MAYBE
proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Left Termination of the query pattern

subset(a,g)

w.r.t. the given Prolog program could not be shown:

(0) Prolog
(1) PrologToPiTRSProof [SOUND, 0 ms]
(2) PiTRS
    (3) DependencyPairsProof [EQUIVALENT, 0 ms]
    (4) PiDP
    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
    (6) AND
        (7) PiDP
            (8) UsableRulesProof [EQUIVALENT, 0 ms]
            (9) PiDP
            (10) PiDPToQDPProof [SOUND, 6 ms]
            (11) QDP
            (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (13) YES
        (14) PiDP
            (15) UsableRulesProof [EQUIVALENT, 0 ms]
            (16) PiDP
            (17) PiDPToQDPProof [SOUND, 0 ms]
            (18) QDP
            (19) TransformationProof [SOUND, 0 ms]
            (20) QDP
            (21) TransformationProof [EQUIVALENT, 0 ms]
            (22) QDP
(23) PrologToPiTRSProof [SOUND, 0 ms]
(24) PiTRS
    (25) DependencyPairsProof [EQUIVALENT, 0 ms]
    (26) PiDP
    (27) DependencyGraphProof [EQUIVALENT, 1 ms]
    (28) AND
        (29) PiDP
            (30) UsableRulesProof [EQUIVALENT, 0 ms]
            (31) PiDP
            (32) PiDPToQDPProof [SOUND, 11 ms]
            (33) QDP
            (34) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (35) YES
        (36) PiDP
            (37) UsableRulesProof [EQUIVALENT, 0 ms]
            (38) PiDP
            (39) PiDPToQDPProof [SOUND, 0 ms]
            (40) QDP
            (41) TransformationProof [SOUND, 0 ms]
            (42) QDP
(43) PrologToTRSTransformerProof [SOUND, 0 ms]
(44) QTRS
    (45) DependencyPairsProof [EQUIVALENT, 0 ms]
    (46) QDP
    (47) DependencyGraphProof [EQUIVALENT, 0 ms]
    (48) AND
        (49) QDP
            (50) UsableRulesProof [EQUIVALENT, 0 ms]
            (51) QDP
            (52) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (53) YES
        (54) QDP
            (55) NonTerminationLoopProof [COMPLETE, 0 ms]
            (56) NO
(57) PrologToIRSwTTransformerProof [SOUND, 0 ms]
(58) AND
    (59) IRSwT
        (60) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 1 ms]
        (61) IRSwT
        (62) IntTRSCompressionProof [EQUIVALENT, 21 ms]
        (63) IRSwT
        (64) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
        (65) IRSwT
        (66) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
        (67) IRSwT
        (68) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 27 ms]
        (69) IRSwT
        (70) TempFilterProof [SOUND, 2 ms]
        (71) IRSwT
        (72) IRSwTToQDPProof [SOUND, 0 ms]
        (73) QDP
        (74) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (75) YES
    (76) IRSwT
        (77) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
        (78) IRSwT
        (79) IntTRSCompressionProof [EQUIVALENT, 6 ms]
        (80) IRSwT
        (81) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
        (82) IRSwT
(83) PrologToDTProblemTransformerProof [SOUND, 0 ms]
(84) TRIPLES
    (85) TriplesToPiDPProof [SOUND, 0 ms]
    (86) PiDP
    (87) DependencyGraphProof [EQUIVALENT, 0 ms]
    (88) AND
        (89) PiDP
            (90) UsableRulesProof [EQUIVALENT, 0 ms]
            (91) PiDP
            (92) PiDPToQDPProof [SOUND, 2 ms]
            (93) QDP
            (94) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (95) YES
        (96) PiDP
            (97) PiDPToQDPProof [SOUND, 0 ms]
            (98) QDP


----------------------------------------

(0)
Obligation:
Clauses:

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


Query: subset(a,g)
----------------------------------------

(1) PrologToPiTRSProof (SOUND)
We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:

subset_in_2: (f,b)

member_in_2: (f,b)

Transforming Prolog into the following Term Rewriting System:

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

   subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys))
   member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs))
   member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs))
   U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs))
   U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
   subset_in_ag([], Ys) -> subset_out_ag([], Ys)
   U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ag(x1, x2)  =  subset_in_ag(x2)

U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)

member_in_ag(x1, x2)  =  member_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)

member_out_ag(x1, x2)  =  member_out_ag(x1)

U3_ag(x1, x2, x3, x4)  =  U3_ag(x1, x4)

subset_out_ag(x1, x2)  =  subset_out_ag(x1)





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_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys))
   member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs))
   member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs))
   U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs))
   U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
   subset_in_ag([], Ys) -> subset_out_ag([], Ys)
   U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ag(x1, x2)  =  subset_in_ag(x2)

U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)

member_in_ag(x1, x2)  =  member_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)

member_out_ag(x1, x2)  =  member_out_ag(x1)

U3_ag(x1, x2, x3, x4)  =  U3_ag(x1, x4)

subset_out_ag(x1, x2)  =  subset_out_ag(x1)



----------------------------------------

(3) DependencyPairsProof (EQUIVALENT)
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_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, member_in_ag(X, Ys))
   SUBSET_IN_AG(.(X, Xs), Ys) -> MEMBER_IN_AG(X, Ys)
   MEMBER_IN_AG(X, .(Y, Xs)) -> U1_AG(X, Y, Xs, member_in_ag(X, Xs))
   MEMBER_IN_AG(X, .(Y, Xs)) -> MEMBER_IN_AG(X, Xs)
   U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_AG(X, Xs, Ys, subset_in_ag(Xs, Ys))
   U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys)

The TRS R consists of the following rules:

   subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys))
   member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs))
   member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs))
   U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs))
   U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
   subset_in_ag([], Ys) -> subset_out_ag([], Ys)
   U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ag(x1, x2)  =  subset_in_ag(x2)

U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)

member_in_ag(x1, x2)  =  member_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)

member_out_ag(x1, x2)  =  member_out_ag(x1)

U3_ag(x1, x2, x3, x4)  =  U3_ag(x1, x4)

subset_out_ag(x1, x2)  =  subset_out_ag(x1)

SUBSET_IN_AG(x1, x2)  =  SUBSET_IN_AG(x2)

U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)

MEMBER_IN_AG(x1, x2)  =  MEMBER_IN_AG(x2)

U1_AG(x1, x2, x3, x4)  =  U1_AG(x4)

U3_AG(x1, x2, x3, x4)  =  U3_AG(x1, 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_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, member_in_ag(X, Ys))
   SUBSET_IN_AG(.(X, Xs), Ys) -> MEMBER_IN_AG(X, Ys)
   MEMBER_IN_AG(X, .(Y, Xs)) -> U1_AG(X, Y, Xs, member_in_ag(X, Xs))
   MEMBER_IN_AG(X, .(Y, Xs)) -> MEMBER_IN_AG(X, Xs)
   U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_AG(X, Xs, Ys, subset_in_ag(Xs, Ys))
   U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys)

The TRS R consists of the following rules:

   subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys))
   member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs))
   member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs))
   U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs))
   U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
   subset_in_ag([], Ys) -> subset_out_ag([], Ys)
   U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ag(x1, x2)  =  subset_in_ag(x2)

U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)

member_in_ag(x1, x2)  =  member_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)

member_out_ag(x1, x2)  =  member_out_ag(x1)

U3_ag(x1, x2, x3, x4)  =  U3_ag(x1, x4)

subset_out_ag(x1, x2)  =  subset_out_ag(x1)

SUBSET_IN_AG(x1, x2)  =  SUBSET_IN_AG(x2)

U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)

MEMBER_IN_AG(x1, x2)  =  MEMBER_IN_AG(x2)

U1_AG(x1, x2, x3, x4)  =  U1_AG(x4)

U3_AG(x1, x2, x3, x4)  =  U3_AG(x1, x4)


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

(5) DependencyGraphProof (EQUIVALENT)
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_AG(X, .(Y, Xs)) -> MEMBER_IN_AG(X, Xs)

The TRS R consists of the following rules:

   subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys))
   member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs))
   member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs))
   U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs))
   U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
   subset_in_ag([], Ys) -> subset_out_ag([], Ys)
   U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ag(x1, x2)  =  subset_in_ag(x2)

