(T67, .(T65, T66)) → U1_AG(T67, T65, T66, member15_in_ag(T67, T66))
MEMBER15_IN_AG(T67, .(T65, T66)) → MEMBER15_IN_AG(T67, T66)
P13_IN_AGAG(T37, T36, T43, T35) → U3_AGAG(T37, T36, T43, T35, memberc15_in_ag(T37, T36))
U3_AGAG(T37, T36, T43, T35, memberc15_out_ag(T37, T36)) → U4_AGAG(T37, T36, T43, T35, subset1_in_ag(T43, .(T35, T36)))
U3_AGAG(T37, T36, T43, T35, memberc15_out_ag(T37, T36)) → SUBSET1_IN_AG(T43, .(T35, T36))
SUBSET1_IN_AG(.(T92, T94), .(T92, T93)) → U7_AG(T92, T94, T93, subset1_in_ag(T94, .(T92, T93)))
SUBSET1_IN_AG(.(T108, T109), .(T106, T107)) → U8_AG(T108, T109, T106, T107, p13_in_agag(T108, T107, T109, T106))

The TRS R consists of the following rules:

memberc15_in_ag(T56, .(T56, T57)) → memberc15_out_ag(T56, .(T56, T57))
memberc15_in_ag(T67, .(T65, T66)) → U14_ag(T67, T65, T66, memberc15_in_ag(T67, T66))
U14_ag(T67, T65, T66, memberc15_out_ag(T67, T66)) → memberc15_out_ag(T67, .(T65, T66))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
p13_in_agag(x1, x2, x3, x4)  =  p13_in_agag(x2, x4)
member15_in_ag(x1, x2)  =  member15_in_ag(x2)
memberc15_in_ag(x1, x2)  =  memberc15_in_ag(x2)
memberc15_out_ag(x1, x2)  =  memberc15_out_ag(x1, x2)
U14_ag(x1, x2, x3, x4)  =  U14_ag(x2, x3, x4)
SUBSET1_IN_AG(x1, x2)  =  SUBSET1_IN_AG(x2)
U5_AG(x1, x2, x3, x4)  =  U5_AG(x1, x3, x4)
U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x3, x4, x5)
P13_IN_AGAG(x1, x2, x3, x4)  =  P13_IN_AGAG(x2, x4)
U2_AGAG(x1, x2, x3, x4, x5)  =  U2_AGAG(x2, x4, x5)
MEMBER15_IN_AG(x1, x2)  =  MEMBER15_IN_AG(x2)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x2, x3, x4)
U3_AGAG(x1, x2, x3, x4, x5)  =  U3_AGAG(x2, x4, x5)
U4_AGAG(x1, x2, x3, x4, x5)  =  U4_AGAG(x1, x2, x4, x5)
U7_AG(x1, x2, x3, x4)  =  U7_AG(x1, x3, x4)
U8_AG(x1, x2, x3, x4, x5)  =  U8_AG(x3, x4, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

SUBSET1_IN_AG(.(T21, T23), .(T21, T22)) → U5_AG(T21, T23, T22, subset1_in_ag(T23, .(T21, T22)))
SUBSET1_IN_AG(.(T21, T23), .(T21, T22)) → SUBSET1_IN_AG(T23, .(T21, T22))
SUBSET1_IN_AG(.(T37, T38), .(T35, T36)) → U6_AG(T37, T38, T35, T36, p13_in_agag(T37, T36, T38, T35))
SUBSET1_IN_AG(.(T37, T38), .(T35, T36)) → P13_IN_AGAG(T37, T36, T38, T35)
P13_IN_AGAG(T37, T36, T38, T35) → U2_AGAG(T37, T36, T38, T35, member15_in_ag(T37, T36))
P13_IN_AGAG(T37, T36, T38, T35) → MEMBER15_IN_AG(T37, T36)
MEMBER15_IN_AG(T67, .(T65, T66)) → U1_AG(T67, T65, T66, member15_in_ag(T67, T66))
MEMBER15_IN_AG(T67, .(T65, T66)) → MEMBER15_IN_AG(T67, T66)
P13_IN_AGAG(T37, T36, T43, T35) → U3_AGAG(T37, T36, T43, T35, memberc15_in_ag(T37, T36))
U3_AGAG(T37, T36, T43, T35, memberc15_out_ag(T37, T36)) → U4_AGAG(T37, T36, T43, T35, subset1_in_ag(T43, .(T35, T36)))
U3_AGAG(T37, T36, T43, T35, memberc15_out_ag(T37, T36)) → SUBSET1_IN_AG(T43, .(T35, T36))
SUBSET1_IN_AG(.(T92, T94), .(T92, T93)) → U7_AG(T92, T94, T93, subset1_in_ag(T94, .(T92, T93)))
SUBSET1_IN_AG(.(T108, T109), .(T106, T107)) → U8_AG(T108, T109, T106, T107, p13_in_agag(T108, T107, T109, T106))

