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 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
    ↳3 PrologToPiTRSProof (⇐)
        ↳4 PiTRS
        ↳5 DependencyPairsProof (⇔)
        ↳6 PiDP
        ↳7 DependencyGraphProof (⇔)
        ↳8 AND
            ↳9 PiDP
                ↳10 UsableRulesProof (⇔)
                ↳11 PiDP
                ↳12 PiDPToQDPProof (⇐)
                ↳13 QDP
                ↳14 QDPSizeChangeProof (⇔)
                ↳15 YES
            ↳16 PiDP
                ↳17 UsableRulesProof (⇔)
                ↳18 PiDP
                ↳19 PiDPToQDPProof (⇐)
                ↳20 QDP
                ↳21 Narrowing (⇐)
                ↳22 QDP
                ↳23 Instantiation (⇔)
                ↳24 QDP
                ↳25 NonTerminationProof (⇔)
                ↳26 NO
    ↳27 PrologToPiTRSProof (⇐)
        ↳28 PiTRS
        ↳29 DependencyPairsProof (⇔)
        ↳30 PiDP
        ↳31 DependencyGraphProof (⇔)
        ↳32 AND
            ↳33 PiDP
                ↳34 UsableRulesProof (⇔)
                ↳35 PiDP
                ↳36 PiDPToQDPProof (⇐)
                ↳37 QDP
                ↳38 QDPSizeChangeProof (⇔)
                ↳39 YES
            ↳40 PiDP
                ↳41 UsableRulesProof (⇔)
                ↳42 PiDP
                ↳43 PiDPToQDPProof (⇐)
                ↳44 QDP
                ↳45 Narrowing (⇐)
                ↳46 QDP
                ↳47 Instantiation (⇔)
                ↳48 QDP
                ↳49 NonTerminationProof (⇔)
                ↳50 NO

(0) Obligation:

Clauses:

member(X, .(Y, Xs)) :- member(X, Xs).
member(X, .(X, Xs)).
subset(.(X, Xs), Ys) :- ','(member(X, Ys), subset(Xs, Ys)).
subset([], Ys).

Queries:

subset(a,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

member10(T50, .(T48, T49)) :- member10(T50, T49).
member10(T58, .(T58, T59)).
subset1(.(T26, T27), .(T24, T25)) :- member10(T26, T25).
subset1(.(T26, T31), .(T24, T25)) :- ','(member10(T26, T25), subset1(T31, .(T24, T25))).
subset1(.(T72, T74), .(T72, T73)) :- subset1(T74, .(T72, T73)).
subset1([], T81).
subset1([], T83).

Queries:

subset1(a,g).

(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:
subset1_in: (f,b)
member10_in: (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

subset1_in_ag(.(T26, T27), .(T24, T25)) → U2_ag(T26, T27, T24, T25, member10_in_ag(T26, T25))
member10_in_ag(T50, .(T48, T49)) → U1_ag(T50, T48, T49, member10_in_ag(T50, T49))
member10_in_ag(T58, .(T58, T59)) → member10_out_ag(T58, .(T58, T59))
U1_ag(T50, T48, T49, member10_out_ag(T50, T49)) → member10_out_ag(T50, .(T48, T49))
U2_ag(T26, T27, T24, T25, member10_out_ag(T26, T25)) → subset1_out_ag(.(T26, T27), .(T24, T25))
subset1_in_ag(.(T26, T31), .(T24, T25)) → U3_ag(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_ag(T26, T31, T24, T25, member10_out_ag(T26, T25)) → U4_ag(T26, T31, T24, T25, subset1_in_ag(T31, .(T24, T25)))
subset1_in_ag(.(T72, T74), .(T72, T73)) → U5_ag(T72, T74, T73, subset1_in_ag(T74, .(T72, T73)))
subset1_in_ag([], T81) → subset1_out_ag([], T81)
U5_ag(T72, T74, T73, subset1_out_ag(T74, .(T72, T73))) → subset1_out_ag(.(T72, T74), .(T72, T73))
U4_ag(T26, T31, T24, T25, subset1_out_ag(T31, .(T24, T25))) → subset1_out_ag(.(T26, T31), .(T24, T25))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x5)
member10_in_ag(x1, x2)  =  member10_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
member10_out_ag(x1, x2)  =  member10_out_ag(x1)
subset1_out_ag(x1, x2)  =  subset1_out_ag
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

subset1_in_ag(.(T26, T27), .(T24, T25)) → U2_ag(T26, T27, T24, T25, member10_in_ag(T26, T25))
member10_in_ag(T50, .(T48, T49)) → U1_ag(T50, T48, T49, member10_in_ag(T50, T49))
member10_in_ag(T58, .(T58, T59)) → member10_out_ag(T58, .(T58, T59))
U1_ag(T50, T48, T49, member10_out_ag(T50, T49)) → member10_out_ag(T50, .(T48, T49))
U2_ag(T26, T27, T24, T25, member10_out_ag(T26, T25)) → subset1_out_ag(.(T26, T27), .(T24, T25))
subset1_in_ag(.(T26, T31), .(T24, T25)) → U3_ag(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_ag(T26, T31, T24, T25, member10_out_ag(T26, T25)) → U4_ag(T26, T31, T24, T25, subset1_in_ag(T31, .(T24, T25)))
subset1_in_ag(.(T72, T74), .(T72, T73)) → U5_ag(T72, T74, T73, subset1_in_ag(T74, .(T72, T73)))
subset1_in_ag([], T81) → subset1_out_ag([], T81)
U5_ag(T72, T74, T73, subset1_out_ag(T74, .(T72, T73))) → subset1_out_ag(.(T72, T74), .(T72, T73))
U4_ag(T26, T31, T24, T25, subset1_out_ag(T31, .(T24, T25))) → subset1_out_ag(.(T26, T31), .(T24, T25))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x5)
member10_in_ag(x1, x2)  =  member10_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
member10_out_ag(x1, x2)  =  member10_out_ag(x1)
subset1_out_ag(x1, x2)  =  subset1_out_ag
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)