U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)

member_in_ag(x1, x2)  =  member_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)

member_out_ag(x1, x2)  =  member_out_ag(x1)

U3_ag(x1, x2, x3, x4)  =  U3_ag(x1, x4)

subset_out_ag(x1, x2)  =  subset_out_ag(x1)

MEMBER_IN_AG(x1, x2)  =  MEMBER_IN_AG(x2)


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

(8) UsableRulesProof (EQUIVALENT)
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_AG(X, .(Y, Xs)) -> MEMBER_IN_AG(X, Xs)

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

MEMBER_IN_AG(x1, x2)  =  MEMBER_IN_AG(x2)


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

(10) PiDPToQDPProof (SOUND)
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_AG(.(Y, Xs)) -> MEMBER_IN_AG(Xs)

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

(12) QDPSizeChangeProof (EQUIVALENT)
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:
*MEMBER_IN_AG(.(Y, Xs)) -> MEMBER_IN_AG(Xs)
The graph contains the following edges 1 > 1


----------------------------------------

(13)
YES

----------------------------------------

(14)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys)
   SUBSET_IN_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, member_in_ag(X, Ys))

The TRS R consists of the following rules:

   subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys))
   member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs))
   member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs))
   U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs))
   U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
   subset_in_ag([], Ys) -> subset_out_ag([], Ys)
   U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ag(x1, x2)  =  subset_in_ag(x2)

U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)

member_in_ag(x1, x2)  =  member_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)

member_out_ag(x1, x2)  =  member_out_ag(x1)

U3_ag(x1, x2, x3, x4)  =  U3_ag(x1, x4)

subset_out_ag(x1, x2)  =  subset_out_ag(x1)

SUBSET_IN_AG(x1, x2)  =  SUBSET_IN_AG(x2)

U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)


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

(15) UsableRulesProof (EQUIVALENT)
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:

   U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys)
   SUBSET_IN_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, member_in_ag(X, Ys))

The TRS R consists of the following rules:

   member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs))
   member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs))
   U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs))

The argument filtering Pi contains the following mapping:
member_in_ag(x1, x2)  =  member_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x4)

member_out_ag(x1, x2)  =  member_out_ag(x1)

SUBSET_IN_AG(x1, x2)  =  SUBSET_IN_AG(x2)

U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)


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

(17) PiDPToQDPProof (SOUND)
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:

   U2_AG(Ys, member_out_ag(X)) -> SUBSET_IN_AG(Ys)
   SUBSET_IN_AG(Ys) -> U2_AG(Ys, member_in_ag(Ys))

The TRS R consists of the following rules:

   member_in_ag(.(Y, Xs)) -> U1_ag(member_in_ag(Xs))
   member_in_ag(.(X, Xs)) -> member_out_ag(X)
   U1_ag(member_out_ag(X)) -> member_out_ag(X)

The set Q consists of the following terms:

   member_in_ag(x0)
   U1_ag(x0)

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

(19) TransformationProof (SOUND)
By narrowing [LPAR04] the rule SUBSET_IN_AG(Ys) -> U2_AG(Ys, member_in_ag(Ys)) at position [1] we obtained the following new rules [LPAR04]:

   (SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), U1_ag(member_in_ag(x1))),SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), U1_ag(member_in_ag(x1))))
   (SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), member_out_ag(x0)),SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), member_out_ag(x0)))


----------------------------------------

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U2_AG(Ys, member_out_ag(X)) -> SUBSET_IN_AG(Ys)
   SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), U1_ag(member_in_ag(x1)))
   SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), member_out_ag(x0))

The TRS R consists of the following rules:

   member_in_ag(.(Y, Xs)) -> U1_ag(member_in_ag(Xs))
   member_in_ag(.(X, Xs)) -> member_out_ag(X)
   U1_ag(member_out_ag(X)) -> member_out_ag(X)

The set Q consists of the following terms:

   member_in_ag(x0)
   U1_ag(x0)

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

(21) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule U2_AG(Ys, member_out_ag(X)) -> SUBSET_IN_AG(Ys) we obtained the following new rules [LPAR04]:

   (U2_AG(.(z0, z1), member_out_ag(x1)) -> SUBSET_IN_AG(.(z0, z1)),U2_AG(.(z0, z1), member_out_ag(x1)) -> SUBSET_IN_AG(.(z0, z1)))
   (U2_AG(.(z0, z1), member_out_ag(z0)) -> SUBSET_IN_AG(.(z0, z1)),U2_AG(.(z0, z1), member_out_ag(z0)) -> SUBSET_IN_AG(.(z0, z1)))


----------------------------------------

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), U1_ag(member_in_ag(x1)))
   SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), member_out_ag(x0))
   U2_AG(.(z0, z1), member_out_ag(x1)) -> SUBSET_IN_AG(.(z0, z1))
   U2_AG(.(z0, z1), member_out_ag(z0)) -> SUBSET_IN_AG(.(z0, z1))

The TRS R consists of the following rules:

   member_in_ag(.(Y, Xs)) -> U1_ag(member_in_ag(Xs))
   member_in_ag(.(X, Xs)) -> member_out_ag(X)
   U1_ag(member_out_ag(X)) -> member_out_ag(X)

The set Q consists of the following terms:

   member_in_ag(x0)
   U1_ag(x0)

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

(23) PrologToPiTRSProof (SOUND)
We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:

subset_in_2: (f,b)

member_in_2: (f,b)

Transforming Prolog into the following Term Rewriting System:

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

   subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys))
   member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs))
   member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs))
   U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs))
   U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
   subset_in_ag([], Ys) -> subset_out_ag([], Ys)
   U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ag(x1, x2)  =  subset_in_ag(x2)

U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)

member_in_ag(x1, x2)  =  member_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)

member_out_ag(x1, x2)  =  member_out_ag(x1, x2)

U3_ag(x1, x2, x3, x4)  =  U3_ag(x1, x3, x4)

subset_out_ag(x1, x2)  =  subset_out_ag(x1, x2)





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



----------------------------------------

(24)
Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:

   subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys))
   member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs))
   member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs))
   U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs))
   U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
   subset_in_ag([], Ys) -> subset_out_ag([], Ys)
   U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ag(x1, x2)  =  subset_in_ag(x2)

U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)

member_in_ag(x1, x2)  =  member_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)

member_out_ag(x1, x2)  =  member_out_ag(x1, x2)

U3_ag(x1, x2, x3, x4)  =  U3_ag(x1, x3, x4)

subset_out_ag(x1, x2)  =  subset_out_ag(x1, x2)



----------------------------------------

(25) DependencyPairsProof (EQUIVALENT)
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_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, member_in_ag(X, Ys))
   SUBSET_IN_AG(.(X, Xs), Ys) -> MEMBER_IN_AG(X, Ys)
   MEMBER_IN_AG(X, .(Y, Xs)) -> U1_AG(X, Y, Xs, member_in_ag(X, Xs))
   MEMBER_IN_AG(X, .(Y, Xs)) -> MEMBER_IN_AG(X, Xs)
   U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_AG(X, Xs, Ys, subset_in_ag(Xs, Ys))
   U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys)

The TRS R consists of the following rules:

   subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys))
   member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs))
   member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs))
   U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs))
   U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
   subset_in_ag([], Ys) -> subset_out_ag([], Ys)
   U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ag(x1, x2)  =  subset_in_ag(x2)

U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)

member_in_ag(x1, x2)  =  member_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)

member_out_ag(x1, x2)  =  member_out_ag(x1, x2)

U3_ag(x1, x2, x3, x4)  =  U3_ag(x1, x3, x4)

subset_out_ag(x1, x2)  =  subset_out_ag(x1, x2)

SUBSET_IN_AG(x1, x2)  =  SUBSET_IN_AG(x2)

U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)

MEMBER_IN_AG(x1, x2)  =  MEMBER_IN_AG(x2)

U1_AG(x1, x2, x3, x4)  =  U1_AG(x2, x3, x4)

U3_AG(x1, x2, x3, x4)  =  U3_AG(x1, x3, x4)


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

(26)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   SUBSET_IN_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, member_in_ag(X, Ys))
   SUBSET_IN_AG(.(X, Xs), Ys) -> MEMBER_IN_AG(X, Ys)
   MEMBER_IN_AG(X, .(Y, Xs)) -> U1_AG(X, Y, Xs, member_in_ag(X, Xs))
   MEMBER_IN_AG(X, .(Y, Xs)) -> MEMBER_IN_AG(X, Xs)
   U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_AG(X, Xs, Ys, subset_in_ag(Xs, Ys))
   U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys)

