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

0 Prolog
↳1 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 NonTerminationProof (⇔)
                ↳15 NO
            ↳16 PiDP
                ↳17 UsableRulesProof (⇔)
                ↳18 PiDP
                ↳19 PiDPToQDPProof (⇐)
                ↳20 QDP
                ↳21 QDPSizeChangeProof (⇔)
                ↳22 YES
    ↳23 PrologToPiTRSProof (⇐)
        ↳24 PiTRS
        ↳25 DependencyPairsProof (⇔)
        ↳26 PiDP
        ↳27 DependencyGraphProof (⇔)
        ↳28 AND
            ↳29 PiDP
                ↳30 UsableRulesProof (⇔)
                ↳31 PiDP
                ↳32 PiDPToQDPProof (⇐)
                ↳33 QDP
                ↳34 NonTerminationProof (⇔)
                ↳35 NO
            ↳36 PiDP
                ↳37 UsableRulesProof (⇔)
                ↳38 PiDP
                ↳39 PiDPToQDPProof (⇐)
                ↳40 QDP
                ↳41 QDPSizeChangeProof (⇔)
                ↳42 YES

(0) Obligation:

Clauses:

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

Queries:

subset(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

member16(T56, .(T56, T57)).
member16(T64, .(T65, T67)) :- member16(T64, T67).
subset1([], T4).
subset1(.(T20, T9), .(T20, T22)) :- subset1(T9, .(T20, T22)).
subset1(.(T33, T9), .(T37, T36)) :- member16(T33, T36).
subset1(.(T33, T9), .(T42, T43)) :- ','(member16(T33, T43), subset1(T9, .(T42, T43))).

Queries:

subset1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
subset1_in: (b,f)
member16_in: (f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

subset1_in_ga([], T4) → subset1_out_ga([], T4)
subset1_in_ga(.(T20, T9), .(T20, T22)) → U2_ga(T20, T9, T22, subset1_in_ga(T9, .(T20, T22)))
subset1_in_ga(.(T33, T9), .(T37, T36)) → U3_ga(T33, T9, T37, T36, member16_in_aa(T33, T36))
member16_in_aa(T56, .(T56, T57)) → member16_out_aa(T56, .(T56, T57))
member16_in_aa(T64, .(T65, T67)) → U1_aa(T64, T65, T67, member16_in_aa(T64, T67))
U1_aa(T64, T65, T67, member16_out_aa(T64, T67)) → member16_out_aa(T64, .(T65, T67))
U3_ga(T33, T9, T37, T36, member16_out_aa(T33, T36)) → subset1_out_ga(.(T33, T9), .(T37, T36))
subset1_in_ga(.(T33, T9), .(T42, T43)) → U4_ga(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_ga(T33, T9, T42, T43, member16_out_aa(T33, T43)) → U5_ga(T33, T9, T42, T43, subset1_in_ga(T9, .(T42, T43)))
U5_ga(T33, T9, T42, T43, subset1_out_ga(T9, .(T42, T43))) → subset1_out_ga(.(T33, T9), .(T42, T43))
U2_ga(T20, T9, T22, subset1_out_ga(T9, .(T20, T22))) → subset1_out_ga(.(T20, T9), .(T20, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ga(x1, x2)  =  subset1_in_ga(x1)
[]  =  []
subset1_out_ga(x1, x2)  =  subset1_out_ga
.(x1, x2)  =  .(x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x2, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
member16_in_aa(x1, x2)  =  member16_in_aa
member16_out_aa(x1, x2)  =  member16_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(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_ga([], T4) → subset1_out_ga([], T4)
subset1_in_ga(.(T20, T9), .(T20, T22)) → U2_ga(T20, T9, T22, subset1_in_ga(T9, .(T20, T22)))
subset1_in_ga(.(T33, T9), .(T37, T36)) → U3_ga(T33, T9, T37, T36, member16_in_aa(T33, T36))
member16_in_aa(T56, .(T56, T57)) → member16_out_aa(T56, .(T56, T57))
member16_in_aa(T64, .(T65, T67)) → U1_aa(T64, T65, T67, member16_in_aa(T64, T67))
U1_aa(T64, T65, T67, member16_out_aa(T64, T67)) → member16_out_aa(T64, .(T65, T67))
U3_ga(T33, T9, T37, T36, member16_out_aa(T33, T36)) → subset1_out_ga(.(T33, T9), .(T37, T36))
subset1_in_ga(.(T33, T9), .(T42, T43)) → U4_ga(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_ga(T33, T9, T42, T43, member16_out_aa(T33, T43)) → U5_ga(T33, T9, T42, T43, subset1_in_ga(T9, .(T42, T43)))
U5_ga(T33, T9, T42, T43, subset1_out_ga(T9, .(T42, T43))) → subset1_out_ga(.(T33, T9), .(T42, T43))
U2_ga(T20, T9, T22, subset1_out_ga(T9, .(T20, T22))) → subset1_out_ga(.(T20, T9), .(T20, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ga(x1, x2)  =  subset1_in_ga(x1)
[]  =  []
subset1_out_ga(x1, x2)  =  subset1_out_ga
.(x1, x2)  =  .(x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x2, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
member16_in_aa(x1, x2)  =  member16_in_aa
member16_out_aa(x1, x2)  =  member16_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(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_GA(.(T20, T9), .(T20, T22)) → U2_GA(T20, T9, T22, subset1_in_ga(T9, .(T20, T22)))
SUBSET1_IN_GA(.(T20, T9), .(T20, T22)) → SUBSET1_IN_GA(T9, .(T20, T22))
SUBSET1_IN_GA(.(T33, T9), .(T37, T36)) → U3_GA(T33, T9, T37, T36, member16_in_aa(T33, T36))
SUBSET1_IN_GA(.(T33, T9), .(T37, T36)) → MEMBER16_IN_AA(T33, T36)
MEMBER16_IN_AA(T64, .(T65, T67)) → U1_AA(T64, T65, T67, member16_in_aa(T64, T67))
MEMBER16_IN_AA(T64, .(T65, T67)) → MEMBER16_IN_AA(T64, T67)
SUBSET1_IN_GA(.(T33, T9), .(T42, T43)) → U4_GA(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_GA(T33, T9, T42, T43, member16_out_aa(T33, T43)) → U5_GA(T33, T9, T42, T43, subset1_in_ga(T9, .(T42, T43)))
U4_GA(T33, T9, T42, T43, member16_out_aa(T33, T43)) → SUBSET1_IN_GA(T9, .(T42, T43))