(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:

SUBSET1_IN_AG(.(T26, T27), .(T24, T25)) → U2_AG(T26, T27, T24, T25, member10_in_ag(T26, T25))
SUBSET1_IN_AG(.(T26, T27), .(T24, T25)) → MEMBER10_IN_AG(T26, T25)
MEMBER10_IN_AG(T50, .(T48, T49)) → U1_AG(T50, T48, T49, member10_in_ag(T50, T49))
MEMBER10_IN_AG(T50, .(T48, T49)) → MEMBER10_IN_AG(T50, T49)
SUBSET1_IN_AG(.(T26, T31), .(T24, T25)) → U3_AG(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_AG(T26, T31, T24, T25, member10_out_ag(T26, T25)) → U4_AG(T26, T31, T24, T25, subset1_in_ag(T31, .(T24, T25)))
U3_AG(T26, T31, T24, T25, member10_out_ag(T26, T25)) → SUBSET1_IN_AG(T31, .(T24, T25))
SUBSET1_IN_AG(.(T72, T74), .(T72, T73)) → U5_AG(T72, T74, T73, subset1_in_ag(T74, .(T72, T73)))
SUBSET1_IN_AG(.(T72, T74), .(T72, T73)) → SUBSET1_IN_AG(T74, .(T72, T73))

The TRS R consists of the following rules:

subset1_in_ag(.(T26, T27), .(T24, T25)) → U2_ag(T26, T27, T24, T25, member10_in_ag(T26, T25))
member10_in_ag(T50, .(T48, T49)) → U1_ag(T50, T48, T49, member10_in_ag(T50, T49))
member10_in_ag(T58, .(T58, T59)) → member10_out_ag(T58, .(T58, T59))
U1_ag(T50, T48, T49, member10_out_ag(T50, T49)) → member10_out_ag(T50, .(T48, T49))
U2_ag(T26, T27, T24, T25, member10_out_ag(T26, T25)) → subset1_out_ag(.(T26, T27), .(T24, T25))
subset1_in_ag(.(T26, T31), .(T24, T25)) → U3_ag(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_ag(T26, T31, T24, T25, member10_out_ag(T26, T25)) → U4_ag(T26, T31, T24, T25, subset1_in_ag(T31, .(T24, T25)))
subset1_in_ag(.(T72, T74), .(T72, T73)) → U5_ag(T72, T74, T73, subset1_in_ag(T74, .(T72, T73)))
subset1_in_ag([], T81) → subset1_out_ag([], T81)
U5_ag(T72, T74, T73, subset1_out_ag(T74, .(T72, T73))) → subset1_out_ag(.(T72, T74), .(T72, T73))
U4_ag(T26, T31, T24, T25, subset1_out_ag(T31, .(T24, T25))) → subset1_out_ag(.(T26, T31), .(T24, T25))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x5)
member10_in_ag(x1, x2)  =  member10_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
member10_out_ag(x1, x2)  =  member10_out_ag(x1)
subset1_out_ag(x1, x2)  =  subset1_out_ag
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
SUBSET1_IN_AG(x1, x2)  =  SUBSET1_IN_AG(x2)
U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x5)
MEMBER10_IN_AG(x1, x2)  =  MEMBER10_IN_AG(x2)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x4)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x5)
U5_AG(x1, x2, x3, x4)  =  U5_AG(x4)

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

(6) Obligation:

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

SUBSET1_IN_AG(.(T26, T27), .(T24, T25)) → U2_AG(T26, T27, T24, T25, member10_in_ag(T26, T25))
SUBSET1_IN_AG(.(T26, T27), .(T24, T25)) → MEMBER10_IN_AG(T26, T25)
MEMBER10_IN_AG(T50, .(T48, T49)) → U1_AG(T50, T48, T49, member10_in_ag(T50, T49))
MEMBER10_IN_AG(T50, .(T48, T49)) → MEMBER10_IN_AG(T50, T49)
SUBSET1_IN_AG(.(T26, T31), .(T24, T25)) → U3_AG(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_AG(T26, T31, T24, T25, member10_out_ag(T26, T25)) → U4_AG(T26, T31, T24, T25, subset1_in_ag(T31, .(T24, T25)))
U3_AG(T26, T31, T24, T25, member10_out_ag(T26, T25)) → SUBSET1_IN_AG(T31, .(T24, T25))
SUBSET1_IN_AG(.(T72, T74), .(T72, T73)) → U5_AG(T72, T74, T73, subset1_in_ag(T74, .(T72, T73)))
SUBSET1_IN_AG(.(T72, T74), .(T72, T73)) → SUBSET1_IN_AG(T74, .(T72, T73))

