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 ForwardInstantiation (⇔)
                ↳26 QDP
                ↳27 NonTerminationProof (⇔)
                ↳28 NO
    ↳29 PrologToPiTRSProof (⇐)
        ↳30 PiTRS
        ↳31 DependencyPairsProof (⇔)
        ↳32 PiDP
        ↳33 DependencyGraphProof (⇔)
        ↳34 AND
            ↳35 PiDP
                ↳36 UsableRulesProof (⇔)
                ↳37 PiDP
                ↳38 PiDPToQDPProof (⇐)
                ↳39 QDP
                ↳40 QDPSizeChangeProof (⇔)
                ↳41 YES
            ↳42 PiDP
                ↳43 UsableRulesProof (⇔)
                ↳44 PiDP
                ↳45 PiDPToQDPProof (⇐)
                ↳46 QDP
                ↳47 Narrowing (⇐)
                ↳48 QDP
                ↳49 Instantiation (⇔)
                ↳50 QDP
                ↳51 ForwardInstantiation (⇔)
                ↳52 QDP
                ↳53 NonTerminationProof (⇔)
                ↳54 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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

member15(T56, .(T56, T57)).
member15(T67, .(T65, T66)) :- member15(T67, T66).
p13(T37, T36, T38, T35) :- member15(T37, T36).
p13(T37, T36, T43, T35) :- ','(member15(T37, T36), subset1(T43, .(T35, T36))).
subset1([], T4).
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).

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)
p13_in: (f,b,f,b)
member15_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([], T4) → subset1_out_ag([], T4)
subset1_in_ag(.(T21, T23), .(T21, T22)) → U5_ag(T21, T23, 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))
p13_in_agag(T37, T36, T38, T35) → U2_agag(T37, T36, T38, T35, member15_in_ag(T37, T36))
member15_in_ag(T56, .(T56, T57)) → member15_out_ag(T56, .(T56, T57))
member15_in_ag(T67, .(T65, T66)) → U1_ag(T67, T65, T66, member15_in_ag(T67, T66))
U1_ag(T67, T65, T66, member15_out_ag(T67, T66)) → member15_out_ag(T67, .(T65, T66))
U2_agag(T37, T36, T38, T35, member15_out_ag(T37, T36)) → p13_out_agag(T37, T36, T38, T35)
p13_in_agag(T37, T36, T43, T35) → U3_agag(T37, T36, T43, T35, member15_in_ag(T37, T36))
U3_agag(T37, T36, T43, T35, member15_out_ag(T37, T36)) → U4_agag(T37, T36, T43, T35, 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))
U8_ag(T108, T109, T106, T107, p13_out_agag(T108, T107, T109, T106)) → subset1_out_ag(.(T108, T109), .(T106, T107))
U7_ag(T92, T94, T93, subset1_out_ag(T94, .(T92, T93))) → subset1_out_ag(.(T92, T94), .(T92, T93))
U4_agag(T37, T36, T43, T35, subset1_out_ag(T43, .(T35, T36))) → p13_out_agag(T37, T36, T43, T35)
U6_ag(T37, T38, T35, T36, p13_out_agag(T37, T36, T38, T35)) → subset1_out_ag(.(T37, T38), .(T35, T36))
U5_ag(T21, T23, T22, subset1_out_ag(T23, .(T21, T22))) → subset1_out_ag(.(T21, T23), .(T21, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
subset1_out_ag(x1, x2)  =  subset1_out_ag
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
p13_in_agag(x1, x2, x3, x4)  =  p13_in_agag(x2, x4)
U2_agag(x1, x2, x3, x4, x5)  =  U2_agag(x5)
member15_in_ag(x1, x2)  =  member15_in_ag(x2)
member15_out_ag(x1, x2)  =  member15_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
p13_out_agag(x1, x2, x3, x4)  =  p13_out_agag(x1)
U3_agag(x1, x2, x3, x4, x5)  =  U3_agag(x2, x4, x5)
U4_agag(x1, x2, x3, x4, x5)  =  U4_agag(x1, x5)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x4)
U8_ag(x1, x2, x3, x4, x5)  =  U8_ag(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:

subset1_in_ag([], T4) → subset1_out_ag([], T4)
subset1_in_ag(.(T21, T23), .(T21, T22)) → U5_ag(T21, T23, 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))
p13_in_agag(T37, T36, T38, T35) → U2_agag(T37, T36, T38, T35, member15_in_ag(T37, T36))
member15_in_ag(T56, .(T56, T57)) → member15_out_ag(T56, .(T56, T57))
member15_in_ag(T67, .(T65, T66)) → U1_ag(T67, T65, T66, member15_in_ag(T67, T66))
U1_ag(T67, T65, T66, member15_out_ag(T67, T66)) → member15_out_ag(T67, .(T65, T66))
U2_agag(T37, T36, T38, T35, member15_out_ag(T37, T36)) → p13_out_agag(T37, T36, T38, T35)
p13_in_agag(T37, T36, T43, T35) → U3_agag(T37, T36, T43, T35, member15_in_ag(T37, T36))
U3_agag(T37, T36, T43, T35, member15_out_ag(T37, T36)) → U4_agag(T37, T36, T43, T35, 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))
U8_ag(T108, T109, T106, T107, p13_out_agag(T108, T107, T109, T106)) → subset1_out_ag(.(T108, T109), .(T106, T107))
U7_ag(T92, T94, T93, subset1_out_ag(T94, .(T92, T93))) → subset1_out_ag(.(T92, T94), .(T92, T93))
U4_agag(T37, T36, T43, T35, subset1_out_ag(T43, .(T35, T36))) → p13_out_agag(T37, T36, T43, T35)
U6_ag(T37, T38, T35, T36, p13_out_agag(T37, T36, T38, T35)) → subset1_out_ag(.(T37, T38), .(T35, T36))
U5_ag(T21, T23, T22, subset1_out_ag(T23, .(T21, T22))) → subset1_out_ag(.(T21, T23), .(T21, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
subset1_out_ag(x1, x2)  =  subset1_out_ag
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
p13_in_agag(x1, x2, x3, x4)  =  p13_in_agag(x2, x4)
U2_agag(x1, x2, x3, x4, x5)  =  U2_agag(x5)
member15_in_ag(x1, x2)  =  member15_in_ag(x2)
member15_out_ag(x1, x2)  =  member15_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
p13_out_agag(x1, x2, x3, x4)  =  p13_out_agag(x1)
U3_agag(x1, x2, x3, x4, x5)  =  U3_agag(x2, x4, x5)
U4_agag(x1, x2, x3, x4, x5)  =  U4_agag(x1, x5)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x4)
U8_ag(x1, x2, x3, x4, x5)  =  U8_ag(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:

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, member15_in_ag(T37, T36))
U3_AGAG(T37, T36, T43, T35, member15_out_ag(T37, T36)) → U4_AGAG(T37, T36, T43, T35, subset1_in_ag(T43, .(T35, T36)))
U3_AGAG(T37, T36, T43, T35, member15_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:

subset1_in_ag([], T4) → subset1_out_ag([], T4)
subset1_in_ag(.(T21, T23), .(T21, T22)) → U5_ag(T21, T23, 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))
p13_in_agag(T37, T36, T38, T35) → U2_agag(T37, T36, T38, T35, member15_in_ag(T37, T36))
member15_in_ag(T56, .(T56, T57)) → member15_out_ag(T56, .(T56, T57))
member15_in_ag(T67, .(T65, T66)) → U1_ag(T67, T65, T66, member15_in_ag(T67, T66))
U1_ag(T67, T65, T66, member15_out_ag(T67, T66)) → member15_out_ag(T67, .(T65, T66))
U2_agag(T37, T36, T38, T35, member15_out_ag(T37, T36)) → p13_out_agag(T37, T36, T38, T35)
p13_in_agag(T37, T36, T43, T35) → U3_agag(T37, T36, T43, T35, member15_in_ag(T37, T36))
U3_agag(T37, T36, T43, T35, member15_out_ag(T37, T36)) → U4_agag(T37, T36, T43, T35, 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))
U8_ag(T108, T109, T106, T107, p13_out_agag(T108, T107, T109, T106)) → subset1_out_ag(.(T108, T109), .(T106, T107))
U7_ag(T92, T94, T93, subset1_out_ag(T94, .(T92, T93))) → subset1_out_ag(.(T92, T94), .(T92, T93))
U4_agag(T37, T36, T43, T35, subset1_out_ag(T43, .(T35, T36))) → p13_out_agag(T37, T36, T43, T35)
U6_ag(T37, T38, T35, T36, p13_out_agag(T37, T36, T38, T35)) → subset1_out_ag(.(T37, T38), .(T35, T36))
U5_ag(T21, T23, T22, subset1_out_ag(T23, .(T21, T22))) → subset1_out_ag(.(T21, T23), .(T21, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
subset1_out_ag(x1, x2)  =  subset1_out_ag
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
p13_in_agag(x1, x2, x3, x4)  =  p13_in_agag(x2, x4)
U2_agag(x1, x2, x3, x4, x5)  =  U2_agag(x5)
member15_in_ag(x1, x2)  =  member15_in_ag(x2)
member15_out_ag(x1, x2)  =  member15_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
p13_out_agag(x1, x2, x3, x4)  =  p13_out_agag(x1)
U3_agag(x1, x2, x3, x4, x5)  =  U3_agag(x2, x4, x5)
U4_agag(x1, x2, x3, x4, x5)  =  U4_agag(x1, x5)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x4)
U8_ag(x1, x2, x3, x4, x5)  =  U8_ag(x5)
SUBSET1_IN_AG(x1, x2)  =  SUBSET1_IN_AG(x2)
U5_AG(x1, x2, x3, x4)  =  U5_AG(x4)
U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x5)
P13_IN_AGAG(x1, x2, x3, x4)  =  P13_IN_AGAG(x2, x4)
U2_AGAG(x1, x2, x3, x4, x5)  =  U2_AGAG(x5)
MEMBER15_IN_AG(x1, x2)  =  MEMBER15_IN_AG(x2)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x4)
U3_AGAG(x1, x2, x3, x4, x5)  =  U3_AGAG(x2, x4, x5)
U4_AGAG(x1, x2, x3, x4, x5)  =  U4_AGAG(x1, x5)
U7_AG(x1, x2, x3, x4)  =  U7_AG(x4)
U8_AG(x1, x2, x3, x4, x5)  =  U8_AG(x5)