The TRS R consists of the following rules:

memberc15_in_ag(T56, .(T56, T57)) → memberc15_out_ag(T56, .(T56, T57))
memberc15_in_ag(T67, .(T65, T66)) → U14_ag(T67, T65, T66, memberc15_in_ag(T67, T66))
U14_ag(T67, T65, T66, memberc15_out_ag(T67, T66)) → memberc15_out_ag(T67, .(T65, T66))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
p13_in_agag(x1, x2, x3, x4)  =  p13_in_agag(x2, x4)
member15_in_ag(x1, x2)  =  member15_in_ag(x2)
memberc15_in_ag(x1, x2)  =  memberc15_in_ag(x2)
memberc15_out_ag(x1, x2)  =  memberc15_out_ag(x1, x2)
U14_ag(x1, x2, x3, x4)  =  U14_ag(x2, x3, x4)
SUBSET1_IN_AG(x1, x2)  =  SUBSET1_IN_AG(x2)
U5_AG(x1, x2, x3, x4)  =  U5_AG(x1, x3, x4)
U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x3, x4, x5)
P13_IN_AGAG(x1, x2, x3, x4)  =  P13_IN_AGAG(x2, x4)
U2_AGAG(x1, x2, x3, x4, x5)  =  U2_AGAG(x2, x4, x5)
MEMBER15_IN_AG(x1, x2)  =  MEMBER15_IN_AG(x2)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x2, x3, x4)
U3_AGAG(x1, x2, x3, x4, x5)  =  U3_AGAG(x2, x4, x5)
U4_AGAG(x1, x2, x3, x4, x5)  =  U4_AGAG(x1, x2, x4, x5)
U7_AG(x1, x2, x3, x4)  =  U7_AG(x1, x3, x4)
U8_AG(x1, x2, x3, x4, x5)  =  U8_AG(x3, x4, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 8 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

MEMBER15_IN_AG(T67, .(T65, T66)) → MEMBER15_IN_AG(T67, T66)

The TRS R consists of the following rules:

memberc15_in_ag(T56, .(T56, T57)) → memberc15_out_ag(T56, .(T56, T57))
memberc15_in_ag(T67, .(T65, T66)) → U14_ag(T67, T65, T66, memberc15_in_ag(T67, T66))
U14_ag(T67, T65, T66, memberc15_out_ag(T67, T66)) → memberc15_out_ag(T67, .(T65, T66))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
memberc15_in_ag(x1, x2)  =  memberc15_in_ag(x2)
memberc15_out_ag(x1, x2)  =  memberc15_out_ag(x1, x2)
U14_ag(x1, x2, x3, x4)  =  U14_ag(x2, x3, x4)
MEMBER15_IN_AG(x1, x2)  =  MEMBER15_IN_AG(x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

MEMBER15_IN_AG(T67, .(T65, T66)) → MEMBER15_IN_AG(T67, T66)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
MEMBER15_IN_AG(x1, x2)  =  MEMBER15_IN_AG(x2)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

MEMBER15_IN_AG(.(T65, T66)) → MEMBER15_IN_AG(T66)

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

(12) 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:

  • MEMBER15_IN_AG(.(T65, T66)) → MEMBER15_IN_AG(T66)
    The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