The TRS R consists of the following rules:

   subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys))
   member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs))
   member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs))
   U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs))
   U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
   subset_in_ag([], Ys) -> subset_out_ag([], Ys)
   U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ag(x1, x2)  =  subset_in_ag(x2)

U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)

member_in_ag(x1, x2)  =  member_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag'(x2, x3, x4)

member_out_ag(x1, x2)  =  member_out_ag(x1, x2)

U3_ag(x1, x2, x3, x4)  =  U3_ag(x1, x3, x4)

subset_out_ag(x1, x2)  =  subset_out_ag(x1, x2)

SUBSET_IN_AG(x1, x2)  =  SUBSET_IN_AG(x2)

U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)

MEMBER_IN_AG(x1, x2)  =  MEMBER_IN_AG(x2)

U1_AG(x1, x2, x3, x4)  =  U1_AG(x2, x3, x4)

U3_AG(x1, x2, x3, x4)  =  U3_AG(x1, x3, x4)


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

(27) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes.
----------------------------------------

(28)
Complex Obligation (AND)

----------------------------------------

(29)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   MEMBER_IN_AG(X, .(Y, Xs)) -> MEMBER_IN_AG(X, Xs)

The TRS R consists of the following rules:

   subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys))
   member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs))
   member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs))
   U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs))
   U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
   subset_in_ag([], Ys) -> subset_out_ag([], Ys)
   U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ag(x1, x2)  =  subset_in_ag(x2)

U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)

member_in_ag(x1, x2)  =  member_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)

member_out_ag(x1, x2)  =  member_out_ag(x1, x2)

U3_ag(x1, x2, x3, x4)  =  U3_ag(x1, x3, x4)

subset_out_ag(x1, x2)  =  subset_out_ag(x1, x2)

MEMBER_IN_AG(x1, x2)  =  MEMBER_IN_AG(x2)


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

(30) UsableRulesProof (EQUIVALENT)
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:

   MEMBER_IN_AG(X, .(Y, Xs)) -> MEMBER_IN_AG(X, Xs)

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

MEMBER_IN_AG(x1, x2)  =  MEMBER_IN_AG(x2)


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

(32) PiDPToQDPProof (SOUND)
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:

   MEMBER_IN_AG(.(Y, Xs)) -> MEMBER_IN_AG(Xs)

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

(34) QDPSizeChangeProof (EQUIVALENT)
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:
*MEMBER_IN_AG(.(Y, Xs)) -> MEMBER_IN_AG(Xs)
The graph contains the following edges 1 > 1


----------------------------------------

(35)
YES

----------------------------------------

(36)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys)
   SUBSET_IN_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, member_in_ag(X, Ys))

The TRS R consists of the following rules:

   subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys))
   member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs))
   member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs))
   U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs))
   U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
   subset_in_ag([], Ys) -> subset_out_ag([], Ys)
   U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_in_ag(x1, x2)  =  subset_in_ag(x2)

U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)

member_in_ag(x1, x2)  =  member_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)

member_out_ag(x1, x2)  =  member_out_ag(x1, x2)

U3_ag(x1, x2, x3, x4)  =  U3_ag(x1, x3, x4)

subset_out_ag(x1, x2)  =  subset_out_ag(x1, x2)

SUBSET_IN_AG(x1, x2)  =  SUBSET_IN_AG(x2)

U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)


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

(37) UsableRulesProof (EQUIVALENT)
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:

   U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys)
   SUBSET_IN_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, member_in_ag(X, Ys))

The TRS R consists of the following rules:

   member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs))
   member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs))
   U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs))

The argument filtering Pi contains the following mapping:
member_in_ag(x1, x2)  =  member_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x2, x3, x4)

member_out_ag(x1, x2)  =  member_out_ag(x1, x2)

SUBSET_IN_AG(x1, x2)  =  SUBSET_IN_AG(x2)

U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)


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

(39) PiDPToQDPProof (SOUND)
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:

   U2_AG(Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Ys)
   SUBSET_IN_AG(Ys) -> U2_AG(Ys, member_in_ag(Ys))

The TRS R consists of the following rules:

   member_in_ag(.(Y, Xs)) -> U1_ag(Y, Xs, member_in_ag(Xs))
   member_in_ag(.(X, Xs)) -> member_out_ag(X, .(X, Xs))
   U1_ag(Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs))

The set Q consists of the following terms:

   member_in_ag(x0)
   U1_ag(x0, x1, x2)

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

(41) TransformationProof (SOUND)
By narrowing [LPAR04] the rule SUBSET_IN_AG(Ys) -> U2_AG(Ys, member_in_ag(Ys)) at position [1] we obtained the following new rules [LPAR04]:

   (SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), U1_ag(x0, x1, member_in_ag(x1))),SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), U1_ag(x0, x1, member_in_ag(x1))))
   (SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), member_out_ag(x0, .(x0, x1))),SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), member_out_ag(x0, .(x0, x1))))


----------------------------------------

(42)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U2_AG(Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Ys)
   SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), U1_ag(x0, x1, member_in_ag(x1)))
   SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), member_out_ag(x0, .(x0, x1)))

The TRS R consists of the following rules:

   member_in_ag(.(Y, Xs)) -> U1_ag(Y, Xs, member_in_ag(Xs))
   member_in_ag(.(X, Xs)) -> member_out_ag(X, .(X, Xs))
   U1_ag(Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs))

The set Q consists of the following terms:

   member_in_ag(x0)
   U1_ag(x0, x1, x2)

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

(43) PrologToTRSTransformerProof (SOUND)
Transformed Prolog program to TRS.