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(.(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, member15_in_ag(T37, T36))
U3_AGAG(T37, T36, T43, T35, member15_out_ag(T37, T36)) → U4_AGAG(T37, T36, T43, T35, subset1_in_ag(T43, .(T35, T36)))
U3_AGAG(T37, T36, T43, T35, member15_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:

subset1_in_ag([], T4) → subset1_out_ag([], T4)
subset1_in_ag(.(T21, T23), .(T21, T22)) → U5_ag(T21, T23, 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))
p13_in_agag(T37, T36, T38, T35) → U2_agag(T37, T36, T38, T35, member15_in_ag(T37, T36))
member15_in_ag(T56, .(T56, T57)) → member15_out_ag(T56, .(T56, T57))
member15_in_ag(T67, .(T65, T66)) → U1_ag(T67, T65, T66, member15_in_ag(T67, T66))
U1_ag(T67, T65, T66, member15_out_ag(T67, T66)) → member15_out_ag(T67, .(T65, T66))
U2_agag(T37, T36, T38, T35, member15_out_ag(T37, T36)) → p13_out_agag(T37, T36, T38, T35)
p13_in_agag(T37, T36, T43, T35) → U3_agag(T37, T36, T43, T35, member15_in_ag(T37, T36))
U3_agag(T37, T36, T43, T35, member15_out_ag(T37, T36)) → U4_agag(T37, T36, T43, T35, 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))
U8_ag(T108, T109, T106, T107, p13_out_agag(T108, T107, T109, T106)) → subset1_out_ag(.(T108, T109), .(T106, T107))
U7_ag(T92, T94, T93, subset1_out_ag(T94, .(T92, T93))) → subset1_out_ag(.(T92, T94), .(T92, T93))
U4_agag(T37, T36, T43, T35, subset1_out_ag(T43, .(T35, T36))) → p13_out_agag(T37, T36, T43, T35)
U6_ag(T37, T38, T35, T36, p13_out_agag(T37, T36, T38, T35)) → subset1_out_ag(.(T37, T38), .(T35, T36))
U5_ag(T21, T23, T22, subset1_out_ag(T23, .(T21, T22))) → subset1_out_ag(.(T21, T23), .(T21, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
subset1_out_ag(x1, x2)  =  subset1_out_ag
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
p13_in_agag(x1, x2, x3, x4)  =  p13_in_agag(x2, x4)
U2_agag(x1, x2, x3, x4, x5)  =  U2_agag(x5)
member15_in_ag(x1, x2)  =  member15_in_ag(x2)
member15_out_ag(x1, x2)  =  member15_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
p13_out_agag(x1, x2, x3, x4)  =  p13_out_agag(x1)
U3_agag(x1, x2, x3, x4, x5)  =  U3_agag(x2, x4, x5)
U4_agag(x1, x2, x3, x4, x5)  =  U4_agag(x1, x5)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x4)
U8_ag(x1, x2, x3, x4, x5)  =  U8_ag(x5)
SUBSET1_IN_AG(x1, x2)  =  SUBSET1_IN_AG(x2)
U5_AG(x1, x2, x3, x4)  =  U5_AG(x4)
U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x5)
P13_IN_AGAG(x1, x2, x3, x4)  =  P13_IN_AGAG(x2, x4)
U2_AGAG(x1, x2, x3, x4, x5)  =  U2_AGAG(x5)
MEMBER15_IN_AG(x1, x2)  =  MEMBER15_IN_AG(x2)
U1_AG(x1, x2, x3, x4)  =  U1_AG(x4)
U3_AGAG(x1, x2, x3, x4, x5)  =  U3_AGAG(x2, x4, x5)
U4_AGAG(x1, x2, x3, x4, x5)  =  U4_AGAG(x1, x5)
U7_AG(x1, x2, x3, x4)  =  U7_AG(x4)
U8_AG(x1, x2, x3, x4, x5)  =  U8_AG(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Complex Obligation (AND)

(9) 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:

subset1_in_ag([], T4) → subset1_out_ag([], T4)
subset1_in_ag(.(T21, T23), .(T21, T22)) → U5_ag(T21, T23, 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))
p13_in_agag(T37, T36, T38, T35) → U2_agag(T37, T36, T38, T35, member15_in_ag(T37, T36))
member15_in_ag(T56, .(T56, T57)) → member15_out_ag(T56, .(T56, T57))
member15_in_ag(T67, .(T65, T66)) → U1_ag(T67, T65, T66, member15_in_ag(T67, T66))
U1_ag(T67, T65, T66, member15_out_ag(T67, T66)) → member15_out_ag(T67, .(T65, T66))
U2_agag(T37, T36, T38, T35, member15_out_ag(T37, T36)) → p13_out_agag(T37, T36, T38, T35)
p13_in_agag(T37, T36, T43, T35) → U3_agag(T37, T36, T43, T35, member15_in_ag(T37, T36))
U3_agag(T37, T36, T43, T35, member15_out_ag(T37, T36)) → U4_agag(T37, T36, T43, T35, 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))
U8_ag(T108, T109, T106, T107, p13_out_agag(T108, T107, T109, T106)) → subset1_out_ag(.(T108, T109), .(T106, T107))
U7_ag(T92, T94, T93, subset1_out_ag(T94, .(T92, T93))) → subset1_out_ag(.(T92, T94), .(T92, T93))
U4_agag(T37, T36, T43, T35, subset1_out_ag(T43, .(T35, T36))) → p13_out_agag(T37, T36, T43, T35)
U6_ag(T37, T38, T35, T36, p13_out_agag(T37, T36, T38, T35)) → subset1_out_ag(.(T37, T38), .(T35, T36))
U5_ag(T21, T23, T22, subset1_out_ag(T23, .(T21, T22))) → subset1_out_ag(.(T21, T23), .(T21, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
subset1_out_ag(x1, x2)  =  subset1_out_ag
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
p13_in_agag(x1, x2, x3, x4)  =  p13_in_agag(x2, x4)
U2_agag(x1, x2, x3, x4, x5)  =  U2_agag(x5)
member15_in_ag(x1, x2)  =  member15_in_ag(x2)
member15_out_ag(x1, x2)  =  member15_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
p13_out_agag(x1, x2, x3, x4)  =  p13_out_agag(x1)
U3_agag(x1, x2, x3, x4, x5)  =  U3_agag(x2, x4, x5)
U4_agag(x1, x2, x3, x4, x5)  =  U4_agag(x1, x5)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x4)
U8_ag(x1, x2, x3, x4, x5)  =  U8_ag(x5)
MEMBER15_IN_AG(x1, x2)  =  MEMBER15_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:

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

(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:

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

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:

  • MEMBER15_IN_AG(.(T65, T66)) → MEMBER15_IN_AG(T66)
    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(.(T37, T38), .(T35, T36)) → P13_IN_AGAG(T37, T36, T38, T35)
P13_IN_AGAG(T37, T36, T43, T35) → U3_AGAG(T37, T36, T43, T35, member15_in_ag(T37, T36))
U3_AGAG(T37, T36, T43, T35, member15_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:

subset1_in_ag([], T4) → subset1_out_ag([], T4)
subset1_in_ag(.(T21, T23), .(T21, T22)) → U5_ag(T21, T23, 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))
p13_in_agag(T37, T36, T38, T35) → U2_agag(T37, T36, T38, T35, member15_in_ag(T37, T36))
member15_in_ag(T56, .(T56, T57)) → member15_out_ag(T56, .(T56, T57))
member15_in_ag(T67, .(T65, T66)) → U1_ag(T67, T65, T66, member15_in_ag(T67, T66))
U1_ag(T67, T65, T66, member15_out_ag(T67, T66)) → member15_out_ag(T67, .(T65, T66))
U2_agag(T37, T36, T38, T35, member15_out_ag(T37, T36)) → p13_out_agag(T37, T36, T38, T35)
p13_in_agag(T37, T36, T43, T35) → U3_agag(T37, T36, T43, T35, member15_in_ag(T37, T36))
U3_agag(T37, T36, T43, T35, member15_out_ag(T37, T36)) → U4_agag(T37, T36, T43, T35, 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))
U8_ag(T108, T109, T106, T107, p13_out_agag(T108, T107, T109, T106)) → subset1_out_ag(.(T108, T109), .(T106, T107))
U7_ag(T92, T94, T93, subset1_out_ag(T94, .(T92, T93))) → subset1_out_ag(.(T92, T94), .(T92, T93))
U4_agag(T37, T36, T43, T35, subset1_out_ag(T43, .(T35, T36))) → p13_out_agag(T37, T36, T43, T35)
U6_ag(T37, T38, T35, T36, p13_out_agag(T37, T36, T38, T35)) → subset1_out_ag(.(T37, T38), .(T35, T36))
U5_ag(T21, T23, T22, subset1_out_ag(T23, .(T21, T22))) → subset1_out_ag(.(T21, T23), .(T21, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
subset1_out_ag(x1, x2)  =  subset1_out_ag
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
p13_in_agag(x1, x2, x3, x4)  =  p13_in_agag(x2, x4)
U2_agag(x1, x2, x3, x4, x5)  =  U2_agag(x5)
member15_in_ag(x1, x2)  =  member15_in_ag(x2)
member15_out_ag(x1, x2)  =  member15_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)
p13_out_agag(x1, x2, x3, x4)  =  p13_out_agag(x1)
U3_agag(x1, x2, x3, x4, x5)  =  U3_agag(x2, x4, x5)
U4_agag(x1, x2, x3, x4, x5)  =  U4_agag(x1, x5)
U7_ag(x1, x2, x3, x4)  =  U7_ag(x4)
U8_ag(x1, x2, x3, x4, x5)  =  U8_ag(x5)
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

(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(.(T37, T38), .(T35, T36)) → P13_IN_AGAG(T37, T36, T38, T35)
P13_IN_AGAG(T37, T36, T43, T35) → U3_AGAG(T37, T36, T43, T35, member15_in_ag(T37, T36))
U3_AGAG(T37, T36, T43, T35, member15_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:

member15_in_ag(T56, .(T56, T57)) → member15_out_ag(T56, .(T56, T57))
member15_in_ag(T67, .(T65, T66)) → U1_ag(T67, T65, T66, member15_in_ag(T67, T66))
U1_ag(T67, T65, T66, member15_out_ag(T67, T66)) → member15_out_ag(T67, .(T65, T66))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
member15_in_ag(x1, x2)  =  member15_in_ag(x2)
member15_out_ag(x1, x2)  =  member15_out_ag(x1)
U1_ag(x1, x2, x3, x4)  =  U1_ag(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

(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(.(T35, T36)) → P13_IN_AGAG(T36, T35)
P13_IN_AGAG(T36, T35) → U3_AGAG(T36, T35, member15_in_ag(T36))
U3_AGAG(T36, T35, member15_out_ag(T37)) → SUBSET1_IN_AG(.(T35, T36))
SUBSET1_IN_AG(.(T21, T22)) → SUBSET1_IN_AG(.(T21, T22))

The TRS R consists of the following rules:

member15_in_ag(.(T56, T57)) → member15_out_ag(T56)
member15_in_ag(.(T65, T66)) → U1_ag(member15_in_ag(T66))
U1_ag(member15_out_ag(T67)) → member15_out_ag(T67)

The set Q consists of the following terms:

member15_in_ag(x0)
U1_ag(x0)

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

(21) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P13_IN_AGAG(T36, T35) → U3_AGAG(T36, T35, member15_in_ag(T36)) at position [2] we obtained the following new rules [LPAR04]:

P13_IN_AGAG(.(x0, x1), y1) → U3_AGAG(.(x0, x1), y1, member15_out_ag(x0))
P13_IN_AGAG(.(x0, x1), y1) → U3_AGAG(.(x0, x1), y1, U1_ag(member15_in_ag(x1)))

(22) 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, member15_out_ag(T37)) → 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, member15_out_ag(x0))
P13_IN_AGAG(.(x0, x1), y1) → U3_AGAG(.(x0, x1), y1, U1_ag(member15_in_ag(x1)))

The TRS R consists of the following rules:

member15_in_ag(.(T56, T57)) → member15_out_ag(T56)
member15_in_ag(.(T65, T66)) → U1_ag(member15_in_ag(T66))
U1_ag(member15_out_ag(T67)) → member15_out_ag(T67)

The set Q consists of the following terms:

member15_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_AGAG(T36, T35, member15_out_ag(T37)) → SUBSET1_IN_AG(.(T35, T36)) we obtained the following new rules [LPAR04]:

U3_AGAG(.(z0, z1), z2, member15_out_ag(z0)) → SUBSET1_IN_AG(.(z2, .(z0, z1)))
U3_AGAG(.(z0, z1), z2, member15_out_ag(x2)) → SUBSET1_IN_AG(.(z2, .(z0, z1)))

(24) 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, member15_out_ag(x0))
P13_IN_AGAG(.(x0, x1), y1) → U3_AGAG(.(x0, x1), y1, U1_ag(member15_in_ag(x1)))
U3_AGAG(.(z0, z1), z2, member15_out_ag(z0)) → SUBSET1_IN_AG(.(z2, .(z0, z1)))
U3_AGAG(.(z0, z1), z2, member15_out_ag(x2)) → SUBSET1_IN_AG(.(z2, .(z0, z1)))

The TRS R consists of the following rules:

member15_in_ag(.(T56, T57)) → member15_out_ag(T56)
member15_in_ag(.(T65, T66)) → U1_ag(member15_in_ag(T66))
U1_ag(member15_out_ag(T67)) → member15_out_ag(T67)

The set Q consists of the following terms:

member15_in_ag(x0)
U1_ag(x0)

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

(25) 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)

(26) 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, member15_out_ag(x0))
P13_IN_AGAG(.(x0, x1), y1) → U3_AGAG(.(x0, x1), y1, U1_ag(member15_in_ag(x1)))
U3_AGAG(.(z0, z1), z2, member15_out_ag(z0)) → SUBSET1_IN_AG(.(z2, .(z0, z1)))
U3_AGAG(.(z0, z1), z2, member15_out_ag(x2)) → 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:

member15_in_ag(.(T56, T57)) → member15_out_ag(T56)
member15_in_ag(.(T65, T66)) → U1_ag(member15_in_ag(T66))
U1_ag(member15_out_ag(T67)) → member15_out_ag(T67)

The set Q consists of the following terms:

member15_in_ag(x0)
U1_ag(x0)

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

(27) 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)).



(28) NO

(29) 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)
p13_in: (f,b,f,b)
member15_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([], T4) → subset1_out_ag([], T4)
subset1_in_ag(.(T21, T23), .(T21, T22)) → U5_ag(T21, T23, 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))
p13_in_agag(T37, T36, T38, T35) → U2_agag(T37, T36, T38, T35, member15_in_ag(T37, T36))
member15_in_ag(T56, .(T56, T57)) → member15_out_ag(T56, .(T56, T57))
member15_in_ag(T67, .(T65, T66)) → U1_ag(T67, T65, T66, member15_in_ag(T67, T66))
U1_ag(T67, T65, T66, member15_out_ag(T67, T66)) → member15_out_ag(T67, .(T65, T66))
U2_agag(T37, T36, T38, T35, member15_out_ag(T37, T36)) → p13_out_agag(T37, T36, T38, T35)
p13_in_agag(T37, T36, T43, T35) → U3_agag(T37, T36, T43, T35, member15_in_ag(T37, T36))
U3_agag(T37, T36, T43, T35, member15_out_ag(T37, T36)) → U4_agag(T37, T36, T43, T35, 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))
U8_ag(T108, T109, T106, T107, p13_out_agag(T108, T107, T109, T106)) → subset1_out_ag(.(T108, T109), .(T106, T107))
U7_ag(T92, T94, T93, subset1_out_ag(T94, .(T92, T93))) → subset1_out_ag(.(T92, T94), .(T92, T93))
U4_agag(T37, T36, T43, T35, subset1_out_ag(T43, .(T35, T36))) → p13_out_agag(T37, T36, T43, T35)
U6_ag(T37, T38, T35, T36, p13_out_agag(T37, T36, T38, T35)) → subset1_out_ag(.(T37, T38), .(T35, T36))
U5_ag(T21, T23, T22, subset1_out_ag(T23, .(T21, T22))) → subset1_out_ag(.(T21, T23), .(T21, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
subset1_out_ag(x1, x2)  =  subset1_out_ag(x2)
.(x1, x2)  =  .(x1, 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)
member15_out_ag(x1, x2)  =  member15_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)
p13_out_agag(x1, x2, x3, x4)  =  p13_out_agag(x1, x2, 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)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(30) Obligation:

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

subset1_in_ag([], T4) → subset1_out_ag([], T4)
subset1_in_ag(.(T21, T23), .(T21, T22)) → U5_ag(T21, T23, 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))
p13_in_agag(T37, T36, T38, T35) → U2_agag(T37, T36, T38, T35, member15_in_ag(T37, T36))
member15_in_ag(T56, .(T56, T57)) → member15_out_ag(T56, .(T56, T57))
member15_in_ag(T67, .(T65, T66)) → U1_ag(T67, T65, T66, member15_in_ag(T67, T66))
U1_ag(T67, T65, T66, member15_out_ag(T67, T66)) → member15_out_ag(T67, .(T65, T66))
U2_agag(T37, T36, T38, T35, member15_out_ag(T37, T36)) → p13_out_agag(T37, T36, T38, T35)
p13_in_agag(T37, T36, T43, T35) → U3_agag(T37, T36, T43, T35, member15_in_ag(T37, T36))
U3_agag(T37, T36, T43, T35, member15_out_ag(T37, T36)) → U4_agag(T37, T36, T43, T35, 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))
U8_ag(T108, T109, T106, T107, p13_out_agag(T108, T107, T109, T106)) → subset1_out_ag(.(T108, T109), .(T106, T107))
U7_ag(T92, T94, T93, subset1_out_ag(T94, .(T92, T93))) → subset1_out_ag(.(T92, T94), .(T92, T93))
U4_agag(T37, T36, T43, T35, subset1_out_ag(T43, .(T35, T36))) → p13_out_agag(T37, T36, T43, T35)
U6_ag(T37, T38, T35, T36, p13_out_agag(T37, T36, T38, T35)) → subset1_out_ag(.(T37, T38), .(T35, T36))
U5_ag(T21, T23, T22, subset1_out_ag(T23, .(T21, T22))) → subset1_out_ag(.(T21, T23), .(T21, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
subset1_out_ag(x1, x2)  =  subset1_out_ag(x2)
.(x1, x2)  =  .(x1, 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)
member15_out_ag(x1, x2)  =  member15_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)
p13_out_agag(x1, x2, x3, x4)  =  p13_out_agag(x1, x2, 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)

(31) 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(.(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, member15_in_ag(T37, T36))
U3_AGAG(T37, T36, T43, T35, member15_out_ag(T37, T36)) → U4_AGAG(T37, T36, T43, T35, subset1_in_ag(T43, .(T35, T36)))
U3_AGAG(T37, T36, T43, T35, member15_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:

subset1_in_ag([], T4) → subset1_out_ag([], T4)
subset1_in_ag(.(T21, T23), .(T21, T22)) → U5_ag(T21, T23, 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))
p13_in_agag(T37, T36, T38, T35) → U2_agag(T37, T36, T38, T35, member15_in_ag(T37, T36))
member15_in_ag(T56, .(T56, T57)) → member15_out_ag(T56, .(T56, T57))
member15_in_ag(T67, .(T65, T66)) → U1_ag(T67, T65, T66, member15_in_ag(T67, T66))
U1_ag(T67, T65, T66, member15_out_ag(T67, T66)) → member15_out_ag(T67, .(T65, T66))
U2_agag(T37, T36, T38, T35, member15_out_ag(T37, T36)) → p13_out_agag(T37, T36, T38, T35)
p13_in_agag(T37, T36, T43, T35) → U3_agag(T37, T36, T43, T35, member15_in_ag(T37, T36))
U3_agag(T37, T36, T43, T35, member15_out_ag(T37, T36)) → U4_agag(T37, T36, T43, T35, 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))
U8_ag(T108, T109, T106, T107, p13_out_agag(T108, T107, T109, T106)) → subset1_out_ag(.(T108, T109), .(T106, T107))
U7_ag(T92, T94, T93, subset1_out_ag(T94, .(T92, T93))) → subset1_out_ag(.(T92, T94), .(T92, T93))
U4_agag(T37, T36, T43, T35, subset1_out_ag(T43, .(T35, T36))) → p13_out_agag(T37, T36, T43, T35)
U6_ag(T37, T38, T35, T36, p13_out_agag(T37, T36, T38, T35)) → subset1_out_ag(.(T37, T38), .(T35, T36))
U5_ag(T21, T23, T22, subset1_out_ag(T23, .(T21, T22))) → subset1_out_ag(.(T21, T23), .(T21, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
subset1_out_ag(x1, x2)  =  subset1_out_ag(x2)
.(x1, x2)  =  .(x1, 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)
member15_out_ag(x1, x2)  =  member15_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)
p13_out_agag(x1, x2, x3, x4)  =  p13_out_agag(x1, x2, 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)
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

(32) 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, member15_in_ag(T37, T36))
U3_AGAG(T37, T36, T43, T35, member15_out_ag(T37, T36)) → U4_AGAG(T37, T36, T43, T35, subset1_in_ag(T43, .(T35, T36)))
U3_AGAG(T37, T36, T43, T35, member15_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:

subset1_in_ag([], T4) → subset1_out_ag([], T4)
subset1_in_ag(.(T21, T23), .(T21, T22)) → U5_ag(T21, T23, 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))
p13_in_agag(T37, T36, T38, T35) → U2_agag(T37, T36, T38, T35, member15_in_ag(T37, T36))
member15_in_ag(T56, .(T56, T57)) → member15_out_ag(T56, .(T56, T57))
member15_in_ag(T67, .(T65, T66)) → U1_ag(T67, T65, T66, member15_in_ag(T67, T66))
U1_ag(T67, T65, T66, member15_out_ag(T67, T66)) → member15_out_ag(T67, .(T65, T66))
U2_agag(T37, T36, T38, T35, member15_out_ag(T37, T36)) → p13_out_agag(T37, T36, T38, T35)
p13_in_agag(T37, T36, T43, T35) → U3_agag(T37, T36, T43, T35, member15_in_ag(T37, T36))
U3_agag(T37, T36, T43, T35, member15_out_ag(T37, T36)) → U4_agag(T37, T36, T43, T35, 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))
U8_ag(T108, T109, T106, T107, p13_out_agag(T108, T107, T109, T106)) → subset1_out_ag(.(T108, T109), .(T106, T107))
U7_ag(T92, T94, T93, subset1_out_ag(T94, .(T92, T93))) → subset1_out_ag(.(T92, T94), .(T92, T93))
U4_agag(T37, T36, T43, T35, subset1_out_ag(T43, .(T35, T36))) → p13_out_agag(T37, T36, T43, T35)
U6_ag(T37, T38, T35, T36, p13_out_agag(T37, T36, T38, T35)) → subset1_out_ag(.(T37, T38), .(T35, T36))
U5_ag(T21, T23, T22, subset1_out_ag(T23, .(T21, T22))) → subset1_out_ag(.(T21, T23), .(T21, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
subset1_out_ag(x1, x2)  =  subset1_out_ag(x2)
.(x1, x2)  =  .(x1, 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)
member15_out_ag(x1, x2)  =  member15_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)
p13_out_agag(x1, x2, x3, x4)  =  p13_out_agag(x1, x2, 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)
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

(33) DependencyGraphProof (EQUIVALENT transformation)

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

(34) Complex Obligation (AND)

(35) 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:

subset1_in_ag([], T4) → subset1_out_ag([], T4)
subset1_in_ag(.(T21, T23), .(T21, T22)) → U5_ag(T21, T23, 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))
p13_in_agag(T37, T36, T38, T35) → U2_agag(T37, T36, T38, T35, member15_in_ag(T37, T36))
member15_in_ag(T56, .(T56, T57)) → member15_out_ag(T56, .(T56, T57))
member15_in_ag(T67, .(T65, T66)) → U1_ag(T67, T65, T66, member15_in_ag(T67, T66))
U1_ag(T67, T65, T66, member15_out_ag(T67, T66)) → member15_out_ag(T67, .(T65, T66))
U2_agag(T37, T36, T38, T35, member15_out_ag(T37, T36)) → p13_out_agag(T37, T36, T38, T35)
p13_in_agag(T37, T36, T43, T35) → U3_agag(T37, T36, T43, T35, member15_in_ag(T37, T36))
U3_agag(T37, T36, T43, T35, member15_out_ag(T37, T36)) → U4_agag(T37, T36, T43, T35, 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))
U8_ag(T108, T109, T106, T107, p13_out_agag(T108, T107, T109, T106)) → subset1_out_ag(.(T108, T109), .(T106, T107))
U7_ag(T92, T94, T93, subset1_out_ag(T94, .(T92, T93))) → subset1_out_ag(.(T92, T94), .(T92, T93))
U4_agag(T37, T36, T43, T35, subset1_out_ag(T43, .(T35, T36))) → p13_out_agag(T37, T36, T43, T35)
U6_ag(T37, T38, T35, T36, p13_out_agag(T37, T36, T38, T35)) → subset1_out_ag(.(T37, T38), .(T35, T36))
U5_ag(T21, T23, T22, subset1_out_ag(T23, .(T21, T22))) → subset1_out_ag(.(T21, T23), .(T21, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
subset1_out_ag(x1, x2)  =  subset1_out_ag(x2)
.(x1, x2)  =  .(x1, 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)
member15_out_ag(x1, x2)  =  member15_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)
p13_out_agag(x1, x2, x3, x4)  =  p13_out_agag(x1, x2, 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)
MEMBER15_IN_AG(x1, x2)  =  MEMBER15_IN_AG(x2)

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

(36) UsableRulesProof (EQUIVALENT transformation)

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

(37) 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

(38) PiDPToQDPProof (SOUND transformation)

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

(39) 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.

(40) 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

(41) YES

(42) 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, member15_in_ag(T37, T36))
U3_AGAG(T37, T36, T43, T35, member15_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:

subset1_in_ag([], T4) → subset1_out_ag([], T4)
subset1_in_ag(.(T21, T23), .(T21, T22)) → U5_ag(T21, T23, 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))
p13_in_agag(T37, T36, T38, T35) → U2_agag(T37, T36, T38, T35, member15_in_ag(T37, T36))
member15_in_ag(T56, .(T56, T57)) → member15_out_ag(T56, .(T56, T57))
member15_in_ag(T67, .(T65, T66)) → U1_ag(T67, T65, T66, member15_in_ag(T67, T66))
U1_ag(T67, T65, T66, member15_out_ag(T67, T66)) → member15_out_ag(T67, .(T65, T66))
U2_agag(T37, T36, T38, T35, member15_out_ag(T37, T36)) → p13_out_agag(T37, T36, T38, T35)
p13_in_agag(T37, T36, T43, T35) → U3_agag(T37, T36, T43, T35, member15_in_ag(T37, T36))
U3_agag(T37, T36, T43, T35, member15_out_ag(T37, T36)) → U4_agag(T37, T36, T43, T35, 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))
U8_ag(T108, T109, T106, T107, p13_out_agag(T108, T107, T109, T106)) → subset1_out_ag(.(T108, T109), .(T106, T107))
U7_ag(T92, T94, T93, subset1_out_ag(T94, .(T92, T93))) → subset1_out_ag(.(T92, T94), .(T92, T93))
U4_agag(T37, T36, T43, T35, subset1_out_ag(T43, .(T35, T36))) → p13_out_agag(T37, T36, T43, T35)
U6_ag(T37, T38, T35, T36, p13_out_agag(T37, T36, T38, T35)) → subset1_out_ag(.(T37, T38), .(T35, T36))
U5_ag(T21, T23, T22, subset1_out_ag(T23, .(T21, T22))) → subset1_out_ag(.(T21, T23), .(T21, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ag(x1, x2)  =  subset1_in_ag(x2)
subset1_out_ag(x1, x2)  =  subset1_out_ag(x2)
.(x1, x2)  =  .(x1, 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)
member15_out_ag(x1, x2)  =  member15_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)
p13_out_agag(x1, x2, x3, x4)  =  p13_out_agag(x1, x2, 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)
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

