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 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 NonTerminationProof (⇔)
        ↳13 FALSE
      ↳14 PiDP
        ↳15 UsableRulesProof (⇔)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇐)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔)
        ↳20 TRUE
  ↳21 PrologToPiTRSProof (⇐)
    ↳22 PiTRS
    ↳23 DependencyPairsProof (⇔)
    ↳24 PiDP
    ↳25 DependencyGraphProof (⇔)
    ↳26 AND
      ↳27 PiDP
        ↳28 UsableRulesProof (⇔)
        ↳29 PiDP
        ↳30 PiDPToQDPProof (⇐)
        ↳31 QDP
        ↳32 NonTerminationProof (⇔)
        ↳33 FALSE
      ↳34 PiDP
        ↳35 UsableRulesProof (⇔)
        ↳36 PiDP
        ↳37 PiDPToQDPProof (⇐)
        ↳38 QDP
        ↳39 QDPSizeChangeProof (⇔)
        ↳40 TRUE

(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) PrologToPiTRSProof (SOUND transformation)

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

subset_in_ga([], X1) → subset_out_ga([], X1)
subset_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, member_in_ga(X, Ys))
member_in_ga(X, .(X, X2)) → member_out_ga(X, .(X, X2))
member_in_ga(X, .(X3, Xs)) → U3_ga(X, X3, Xs, member_in_ga(X, Xs))
U3_ga(X, X3, Xs, member_out_ga(X, Xs)) → member_out_ga(X, .(X3, Xs))
U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) → U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys))
U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) → subset_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ga(x1, x2)  =  subset_in_ga(x1)
[]  =  []
subset_out_ga(x1, x2)  =  subset_out_ga
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x2, x4)
member_in_ga(x1, x2)  =  member_in_ga(x1)
member_out_ga(x1, x2)  =  member_out_ga
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

subset_in_ga([], X1) → subset_out_ga([], X1)
subset_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, member_in_ga(X, Ys))
member_in_ga(X, .(X, X2)) → member_out_ga(X, .(X, X2))
member_in_ga(X, .(X3, Xs)) → U3_ga(X, X3, Xs, member_in_ga(X, Xs))
U3_ga(X, X3, Xs, member_out_ga(X, Xs)) → member_out_ga(X, .(X3, Xs))
U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) → U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys))
U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) → subset_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ga(x1, x2)  =  subset_in_ga(x1)
[]  =  []
subset_out_ga(x1, x2)  =  subset_out_ga
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x2, x4)
member_in_ga(x1, x2)  =  member_in_ga(x1)
member_out_ga(x1, x2)  =  member_out_ga
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)

(3) 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:

SUBSET_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, member_in_ga(X, Ys))
SUBSET_IN_GA(.(X, Xs), Ys) → MEMBER_IN_GA(X, Ys)
MEMBER_IN_GA(X, .(X3, Xs)) → U3_GA(X, X3, Xs, member_in_ga(X, Xs))
MEMBER_IN_GA(X, .(X3, Xs)) → MEMBER_IN_GA(X, Xs)
U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) → U2_GA(X, Xs, Ys, subset_in_ga(Xs, Ys))
U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) → SUBSET_IN_GA(Xs, Ys)

The TRS R consists of the following rules:

subset_in_ga([], X1) → subset_out_ga([], X1)
subset_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, member_in_ga(X, Ys))
member_in_ga(X, .(X, X2)) → member_out_ga(X, .(X, X2))
member_in_ga(X, .(X3, Xs)) → U3_ga(X, X3, Xs, member_in_ga(X, Xs))
U3_ga(X, X3, Xs, member_out_ga(X, Xs)) → member_out_ga(X, .(X3, Xs))
U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) → U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys))
U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) → subset_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ga(x1, x2)  =  subset_in_ga(x1)
[]  =  []
subset_out_ga(x1, x2)  =  subset_out_ga
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x2, x4)
member_in_ga(x1, x2)  =  member_in_ga(x1)
member_out_ga(x1, x2)  =  member_out_ga
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
SUBSET_IN_GA(x1, x2)  =  SUBSET_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x2, x4)
MEMBER_IN_GA(x1, x2)  =  MEMBER_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)

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

(4) Obligation:

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

SUBSET_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, member_in_ga(X, Ys))
SUBSET_IN_GA(.(X, Xs), Ys) → MEMBER_IN_GA(X, Ys)
MEMBER_IN_GA(X, .(X3, Xs)) → U3_GA(X, X3, Xs, member_in_ga(X, Xs))
MEMBER_IN_GA(X, .(X3, Xs)) → MEMBER_IN_GA(X, Xs)
U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) → U2_GA(X, Xs, Ys, subset_in_ga(Xs, Ys))
U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) → SUBSET_IN_GA(Xs, Ys)

