Left Termination of the query pattern subset_in_2(g, a) 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 NonTerminationProof (⇔)
        ↳13 NO
    ↳14 PiDP
        ↳15 PiDPToQDPProof (⇐)
        ↳16 QDP
        ↳17 QDPSizeChangeProof (⇔)
        ↳18 YES

(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(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

member16(T64, .(T65, T67)) :- member16(T64, T67).
subset1(.(T20, T9), .(T20, T22)) :- subset1(T9, .(T20, T22)).
subset1(.(T33, T9), .(T37, T36)) :- member16(T33, T36).
subset1(.(T33, T9), .(T42, T43)) :- ','(memberc16(T33, T43), subset1(T9, .(T42, T43))).

Clauses:

subsetc1([], T4).
subsetc1(.(T20, T9), .(T20, T22)) :- subsetc1(T9, .(T20, T22)).
subsetc1(.(T33, T9), .(T42, T43)) :- ','(memberc16(T33, T43), subsetc1(T9, .(T42, T43))).
memberc16(T56, .(T56, T57)).
memberc16(T64, .(T65, T67)) :- memberc16(T64, T67).

Afs:

subset1(x1, x2)  =  subset1(x1)

(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: (b,f)
member16_in: (f,f)
memberc16_in: (f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

SUBSET1_IN_GA(.(T20, T9), .(T20, T22)) → U2_GA(T20, T9, T22, subset1_in_ga(T9, .(T20, T22)))
SUBSET1_IN_GA(.(T20, T9), .(T20, T22)) → SUBSET1_IN_GA(T9, .(T20, T22))
SUBSET1_IN_GA(.(T33, T9), .(T37, T36)) → U3_GA(T33, T9, T37, T36, member16_in_aa(T33, T36))
SUBSET1_IN_GA(.(T33, T9), .(T37, T36)) → MEMBER16_IN_AA(T33, T36)
MEMBER16_IN_AA(T64, .(T65, T67)) → U1_AA(T64, T65, T67, member16_in_aa(T64, T67))
MEMBER16_IN_AA(T64, .(T65, T67)) → MEMBER16_IN_AA(T64, T67)
SUBSET1_IN_GA(.(T33, T9), .(T42, T43)) → U4_GA(T33, T9, T42, T43, memberc16_in_aa(T33, T43))
U4_GA(T33, T9, T42, T43, memberc16_out_aa(T33, T43)) → U5_GA(T33, T9, T42, T43, subset1_in_ga(T9, .(T42, T43)))
U4_GA(T33, T9, T42, T43, memberc16_out_aa(T33, T43)) → SUBSET1_IN_GA(T9, .(T42, T43))

The TRS R consists of the following rules:

memberc16_in_aa(T56, .(T56, T57)) → memberc16_out_aa(T56, .(T56, T57))
memberc16_in_aa(T64, .(T65, T67)) → U10_aa(T64, T65, T67, memberc16_in_aa(T64, T67))
U10_aa(T64, T65, T67, memberc16_out_aa(T64, T67)) → memberc16_out_aa(T64, .(T65, T67))

The argument filtering Pi contains the following mapping:
subset1_in_ga(x1, x2)  =  subset1_in_ga(x1)
.(x1, x2)  =  .(x2)
member16_in_aa(x1, x2)  =  member16_in_aa
memberc16_in_aa(x1, x2)  =  memberc16_in_aa
memberc16_out_aa(x1, x2)  =  memberc16_out_aa
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
SUBSET1_IN_GA(x1, x2)  =  SUBSET1_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x2, x4)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x2, x5)
MEMBER16_IN_AA(x1, x2)  =  MEMBER16_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x2, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x2, 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_GA(.(T20, T9), .(T20, T22)) → U2_GA(T20, T9, T22, subset1_in_ga(T9, .(T20, T22)))
SUBSET1_IN_GA(.(T20, T9), .(T20, T22)) → SUBSET1_IN_GA(T9, .(T20, T22))
SUBSET1_IN_GA(.(T33, T9), .(T37, T36)) → U3_GA(T33, T9, T37, T36, member16_in_aa(T33, T36))
SUBSET1_IN_GA(.(T33, T9), .(T37, T36)) → MEMBER16_IN_AA(T33, T36)
MEMBER16_IN_AA(T64, .(T65, T67)) → U1_AA(T64, T65, T67, member16_in_aa(T64, T67))
MEMBER16_IN_AA(T64, .(T65, T67)) → MEMBER16_IN_AA(T64, T67)
SUBSET1_IN_GA(.(T33, T9), .(T42, T43)) → U4_GA(T33, T9, T42, T43, memberc16_in_aa(T33, T43))
U4_GA(T33, T9, T42, T43, memberc16_out_aa(T33, T43)) → U5_GA(T33, T9, T42, T43, subset1_in_ga(T9, .(T42, T43)))
U4_GA(T33, T9, T42, T43, memberc16_out_aa(T33, T43)) → SUBSET1_IN_GA(T9, .(T42, T43))