The TRS R consists of the following rules:

subset1_in_ga([], T4) → subset1_out_ga([], T4)
subset1_in_ga(.(T20, T9), .(T20, T22)) → U2_ga(T20, T9, T22, subset1_in_ga(T9, .(T20, T22)))
subset1_in_ga(.(T33, T9), .(T37, T36)) → U3_ga(T33, T9, T37, T36, member16_in_aa(T33, T36))
member16_in_aa(T56, .(T56, T57)) → member16_out_aa(T56, .(T56, T57))
member16_in_aa(T64, .(T65, T67)) → U1_aa(T64, T65, T67, member16_in_aa(T64, T67))
U1_aa(T64, T65, T67, member16_out_aa(T64, T67)) → member16_out_aa(T64, .(T65, T67))
U3_ga(T33, T9, T37, T36, member16_out_aa(T33, T36)) → subset1_out_ga(.(T33, T9), .(T37, T36))
subset1_in_ga(.(T33, T9), .(T42, T43)) → U4_ga(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_ga(T33, T9, T42, T43, member16_out_aa(T33, T43)) → U5_ga(T33, T9, T42, T43, subset1_in_ga(T9, .(T42, T43)))
U5_ga(T33, T9, T42, T43, subset1_out_ga(T9, .(T42, T43))) → subset1_out_ga(.(T33, T9), .(T42, T43))
U2_ga(T20, T9, T22, subset1_out_ga(T9, .(T20, T22))) → subset1_out_ga(.(T20, T9), .(T20, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ga(x1, x2)  =  subset1_in_ga(x1)
[]  =  []
subset1_out_ga(x1, x2)  =  subset1_out_ga
.(x1, x2)  =  .(x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x2, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
member16_in_aa(x1, x2)  =  member16_in_aa
member16_out_aa(x1, x2)  =  member16_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
SUBSET1_IN_GA(x1, x2)  =  SUBSET1_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
MEMBER16_IN_AA(x1, x2)  =  MEMBER16_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x2, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(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_GA(.(T20, T9), .(T20, T22)) → U2_GA(T20, T9, T22, subset1_in_ga(T9, .(T20, T22)))
SUBSET1_IN_GA(.(T20, T9), .(T20, T22)) → SUBSET1_IN_GA(T9, .(T20, T22))
SUBSET1_IN_GA(.(T33, T9), .(T37, T36)) → U3_GA(T33, T9, T37, T36, member16_in_aa(T33, T36))
SUBSET1_IN_GA(.(T33, T9), .(T37, T36)) → MEMBER16_IN_AA(T33, T36)
MEMBER16_IN_AA(T64, .(T65, T67)) → U1_AA(T64, T65, T67, member16_in_aa(T64, T67))
MEMBER16_IN_AA(T64, .(T65, T67)) → MEMBER16_IN_AA(T64, T67)
SUBSET1_IN_GA(.(T33, T9), .(T42, T43)) → U4_GA(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_GA(T33, T9, T42, T43, member16_out_aa(T33, T43)) → U5_GA(T33, T9, T42, T43, subset1_in_ga(T9, .(T42, T43)))
U4_GA(T33, T9, T42, T43, member16_out_aa(T33, T43)) → SUBSET1_IN_GA(T9, .(T42, T43))