{
    "root": 11,
    "program": {
        "directives": [],
        "clauses": [
            [
                "(member X (. Y Xs))",
                "(member X Xs)"
            ],
            [
                "(member X (. X Xs))",
                null
            ],
            [
                "(subset (. X Xs) Ys)",
                "(',' (member X Ys) (subset Xs Ys))"
            ],
            [
                "(subset ([]) Ys)",
                null
            ]
        ]
    },
    "graph": {
        "nodes": {
            "11": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(subset T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T2"],
                    "free": [],
                    "exprvars": []
                }
            },
            "12": {
                "goal": [
                    {
                        "clause": 2,
                        "scope": 1,
                        "term": "(subset T1 T2)"
                    },
                    {
                        "clause": 3,
                        "scope": 1,
                        "term": "(subset T1 T2)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T2"],
                    "free": [],
                    "exprvars": []
                }
            },
            "13": {
                "goal": [{
                    "clause": 2,
                    "scope": 1,
                    "term": "(subset T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T2"],
                    "free": [],
                    "exprvars": []
                }
            },
            "14": {
                "goal": [{
                    "clause": 3,
                    "scope": 1,
                    "term": "(subset T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T2"],
                    "free": [],
                    "exprvars": []
                }
            },
            "15": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (member T18 T17) (subset T19 T17))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T17"],
                    "free": [],
                    "exprvars": []
                }
            },
            "16": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "type": "Nodes",
            "151": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "152": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "153": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "143": {
                "goal": [{
                    "clause": 0,
                    "scope": 2,
                    "term": "(member T18 T17)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T17"],
                    "free": [],
                    "exprvars": []
                }
            },
            "154": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "122": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(member T18 T17)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T17"],
                    "free": [],
                    "exprvars": []
                }
            },
            "144": {
                "goal": [{
                    "clause": 1,
                    "scope": 2,
                    "term": "(member T18 T17)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T17"],
                    "free": [],
                    "exprvars": []
                }
            },
            "155": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "123": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(subset T23 T17)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T17"],
                    "free": [],
                    "exprvars": []
                }
            },
            "145": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(member T42 T41)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T41"],
                    "free": [],
                    "exprvars": []
                }
            },
            "156": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "146": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "136": {
                "goal": [
                    {
                        "clause": 0,
                        "scope": 2,
                        "term": "(member T18 T17)"
                    },
                    {
                        "clause": 1,
                        "scope": 2,
                        "term": "(member T18 T17)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T17"],
                    "free": [],
                    "exprvars": []
                }
            }
        },
        "edges": [
            {
                "from": 11,
                "to": 12,
                "label": "CASE"
            },
            {
                "from": 12,
                "to": 13,
                "label": "PARALLEL"
            },
            {
                "from": 12,
                "to": 14,
                "label": "PARALLEL"
            },
            {
                "from": 13,
                "to": 15,
                "label": "EVAL with clause\nsubset(.(X13, X14), X15) :- ','(member(X13, X15), subset(X14, X15)).\nand substitutionX13 -> T18,\nX14 -> T19,\nT1 -> .(T18, T19),\nT2 -> T17,\nX15 -> T17,\nT15 -> T18,\nT16 -> T19"
            },
            {
                "from": 13,
                "to": 16,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 14,
                "to": 154,
                "label": "EVAL with clause\nsubset([], X51).\nand substitutionT1 -> [],\nT2 -> T57,\nX51 -> T57"
            },
            {
                "from": 14,
                "to": 155,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 15,
                "to": 122,
                "label": "SPLIT 1"
            },
            {
                "from": 15,
                "to": 123,
                "label": "SPLIT 2\nnew knowledge:\nT18 is ground\nT17 is ground\nreplacements:T19 -> T23"
            },
            {
                "from": 122,
                "to": 136,
                "label": "CASE"
            },
            {
                "from": 123,
                "to": 11,
                "label": "INSTANCE with matching:\nT1 -> T23\nT2 -> T17"
            },
            {
                "from": 136,
                "to": 143,
                "label": "PARALLEL"
            },
            {
                "from": 136,
                "to": 144,
                "label": "PARALLEL"
            },
            {
                "from": 143,
                "to": 145,
                "label": "EVAL with clause\nmember(X34, .(X35, X36)) :- member(X34, X36).\nand substitutionT18 -> T42,\nX34 -> T42,\nX35 -> T40,\nX36 -> T41,\nT17 -> .(T40, T41),\nT39 -> T42"
            },
            {
                "from": 143,
                "to": 146,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 144,
                "to": 151,
                "label": "EVAL with clause\nmember(X44, .(X44, X45)).\nand substitutionT18 -> T50,\nX44 -> T50,\nX45 -> T51,\nT17 -> .(T50, T51)"
            },
            {
                "from": 144,
                "to": 152,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 145,
                "to": 122,
                "label": "INSTANCE with matching:\nT18 -> T42\nT17 -> T41"
            },
            {
                "from": 151,
                "to": 153,
                "label": "SUCCESS"
            },
            {
                "from": 154,
                "to": 156,
                "label": "SUCCESS"
            }
        ],
        "type": "Graph"
    }
}

----------------------------------------

(44)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f11_in(T17) -> U1(f15_in(T17), T17)
   U1(f15_out1(T18, T19), T17) -> f11_out1(.(T18, T19))
   f11_in(T57) -> f11_out1([])
   f122_in(.(T40, T41)) -> U2(f122_in(T41), .(T40, T41))
   U2(f122_out1(T42), .(T40, T41)) -> f122_out1(T42)
   f122_in(.(T50, T51)) -> f122_out1(T50)
   f15_in(T17) -> U3(f122_in(T17), T17)
   U3(f122_out1(T18), T17) -> U4(f11_in(T17), T17, T18)
   U4(f11_out1(T23), T17, T18) -> f15_out1(T18, T23)

Q is empty.

----------------------------------------

(45) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(46)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F11_IN(T17) -> U1^1(f15_in(T17), T17)
   F11_IN(T17) -> F15_IN(T17)
   F122_IN(.(T40, T41)) -> U2^1(f122_in(T41), .(T40, T41))
   F122_IN(.(T40, T41)) -> F122_IN(T41)
   F15_IN(T17) -> U3^1(f122_in(T17), T17)
   F15_IN(T17) -> F122_IN(T17)
   U3^1(f122_out1(T18), T17) -> U4^1(f11_in(T17), T17, T18)
   U3^1(f122_out1(T18), T17) -> F11_IN(T17)

The TRS R consists of the following rules:

   f11_in(T17) -> U1(f15_in(T17), T17)
   U1(f15_out1(T18, T19), T17) -> f11_out1(.(T18, T19))
   f11_in(T57) -> f11_out1([])
   f122_in(.(T40, T41)) -> U2(f122_in(T41), .(T40, T41))
   U2(f122_out1(T42), .(T40, T41)) -> f122_out1(T42)
   f122_in(.(T50, T51)) -> f122_out1(T50)
   f15_in(T17) -> U3(f122_in(T17), T17)
   U3(f122_out1(T18), T17) -> U4(f11_in(T17), T17, T18)
   U4(f11_out1(T23), T17, T18) -> f15_out1(T18, T23)

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

(47) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.
----------------------------------------

(48)
Complex Obligation (AND)

----------------------------------------

(49)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F122_IN(.(T40, T41)) -> F122_IN(T41)

The TRS R consists of the following rules:

   f11_in(T17) -> U1(f15_in(T17), T17)
   U1(f15_out1(T18, T19), T17) -> f11_out1(.(T18, T19))
   f11_in(T57) -> f11_out1([])
   f122_in(.(T40, T41)) -> U2(f122_in(T41), .(T40, T41))
   U2(f122_out1(T42), .(T40, T41)) -> f122_out1(T42)
   f122_in(.(T50, T51)) -> f122_out1(T50)
   f15_in(T17) -> U3(f122_in(T17), T17)
   U3(f122_out1(T18), T17) -> U4(f11_in(T17), T17, T18)
   U4(f11_out1(T23), T17, T18) -> f15_out1(T18, T23)

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

(50) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(51)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F122_IN(.(T40, T41)) -> F122_IN(T41)

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

(52) QDPSizeChangeProof (EQUIVALENT)
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:
*F122_IN(.(T40, T41)) -> F122_IN(T41)
The graph contains the following edges 1 > 1


----------------------------------------

(53)
YES

----------------------------------------

(54)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F11_IN(T17) -> F15_IN(T17)
   F15_IN(T17) -> U3^1(f122_in(T17), T17)
   U3^1(f122_out1(T18), T17) -> F11_IN(T17)

The TRS R consists of the following rules:

   f11_in(T17) -> U1(f15_in(T17), T17)
   U1(f15_out1(T18, T19), T17) -> f11_out1(.(T18, T19))
   f11_in(T57) -> f11_out1([])
   f122_in(.(T40, T41)) -> U2(f122_in(T41), .(T40, T41))
   U2(f122_out1(T42), .(T40, T41)) -> f122_out1(T42)
   f122_in(.(T50, T51)) -> f122_out1(T50)
   f15_in(T17) -> U3(f122_in(T17), T17)
   U3(f122_out1(T18), T17) -> U4(f11_in(T17), T17, T18)
   U4(f11_out1(T23), T17, T18) -> f15_out1(T18, T23)

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

(55) NonTerminationLoopProof (COMPLETE)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = F15_IN(.(T50, T51)) evaluates to  t =F15_IN(.(T50, T51))

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

--------------------------------------------------------------------------------
Rewriting sequence

F15_IN(.(T50, T51)) -> U3^1(f122_in(.(T50, T51)), .(T50, T51))
with rule F15_IN(T17) -> U3^1(f122_in(T17), T17) at position [] and matcher [T17 / .(T50, T51)]

U3^1(f122_in(.(T50, T51)), .(T50, T51)) -> U3^1(f122_out1(T50), .(T50, T51))
with rule f122_in(.(T50', T51')) -> f122_out1(T50') at position [0] and matcher [T50' / T50, T51' / T51]

U3^1(f122_out1(T50), .(T50, T51)) -> F11_IN(.(T50, T51))
with rule U3^1(f122_out1(T18), T17') -> F11_IN(T17') at position [] and matcher [T18 / T50, T17' / .(T50, T51)]

F11_IN(.(T50, T51)) -> F15_IN(.(T50, T51))
with rule F11_IN(T17) -> F15_IN(T17)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.