The TRS R consists of the following rules:

subset_in_ga([], X1) → subset_out_ga([], X1)
subset_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, member_in_ga(X, Ys))
member_in_ga(X, .(X, X2)) → member_out_ga(X, .(X, X2))
member_in_ga(X, .(X3, Xs)) → U3_ga(X, X3, Xs, member_in_ga(X, Xs))
U3_ga(X, X3, Xs, member_out_ga(X, Xs)) → member_out_ga(X, .(X3, Xs))
U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) → U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys))
U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) → subset_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ga(x1, x2)  =  subset_in_ga(x1)
[]  =  []
subset_out_ga(x1, x2)  =  subset_out_ga
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x2, x4)
member_in_ga(x1, x2)  =  member_in_ga(x1)
member_out_ga(x1, x2)  =  member_out_ga
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
SUBSET_IN_GA(x1, x2)  =  SUBSET_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x2, x4)
MEMBER_IN_GA(x1, x2)  =  MEMBER_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

MEMBER_IN_GA(X, .(X3, Xs)) → MEMBER_IN_GA(X, Xs)

The TRS R consists of the following rules:

subset_in_ga([], X1) → subset_out_ga([], X1)
subset_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, member_in_ga(X, Ys))
member_in_ga(X, .(X, X2)) → member_out_ga(X, .(X, X2))
member_in_ga(X, .(X3, Xs)) → U3_ga(X, X3, Xs, member_in_ga(X, Xs))
U3_ga(X, X3, Xs, member_out_ga(X, Xs)) → member_out_ga(X, .(X3, Xs))
U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) → U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys))
U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) → subset_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ga(x1, x2)  =  subset_in_ga(x1)
[]  =  []
subset_out_ga(x1, x2)  =  subset_out_ga
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x2, x4)
member_in_ga(x1, x2)  =  member_in_ga(x1)
member_out_ga(x1, x2)  =  member_out_ga
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
MEMBER_IN_GA(x1, x2)  =  MEMBER_IN_GA(x1)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

MEMBER_IN_GA(X, .(X3, Xs)) → MEMBER_IN_GA(X, Xs)

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

MEMBER_IN_GA(X) → MEMBER_IN_GA(X)

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

(12) NonTerminationProof (EQUIVALENT transformation)

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

s = MEMBER_IN_GA(X) evaluates to t =MEMBER_IN_GA(X)

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 MEMBER_IN_GA(X) to MEMBER_IN_GA(X).



(13) FALSE

(14) Obligation:

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

U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) → SUBSET_IN_GA(Xs, Ys)
SUBSET_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, member_in_ga(X, Ys))

The TRS R consists of the following rules:

subset_in_ga([], X1) → subset_out_ga([], X1)
subset_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, member_in_ga(X, Ys))
member_in_ga(X, .(X, X2)) → member_out_ga(X, .(X, X2))
member_in_ga(X, .(X3, Xs)) → U3_ga(X, X3, Xs, member_in_ga(X, Xs))
U3_ga(X, X3, Xs, member_out_ga(X, Xs)) → member_out_ga(X, .(X3, Xs))
U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) → U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys))
U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) → subset_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ga(x1, x2)  =  subset_in_ga(x1)
[]  =  []
subset_out_ga(x1, x2)  =  subset_out_ga
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x2, x4)
member_in_ga(x1, x2)  =  member_in_ga(x1)
member_out_ga(x1, x2)  =  member_out_ga
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
SUBSET_IN_GA(x1, x2)  =  SUBSET_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x2, x4)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) → SUBSET_IN_GA(Xs, Ys)
SUBSET_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, member_in_ga(X, Ys))

The TRS R consists of the following rules:

member_in_ga(X, .(X, X2)) → member_out_ga(X, .(X, X2))
member_in_ga(X, .(X3, Xs)) → U3_ga(X, X3, Xs, member_in_ga(X, Xs))
U3_ga(X, X3, Xs, member_out_ga(X, Xs)) → member_out_ga(X, .(X3, Xs))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
member_in_ga(x1, x2)  =  member_in_ga(x1)
member_out_ga(x1, x2)  =  member_out_ga
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
SUBSET_IN_GA(x1, x2)  =  SUBSET_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x2, x4)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

U1_GA(Xs, member_out_ga) → SUBSET_IN_GA(Xs)
SUBSET_IN_GA(.(X, Xs)) → U1_GA(Xs, member_in_ga(X))

The TRS R consists of the following rules:

member_in_ga(X) → member_out_ga
member_in_ga(X) → U3_ga(member_in_ga(X))
U3_ga(member_out_ga) → member_out_ga

The set Q consists of the following terms:

member_in_ga(x0)
U3_ga(x0)