SUBSET1_IN_AG(.(T37, T38), .(T35, T36)) → P13_IN_AGAG(T37, T36, T38, T35)
P13_IN_AGAG(T37, T36, T43, T35) → U3_AGAG(T37, T36, T43, T35, memberc15_in_ag(T37, T36))
U3_AGAG(T37, T36, T43, T35, memberc15_out_ag(T37, T36)) → SUBSET1_IN_AG(T43, .(T35, T36))
SUBSET1_IN_AG(.(T21, T23), .(T21, T22)) → SUBSET1_IN_AG(T23, .(T21, T22))

The TRS R consists of the following rules:

memberc15_in_ag(T56, .(T56, T57)) → memberc15_out_ag(T56, .(T56, T57))
memberc15_in_ag(T67, .(T65, T66)) → U14_ag(T67, T65, T66, memberc15_in_ag(T67, T66))
U14_ag(T67, T65, T66, memberc15_out_ag(T67, T66)) → memberc15_out_ag(T67, .(T65, T66))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
memberc15_in_ag(x1, x2)  =  memberc15_in_ag(x2)
memberc15_out_ag(x1, x2)  =  memberc15_out_ag(x1, x2)
U14_ag(x1, x2, x3, x4)  =  U14_ag(x2, x3, x4)
SUBSET1_IN_AG(x1, x2)  =  SUBSET1_IN_AG(x2)
P13_IN_AGAG(x1, x2, x3, x4)  =  P13_IN_AGAG(x2, x4)
U3_AGAG(x1, x2, x3, x4, x5)  =  U3_AGAG(x2, x4, x5)

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

(15) PiDPToQDPProof (SOUND transformation)

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

(16) Obligation:

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

SUBSET1_IN_AG(.(T35, T36)) → P13_IN_AGAG(T36, T35)
P13_IN_AGAG(T36, T35) → U3_AGAG(T36, T35, memberc15_in_ag(T36))
U3_AGAG(T36, T35, memberc15_out_ag(T37, T36)) → SUBSET1_IN_AG(.(T35, T36))
SUBSET1_IN_AG(.(T21, T22)) → SUBSET1_IN_AG(.(T21, T22))

The TRS R consists of the following rules:

memberc15_in_ag(.(T56, T57)) → memberc15_out_ag(T56, .(T56, T57))
memberc15_in_ag(.(T65, T66)) → U14_ag(T65, T66, memberc15_in_ag(T66))
U14_ag(T65, T66, memberc15_out_ag(T67, T66)) → memberc15_out_ag(T67, .(T65, T66))

The set Q consists of the following terms:

memberc15_in_ag(x0)
U14_ag(x0, x1, x2)

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

(17) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P13_IN_AGAG(T36, T35) → U3_AGAG(T36, T35, memberc15_in_ag(T36)) at position [2] we obtained the following new rules [LPAR04]:

P13_IN_AGAG(.(x0, x1), y1) → U3_AGAG(.(x0, x1), y1, memberc15_out_ag(x0, .(x0, x1)))
P13_IN_AGAG(.(x0, x1), y1) → U3_AGAG(.(x0, x1), y1, U14_ag(x0, x1, memberc15_in_ag(x1)))

(18) Obligation:

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

SUBSET1_IN_AG(.(T35, T36)) → P13_IN_AGAG(T36, T35)
U3_AGAG(T36, T35, memberc15_out_ag(T37, T36)) → SUBSET1_IN_AG(.(T35, T36))
SUBSET1_IN_AG(.(T21, T22)) → SUBSET1_IN_AG(.(T21, T22))
P13_IN_AGAG(.(x0, x1), y1) → U3_AGAG(.(x0, x1), y1, memberc15_out_ag(x0, .(x0, x1)))
P13_IN_AGAG(.(x0, x1), y1) → U3_AGAG(.(x0, x1), y1, U14_ag(x0, x1, memberc15_in_ag(x1)))

The TRS R consists of the following rules:

memberc15_in_ag(.(T56, T57)) → memberc15_out_ag(T56, .(T56, T57))
memberc15_in_ag(.(T65, T66)) → U14_ag(T65, T66, memberc15_in_ag(T66))
U14_ag(T65, T66, memberc15_out_ag(T67, T66)) → memberc15_out_ag(T67, .(T65, T66))