----------------------------------------

(56)
NO

----------------------------------------

(57) PrologToIRSwTTransformerProof (SOUND)
Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert

{
    "root": 1,
    "program": {
        "directives": [],
        "clauses": [
            [
                "(member X (. Y Xs))",
                "(member X Xs)"
            ],
            [
                "(member X (. X Xs))",
                null
            ],
            [
                "(subset (. X Xs) Ys)",
                "(',' (member X Ys) (subset Xs Ys))"
            ],
            [
                "(subset ([]) Ys)",
                null
            ]
        ]
    },
    "graph": {
        "nodes": {
            "22": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "77": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(subset T23 T17)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T17"],
                    "free": [],
                    "exprvars": []
                }
            },
            "19": {
                "goal": [{
                    "clause": 2,
                    "scope": 1,
                    "term": "(subset T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T2"],
                    "free": [],
                    "exprvars": []
                }
            },
            "182": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "type": "Nodes",
            "150": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "183": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(subset T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T2"],
                    "free": [],
                    "exprvars": []
                }
            },
            "2": {
                "goal": [
                    {
                        "clause": 2,
                        "scope": 1,
                        "term": "(subset T1 T2)"
                    },
                    {
                        "clause": 3,
                        "scope": 1,
                        "term": "(subset T1 T2)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T2"],
                    "free": [],
                    "exprvars": []
                }
            },
            "112": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "147": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "148": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "105": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(member T42 T41)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T41"],
                    "free": [],
                    "exprvars": []
                }
            },
            "149": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "80": {
                "goal": [
                    {
                        "clause": 0,
                        "scope": 2,
                        "term": "(member T18 T17)"
                    },
                    {
                        "clause": 1,
                        "scope": 2,
                        "term": "(member T18 T17)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T17"],
                    "free": [],
                    "exprvars": []
                }
            },
            "81": {
                "goal": [{
                    "clause": 0,
                    "scope": 2,
                    "term": "(member T18 T17)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T17"],
                    "free": [],
                    "exprvars": []
                }
            },
            "82": {
                "goal": [{
                    "clause": 1,
                    "scope": 2,
                    "term": "(member T18 T17)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T17"],
                    "free": [],
                    "exprvars": []
                }
            },
            "20": {
                "goal": [{
                    "clause": 3,
                    "scope": 1,
                    "term": "(subset T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T2"],
                    "free": [],
                    "exprvars": []
                }
            },
            "21": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (member T18 T17) (subset T19 T17))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T17"],
                    "free": [],
                    "exprvars": []
                }
            },
            "76": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(member T18 T17)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T17"],
                    "free": [],
                    "exprvars": []
                }
            }
        },
        "edges": [
            {
                "from": 1,
                "to": 2,
                "label": "CASE"
            },
            {
                "from": 2,
                "to": 19,
                "label": "PARALLEL"
            },
            {
                "from": 2,
                "to": 20,
                "label": "PARALLEL"
            },
            {
                "from": 19,
                "to": 21,
                "label": "EVAL with clause\nsubset(.(X13, X14), X15) :- ','(member(X13, X15), subset(X14, X15)).\nand substitutionX13 -> T18,\nX14 -> T19,\nT1 -> .(T18, T19),\nT2 -> T17,\nX15 -> T17,\nT15 -> T18,\nT16 -> T19"
            },
            {
                "from": 19,
                "to": 22,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 20,
                "to": 150,
                "label": "EVAL with clause\nsubset([], X51).\nand substitutionT1 -> [],\nT2 -> T57,\nX51 -> T57"
            },
            {
                "from": 20,
                "to": 182,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 21,
                "to": 76,
                "label": "SPLIT 1"
            },
            {
                "from": 21,
                "to": 77,
                "label": "SPLIT 2\nnew knowledge:\nT18 is ground\nT17 is ground\nreplacements:T19 -> T23"
            },
            {
                "from": 76,
                "to": 80,
                "label": "CASE"
            },
            {
                "from": 77,
                "to": 1,
                "label": "INSTANCE with matching:\nT1 -> T23\nT2 -> T17"
            },
            {
                "from": 80,
                "to": 81,
                "label": "PARALLEL"
            },
            {
                "from": 80,
                "to": 82,
                "label": "PARALLEL"
            },
            {
                "from": 81,
                "to": 105,
                "label": "EVAL with clause\nmember(X34, .(X35, X36)) :- member(X34, X36).\nand substitutionT18 -> T42,\nX34 -> T42,\nX35 -> T40,\nX36 -> T41,\nT17 -> .(T40, T41),\nT39 -> T42"
            },
            {
                "from": 81,
                "to": 112,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 82,
                "to": 147,
                "label": "EVAL with clause\nmember(X44, .(X44, X45)).\nand substitutionT18 -> T50,\nX44 -> T50,\nX45 -> T51,\nT17 -> .(T50, T51)"
            },
            {
                "from": 82,
                "to": 148,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 105,
                "to": 76,
                "label": "INSTANCE with matching:\nT18 -> T42\nT17 -> T41"
            },
            {
                "from": 147,
                "to": 149,
                "label": "SUCCESS"
            },
            {
                "from": 150,
                "to": 183,
                "label": "SUCCESS"
            }
        ],
        "type": "Graph"
    }
}

----------------------------------------

(58)
Complex Obligation (AND)

----------------------------------------

(59)
Obligation:
Rules:
f80_in(T17) -> f82_in(T17) :|: TRUE
f81_out(x) -> f80_out(x) :|: TRUE
f82_out(x1) -> f80_out(x1) :|: TRUE
f80_in(x2) -> f81_in(x2) :|: TRUE
f112_out -> f81_out(x3) :|: TRUE
f81_in(x4) -> f112_in :|: TRUE
f105_out(T41) -> f81_out(.(T40, T41)) :|: TRUE
f81_in(.(x5, x6)) -> f105_in(x6) :|: TRUE
f76_in(x7) -> f80_in(x7) :|: TRUE
f80_out(x8) -> f76_out(x8) :|: TRUE
f105_in(x9) -> f76_in(x9) :|: TRUE
f76_out(x10) -> f105_out(x10) :|: TRUE
f2_out(T2) -> f1_out(T2) :|: TRUE
f1_in(x11) -> f2_in(x11) :|: TRUE
f2_in(x12) -> f20_in(x12) :|: TRUE
f19_out(x13) -> f2_out(x13) :|: TRUE
f20_out(x14) -> f2_out(x14) :|: TRUE
f2_in(x15) -> f19_in(x15) :|: TRUE
f22_out -> f19_out(x16) :|: TRUE
f21_out(x17) -> f19_out(x17) :|: TRUE
f19_in(x18) -> f22_in :|: TRUE
f19_in(x19) -> f21_in(x19) :|: TRUE
f21_in(x20) -> f76_in(x20) :|: TRUE
f76_out(x21) -> f77_in(x21) :|: TRUE
f77_out(x22) -> f21_out(x22) :|: TRUE
Start term: f1_in(T2)

----------------------------------------

(60) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
Constructed simple dependency graph.

Simplified to the following IRSwTs:

intTRSProblem:
f80_in(x2) -> f81_in(x2) :|: TRUE
f81_in(.(x5, x6)) -> f105_in(x6) :|: TRUE
f76_in(x7) -> f80_in(x7) :|: TRUE
f105_in(x9) -> f76_in(x9) :|: TRUE


----------------------------------------

(61)
Obligation:
Rules:
f80_in(x2) -> f81_in(x2) :|: TRUE
f81_in(.(x5, x6)) -> f105_in(x6) :|: TRUE
f76_in(x7) -> f80_in(x7) :|: TRUE
f105_in(x9) -> f76_in(x9) :|: TRUE

----------------------------------------

(62) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(63)
Obligation:
Rules:
f76_in(.(x5:0, x6:0)) -> f76_in(x6:0) :|: TRUE

----------------------------------------

(64) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
----------------------------------------

(65)
Obligation:
Rules:
f76_in(.(x5:0, x6:0)) -> f76_in(x6:0) :|: TRUE

----------------------------------------

(66) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f76_in(.(x5:0, x6:0)) -> f76_in(x6:0) :|: TRUE

Arcs:
(1) -> (1)

This digraph is fully evaluated!
----------------------------------------

(67)
Obligation:

Termination digraph:
Nodes:
(1) f76_in(.(x5:0, x6:0)) -> f76_in(x6:0) :|: TRUE

Arcs:
(1) -> (1)

This digraph is fully evaluated!