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

(19) 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:

  • SUBSET_IN_GA(.(X, Xs)) → U1_GA(Xs, member_in_ga(X))
    The graph contains the following edges 1 > 1

  • U1_GA(Xs, member_out_ga) → SUBSET_IN_GA(Xs)
    The graph contains the following edges 1 >= 1

(20) TRUE

(21) PrologToPiTRSProof (SOUND transformation)

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

subset_in_ga([], X1) → subset_out_ga([], X1)
subset_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, member_in_ga(X, Ys))
member_in_ga(X, .(X, X2)) → member_out_ga(X, .(X, X2))
member_in_ga(X, .(X3, Xs)) → U3_ga(X, X3, Xs, member_in_ga(X, Xs))
U3_ga(X, X3, Xs, member_out_ga(X, Xs)) → member_out_ga(X, .(X3, Xs))
U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) → U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys))
U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) → subset_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ga(x1, x2)  =  subset_in_ga(x1)
[]  =  []
subset_out_ga(x1, x2)  =  subset_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
member_in_ga(x1, x2)  =  member_in_ga(x1)
member_out_ga(x1, x2)  =  member_out_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(22) Obligation:

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

subset_in_ga([], X1) → subset_out_ga([], X1)
subset_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, member_in_ga(X, Ys))
member_in_ga(X, .(X, X2)) → member_out_ga(X, .(X, X2))
member_in_ga(X, .(X3, Xs)) → U3_ga(X, X3, Xs, member_in_ga(X, Xs))
U3_ga(X, X3, Xs, member_out_ga(X, Xs)) → member_out_ga(X, .(X3, Xs))
U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) → U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys))
U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) → subset_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ga(x1, x2)  =  subset_in_ga(x1)
[]  =  []
subset_out_ga(x1, x2)  =  subset_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
member_in_ga(x1, x2)  =  member_in_ga(x1)
member_out_ga(x1, x2)  =  member_out_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)

(23) 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:

SUBSET_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, member_in_ga(X, Ys))
SUBSET_IN_GA(.(X, Xs), Ys) → MEMBER_IN_GA(X, Ys)
MEMBER_IN_GA(X, .(X3, Xs)) → U3_GA(X, X3, Xs, member_in_ga(X, Xs))
MEMBER_IN_GA(X, .(X3, Xs)) → MEMBER_IN_GA(X, Xs)
U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) → U2_GA(X, Xs, Ys, subset_in_ga(Xs, Ys))
U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) → SUBSET_IN_GA(Xs, Ys)

The TRS R consists of the following rules:

subset_in_ga([], X1) → subset_out_ga([], X1)
subset_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, member_in_ga(X, Ys))
member_in_ga(X, .(X, X2)) → member_out_ga(X, .(X, X2))
member_in_ga(X, .(X3, Xs)) → U3_ga(X, X3, Xs, member_in_ga(X, Xs))
U3_ga(X, X3, Xs, member_out_ga(X, Xs)) → member_out_ga(X, .(X3, Xs))
U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) → U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys))
U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) → subset_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ga(x1, x2)  =  subset_in_ga(x1)
[]  =  []
subset_out_ga(x1, x2)  =  subset_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
member_in_ga(x1, x2)  =  member_in_ga(x1)
member_out_ga(x1, x2)  =  member_out_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
SUBSET_IN_GA(x1, x2)  =  SUBSET_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
MEMBER_IN_GA(x1, x2)  =  MEMBER_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)

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

(24) Obligation:

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

SUBSET_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, member_in_ga(X, Ys))
SUBSET_IN_GA(.(X, Xs), Ys) → MEMBER_IN_GA(X, Ys)
MEMBER_IN_GA(X, .(X3, Xs)) → U3_GA(X, X3, Xs, member_in_ga(X, Xs))
MEMBER_IN_GA(X, .(X3, Xs)) → MEMBER_IN_GA(X, Xs)
U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) → U2_GA(X, Xs, Ys, subset_in_ga(Xs, Ys))
U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) → SUBSET_IN_GA(Xs, Ys)

The TRS R consists of the following rules:

subset_in_ga([], X1) → subset_out_ga([], X1)
subset_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, member_in_ga(X, Ys))
member_in_ga(X, .(X, X2)) → member_out_ga(X, .(X, X2))
member_in_ga(X, .(X3, Xs)) → U3_ga(X, X3, Xs, member_in_ga(X, Xs))
U3_ga(X, X3, Xs, member_out_ga(X, Xs)) → member_out_ga(X, .(X3, Xs))
U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) → U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys))
U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) → subset_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ga(x1, x2)  =  subset_in_ga(x1)
[]  =  []
subset_out_ga(x1, x2)  =  subset_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
member_in_ga(x1, x2)  =  member_in_ga(x1)
member_out_ga(x1, x2)  =  member_out_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
SUBSET_IN_GA(x1, x2)  =  SUBSET_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
MEMBER_IN_GA(x1, x2)  =  MEMBER_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes.