The TRS R consists of the following rules:

subset1_in_ga([], T4) → subset1_out_ga([], T4)
subset1_in_ga(.(T20, T9), .(T20, T22)) → U2_ga(T20, T9, T22, subset1_in_ga(T9, .(T20, T22)))
subset1_in_ga(.(T33, T9), .(T37, T36)) → U3_ga(T33, T9, T37, T36, member16_in_aa(T33, T36))
member16_in_aa(T56, .(T56, T57)) → member16_out_aa(T56, .(T56, T57))
member16_in_aa(T64, .(T65, T67)) → U1_aa(T64, T65, T67, member16_in_aa(T64, T67))
U1_aa(T64, T65, T67, member16_out_aa(T64, T67)) → member16_out_aa(T64, .(T65, T67))
U3_ga(T33, T9, T37, T36, member16_out_aa(T33, T36)) → subset1_out_ga(.(T33, T9), .(T37, T36))
subset1_in_ga(.(T33, T9), .(T42, T43)) → U4_ga(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_ga(T33, T9, T42, T43, member16_out_aa(T33, T43)) → U5_ga(T33, T9, T42, T43, subset1_in_ga(T9, .(T42, T43)))
U5_ga(T33, T9, T42, T43, subset1_out_ga(T9, .(T42, T43))) → subset1_out_ga(.(T33, T9), .(T42, T43))
U2_ga(T20, T9, T22, subset1_out_ga(T9, .(T20, T22))) → subset1_out_ga(.(T20, T9), .(T20, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ga(x1, x2)  =  subset1_in_ga(x1)
[]  =  []
subset1_out_ga(x1, x2)  =  subset1_out_ga
.(x1, x2)  =  .(x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x2, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
member16_in_aa(x1, x2)  =  member16_in_aa
member16_out_aa(x1, x2)  =  member16_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
SUBSET1_IN_GA(x1, x2)  =  SUBSET1_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
MEMBER16_IN_AA(x1, x2)  =  MEMBER16_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x2, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(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 5 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

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

The TRS R consists of the following rules:

subset1_in_ga([], T4) → subset1_out_ga([], T4)
subset1_in_ga(.(T20, T9), .(T20, T22)) → U2_ga(T20, T9, T22, subset1_in_ga(T9, .(T20, T22)))
subset1_in_ga(.(T33, T9), .(T37, T36)) → U3_ga(T33, T9, T37, T36, member16_in_aa(T33, T36))
member16_in_aa(T56, .(T56, T57)) → member16_out_aa(T56, .(T56, T57))
member16_in_aa(T64, .(T65, T67)) → U1_aa(T64, T65, T67, member16_in_aa(T64, T67))
U1_aa(T64, T65, T67, member16_out_aa(T64, T67)) → member16_out_aa(T64, .(T65, T67))
U3_ga(T33, T9, T37, T36, member16_out_aa(T33, T36)) → subset1_out_ga(.(T33, T9), .(T37, T36))
subset1_in_ga(.(T33, T9), .(T42, T43)) → U4_ga(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_ga(T33, T9, T42, T43, member16_out_aa(T33, T43)) → U5_ga(T33, T9, T42, T43, subset1_in_ga(T9, .(T42, T43)))
U5_ga(T33, T9, T42, T43, subset1_out_ga(T9, .(T42, T43))) → subset1_out_ga(.(T33, T9), .(T42, T43))
U2_ga(T20, T9, T22, subset1_out_ga(T9, .(T20, T22))) → subset1_out_ga(.(T20, T9), .(T20, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ga(x1, x2)  =  subset1_in_ga(x1)
[]  =  []
subset1_out_ga(x1, x2)  =  subset1_out_ga
.(x1, x2)  =  .(x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x2, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
member16_in_aa(x1, x2)  =  member16_in_aa
member16_out_aa(x1, x2)  =  member16_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
MEMBER16_IN_AA(x1, x2)  =  MEMBER16_IN_AA

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:

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

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

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

(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:

MEMBER16_IN_AA → MEMBER16_IN_AA

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

(14) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = MEMBER16_IN_AA evaluates to t =MEMBER16_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from MEMBER16_IN_AA to MEMBER16_IN_AA.