The TRS R consists of the following rules:

subset1_in_ag(.(T26, T27), .(T24, T25)) → U2_ag(T26, T27, T24, T25, member10_in_ag(T26, T25))
member10_in_ag(T50, .(T48, T49)) → U1_ag(T50, T48, T49, member10_in_ag(T50, T49))
member10_in_ag(T58, .(T58, T59)) → member10_out_ag(T58, .(T58, T59))
U1_ag(T50, T48, T49, member10_out_ag(T50, T49)) → member10_out_ag(T50, .(T48, T49))
U2_ag(T26, T27, T24, T25, member10_out_ag(T26, T25)) → subset1_out_ag(.(T26, T27), .(T24, T25))
subset1_in_ag(.(T26, T31), .(T24, T25)) → U3_ag(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_ag(T26, T31, T24, T25, member10_out_ag(T26, T25)) → U4_ag(T26, T31, T24, T25, subset1_in_ag(T31, .(T24, T25)))
subset1_in_ag(.(T72, T74), .(T72, T73)) → U5_ag(T72, T74, T73, subset1_in_ag(T74, .(T72, T73)))
subset1_in_ag([], T81) → subset1_out_ag([], T81)
U5_ag(T72, T74, T73, subset1_out_ag(T74, .(T72, T73))) → subset1_out_ag(.(T72, T74), .(T72, T73))
U4_ag(T26, T31, T24, T25, subset1_out_ag(T31, .(T24, T25))) → subset1_out_ag(.(T26, T31), .(T24, T25))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x5)
member10_in_ag(x1, x2)  =  member10_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
member10_out_ag(x1, x2)  =  member10_out_ag(x1)
subset1_out_ag(x1, x2)  =  subset1_out_ag
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
SUBSET1_IN_AG(x1, x2)  =  SUBSET1_IN_AG(x2)
U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x5)
MEMBER10_IN_AG(x1, x2)  =  MEMBER10_IN_AG(x2)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x4)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x5)
U5_AG(x1, x2, x3, x4)  =  U5_AG(x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

MEMBER10_IN_AG(T50, .(T48, T49)) → MEMBER10_IN_AG(T50, T49)

The TRS R consists of the following rules:

subset1_in_ag(.(T26, T27), .(T24, T25)) → U2_ag(T26, T27, T24, T25, member10_in_ag(T26, T25))
member10_in_ag(T50, .(T48, T49)) → U1_ag(T50, T48, T49, member10_in_ag(T50, T49))
member10_in_ag(T58, .(T58, T59)) → member10_out_ag(T58, .(T58, T59))
U1_ag(T50, T48, T49, member10_out_ag(T50, T49)) → member10_out_ag(T50, .(T48, T49))
U2_ag(T26, T27, T24, T25, member10_out_ag(T26, T25)) → subset1_out_ag(.(T26, T27), .(T24, T25))
subset1_in_ag(.(T26, T31), .(T24, T25)) → U3_ag(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_ag(T26, T31, T24, T25, member10_out_ag(T26, T25)) → U4_ag(T26, T31, T24, T25, subset1_in_ag(T31, .(T24, T25)))
subset1_in_ag(.(T72, T74), .(T72, T73)) → U5_ag(T72, T74, T73, subset1_in_ag(T74, .(T72, T73)))
subset1_in_ag([], T81) → subset1_out_ag([], T81)
U5_ag(T72, T74, T73, subset1_out_ag(T74, .(T72, T73))) → subset1_out_ag(.(T72, T74), .(T72, T73))
U4_ag(T26, T31, T24, T25, subset1_out_ag(T31, .(T24, T25))) → subset1_out_ag(.(T26, T31), .(T24, T25))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x5)
member10_in_ag(x1, x2)  =  member10_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
member10_out_ag(x1, x2)  =  member10_out_ag(x1)
subset1_out_ag(x1, x2)  =  subset1_out_ag
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
MEMBER10_IN_AG(x1, x2)  =  MEMBER10_IN_AG(x2)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

MEMBER10_IN_AG(T50, .(T48, T49)) → MEMBER10_IN_AG(T50, T49)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
MEMBER10_IN_AG(x1, x2)  =  MEMBER10_IN_AG(x2)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

MEMBER10_IN_AG(.(T48, T49)) → MEMBER10_IN_AG(T49)

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

(14) 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:

  • MEMBER10_IN_AG(.(T48, T49)) → MEMBER10_IN_AG(T49)
    The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

SUBSET1_IN_AG(.(T26, T31), .(T24, T25)) → U3_AG(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_AG(T26, T31, T24, T25, member10_out_ag(T26, T25)) → SUBSET1_IN_AG(T31, .(T24, T25))
SUBSET1_IN_AG(.(T72, T74), .(T72, T73)) → SUBSET1_IN_AG(T74, .(T72, T73))

The TRS R consists of the following rules:

subset1_in_ag(.(T26, T27), .(T24, T25)) → U2_ag(T26, T27, T24, T25, member10_in_ag(T26, T25))
member10_in_ag(T50, .(T48, T49)) → U1_ag(T50, T48, T49, member10_in_ag(T50, T49))
member10_in_ag(T58, .(T58, T59)) → member10_out_ag(T58, .(T58, T59))
U1_ag(T50, T48, T49, member10_out_ag(T50, T49)) → member10_out_ag(T50, .(T48, T49))
U2_ag(T26, T27, T24, T25, member10_out_ag(T26, T25)) → subset1_out_ag(.(T26, T27), .(T24, T25))
subset1_in_ag(.(T26, T31), .(T24, T25)) → U3_ag(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_ag(T26, T31, T24, T25, member10_out_ag(T26, T25)) → U4_ag(T26, T31, T24, T25, subset1_in_ag(T31, .(T24, T25)))
subset1_in_ag(.(T72, T74), .(T72, T73)) → U5_ag(T72, T74, T73, subset1_in_ag(T74, .(T72, T73)))
subset1_in_ag([], T81) → subset1_out_ag([], T81)
U5_ag(T72, T74, T73, subset1_out_ag(T74, .(T72, T73))) → subset1_out_ag(.(T72, T74), .(T72, T73))
U4_ag(T26, T31, T24, T25, subset1_out_ag(T31, .(T24, T25))) → subset1_out_ag(.(T26, T31), .(T24, T25))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x5)
member10_in_ag(x1, x2)  =  member10_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
member10_out_ag(x1, x2)  =  member10_out_ag(x1)
subset1_out_ag(x1, x2)  =  subset1_out_ag
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
SUBSET1_IN_AG(x1, x2)  =  SUBSET1_IN_AG(x2)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

SUBSET1_IN_AG(.(T26, T31), .(T24, T25)) → U3_AG(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_AG(T26, T31, T24, T25, member10_out_ag(T26, T25)) → SUBSET1_IN_AG(T31, .(T24, T25))
SUBSET1_IN_AG(.(T72, T74), .(T72, T73)) → SUBSET1_IN_AG(T74, .(T72, T73))

The TRS R consists of the following rules:

member10_in_ag(T50, .(T48, T49)) → U1_ag(T50, T48, T49, member10_in_ag(T50, T49))
member10_in_ag(T58, .(T58, T59)) → member10_out_ag(T58, .(T58, T59))
U1_ag(T50, T48, T49, member10_out_ag(T50, T49)) → member10_out_ag(T50, .(T48, T49))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
member10_in_ag(x1, x2)  =  member10_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
member10_out_ag(x1, x2)  =  member10_out_ag(x1)
SUBSET1_IN_AG(x1, x2)  =  SUBSET1_IN_AG(x2)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

SUBSET1_IN_AG(.(T24, T25)) → U3_AG(T24, T25, member10_in_ag(T25))
U3_AG(T24, T25, member10_out_ag(T26)) → SUBSET1_IN_AG(.(T24, T25))
SUBSET1_IN_AG(.(T72, T73)) → SUBSET1_IN_AG(.(T72, T73))

The TRS R consists of the following rules:

member10_in_ag(.(T48, T49)) → U1_ag(member10_in_ag(T49))
member10_in_ag(.(T58, T59)) → member10_out_ag(T58)
U1_ag(member10_out_ag(T50)) → member10_out_ag(T50)

The set Q consists of the following terms:

member10_in_ag(x0)
U1_ag(x0)

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

(21) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SUBSET1_IN_AG(.(T24, T25)) → U3_AG(T24, T25, member10_in_ag(T25)) at position [2] we obtained the following new rules [LPAR04]:

SUBSET1_IN_AG(.(y0, .(x0, x1))) → U3_AG(y0, .(x0, x1), U1_ag(member10_in_ag(x1)))
SUBSET1_IN_AG(.(y0, .(x0, x1))) → U3_AG(y0, .(x0, x1), member10_out_ag(x0))

(22) Obligation:

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

U3_AG(T24, T25, member10_out_ag(T26)) → SUBSET1_IN_AG(.(T24, T25))
SUBSET1_IN_AG(.(T72, T73)) → SUBSET1_IN_AG(.(T72, T73))
SUBSET1_IN_AG(.(y0, .(x0, x1))) → U3_AG(y0, .(x0, x1), U1_ag(member10_in_ag(x1)))
SUBSET1_IN_AG(.(y0, .(x0, x1))) → U3_AG(y0, .(x0, x1), member10_out_ag(x0))

The TRS R consists of the following rules:

member10_in_ag(.(T48, T49)) → U1_ag(member10_in_ag(T49))
member10_in_ag(.(T58, T59)) → member10_out_ag(T58)
U1_ag(member10_out_ag(T50)) → member10_out_ag(T50)

The set Q consists of the following terms:

member10_in_ag(x0)
U1_ag(x0)

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

(23) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_AG(T24, T25, member10_out_ag(T26)) → SUBSET1_IN_AG(.(T24, T25)) we obtained the following new rules [LPAR04]:

U3_AG(z0, .(z1, z2), member10_out_ag(x2)) → SUBSET1_IN_AG(.(z0, .(z1, z2)))
U3_AG(z0, .(z1, z2), member10_out_ag(z1)) → SUBSET1_IN_AG(.(z0, .(z1, z2)))

(24) Obligation:

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

SUBSET1_IN_AG(.(T72, T73)) → SUBSET1_IN_AG(.(T72, T73))
SUBSET1_IN_AG(.(y0, .(x0, x1))) → U3_AG(y0, .(x0, x1), U1_ag(member10_in_ag(x1)))
SUBSET1_IN_AG(.(y0, .(x0, x1))) → U3_AG(y0, .(x0, x1), member10_out_ag(x0))
U3_AG(z0, .(z1, z2), member10_out_ag(x2)) → SUBSET1_IN_AG(.(z0, .(z1, z2)))
U3_AG(z0, .(z1, z2), member10_out_ag(z1)) → SUBSET1_IN_AG(.(z0, .(z1, z2)))

The TRS R consists of the following rules:

member10_in_ag(.(T48, T49)) → U1_ag(member10_in_ag(T49))
member10_in_ag(.(T58, T59)) → member10_out_ag(T58)
U1_ag(member10_out_ag(T50)) → member10_out_ag(T50)

The set Q consists of the following terms:

member10_in_ag(x0)
U1_ag(x0)

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 = SUBSET1_IN_AG(.(T72, T73)) evaluates to t =SUBSET1_IN_AG(.(T72, T73))

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(.(T72, T73)) to SUBSET1_IN_AG(.(T72, T73)).