(26) Complex Obligation (AND)

(27) Obligation:

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

MEMBER_IN_GA(X, .(X3, Xs)) → MEMBER_IN_GA(X, Xs)

The TRS R consists of the following rules:

subset_in_ga([], X1) → subset_out_ga([], X1)
subset_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, member_in_ga(X, Ys))
member_in_ga(X, .(X, X2)) → member_out_ga(X, .(X, X2))
member_in_ga(X, .(X3, Xs)) → U3_ga(X, X3, Xs, member_in_ga(X, Xs))
U3_ga(X, X3, Xs, member_out_ga(X, Xs)) → member_out_ga(X, .(X3, Xs))
U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) → U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys))
U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) → subset_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ga(x1, x2)  =  subset_in_ga(x1)
[]  =  []
subset_out_ga(x1, x2)  =  subset_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
member_in_ga(x1, x2)  =  member_in_ga(x1)
member_out_ga(x1, x2)  =  member_out_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
MEMBER_IN_GA(x1, x2)  =  MEMBER_IN_GA(x1)

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

(28) UsableRulesProof (EQUIVALENT transformation)

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

(29) Obligation:

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

MEMBER_IN_GA(X, .(X3, Xs)) → MEMBER_IN_GA(X, Xs)

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

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

(30) PiDPToQDPProof (SOUND transformation)

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

(31) Obligation:

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

MEMBER_IN_GA(X) → MEMBER_IN_GA(X)

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

(32) 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 = MEMBER_IN_GA(X) evaluates to t =MEMBER_IN_GA(X)

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 MEMBER_IN_GA(X) to MEMBER_IN_GA(X).



(33) FALSE

(34) Obligation:

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

U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) → SUBSET_IN_GA(Xs, Ys)
SUBSET_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, member_in_ga(X, Ys))

The TRS R consists of the following rules:

subset_in_ga([], X1) → subset_out_ga([], X1)
subset_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, member_in_ga(X, Ys))
member_in_ga(X, .(X, X2)) → member_out_ga(X, .(X, X2))
member_in_ga(X, .(X3, Xs)) → U3_ga(X, X3, Xs, member_in_ga(X, Xs))
U3_ga(X, X3, Xs, member_out_ga(X, Xs)) → member_out_ga(X, .(X3, Xs))
U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) → U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys))
U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) → subset_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ga(x1, x2)  =  subset_in_ga(x1)
[]  =  []
subset_out_ga(x1, x2)  =  subset_out_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
member_in_ga(x1, x2)  =  member_in_ga(x1)
member_out_ga(x1, x2)  =  member_out_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
SUBSET_IN_GA(x1, x2)  =  SUBSET_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)

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

(35) UsableRulesProof (EQUIVALENT transformation)

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

(36) Obligation:

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

U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) → SUBSET_IN_GA(Xs, Ys)
SUBSET_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, member_in_ga(X, Ys))

The TRS R consists of the following rules:

member_in_ga(X, .(X, X2)) → member_out_ga(X, .(X, X2))
member_in_ga(X, .(X3, Xs)) → U3_ga(X, X3, Xs, member_in_ga(X, Xs))
U3_ga(X, X3, Xs, member_out_ga(X, Xs)) → member_out_ga(X, .(X3, Xs))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
member_in_ga(x1, x2)  =  member_in_ga(x1)
member_out_ga(x1, x2)  =  member_out_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
SUBSET_IN_GA(x1, x2)  =  SUBSET_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)

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

(37) PiDPToQDPProof (SOUND transformation)

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

(38) Obligation:

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

U1_GA(X, Xs, member_out_ga(X)) → SUBSET_IN_GA(Xs)
SUBSET_IN_GA(.(X, Xs)) → U1_GA(X, Xs, member_in_ga(X))

The TRS R consists of the following rules:

member_in_ga(X) → member_out_ga(X)
member_in_ga(X) → U3_ga(X, member_in_ga(X))
U3_ga(X, member_out_ga(X)) → member_out_ga(X)

The set Q consists of the following terms:

member_in_ga(x0)
U3_ga(x0, x1)

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

(39) 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:

  • SUBSET_IN_GA(.(X, Xs)) → U1_GA(X, Xs, member_in_ga(X))
    The graph contains the following edges 1 > 1, 1 > 2

  • U1_GA(X, Xs, member_out_ga(X)) → SUBSET_IN_GA(Xs)
    The graph contains the following edges 2 >= 1

(40) TRUE