(15) NO

(16) Obligation:

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

SUBSET1_IN_GA(.(T33, T9), .(T42, T43)) → U4_GA(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_GA(T33, T9, T42, T43, member16_out_aa(T33, T43)) → SUBSET1_IN_GA(T9, .(T42, T43))
SUBSET1_IN_GA(.(T20, T9), .(T20, T22)) → SUBSET1_IN_GA(T9, .(T20, T22))

The TRS R consists of the following rules:

subset1_in_ga([], T4) → subset1_out_ga([], T4)
subset1_in_ga(.(T20, T9), .(T20, T22)) → U2_ga(T20, T9, T22, subset1_in_ga(T9, .(T20, T22)))
subset1_in_ga(.(T33, T9), .(T37, T36)) → U3_ga(T33, T9, T37, T36, member16_in_aa(T33, T36))
member16_in_aa(T56, .(T56, T57)) → member16_out_aa(T56, .(T56, T57))
member16_in_aa(T64, .(T65, T67)) → U1_aa(T64, T65, T67, member16_in_aa(T64, T67))
U1_aa(T64, T65, T67, member16_out_aa(T64, T67)) → member16_out_aa(T64, .(T65, T67))
U3_ga(T33, T9, T37, T36, member16_out_aa(T33, T36)) → subset1_out_ga(.(T33, T9), .(T37, T36))
subset1_in_ga(.(T33, T9), .(T42, T43)) → U4_ga(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_ga(T33, T9, T42, T43, member16_out_aa(T33, T43)) → U5_ga(T33, T9, T42, T43, subset1_in_ga(T9, .(T42, T43)))
U5_ga(T33, T9, T42, T43, subset1_out_ga(T9, .(T42, T43))) → subset1_out_ga(.(T33, T9), .(T42, T43))
U2_ga(T20, T9, T22, subset1_out_ga(T9, .(T20, T22))) → subset1_out_ga(.(T20, T9), .(T20, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ga(x1, x2)  =  subset1_in_ga(x1)
[]  =  []
subset1_out_ga(x1, x2)  =  subset1_out_ga
.(x1, x2)  =  .(x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x2, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
member16_in_aa(x1, x2)  =  member16_in_aa
member16_out_aa(x1, x2)  =  member16_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
SUBSET1_IN_GA(x1, x2)  =  SUBSET1_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x2, x5)

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

(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_GA(.(T33, T9), .(T42, T43)) → U4_GA(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_GA(T33, T9, T42, T43, member16_out_aa(T33, T43)) → SUBSET1_IN_GA(T9, .(T42, T43))
SUBSET1_IN_GA(.(T20, T9), .(T20, T22)) → SUBSET1_IN_GA(T9, .(T20, T22))

The TRS R consists of the following rules:

member16_in_aa(T56, .(T56, T57)) → member16_out_aa(T56, .(T56, T57))
member16_in_aa(T64, .(T65, T67)) → U1_aa(T64, T65, T67, member16_in_aa(T64, T67))
U1_aa(T64, T65, T67, member16_out_aa(T64, T67)) → member16_out_aa(T64, .(T65, T67))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
member16_in_aa(x1, x2)  =  member16_in_aa
member16_out_aa(x1, x2)  =  member16_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
SUBSET1_IN_GA(x1, x2)  =  SUBSET1_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x2, x5)

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

(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_GA(.(T9)) → U4_GA(T9, member16_in_aa)
U4_GA(T9, member16_out_aa) → SUBSET1_IN_GA(T9)
SUBSET1_IN_GA(.(T9)) → SUBSET1_IN_GA(T9)

The TRS R consists of the following rules:

member16_in_aa → member16_out_aa
member16_in_aa → U1_aa(member16_in_aa)
U1_aa(member16_out_aa) → member16_out_aa