(26) NO

(27) PrologToPiTRSProof (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)
member10_in: (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

subset1_in_ag(.(T26, T27), .(T24, T25)) → U2_ag(T26, T27, T24, T25, member10_in_ag(T26, T25))
member10_in_ag(T50, .(T48, T49)) → U1_ag(T50, T48, T49, member10_in_ag(T50, T49))
member10_in_ag(T58, .(T58, T59)) → member10_out_ag(T58, .(T58, T59))
U1_ag(T50, T48, T49, member10_out_ag(T50, T49)) → member10_out_ag(T50, .(T48, T49))
U2_ag(T26, T27, T24, T25, member10_out_ag(T26, T25)) → subset1_out_ag(.(T26, T27), .(T24, T25))
subset1_in_ag(.(T26, T31), .(T24, T25)) → U3_ag(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_ag(T26, T31, T24, T25, member10_out_ag(T26, T25)) → U4_ag(T26, T31, T24, T25, subset1_in_ag(T31, .(T24, T25)))
subset1_in_ag(.(T72, T74), .(T72, T73)) → U5_ag(T72, T74, T73, subset1_in_ag(T74, .(T72, T73)))
subset1_in_ag([], T81) → subset1_out_ag([], T81)
U5_ag(T72, T74, T73, subset1_out_ag(T74, .(T72, T73))) → subset1_out_ag(.(T72, T74), .(T72, T73))
U4_ag(T26, T31, T24, T25, subset1_out_ag(T31, .(T24, T25))) → subset1_out_ag(.(T26, T31), .(T24, T25))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x4, x5)
member10_in_ag(x1, x2)  =  member10_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)
member10_out_ag(x1, x2)  =  member10_out_ag(x1, x2)
subset1_out_ag(x1, x2)  =  subset1_out_ag(x2)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x3, x4, x5)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x1, x3, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(28) Obligation:

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

subset1_in_ag(.(T26, T27), .(T24, T25)) → U2_ag(T26, T27, T24, T25, member10_in_ag(T26, T25))
member10_in_ag(T50, .(T48, T49)) → U1_ag(T50, T48, T49, member10_in_ag(T50, T49))
member10_in_ag(T58, .(T58, T59)) → member10_out_ag(T58, .(T58, T59))
U1_ag(T50, T48, T49, member10_out_ag(T50, T49)) → member10_out_ag(T50, .(T48, T49))
U2_ag(T26, T27, T24, T25, member10_out_ag(T26, T25)) → subset1_out_ag(.(T26, T27), .(T24, T25))
subset1_in_ag(.(T26, T31), .(T24, T25)) → U3_ag(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_ag(T26, T31, T24, T25, member10_out_ag(T26, T25)) → U4_ag(T26, T31, T24, T25, subset1_in_ag(T31, .(T24, T25)))
subset1_in_ag(.(T72, T74), .(T72, T73)) → U5_ag(T72, T74, T73, subset1_in_ag(T74, .(T72, T73)))
subset1_in_ag([], T81) → subset1_out_ag([], T81)
U5_ag(T72, T74, T73, subset1_out_ag(T74, .(T72, T73))) → subset1_out_ag(.(T72, T74), .(T72, T73))
U4_ag(T26, T31, T24, T25, subset1_out_ag(T31, .(T24, T25))) → subset1_out_ag(.(T26, T31), .(T24, T25))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x4, x5)
member10_in_ag(x1, x2)  =  member10_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)
member10_out_ag(x1, x2)  =  member10_out_ag(x1, x2)
subset1_out_ag(x1, x2)  =  subset1_out_ag(x2)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x3, x4, x5)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x1, x3, x4)

(29) 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:

SUBSET1_IN_AG(.(T26, T27), .(T24, T25)) → U2_AG(T26, T27, T24, T25, member10_in_ag(T26, T25))
SUBSET1_IN_AG(.(T26, T27), .(T24, T25)) → MEMBER10_IN_AG(T26, T25)
MEMBER10_IN_AG(T50, .(T48, T49)) → U1_AG(T50, T48, T49, member10_in_ag(T50, T49))
MEMBER10_IN_AG(T50, .(T48, T49)) → MEMBER10_IN_AG(T50, T49)
SUBSET1_IN_AG(.(T26, T31), .(T24, T25)) → U3_AG(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_AG(T26, T31, T24, T25, member10_out_ag(T26, T25)) → U4_AG(T26, T31, T24, T25, subset1_in_ag(T31, .(T24, T25)))
U3_AG(T26, T31, T24, T25, member10_out_ag(T26, T25)) → SUBSET1_IN_AG(T31, .(T24, T25))
SUBSET1_IN_AG(.(T72, T74), .(T72, T73)) → U5_AG(T72, T74, T73, subset1_in_ag(T74, .(T72, T73)))
SUBSET1_IN_AG(.(T72, T74), .(T72, T73)) → SUBSET1_IN_AG(T74, .(T72, T73))