----------------------------------------

(68) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:

   .(x1, x2) -> .(x2)

----------------------------------------

(69)
Obligation:
Rules:
f76_in(.(x6:0)) -> f76_in(x6:0) :|: TRUE

----------------------------------------

(70) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f76_in(VARIABLE)
.(VARIABLE)
Removed predefined arithmetic.
----------------------------------------

(71)
Obligation:
Rules:
f76_in(.(x6:0)) -> f76_in(x6:0)

----------------------------------------

(72) IRSwTToQDPProof (SOUND)
Removed the integers and created a QDP-Problem.
----------------------------------------

(73)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   f76_in(.(x6:0)) -> f76_in(x6:0)

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

(74) QDPSizeChangeProof (EQUIVALENT)
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:
*f76_in(.(x6:0)) -> f76_in(x6:0)
The graph contains the following edges 1 > 1


----------------------------------------

(75)
YES

----------------------------------------

(76)
Obligation:
Rules:
f2_in(T2) -> f20_in(T2) :|: TRUE
f19_out(x) -> f2_out(x) :|: TRUE
f20_out(x1) -> f2_out(x1) :|: TRUE
f2_in(x2) -> f19_in(x2) :|: TRUE
f2_out(x3) -> f1_out(x3) :|: TRUE
f1_in(x4) -> f2_in(x4) :|: TRUE
f80_in(T17) -> f82_in(T17) :|: TRUE
f81_out(x5) -> f80_out(x5) :|: TRUE
f82_out(x6) -> f80_out(x6) :|: TRUE
f80_in(x7) -> f81_in(x7) :|: TRUE
f1_out(x8) -> f77_out(x8) :|: TRUE
f77_in(x9) -> f1_in(x9) :|: TRUE
f76_in(x10) -> f80_in(x10) :|: TRUE
f80_out(x11) -> f76_out(x11) :|: TRUE
f105_in(T41) -> f76_in(T41) :|: TRUE
f76_out(x12) -> f105_out(x12) :|: TRUE
f21_in(x13) -> f76_in(x13) :|: TRUE
f76_out(x14) -> f77_in(x14) :|: TRUE
f77_out(x15) -> f21_out(x15) :|: TRUE
f112_out -> f81_out(x16) :|: TRUE
f81_in(x17) -> f112_in :|: TRUE
f105_out(x18) -> f81_out(.(x19, x18)) :|: TRUE
f81_in(.(x20, x21)) -> f105_in(x21) :|: TRUE
f147_in -> f147_out :|: TRUE
f22_out -> f19_out(x22) :|: TRUE
f21_out(x23) -> f19_out(x23) :|: TRUE
f19_in(x24) -> f22_in :|: TRUE
f19_in(x25) -> f21_in(x25) :|: TRUE
f148_out -> f82_out(x26) :|: TRUE
f82_in(.(T50, T51)) -> f147_in :|: TRUE
f82_in(x27) -> f148_in :|: TRUE
f147_out -> f82_out(.(x28, x29)) :|: TRUE
Start term: f1_in(T2)

----------------------------------------

(77) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
Constructed simple dependency graph.

Simplified to the following IRSwTs:

intTRSProblem:
f2_in(x2) -> f19_in(x2) :|: TRUE
f1_in(x4) -> f2_in(x4) :|: TRUE
f80_in(T17) -> f82_in(T17) :|: TRUE
f81_out(x5) -> f80_out(x5) :|: TRUE
f82_out(x6) -> f80_out(x6) :|: TRUE
f80_in(x7) -> f81_in(x7) :|: TRUE
f77_in(x9) -> f1_in(x9) :|: TRUE
f76_in(x10) -> f80_in(x10) :|: TRUE
f80_out(x11) -> f76_out(x11) :|: TRUE
f105_in(T41) -> f76_in(T41) :|: TRUE
f76_out(x12) -> f105_out(x12) :|: TRUE
f21_in(x13) -> f76_in(x13) :|: TRUE
f76_out(x14) -> f77_in(x14) :|: TRUE
f105_out(x18) -> f81_out(.(x19, x18)) :|: TRUE
f81_in(.(x20, x21)) -> f105_in(x21) :|: TRUE
f147_in -> f147_out :|: TRUE
f19_in(x25) -> f21_in(x25) :|: TRUE
f82_in(.(T50, T51)) -> f147_in :|: TRUE
f147_out -> f82_out(.(x28, x29)) :|: TRUE


----------------------------------------

(78)
Obligation:
Rules:
f2_in(x2) -> f19_in(x2) :|: TRUE
f1_in(x4) -> f2_in(x4) :|: TRUE
f80_in(T17) -> f82_in(T17) :|: TRUE
f81_out(x5) -> f80_out(x5) :|: TRUE
f82_out(x6) -> f80_out(x6) :|: TRUE
f80_in(x7) -> f81_in(x7) :|: TRUE
f77_in(x9) -> f1_in(x9) :|: TRUE
f76_in(x10) -> f80_in(x10) :|: TRUE
f80_out(x11) -> f76_out(x11) :|: TRUE
f105_in(T41) -> f76_in(T41) :|: TRUE
f76_out(x12) -> f105_out(x12) :|: TRUE
f21_in(x13) -> f76_in(x13) :|: TRUE
f76_out(x14) -> f77_in(x14) :|: TRUE
f105_out(x18) -> f81_out(.(x19, x18)) :|: TRUE
f81_in(.(x20, x21)) -> f105_in(x21) :|: TRUE
f147_in -> f147_out :|: TRUE
f19_in(x25) -> f21_in(x25) :|: TRUE
f82_in(.(T50, T51)) -> f147_in :|: TRUE
f147_out -> f82_out(.(x28, x29)) :|: TRUE

----------------------------------------

(79) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(80)
Obligation:
Rules:
f76_in(.(x20:0, x21:0)) -> f76_in(x21:0) :|: TRUE
f76_in(.(T50:0, T51:0)) -> f76_out(.(x28:0, x29:0)) :|: TRUE
f76_out(x12:0) -> f76_out(.(x19:0, x12:0)) :|: TRUE
f76_out(x14:0) -> f76_in(x14:0) :|: TRUE

----------------------------------------

(81) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
----------------------------------------

(82)
Obligation:
Rules:
f76_in(.(x20:0, x21:0)) -> f76_in(x21:0) :|: TRUE
f76_in(.(T50:0, T51:0)) -> f76_out(.(x28:0, x29:0)) :|: TRUE
f76_out(x12:0) -> f76_out(.(x19:0, x12:0)) :|: TRUE
f76_out(x14:0) -> f76_in(x14:0) :|: TRUE

----------------------------------------

(83) PrologToDTProblemTransformerProof (SOUND)
Built DT problem from termination graph DT10.