The set Q consists of the following terms:

member16_in_aa
U1_aa(x0)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • U4_GA(T9, member16_out_aa) → SUBSET1_IN_GA(T9)
    The graph contains the following edges 1 >= 1

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

  • SUBSET1_IN_GA(.(T9)) → U4_GA(T9, member16_in_aa)
    The graph contains the following edges 1 > 1

(22) YES

(23) 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: (b,f)
member16_in: (f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

subset1_in_ga([], T4) → subset1_out_ga([], T4)
subset1_in_ga(.(T20, T9), .(T20, T22)) → U2_ga(T20, T9, T22, subset1_in_ga(T9, .(T20, T22)))
subset1_in_ga(.(T33, T9), .(T37, T36)) → U3_ga(T33, T9, T37, T36, member16_in_aa(T33, T36))
member16_in_aa(T56, .(T56, T57)) → member16_out_aa(T56, .(T56, T57))
member16_in_aa(T64, .(T65, T67)) → U1_aa(T64, T65, T67, member16_in_aa(T64, T67))
U1_aa(T64, T65, T67, member16_out_aa(T64, T67)) → member16_out_aa(T64, .(T65, T67))
U3_ga(T33, T9, T37, T36, member16_out_aa(T33, T36)) → subset1_out_ga(.(T33, T9), .(T37, T36))
subset1_in_ga(.(T33, T9), .(T42, T43)) → U4_ga(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_ga(T33, T9, T42, T43, member16_out_aa(T33, T43)) → U5_ga(T33, T9, T42, T43, subset1_in_ga(T9, .(T42, T43)))
U5_ga(T33, T9, T42, T43, subset1_out_ga(T9, .(T42, T43))) → subset1_out_ga(.(T33, T9), .(T42, T43))
U2_ga(T20, T9, T22, subset1_out_ga(T9, .(T20, T22))) → subset1_out_ga(.(T20, T9), .(T20, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ga(x1, x2)  =  subset1_in_ga(x1)
[]  =  []
subset1_out_ga(x1, x2)  =  subset1_out_ga(x1)
.(x1, x2)  =  .(x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x2, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x2, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x2, x5)
member16_in_aa(x1, x2)  =  member16_in_aa
member16_out_aa(x1, x2)  =  member16_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(24) Obligation:

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

subset1_in_ga([], T4) → subset1_out_ga([], T4)
subset1_in_ga(.(T20, T9), .(T20, T22)) → U2_ga(T20, T9, T22, subset1_in_ga(T9, .(T20, T22)))
subset1_in_ga(.(T33, T9), .(T37, T36)) → U3_ga(T33, T9, T37, T36, member16_in_aa(T33, T36))
member16_in_aa(T56, .(T56, T57)) → member16_out_aa(T56, .(T56, T57))
member16_in_aa(T64, .(T65, T67)) → U1_aa(T64, T65, T67, member16_in_aa(T64, T67))
U1_aa(T64, T65, T67, member16_out_aa(T64, T67)) → member16_out_aa(T64, .(T65, T67))
U3_ga(T33, T9, T37, T36, member16_out_aa(T33, T36)) → subset1_out_ga(.(T33, T9), .(T37, T36))
subset1_in_ga(.(T33, T9), .(T42, T43)) → U4_ga(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_ga(T33, T9, T42, T43, member16_out_aa(T33, T43)) → U5_ga(T33, T9, T42, T43, subset1_in_ga(T9, .(T42, T43)))
U5_ga(T33, T9, T42, T43, subset1_out_ga(T9, .(T42, T43))) → subset1_out_ga(.(T33, T9), .(T42, T43))
U2_ga(T20, T9, T22, subset1_out_ga(T9, .(T20, T22))) → subset1_out_ga(.(T20, T9), .(T20, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ga(x1, x2)  =  subset1_in_ga(x1)
[]  =  []
subset1_out_ga(x1, x2)  =  subset1_out_ga(x1)
.(x1, x2)  =  .(x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x2, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x2, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x2, x5)
member16_in_aa(x1, x2)  =  member16_in_aa
member16_out_aa(x1, x2)  =  member16_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