The TRS R consists of the following rules:

subset1_in_ag(.(T26, T27), .(T24, T25)) → U2_ag(T26, T27, T24, T25, member10_in_ag(T26, T25))
member10_in_ag(T50, .(T48, T49)) → U1_ag(T50, T48, T49, member10_in_ag(T50, T49))
member10_in_ag(T58, .(T58, T59)) → member10_out_ag(T58, .(T58, T59))
U1_ag(T50, T48, T49, member10_out_ag(T50, T49)) → member10_out_ag(T50, .(T48, T49))
U2_ag(T26, T27, T24, T25, member10_out_ag(T26, T25)) → subset1_out_ag(.(T26, T27), .(T24, T25))
subset1_in_ag(.(T26, T31), .(T24, T25)) → U3_ag(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_ag(T26, T31, T24, T25, member10_out_ag(T26, T25)) → U4_ag(T26, T31, T24, T25, subset1_in_ag(T31, .(T24, T25)))
subset1_in_ag(.(T72, T74), .(T72, T73)) → U5_ag(T72, T74, T73, subset1_in_ag(T74, .(T72, T73)))
subset1_in_ag([], T81) → subset1_out_ag([], T81)
U5_ag(T72, T74, T73, subset1_out_ag(T74, .(T72, T73))) → subset1_out_ag(.(T72, T74), .(T72, T73))
U4_ag(T26, T31, T24, T25, subset1_out_ag(T31, .(T24, T25))) → subset1_out_ag(.(T26, T31), .(T24, T25))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x4, x5)
member10_in_ag(x1, x2)  =  member10_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)
member10_out_ag(x1, x2)  =  member10_out_ag(x1, x2)
subset1_out_ag(x1, x2)  =  subset1_out_ag(x2)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x3, x4, x5)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x1, x3, x4)
SUBSET1_IN_AG(x1, x2)  =  SUBSET1_IN_AG(x2)
U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x3, x4, x5)
MEMBER10_IN_AG(x1, x2)  =  MEMBER10_IN_AG(x2)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x2, x3, x4)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x3, x4, x5)
U5_AG(x1, x2, x3, x4)  =  U5_AG(x1, x3, x4)

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

(30) Obligation:

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

SUBSET1_IN_AG(.(T26, T27), .(T24, T25)) → U2_AG(T26, T27, T24, T25, member10_in_ag(T26, T25))
SUBSET1_IN_AG(.(T26, T27), .(T24, T25)) → MEMBER10_IN_AG(T26, T25)
MEMBER10_IN_AG(T50, .(T48, T49)) → U1_AG(T50, T48, T49, member10_in_ag(T50, T49))
MEMBER10_IN_AG(T50, .(T48, T49)) → MEMBER10_IN_AG(T50, T49)
SUBSET1_IN_AG(.(T26, T31), .(T24, T25)) → U3_AG(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_AG(T26, T31, T24, T25, member10_out_ag(T26, T25)) → U4_AG(T26, T31, T24, T25, subset1_in_ag(T31, .(T24, T25)))
U3_AG(T26, T31, T24, T25, member10_out_ag(T26, T25)) → SUBSET1_IN_AG(T31, .(T24, T25))
SUBSET1_IN_AG(.(T72, T74), .(T72, T73)) → U5_AG(T72, T74, T73, subset1_in_ag(T74, .(T72, T73)))
SUBSET1_IN_AG(.(T72, T74), .(T72, T73)) → SUBSET1_IN_AG(T74, .(T72, T73))

The TRS R consists of the following rules:

subset1_in_ag(.(T26, T27), .(T24, T25)) → U2_ag(T26, T27, T24, T25, member10_in_ag(T26, T25))
member10_in_ag(T50, .(T48, T49)) → U1_ag(T50, T48, T49, member10_in_ag(T50, T49))
member10_in_ag(T58, .(T58, T59)) → member10_out_ag(T58, .(T58, T59))
U1_ag(T50, T48, T49, member10_out_ag(T50, T49)) → member10_out_ag(T50, .(T48, T49))
U2_ag(T26, T27, T24, T25, member10_out_ag(T26, T25)) → subset1_out_ag(.(T26, T27), .(T24, T25))
subset1_in_ag(.(T26, T31), .(T24, T25)) → U3_ag(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_ag(T26, T31, T24, T25, member10_out_ag(T26, T25)) → U4_ag(T26, T31, T24, T25, subset1_in_ag(T31, .(T24, T25)))
subset1_in_ag(.(T72, T74), .(T72, T73)) → U5_ag(T72, T74, T73, subset1_in_ag(T74, .(T72, T73)))
subset1_in_ag([], T81) → subset1_out_ag([], T81)
U5_ag(T72, T74, T73, subset1_out_ag(T74, .(T72, T73))) → subset1_out_ag(.(T72, T74), .(T72, T73))
U4_ag(T26, T31, T24, T25, subset1_out_ag(T31, .(T24, T25))) → subset1_out_ag(.(T26, T31), .(T24, T25))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x4, x5)
member10_in_ag(x1, x2)  =  member10_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)
member10_out_ag(x1, x2)  =  member10_out_ag(x1, x2)
subset1_out_ag(x1, x2)  =  subset1_out_ag(x2)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x3, x4, x5)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x1, x3, x4)
SUBSET1_IN_AG(x1, x2)  =  SUBSET1_IN_AG(x2)
U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x3, x4, x5)
MEMBER10_IN_AG(x1, x2)  =  MEMBER10_IN_AG(x2)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x2, x3, x4)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x3, x4, x5)
U5_AG(x1, x2, x3, x4)  =  U5_AG(x1, x3, x4)