(43) UsableRulesProof (EQUIVALENT transformation)

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

(44) 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, member15_in_ag(T37, T36))
U3_AGAG(T37, T36, T43, T35, member15_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:

member15_in_ag(T56, .(T56, T57)) → member15_out_ag(T56, .(T56, T57))
member15_in_ag(T67, .(T65, T66)) → U1_ag(T67, T65, T66, member15_in_ag(T67, T66))
U1_ag(T67, T65, T66, member15_out_ag(T67, T66)) → member15_out_ag(T67, .(T65, T66))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
member15_in_ag(x1, x2)  =  member15_in_ag(x2)
member15_out_ag(x1, x2)  =  member15_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4)  =  U1_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

(45) PiDPToQDPProof (SOUND transformation)

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

(46) 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, member15_in_ag(T36))
U3_AGAG(T36, T35, member15_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:

member15_in_ag(.(T56, T57)) → member15_out_ag(T56, .(T56, T57))
member15_in_ag(.(T65, T66)) → U1_ag(T65, T66, member15_in_ag(T66))
U1_ag(T65, T66, member15_out_ag(T67, T66)) → member15_out_ag(T67, .(T65, T66))

The set Q consists of the following terms:

member15_in_ag(x0)
U1_ag(x0, x1, x2)

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

(47) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P13_IN_AGAG(T36, T35) → U3_AGAG(T36, T35, member15_in_ag(T36)) at position [2] we obtained the following new rules [LPAR04]:

P13_IN_AGAG(.(x0, x1), y1) → U3_AGAG(.(x0, x1), y1, member15_out_ag(x0, .(x0, x1)))
P13_IN_AGAG(.(x0, x1), y1) → U3_AGAG(.(x0, x1), y1, U1_ag(x0, x1, member15_in_ag(x1)))

(48) 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, member15_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, member15_out_ag(x0, .(x0, x1)))
P13_IN_AGAG(.(x0, x1), y1) → U3_AGAG(.(x0, x1), y1, U1_ag(x0, x1, member15_in_ag(x1)))

The TRS R consists of the following rules:

member15_in_ag(.(T56, T57)) → member15_out_ag(T56, .(T56, T57))
member15_in_ag(.(T65, T66)) → U1_ag(T65, T66, member15_in_ag(T66))
U1_ag(T65, T66, member15_out_ag(T67, T66)) → member15_out_ag(T67, .(T65, T66))

The set Q consists of the following terms:

member15_in_ag(x0)
U1_ag(x0, x1, x2)

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

(49) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_AGAG(T36, T35, member15_out_ag(T37, T36)) → SUBSET1_IN_AG(.(T35, T36)) we obtained the following new rules [LPAR04]:

U3_AGAG(.(z0, z1), z2, member15_out_ag(z0, .(z0, z1))) → SUBSET1_IN_AG(.(z2, .(z0, z1)))
U3_AGAG(.(z0, z1), z2, member15_out_ag(x2, .(z0, z1))) → SUBSET1_IN_AG(.(z2, .(z0, z1)))

(50) 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, member15_out_ag(x0, .(x0, x1)))
P13_IN_AGAG(.(x0, x1), y1) → U3_AGAG(.(x0, x1), y1, U1_ag(x0, x1, member15_in_ag(x1)))
U3_AGAG(.(z0, z1), z2, member15_out_ag(z0, .(z0, z1))) → SUBSET1_IN_AG(.(z2, .(z0, z1)))
U3_AGAG(.(z0, z1), z2, member15_out_ag(x2, .(z0, z1))) → SUBSET1_IN_AG(.(z2, .(z0, z1)))

The TRS R consists of the following rules:

member15_in_ag(.(T56, T57)) → member15_out_ag(T56, .(T56, T57))
member15_in_ag(.(T65, T66)) → U1_ag(T65, T66, member15_in_ag(T66))
U1_ag(T65, T66, member15_out_ag(T67, T66)) → member15_out_ag(T67, .(T65, T66))

The set Q consists of the following terms:

member15_in_ag(x0)
U1_ag(x0, x1, x2)

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

(51) 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)

(52) 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, member15_out_ag(x0, .(x0, x1)))
P13_IN_AGAG(.(x0, x1), y1) → U3_AGAG(.(x0, x1), y1, U1_ag(x0, x1, member15_in_ag(x1)))
U3_AGAG(.(z0, z1), z2, member15_out_ag(z0, .(z0, z1))) → SUBSET1_IN_AG(.(z2, .(z0, z1)))
U3_AGAG(.(z0, z1), z2, member15_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:

member15_in_ag(.(T56, T57)) → member15_out_ag(T56, .(T56, T57))
member15_in_ag(.(T65, T66)) → U1_ag(T65, T66, member15_in_ag(T66))
U1_ag(T65, T66, member15_out_ag(T67, T66)) → member15_out_ag(T67, .(T65, T66))

The set Q consists of the following terms:

member15_in_ag(x0)
U1_ag(x0, x1, x2)

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

(53) 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:
  • Matcher: [ ]
  • Semiunifier: [ ]



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)).



(54) NO