(25) 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_GA(.(T20, T9), .(T20, T22)) → U2_GA(T20, T9, T22, subset1_in_ga(T9, .(T20, T22)))
SUBSET1_IN_GA(.(T20, T9), .(T20, T22)) → SUBSET1_IN_GA(T9, .(T20, T22))
SUBSET1_IN_GA(.(T33, T9), .(T37, T36)) → U3_GA(T33, T9, T37, T36, member16_in_aa(T33, T36))
SUBSET1_IN_GA(.(T33, T9), .(T37, T36)) → MEMBER16_IN_AA(T33, T36)
MEMBER16_IN_AA(T64, .(T65, T67)) → U1_AA(T64, T65, T67, member16_in_aa(T64, T67))
MEMBER16_IN_AA(T64, .(T65, T67)) → MEMBER16_IN_AA(T64, T67)
SUBSET1_IN_GA(.(T33, T9), .(T42, T43)) → U4_GA(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_GA(T33, T9, T42, T43, member16_out_aa(T33, T43)) → U5_GA(T33, T9, T42, T43, subset1_in_ga(T9, .(T42, T43)))
U4_GA(T33, T9, T42, T43, member16_out_aa(T33, T43)) → SUBSET1_IN_GA(T9, .(T42, T43))

The TRS R consists of the following rules:

subset1_in_ga([], T4) → subset1_out_ga([], T4)
subset1_in_ga(.(T20, T9), .(T20, T22)) → U2_ga(T20, T9, T22, subset1_in_ga(T9, .(T20, T22)))
subset1_in_ga(.(T33, T9), .(T37, T36)) → U3_ga(T33, T9, T37, T36, member16_in_aa(T33, T36))
member16_in_aa(T56, .(T56, T57)) → member16_out_aa(T56, .(T56, T57))
member16_in_aa(T64, .(T65, T67)) → U1_aa(T64, T65, T67, member16_in_aa(T64, T67))
U1_aa(T64, T65, T67, member16_out_aa(T64, T67)) → member16_out_aa(T64, .(T65, T67))
U3_ga(T33, T9, T37, T36, member16_out_aa(T33, T36)) → subset1_out_ga(.(T33, T9), .(T37, T36))
subset1_in_ga(.(T33, T9), .(T42, T43)) → U4_ga(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_ga(T33, T9, T42, T43, member16_out_aa(T33, T43)) → U5_ga(T33, T9, T42, T43, subset1_in_ga(T9, .(T42, T43)))
U5_ga(T33, T9, T42, T43, subset1_out_ga(T9, .(T42, T43))) → subset1_out_ga(.(T33, T9), .(T42, T43))
U2_ga(T20, T9, T22, subset1_out_ga(T9, .(T20, T22))) → subset1_out_ga(.(T20, T9), .(T20, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ga(x1, x2)  =  subset1_in_ga(x1)
[]  =  []
subset1_out_ga(x1, x2)  =  subset1_out_ga(x1)
.(x1, x2)  =  .(x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x2, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x2, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x2, x5)
member16_in_aa(x1, x2)  =  member16_in_aa
member16_out_aa(x1, x2)  =  member16_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
SUBSET1_IN_GA(x1, x2)  =  SUBSET1_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x2, x4)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x2, x5)
MEMBER16_IN_AA(x1, x2)  =  MEMBER16_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x2, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x2, x5)

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

(26) Obligation:

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

SUBSET1_IN_GA(.(T20, T9), .(T20, T22)) → U2_GA(T20, T9, T22, subset1_in_ga(T9, .(T20, T22)))
SUBSET1_IN_GA(.(T20, T9), .(T20, T22)) → SUBSET1_IN_GA(T9, .(T20, T22))
SUBSET1_IN_GA(.(T33, T9), .(T37, T36)) → U3_GA(T33, T9, T37, T36, member16_in_aa(T33, T36))
SUBSET1_IN_GA(.(T33, T9), .(T37, T36)) → MEMBER16_IN_AA(T33, T36)
MEMBER16_IN_AA(T64, .(T65, T67)) → U1_AA(T64, T65, T67, member16_in_aa(T64, T67))
MEMBER16_IN_AA(T64, .(T65, T67)) → MEMBER16_IN_AA(T64, T67)
SUBSET1_IN_GA(.(T33, T9), .(T42, T43)) → U4_GA(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_GA(T33, T9, T42, T43, member16_out_aa(T33, T43)) → U5_GA(T33, T9, T42, T43, subset1_in_ga(T9, .(T42, T43)))
U4_GA(T33, T9, T42, T43, member16_out_aa(T33, T43)) → SUBSET1_IN_GA(T9, .(T42, T43))