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

(31) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(32) Complex Obligation (AND)

(33) Obligation:

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

MEMBER10_IN_AG(T50, .(T48, T49)) → MEMBER10_IN_AG(T50, T49)

The TRS R consists of the following rules:

subset1_in_ag(.(T26, T27), .(T24, T25)) → U2_ag(T26, T27, T24, T25, member10_in_ag(T26, T25))
member10_in_ag(T50, .(T48, T49)) → U1_ag(T50, T48, T49, member10_in_ag(T50, T49))
member10_in_ag(T58, .(T58, T59)) → member10_out_ag(T58, .(T58, T59))
U1_ag(T50, T48, T49, member10_out_ag(T50, T49)) → member10_out_ag(T50, .(T48, T49))
U2_ag(T26, T27, T24, T25, member10_out_ag(T26, T25)) → subset1_out_ag(.(T26, T27), .(T24, T25))
subset1_in_ag(.(T26, T31), .(T24, T25)) → U3_ag(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_ag(T26, T31, T24, T25, member10_out_ag(T26, T25)) → U4_ag(T26, T31, T24, T25, subset1_in_ag(T31, .(T24, T25)))
subset1_in_ag(.(T72, T74), .(T72, T73)) → U5_ag(T72, T74, T73, subset1_in_ag(T74, .(T72, T73)))
subset1_in_ag([], T81) → subset1_out_ag([], T81)
U5_ag(T72, T74, T73, subset1_out_ag(T74, .(T72, T73))) → subset1_out_ag(.(T72, T74), .(T72, T73))
U4_ag(T26, T31, T24, T25, subset1_out_ag(T31, .(T24, T25))) → subset1_out_ag(.(T26, T31), .(T24, T25))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x4, x5)
member10_in_ag(x1, x2)  =  member10_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)
member10_out_ag(x1, x2)  =  member10_out_ag(x1, x2)
subset1_out_ag(x1, x2)  =  subset1_out_ag(x2)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x3, x4, x5)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x1, x3, x4)
MEMBER10_IN_AG(x1, x2)  =  MEMBER10_IN_AG(x2)

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

(34) UsableRulesProof (EQUIVALENT transformation)

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

(35) Obligation:

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

MEMBER10_IN_AG(T50, .(T48, T49)) → MEMBER10_IN_AG(T50, T49)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
MEMBER10_IN_AG(x1, x2)  =  MEMBER10_IN_AG(x2)

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

(36) PiDPToQDPProof (SOUND transformation)

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

(37) Obligation:

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

MEMBER10_IN_AG(.(T48, T49)) → MEMBER10_IN_AG(T49)

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

(38) 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:

  • MEMBER10_IN_AG(.(T48, T49)) → MEMBER10_IN_AG(T49)
    The graph contains the following edges 1 > 1

(39) YES

(40) Obligation:

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

SUBSET1_IN_AG(.(T26, T31), .(T24, T25)) → U3_AG(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_AG(T26, T31, T24, T25, member10_out_ag(T26, T25)) → SUBSET1_IN_AG(T31, .(T24, T25))
SUBSET1_IN_AG(.(T72, T74), .(T72, T73)) → SUBSET1_IN_AG(T74, .(T72, T73))

The TRS R consists of the following rules:

subset1_in_ag(.(T26, T27), .(T24, T25)) → U2_ag(T26, T27, T24, T25, member10_in_ag(T26, T25))
member10_in_ag(T50, .(T48, T49)) → U1_ag(T50, T48, T49, member10_in_ag(T50, T49))
member10_in_ag(T58, .(T58, T59)) → member10_out_ag(T58, .(T58, T59))
U1_ag(T50, T48, T49, member10_out_ag(T50, T49)) → member10_out_ag(T50, .(T48, T49))
U2_ag(T26, T27, T24, T25, member10_out_ag(T26, T25)) → subset1_out_ag(.(T26, T27), .(T24, T25))
subset1_in_ag(.(T26, T31), .(T24, T25)) → U3_ag(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_ag(T26, T31, T24, T25, member10_out_ag(T26, T25)) → U4_ag(T26, T31, T24, T25, subset1_in_ag(T31, .(T24, T25)))
subset1_in_ag(.(T72, T74), .(T72, T73)) → U5_ag(T72, T74, T73, subset1_in_ag(T74, .(T72, T73)))
subset1_in_ag([], T81) → subset1_out_ag([], T81)
U5_ag(T72, T74, T73, subset1_out_ag(T74, .(T72, T73))) → subset1_out_ag(.(T72, T74), .(T72, T73))
U4_ag(T26, T31, T24, T25, subset1_out_ag(T31, .(T24, T25))) → subset1_out_ag(.(T26, T31), .(T24, T25))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
.(x1, x2)  =  .(x1, x2)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x4, x5)
member10_in_ag(x1, x2)  =  member10_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)
member10_out_ag(x1, x2)  =  member10_out_ag(x1, x2)
subset1_out_ag(x1, x2)  =  subset1_out_ag(x2)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x3, x4, x5)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x1, x3, x4)
SUBSET1_IN_AG(x1, x2)  =  SUBSET1_IN_AG(x2)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)

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

(41) UsableRulesProof (EQUIVALENT transformation)

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

(42) Obligation:

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

SUBSET1_IN_AG(.(T26, T31), .(T24, T25)) → U3_AG(T26, T31, T24, T25, member10_in_ag(T26, T25))
U3_AG(T26, T31, T24, T25, member10_out_ag(T26, T25)) → SUBSET1_IN_AG(T31, .(T24, T25))
SUBSET1_IN_AG(.(T72, T74), .(T72, T73)) → SUBSET1_IN_AG(T74, .(T72, T73))

The TRS R consists of the following rules:

member10_in_ag(T50, .(T48, T49)) → U1_ag(T50, T48, T49, member10_in_ag(T50, T49))
member10_in_ag(T58, .(T58, T59)) → member10_out_ag(T58, .(T58, T59))
U1_ag(T50, T48, T49, member10_out_ag(T50, T49)) → member10_out_ag(T50, .(T48, T49))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
member10_in_ag(x1, x2)  =  member10_in_ag(x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)
member10_out_ag(x1, x2)  =  member10_out_ag(x1, x2)
SUBSET1_IN_AG(x1, x2)  =  SUBSET1_IN_AG(x2)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)

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

(43) PiDPToQDPProof (SOUND transformation)

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

(44) Obligation:

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

SUBSET1_IN_AG(.(T24, T25)) → U3_AG(T24, T25, member10_in_ag(T25))
U3_AG(T24, T25, member10_out_ag(T26, T25)) → SUBSET1_IN_AG(.(T24, T25))
SUBSET1_IN_AG(.(T72, T73)) → SUBSET1_IN_AG(.(T72, T73))

The TRS R consists of the following rules:

member10_in_ag(.(T48, T49)) → U1_ag(T48, T49, member10_in_ag(T49))
member10_in_ag(.(T58, T59)) → member10_out_ag(T58, .(T58, T59))
U1_ag(T48, T49, member10_out_ag(T50, T49)) → member10_out_ag(T50, .(T48, T49))

The set Q consists of the following terms:

member10_in_ag(x0)
U1_ag(x0, x1, x2)

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

(45) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule SUBSET1_IN_AG(.(T24, T25)) → U3_AG(T24, T25, member10_in_ag(T25)) at position [2] we obtained the following new rules [LPAR04]:

SUBSET1_IN_AG(.(y0, .(x0, x1))) → U3_AG(y0, .(x0, x1), U1_ag(x0, x1, member10_in_ag(x1)))
SUBSET1_IN_AG(.(y0, .(x0, x1))) → U3_AG(y0, .(x0, x1), member10_out_ag(x0, .(x0, x1)))

(46) Obligation:

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

U3_AG(T24, T25, member10_out_ag(T26, T25)) → SUBSET1_IN_AG(.(T24, T25))
SUBSET1_IN_AG(.(T72, T73)) → SUBSET1_IN_AG(.(T72, T73))
SUBSET1_IN_AG(.(y0, .(x0, x1))) → U3_AG(y0, .(x0, x1), U1_ag(x0, x1, member10_in_ag(x1)))
SUBSET1_IN_AG(.(y0, .(x0, x1))) → U3_AG(y0, .(x0, x1), member10_out_ag(x0, .(x0, x1)))

The TRS R consists of the following rules:

member10_in_ag(.(T48, T49)) → U1_ag(T48, T49, member10_in_ag(T49))
member10_in_ag(.(T58, T59)) → member10_out_ag(T58, .(T58, T59))
U1_ag(T48, T49, member10_out_ag(T50, T49)) → member10_out_ag(T50, .(T48, T49))

The set Q consists of the following terms:

member10_in_ag(x0)
U1_ag(x0, x1, x2)

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

(47) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_AG(T24, T25, member10_out_ag(T26, T25)) → SUBSET1_IN_AG(.(T24, T25)) we obtained the following new rules [LPAR04]:

U3_AG(z0, .(z1, z2), member10_out_ag(x2, .(z1, z2))) → SUBSET1_IN_AG(.(z0, .(z1, z2)))
U3_AG(z0, .(z1, z2), member10_out_ag(z1, .(z1, z2))) → SUBSET1_IN_AG(.(z0, .(z1, z2)))

(48) Obligation:

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

SUBSET1_IN_AG(.(T72, T73)) → SUBSET1_IN_AG(.(T72, T73))
SUBSET1_IN_AG(.(y0, .(x0, x1))) → U3_AG(y0, .(x0, x1), U1_ag(x0, x1, member10_in_ag(x1)))
SUBSET1_IN_AG(.(y0, .(x0, x1))) → U3_AG(y0, .(x0, x1), member10_out_ag(x0, .(x0, x1)))
U3_AG(z0, .(z1, z2), member10_out_ag(x2, .(z1, z2))) → SUBSET1_IN_AG(.(z0, .(z1, z2)))
U3_AG(z0, .(z1, z2), member10_out_ag(z1, .(z1, z2))) → SUBSET1_IN_AG(.(z0, .(z1, z2)))

The TRS R consists of the following rules:

member10_in_ag(.(T48, T49)) → U1_ag(T48, T49, member10_in_ag(T49))
member10_in_ag(.(T58, T59)) → member10_out_ag(T58, .(T58, T59))
U1_ag(T48, T49, member10_out_ag(T50, T49)) → member10_out_ag(T50, .(T48, T49))

The set Q consists of the following terms:

member10_in_ag(x0)
U1_ag(x0, x1, x2)

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

(49) 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(.(T72, T73)) evaluates to t =SUBSET1_IN_AG(.(T72, T73))

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(.(T72, T73)) to SUBSET1_IN_AG(.(T72, T73)).



(50) NO