The set Q consists of the following terms:

memberc15_in_ag(x0)
U14_ag(x0, x1, x2)

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

(19) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_AGAG(T36, T35, memberc15_out_ag(T37, T36)) → SUBSET1_IN_AG(.(T35, T36)) we obtained the following new rules [LPAR04]:

U3_AGAG(.(z0, z1), z2, memberc15_out_ag(z0, .(z0, z1))) → SUBSET1_IN_AG(.(z2, .(z0, z1)))
U3_AGAG(.(z0, z1), z2, memberc15_out_ag(x2, .(z0, z1))) → SUBSET1_IN_AG(.(z2, .(z0, z1)))

(20) Obligation:

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

SUBSET1_IN_AG(.(T35, T36)) → P13_IN_AGAG(T36, T35)
SUBSET1_IN_AG(.(T21, T22)) → SUBSET1_IN_AG(.(T21, T22))
P13_IN_AGAG(.(x0, x1), y1) → U3_AGAG(.(x0, x1), y1, memberc15_out_ag(x0, .(x0, x1)))
P13_IN_AGAG(.(x0, x1), y1) → U3_AGAG(.(x0, x1), y1, U14_ag(x0, x1, memberc15_in_ag(x1)))
U3_AGAG(.(z0, z1), z2, memberc15_out_ag(z0, .(z0, z1))) → SUBSET1_IN_AG(.(z2, .(z0, z1)))
U3_AGAG(.(z0, z1), z2, memberc15_out_ag(x2, .(z0, z1))) → SUBSET1_IN_AG(.(z2, .(z0, z1)))

The TRS R consists of the following rules:

memberc15_in_ag(.(T56, T57)) → memberc15_out_ag(T56, .(T56, T57))
memberc15_in_ag(.(T65, T66)) → U14_ag(T65, T66, memberc15_in_ag(T66))
U14_ag(T65, T66, memberc15_out_ag(T67, T66)) → memberc15_out_ag(T67, .(T65, T66))

The set Q consists of the following terms:

memberc15_in_ag(x0)
U14_ag(x0, x1, x2)

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

(21) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule SUBSET1_IN_AG(.(T35, T36)) → P13_IN_AGAG(T36, T35) we obtained the following new rules [LPAR04]:

SUBSET1_IN_AG(.(x0, .(y_0, y_1))) → P13_IN_AGAG(.(y_0, y_1), x0)

(22) Obligation:

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

SUBSET1_IN_AG(.(T21, T22)) → SUBSET1_IN_AG(.(T21, T22))
P13_IN_AGAG(.(x0, x1), y1) → U3_AGAG(.(x0, x1), y1, memberc15_out_ag(x0, .(x0, x1)))
P13_IN_AGAG(.(x0, x1), y1) → U3_AGAG(.(x0, x1), y1, U14_ag(x0, x1, memberc15_in_ag(x1)))
U3_AGAG(.(z0, z1), z2, memberc15_out_ag(z0, .(z0, z1))) → SUBSET1_IN_AG(.(z2, .(z0, z1)))
U3_AGAG(.(z0, z1), z2, memberc15_out_ag(x2, .(z0, z1))) → SUBSET1_IN_AG(.(z2, .(z0, z1)))
SUBSET1_IN_AG(.(x0, .(y_0, y_1))) → P13_IN_AGAG(.(y_0, y_1), x0)

The TRS R consists of the following rules:

memberc15_in_ag(.(T56, T57)) → memberc15_out_ag(T56, .(T56, T57))
memberc15_in_ag(.(T65, T66)) → U14_ag(T65, T66, memberc15_in_ag(T66))
U14_ag(T65, T66, memberc15_out_ag(T67, T66)) → memberc15_out_ag(T67, .(T65, T66))

The set Q consists of the following terms:

memberc15_in_ag(x0)
U14_ag(x0, x1, x2)

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 = SUBSET1_IN_AG(.(T21, T22)) evaluates to t =SUBSET1_IN_AG(.(T21, T22))

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 SUBSET1_IN_AG(.(T21, T22)) to SUBSET1_IN_AG(.(T21, T22)).



(24) NO