The TRS R consists of the following rules:

memberc16_in_aa(T56, .(T56, T57)) → memberc16_out_aa(T56, .(T56, T57))
memberc16_in_aa(T64, .(T65, T67)) → U10_aa(T64, T65, T67, memberc16_in_aa(T64, T67))
U10_aa(T64, T65, T67, memberc16_out_aa(T64, T67)) → memberc16_out_aa(T64, .(T65, T67))

The argument filtering Pi contains the following mapping:
subset1_in_ga(x1, x2)  =  subset1_in_ga(x1)
.(x1, x2)  =  .(x2)
member16_in_aa(x1, x2)  =  member16_in_aa
memberc16_in_aa(x1, x2)  =  memberc16_in_aa
memberc16_out_aa(x1, x2)  =  memberc16_out_aa
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
SUBSET1_IN_GA(x1, x2)  =  SUBSET1_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x2, x4)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x2, x5)
MEMBER16_IN_AA(x1, x2)  =  MEMBER16_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x2, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x2, 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 5 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

MEMBER16_IN_AA(T64, .(T65, T67)) → MEMBER16_IN_AA(T64, T67)

The TRS R consists of the following rules:

memberc16_in_aa(T56, .(T56, T57)) → memberc16_out_aa(T56, .(T56, T57))
memberc16_in_aa(T64, .(T65, T67)) → U10_aa(T64, T65, T67, memberc16_in_aa(T64, T67))
U10_aa(T64, T65, T67, memberc16_out_aa(T64, T67)) → memberc16_out_aa(T64, .(T65, T67))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
memberc16_in_aa(x1, x2)  =  memberc16_in_aa
memberc16_out_aa(x1, x2)  =  memberc16_out_aa
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
MEMBER16_IN_AA(x1, x2)  =  MEMBER16_IN_AA

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:

MEMBER16_IN_AA(T64, .(T65, T67)) → MEMBER16_IN_AA(T64, T67)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
MEMBER16_IN_AA(x1, x2)  =  MEMBER16_IN_AA

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:

MEMBER16_IN_AA → MEMBER16_IN_AA

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

(12) 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 = MEMBER16_IN_AA evaluates to t =MEMBER16_IN_AA

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 MEMBER16_IN_AA to MEMBER16_IN_AA.



(13) NO

(14) Obligation:

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

SUBSET1_IN_GA(.(T33, T9), .(T42, T43)) → U4_GA(T33, T9, T42, T43, memberc16_in_aa(T33, T43))
U4_GA(T33, T9, T42, T43, memberc16_out_aa(T33, T43)) → SUBSET1_IN_GA(T9, .(T42, T43))
SUBSET1_IN_GA(.(T20, T9), .(T20, T22)) → SUBSET1_IN_GA(T9, .(T20, T22))

The TRS R consists of the following rules:

memberc16_in_aa(T56, .(T56, T57)) → memberc16_out_aa(T56, .(T56, T57))
memberc16_in_aa(T64, .(T65, T67)) → U10_aa(T64, T65, T67, memberc16_in_aa(T64, T67))
U10_aa(T64, T65, T67, memberc16_out_aa(T64, T67)) → memberc16_out_aa(T64, .(T65, T67))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
memberc16_in_aa(x1, x2)  =  memberc16_in_aa
memberc16_out_aa(x1, x2)  =  memberc16_out_aa
U10_aa(x1, x2, x3, x4)  =  U10_aa(x4)
SUBSET1_IN_GA(x1, x2)  =  SUBSET1_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x2, 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_GA(.(T9)) → U4_GA(T9, memberc16_in_aa)
U4_GA(T9, memberc16_out_aa) → SUBSET1_IN_GA(T9)
SUBSET1_IN_GA(.(T9)) → SUBSET1_IN_GA(T9)

The TRS R consists of the following rules:

memberc16_in_aa → memberc16_out_aa
memberc16_in_aa → U10_aa(memberc16_in_aa)
U10_aa(memberc16_out_aa) → memberc16_out_aa

The set Q consists of the following terms:

memberc16_in_aa
U10_aa(x0)

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

(17) 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:

  • U4_GA(T9, memberc16_out_aa) → SUBSET1_IN_GA(T9)
    The graph contains the following edges 1 >= 1

  • SUBSET1_IN_GA(.(T9)) → SUBSET1_IN_GA(T9)
    The graph contains the following edges 1 > 1

  • SUBSET1_IN_GA(.(T9)) → U4_GA(T9, memberc16_in_aa)
    The graph contains the following edges 1 > 1

(18) YES