{
    "root": 23,
    "program": {
        "directives": [],
        "clauses": [
            [
                "(member X (. Y Xs))",
                "(member X Xs)"
            ],
            [
                "(member X (. X Xs))",
                null
            ],
            [
                "(subset (. X Xs) Ys)",
                "(',' (member X Ys) (subset Xs Ys))"
            ],
            [
                "(subset ([]) Ys)",
                null
            ]
        ]
    },
    "graph": {
        "nodes": {
            "23": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(subset T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T2"],
                    "free": [],
                    "exprvars": []
                }
            },
            "24": {
                "goal": [
                    {
                        "clause": 2,
                        "scope": 1,
                        "term": "(subset T1 T2)"
                    },
                    {
                        "clause": 3,
                        "scope": 1,
                        "term": "(subset T1 T2)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T2"],
                    "free": [],
                    "exprvars": []
                }
            },
            "27": {
                "goal": [
                    {
                        "clause": -1,
                        "scope": -1,
                        "term": "(',' (member T9 T8) (subset T10 T8))"
                    },
                    {
                        "clause": 3,
                        "scope": 1,
                        "term": "(subset T1 T8)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T8"],
                    "free": [],
                    "exprvars": []
                }
            },
            "28": {
                "goal": [{
                    "clause": 3,
                    "scope": 1,
                    "term": "(subset T1 T2)"
                }],
                "kb": {
                    "nonunifying": [[
                        "(subset T1 T2)",
                        "(subset (. X4 X5) X6)"
                    ]],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T2"],
                    "free": [
                        "X4",
                        "X5",
                        "X6"
                    ],
                    "exprvars": []
                }
            },
            "190": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "191": {
                "goal": [{
                    "clause": 3,
                    "scope": 1,
                    "term": "(subset T1 T8)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T8"],
                    "free": [],
                    "exprvars": []
                }
            },
            "192": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "193": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "type": "Nodes",
            "194": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "195": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "196": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "197": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "115": {
                "goal": [{
                    "clause": 0,
                    "scope": 3,
                    "term": "(member T26 T25)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T25"],
                    "free": [],
                    "exprvars": []
                }
            },
            "116": {
                "goal": [{
                    "clause": 1,
                    "scope": 3,
                    "term": "(member T26 T25)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T25"],
                    "free": [],
                    "exprvars": []
                }
            },
            "117": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(member T50 T49)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T49"],
                    "free": [],
                    "exprvars": []
                }
            },
            "118": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "73": {
                "goal": [
                    {
                        "clause": 0,
                        "scope": 2,
                        "term": "(',' (member T9 T8) (subset T10 T8))"
                    },
                    {
                        "clause": 1,
                        "scope": 2,
                        "term": "(',' (member T9 T8) (subset T10 T8))"
                    },
                    {
                        "clause": -1,
                        "scope": 2,
                        "term": null
                    },
                    {
                        "clause": 3,
                        "scope": 1,
                        "term": "(subset T1 T8)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T8"],
                    "free": [],
                    "exprvars": []
                }
            },
            "74": {
                "goal": [{
                    "clause": 0,
                    "scope": 2,
                    "term": "(',' (member T9 T8) (subset T10 T8))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T8"],
                    "free": [],
                    "exprvars": []
                }
            },
            "75": {
                "goal": [
                    {
                        "clause": 1,
                        "scope": 2,
                        "term": "(',' (member T9 T8) (subset T10 T8))"
                    },
                    {
                        "clause": -1,
                        "scope": 2,
                        "term": null
                    },
                    {
                        "clause": 3,
                        "scope": 1,
                        "term": "(subset T1 T8)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T8"],
                    "free": [],
                    "exprvars": []
                }
            },
            "78": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (member T26 T25) (subset T27 (. T24 T25)))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T24",
                        "T25"
                    ],
                    "free": [],
                    "exprvars": []
                }
            },
            "79": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "184": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "185": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "186": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "187": {
                "goal": [{
                    "clause": 1,
                    "scope": 2,
                    "term": "(',' (member T9 T8) (subset T10 T8))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T8"],
                    "free": [],
                    "exprvars": []
                }
            },
            "188": {
                "goal": [
                    {
                        "clause": -1,
                        "scope": 2,
                        "term": null
                    },
                    {
                        "clause": 3,
                        "scope": 1,
                        "term": "(subset T1 T8)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T8"],
                    "free": [],
                    "exprvars": []
                }
            },
            "189": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(subset T74 (. T72 T73))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T72",
                        "T73"
                    ],
                    "free": [],
                    "exprvars": []
                }
            },
            "104": {
                "goal": [
                    {
                        "clause": 0,
                        "scope": 3,
                        "term": "(member T26 T25)"
                    },
                    {
                        "clause": 1,
                        "scope": 3,
                        "term": "(member T26 T25)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T25"],
                    "free": [],
                    "exprvars": []
                }
            },
            "83": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(member T26 T25)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T25"],
                    "free": [],
                    "exprvars": []
                }
            },
            "84": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(subset T31 (. T24 T25))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T24",
                        "T25"
                    ],
                    "free": [],
                    "exprvars": []
                }
            }
        },
        "edges": [
            {
                "from": 23,
                "to": 24,
                "label": "CASE"
            },
            {
                "from": 24,
                "to": 27,
                "label": "EVAL with clause\nsubset(.(X4, X5), X6) :- ','(member(X4, X6), subset(X5, X6)).\nand substitutionX4 -> T9,\nX5 -> T10,\nT1 -> .(T9, T10),\nT2 -> T8,\nX6 -> T8,\nT6 -> T9,\nT7 -> T10"
            },
            {
                "from": 24,
                "to": 28,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 27,
                "to": 73,
                "label": "CASE"
            },
            {
                "from": 28,
                "to": 195,
                "label": "EVAL with clause\nsubset([], X72).\nand substitutionT1 -> [],\nT2 -> T83,\nX72 -> T83"
            },
            {
                "from": 28,
                "to": 196,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 73,
                "to": 74,
                "label": "PARALLEL"
            },
            {
                "from": 73,
                "to": 75,
                "label": "PARALLEL"
            },
            {
                "from": 74,
                "to": 78,
                "label": "EVAL with clause\nmember(X19, .(X20, X21)) :- member(X19, X21).\nand substitutionT9 -> T26,\nX19 -> T26,\nX20 -> T24,\nX21 -> T25,\nT8 -> .(T24, T25),\nT23 -> T26,\nT10 -> T27"
            },
            {
                "from": 74,
                "to": 79,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 75,
                "to": 187,
                "label": "PARALLEL"
            },
            {
                "from": 75,
                "to": 188,
                "label": "PARALLEL"
            },
            {
                "from": 78,
                "to": 83,
                "label": "SPLIT 1"
            },
            {
                "from": 78,
                "to": 84,
                "label": "SPLIT 2\nnew knowledge:\nT26 is ground\nT25 is ground\nreplacements:T27 -> T31"
            },
            {
                "from": 83,
                "to": 104,
                "label": "CASE"
            },
            {
                "from": 84,
                "to": 23,
                "label": "INSTANCE with matching:\nT1 -> T31\nT2 -> .(T24, T25)"
            },
            {
                "from": 104,
                "to": 115,
                "label": "PARALLEL"
            },
            {
                "from": 104,
                "to": 116,
                "label": "PARALLEL"
            },
            {
                "from": 115,
                "to": 117,
                "label": "EVAL with clause\nmember(X40, .(X41, X42)) :- member(X40, X42).\nand substitutionT26 -> T50,\nX40 -> T50,\nX41 -> T48,\nX42 -> T49,\nT25 -> .(T48, T49),\nT47 -> T50"
            },
            {
                "from": 115,
                "to": 118,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 116,
                "to": 184,
                "label": "EVAL with clause\nmember(X50, .(X50, X51)).\nand substitutionT26 -> T58,\nX50 -> T58,\nX51 -> T59,\nT25 -> .(T58, T59)"
            },
            {
                "from": 116,
                "to": 185,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 117,
                "to": 83,
                "label": "INSTANCE with matching:\nT26 -> T50\nT25 -> T49"
            },
            {
                "from": 184,
                "to": 186,
                "label": "SUCCESS"
            },
            {
                "from": 187,
                "to": 189,
                "label": "EVAL with clause\nmember(X63, .(X63, X64)).\nand substitutionT9 -> T72,\nX63 -> T72,\nX64 -> T73,\nT8 -> .(T72, T73),\nT10 -> T74"
            },
            {
                "from": 187,
                "to": 190,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 188,
                "to": 191,
                "label": "FAILURE"
            },
            {
                "from": 189,
                "to": 23,
                "label": "INSTANCE with matching:\nT1 -> T74\nT2 -> .(T72, T73)"
            },
            {
                "from": 191,
                "to": 192,
                "label": "EVAL with clause\nsubset([], X70).\nand substitutionT1 -> [],\nT8 -> T81,\nX70 -> T81"
            },
            {
                "from": 191,
                "to": 193,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 192,
                "to": 194,
                "label": "SUCCESS"
            },
            {
                "from": 195,
                "to": 197,
                "label": "SUCCESS"
            }
        ],
        "type": "Graph"
    }
}

----------------------------------------

(84)
Obligation:
Triples:

memberB(X1, .(X2, X3)) :- memberB(X1, X3).
subsetA(.(X1, X2), .(X3, X4)) :- memberB(X1, X4).
subsetA(.(X1, X2), .(X3, X4)) :- ','(membercB(X1, X4), subsetA(X2, .(X3, X4))).
subsetA(.(X1, X2), .(X1, X3)) :- subsetA(X2, .(X1, X3)).

Clauses:

subsetcA(.(X1, X2), .(X3, X4)) :- ','(membercB(X1, X4), subsetcA(X2, .(X3, X4))).
subsetcA(.(X1, X2), .(X1, X3)) :- subsetcA(X2, .(X1, X3)).
subsetcA([], X1).
subsetcA([], X1).
membercB(X1, .(X2, X3)) :- membercB(X1, X3).
membercB(X1, .(X1, X2)).