The TRS R consists of the following rules:

subset1_in_ga([], T4) → subset1_out_ga([], T4)
subset1_in_ga(.(T20, T9), .(T20, T22)) → U2_ga(T20, T9, T22, subset1_in_ga(T9, .(T20, T22)))
subset1_in_ga(.(T33, T9), .(T37, T36)) → U3_ga(T33, T9, T37, T36, member16_in_aa(T33, T36))
member16_in_aa(T56, .(T56, T57)) → member16_out_aa(T56, .(T56, T57))
member16_in_aa(T64, .(T65, T67)) → U1_aa(T64, T65, T67, member16_in_aa(T64, T67))
U1_aa(T64, T65, T67, member16_out_aa(T64, T67)) → member16_out_aa(T64, .(T65, T67))
U3_ga(T33, T9, T37, T36, member16_out_aa(T33, T36)) → subset1_out_ga(.(T33, T9), .(T37, T36))
subset1_in_ga(.(T33, T9), .(T42, T43)) → U4_ga(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_ga(T33, T9, T42, T43, member16_out_aa(T33, T43)) → U5_ga(T33, T9, T42, T43, subset1_in_ga(T9, .(T42, T43)))
U5_ga(T33, T9, T42, T43, subset1_out_ga(T9, .(T42, T43))) → subset1_out_ga(.(T33, T9), .(T42, T43))
U2_ga(T20, T9, T22, subset1_out_ga(T9, .(T20, T22))) → subset1_out_ga(.(T20, T9), .(T20, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ga(x1, x2)  =  subset1_in_ga(x1)
[]  =  []
subset1_out_ga(x1, x2)  =  subset1_out_ga(x1)
.(x1, x2)  =  .(x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x2, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x2, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x2, x5)
member16_in_aa(x1, x2)  =  member16_in_aa
member16_out_aa(x1, x2)  =  member16_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
SUBSET1_IN_GA(x1, x2)  =  SUBSET1_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x2, x4)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x2, x5)
MEMBER16_IN_AA(x1, x2)  =  MEMBER16_IN_AA
U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x2, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x2, x5)

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

(27) DependencyGraphProof (EQUIVALENT transformation)

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

(28) Complex Obligation (AND)

(29) Obligation:

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

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

The TRS R consists of the following rules:

subset1_in_ga([], T4) → subset1_out_ga([], T4)
subset1_in_ga(.(T20, T9), .(T20, T22)) → U2_ga(T20, T9, T22, subset1_in_ga(T9, .(T20, T22)))
subset1_in_ga(.(T33, T9), .(T37, T36)) → U3_ga(T33, T9, T37, T36, member16_in_aa(T33, T36))
member16_in_aa(T56, .(T56, T57)) → member16_out_aa(T56, .(T56, T57))
member16_in_aa(T64, .(T65, T67)) → U1_aa(T64, T65, T67, member16_in_aa(T64, T67))
U1_aa(T64, T65, T67, member16_out_aa(T64, T67)) → member16_out_aa(T64, .(T65, T67))
U3_ga(T33, T9, T37, T36, member16_out_aa(T33, T36)) → subset1_out_ga(.(T33, T9), .(T37, T36))
subset1_in_ga(.(T33, T9), .(T42, T43)) → U4_ga(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_ga(T33, T9, T42, T43, member16_out_aa(T33, T43)) → U5_ga(T33, T9, T42, T43, subset1_in_ga(T9, .(T42, T43)))
U5_ga(T33, T9, T42, T43, subset1_out_ga(T9, .(T42, T43))) → subset1_out_ga(.(T33, T9), .(T42, T43))
U2_ga(T20, T9, T22, subset1_out_ga(T9, .(T20, T22))) → subset1_out_ga(.(T20, T9), .(T20, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ga(x1, x2)  =  subset1_in_ga(x1)
[]  =  []
subset1_out_ga(x1, x2)  =  subset1_out_ga(x1)
.(x1, x2)  =  .(x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x2, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x2, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x2, x5)
member16_in_aa(x1, x2)  =  member16_in_aa
member16_out_aa(x1, x2)  =  member16_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
MEMBER16_IN_AA(x1, x2)  =  MEMBER16_IN_AA

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

(30) UsableRulesProof (EQUIVALENT transformation)

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

(31) Obligation:

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

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

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

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

(32) PiDPToQDPProof (SOUND transformation)

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

(33) Obligation:

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

MEMBER16_IN_AA → MEMBER16_IN_AA

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

(34) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = MEMBER16_IN_AA evaluates to t =MEMBER16_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from MEMBER16_IN_AA to MEMBER16_IN_AA.



(35) NO

(36) Obligation:

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

SUBSET1_IN_GA(.(T33, T9), .(T42, T43)) → U4_GA(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_GA(T33, T9, T42, T43, member16_out_aa(T33, T43)) → SUBSET1_IN_GA(T9, .(T42, T43))
SUBSET1_IN_GA(.(T20, T9), .(T20, T22)) → SUBSET1_IN_GA(T9, .(T20, T22))

The TRS R consists of the following rules:

subset1_in_ga([], T4) → subset1_out_ga([], T4)
subset1_in_ga(.(T20, T9), .(T20, T22)) → U2_ga(T20, T9, T22, subset1_in_ga(T9, .(T20, T22)))
subset1_in_ga(.(T33, T9), .(T37, T36)) → U3_ga(T33, T9, T37, T36, member16_in_aa(T33, T36))
member16_in_aa(T56, .(T56, T57)) → member16_out_aa(T56, .(T56, T57))
member16_in_aa(T64, .(T65, T67)) → U1_aa(T64, T65, T67, member16_in_aa(T64, T67))
U1_aa(T64, T65, T67, member16_out_aa(T64, T67)) → member16_out_aa(T64, .(T65, T67))
U3_ga(T33, T9, T37, T36, member16_out_aa(T33, T36)) → subset1_out_ga(.(T33, T9), .(T37, T36))
subset1_in_ga(.(T33, T9), .(T42, T43)) → U4_ga(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_ga(T33, T9, T42, T43, member16_out_aa(T33, T43)) → U5_ga(T33, T9, T42, T43, subset1_in_ga(T9, .(T42, T43)))
U5_ga(T33, T9, T42, T43, subset1_out_ga(T9, .(T42, T43))) → subset1_out_ga(.(T33, T9), .(T42, T43))
U2_ga(T20, T9, T22, subset1_out_ga(T9, .(T20, T22))) → subset1_out_ga(.(T20, T9), .(T20, T22))

The argument filtering Pi contains the following mapping:
subset1_in_ga(x1, x2)  =  subset1_in_ga(x1)
[]  =  []
subset1_out_ga(x1, x2)  =  subset1_out_ga(x1)
.(x1, x2)  =  .(x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x2, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x2, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x2, x5)
member16_in_aa(x1, x2)  =  member16_in_aa
member16_out_aa(x1, x2)  =  member16_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
SUBSET1_IN_GA(x1, x2)  =  SUBSET1_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x2, x5)

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

(37) UsableRulesProof (EQUIVALENT transformation)

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

(38) Obligation:

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

SUBSET1_IN_GA(.(T33, T9), .(T42, T43)) → U4_GA(T33, T9, T42, T43, member16_in_aa(T33, T43))
U4_GA(T33, T9, T42, T43, member16_out_aa(T33, T43)) → SUBSET1_IN_GA(T9, .(T42, T43))
SUBSET1_IN_GA(.(T20, T9), .(T20, T22)) → SUBSET1_IN_GA(T9, .(T20, T22))

The TRS R consists of the following rules:

member16_in_aa(T56, .(T56, T57)) → member16_out_aa(T56, .(T56, T57))
member16_in_aa(T64, .(T65, T67)) → U1_aa(T64, T65, T67, member16_in_aa(T64, T67))
U1_aa(T64, T65, T67, member16_out_aa(T64, T67)) → member16_out_aa(T64, .(T65, T67))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
member16_in_aa(x1, x2)  =  member16_in_aa
member16_out_aa(x1, x2)  =  member16_out_aa
U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
SUBSET1_IN_GA(x1, x2)  =  SUBSET1_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x2, x5)

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

(39) PiDPToQDPProof (SOUND transformation)

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

(40) Obligation:

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

SUBSET1_IN_GA(.(T9)) → U4_GA(T9, member16_in_aa)
U4_GA(T9, member16_out_aa) → SUBSET1_IN_GA(T9)
SUBSET1_IN_GA(.(T9)) → SUBSET1_IN_GA(T9)

The TRS R consists of the following rules:

member16_in_aa → member16_out_aa
member16_in_aa → U1_aa(member16_in_aa)
U1_aa(member16_out_aa) → member16_out_aa

The set Q consists of the following terms:

member16_in_aa
U1_aa(x0)

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

(41) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • U4_GA(T9, member16_out_aa) → SUBSET1_IN_GA(T9)
    The graph contains the following edges 1 >= 1

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

  • SUBSET1_IN_GA(.(T9)) → U4_GA(T9, member16_in_aa)
    The graph contains the following edges 1 > 1

(42) YES