Afs:

subsetA(x1, x2)  =  subsetA(x2)


----------------------------------------

(85) TriplesToPiDPProof (SOUND)
We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:

subsetA_in_2: (f,b)

memberB_in_2: (f,b)

membercB_in_2: (f,b)

Transforming TRIPLES into the following Term Rewriting System:

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

   SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> U2_AG(X1, X2, X3, X4, memberB_in_ag(X1, X4))
   SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> MEMBERB_IN_AG(X1, X4)
   MEMBERB_IN_AG(X1, .(X2, X3)) -> U1_AG(X1, X2, X3, memberB_in_ag(X1, X3))
   MEMBERB_IN_AG(X1, .(X2, X3)) -> MEMBERB_IN_AG(X1, X3)
   SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> U3_AG(X1, X2, X3, X4, membercB_in_ag(X1, X4))
   U3_AG(X1, X2, X3, X4, membercB_out_ag(X1, X4)) -> U4_AG(X1, X2, X3, X4, subsetA_in_ag(X2, .(X3, X4)))
   U3_AG(X1, X2, X3, X4, membercB_out_ag(X1, X4)) -> SUBSETA_IN_AG(X2, .(X3, X4))
   SUBSETA_IN_AG(.(X1, X2), .(X1, X3)) -> U5_AG(X1, X2, X3, subsetA_in_ag(X2, .(X1, X3)))
   SUBSETA_IN_AG(.(X1, X2), .(X1, X3)) -> SUBSETA_IN_AG(X2, .(X1, X3))

The TRS R consists of the following rules:

   membercB_in_ag(X1, .(X2, X3)) -> U10_ag(X1, X2, X3, membercB_in_ag(X1, X3))
   membercB_in_ag(X1, .(X1, X2)) -> membercB_out_ag(X1, .(X1, X2))
   U10_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3))

The argument filtering Pi contains the following mapping:
subsetA_in_ag(x1, x2)  =  subsetA_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

memberB_in_ag(x1, x2)  =  memberB_in_ag(x2)

membercB_in_ag(x1, x2)  =  membercB_in_ag(x2)

U10_ag(x1, x2, x3, x4)  =  U10_ag(x2, x3, x4)

membercB_out_ag(x1, x2)  =  membercB_out_ag(x1, x2)

SUBSETA_IN_AG(x1, x2)  =  SUBSETA_IN_AG(x2)

U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x3, x4, x5)

MEMBERB_IN_AG(x1, x2)  =  MEMBERB_IN_AG(x2)

U1_AG(x1, x2, x3, x4)  =  U1_AG(x2, x3, x4)

U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)

U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x3, x4, x5)

U5_AG(x1, x2, x3, x4)  =  U5_AG(x1, x3, x4)


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


Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES



----------------------------------------

(86)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> U2_AG(X1, X2, X3, X4, memberB_in_ag(X1, X4))
   SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> MEMBERB_IN_AG(X1, X4)
   MEMBERB_IN_AG(X1, .(X2, X3)) -> U1_AG(X1, X2, X3, memberB_in_ag(X1, X3))
   MEMBERB_IN_AG(X1, .(X2, X3)) -> MEMBERB_IN_AG(X1, X3)
   SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> U3_AG(X1, X2, X3, X4, membercB_in_ag(X1, X4))
   U3_AG(X1, X2, X3, X4, membercB_out_ag(X1, X4)) -> U4_AG(X1, X2, X3, X4, subsetA_in_ag(X2, .(X3, X4)))
   U3_AG(X1, X2, X3, X4, membercB_out_ag(X1, X4)) -> SUBSETA_IN_AG(X2, .(X3, X4))
   SUBSETA_IN_AG(.(X1, X2), .(X1, X3)) -> U5_AG(X1, X2, X3, subsetA_in_ag(X2, .(X1, X3)))
   SUBSETA_IN_AG(.(X1, X2), .(X1, X3)) -> SUBSETA_IN_AG(X2, .(X1, X3))

The TRS R consists of the following rules:

   membercB_in_ag(X1, .(X2, X3)) -> U10_ag(X1, X2, X3, membercB_in_ag(X1, X3))
   membercB_in_ag(X1, .(X1, X2)) -> membercB_out_ag(X1, .(X1, X2))
   U10_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3))

The argument filtering Pi contains the following mapping:
subsetA_in_ag(x1, x2)  =  subsetA_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

memberB_in_ag(x1, x2)  =  memberB_in_ag(x2)

membercB_in_ag(x1, x2)  =  membercB_in_ag(x2)

U10_ag(x1, x2, x3, x4)  =  U10_ag(x2, x3, x4)

membercB_out_ag(x1, x2)  =  membercB_out_ag(x1, x2)

SUBSETA_IN_AG(x1, x2)  =  SUBSETA_IN_AG(x2)

U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x3, x4, x5)

MEMBERB_IN_AG(x1, x2)  =  MEMBERB_IN_AG(x2)

U1_AG(x1, x2, x3, x4)  =  U1_AG(x2, x3, x4)

U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)

U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x3, x4, x5)

U5_AG(x1, x2, x3, x4)  =  U5_AG(x1, x3, x4)


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

(87) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.
----------------------------------------

(88)
Complex Obligation (AND)

----------------------------------------

(89)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   MEMBERB_IN_AG(X1, .(X2, X3)) -> MEMBERB_IN_AG(X1, X3)

The TRS R consists of the following rules:

   membercB_in_ag(X1, .(X2, X3)) -> U10_ag(X1, X2, X3, membercB_in_ag(X1, X3))
   membercB_in_ag(X1, .(X1, X2)) -> membercB_out_ag(X1, .(X1, X2))
   U10_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3))

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

membercB_in_ag(x1, x2)  =  membercB_in_ag(x2)

U10_ag(x1, x2, x3, x4)  =  U10_ag(x2, x3, x4)

membercB_out_ag(x1, x2)  =  membercB_out_ag(x1, x2)

MEMBERB_IN_AG(x1, x2)  =  MEMBERB_IN_AG(x2)


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

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

(91)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   MEMBERB_IN_AG(X1, .(X2, X3)) -> MEMBERB_IN_AG(X1, X3)

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

MEMBERB_IN_AG(x1, x2)  =  MEMBERB_IN_AG(x2)


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

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

(93)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MEMBERB_IN_AG(.(X2, X3)) -> MEMBERB_IN_AG(X3)

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

(94) QDPSizeChangeProof (EQUIVALENT)
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:
*MEMBERB_IN_AG(.(X2, X3)) -> MEMBERB_IN_AG(X3)
The graph contains the following edges 1 > 1


----------------------------------------

(95)
YES

----------------------------------------

(96)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> U3_AG(X1, X2, X3, X4, membercB_in_ag(X1, X4))
   U3_AG(X1, X2, X3, X4, membercB_out_ag(X1, X4)) -> SUBSETA_IN_AG(X2, .(X3, X4))
   SUBSETA_IN_AG(.(X1, X2), .(X1, X3)) -> SUBSETA_IN_AG(X2, .(X1, X3))

The TRS R consists of the following rules:

   membercB_in_ag(X1, .(X2, X3)) -> U10_ag(X1, X2, X3, membercB_in_ag(X1, X3))
   membercB_in_ag(X1, .(X1, X2)) -> membercB_out_ag(X1, .(X1, X2))
   U10_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3))

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

membercB_in_ag(x1, x2)  =  membercB_in_ag(x2)

U10_ag(x1, x2, x3, x4)  =  U10_ag(x2, x3, x4)

membercB_out_ag(x1, x2)  =  membercB_out_ag(x1, x2)

SUBSETA_IN_AG(x1, x2)  =  SUBSETA_IN_AG(x2)

U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)


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

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

(98)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   SUBSETA_IN_AG(.(X3, X4)) -> U3_AG(X3, X4, membercB_in_ag(X4))
   U3_AG(X3, X4, membercB_out_ag(X1, X4)) -> SUBSETA_IN_AG(.(X3, X4))
   SUBSETA_IN_AG(.(X1, X3)) -> SUBSETA_IN_AG(.(X1, X3))

The TRS R consists of the following rules:

   membercB_in_ag(.(X2, X3)) -> U10_ag(X2, X3, membercB_in_ag(X3))
   membercB_in_ag(.(X1, X2)) -> membercB_out_ag(X1, .(X1, X2))
   U10_ag(X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3))

The set Q consists of the following terms:

   membercB_in_ag(x0)
   U10_ag(x0, x1, x2)

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