/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files


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


MAYBE
proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Left Termination of the query pattern

len(g,a)

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

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


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

(0)
Obligation:
Clauses:

len([], 0) :- !.
len(Xs, s(N)) :- ','(tail(Xs, Ys), len(Ys, N)).
tail([], []).
tail(.(X, Xs), Xs).


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

(1) CutEliminatorProof (SOUND)
Eliminated all cuts by simply ignoring them[PROLOG].
----------------------------------------

(2)
Obligation:
Clauses:

len([], 0).
len(Xs, s(N)) :- ','(tail(Xs, Ys), len(Ys, N)).
tail([], []).
tail(.(X, Xs), Xs).


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

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

len_in_2: (b,f)

Transforming Prolog into the following Term Rewriting System:

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

   len_in_ga([], 0) -> len_out_ga([], 0)
   len_in_ga(Xs, s(N)) -> U1_ga(Xs, N, tail_in_ga(Xs, Ys))
   tail_in_ga([], []) -> tail_out_ga([], [])
   tail_in_ga(.(X, Xs), Xs) -> tail_out_ga(.(X, Xs), Xs)
   U1_ga(Xs, N, tail_out_ga(Xs, Ys)) -> U2_ga(Xs, N, Ys, len_in_ga(Ys, N))
   U2_ga(Xs, N, Ys, len_out_ga(Ys, N)) -> len_out_ga(Xs, s(N))

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

[]  =  []

len_out_ga(x1, x2)  =  len_out_ga(x2)

U1_ga(x1, x2, x3)  =  U1_ga(x3)

tail_in_ga(x1, x2)  =  tail_in_ga(x1)

tail_out_ga(x1, x2)  =  tail_out_ga(x2)

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

U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)

s(x1)  =  s(x1)





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



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

(4)
Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:

   len_in_ga([], 0) -> len_out_ga([], 0)
   len_in_ga(Xs, s(N)) -> U1_ga(Xs, N, tail_in_ga(Xs, Ys))
   tail_in_ga([], []) -> tail_out_ga([], [])
   tail_in_ga(.(X, Xs), Xs) -> tail_out_ga(.(X, Xs), Xs)
   U1_ga(Xs, N, tail_out_ga(Xs, Ys)) -> U2_ga(Xs, N, Ys, len_in_ga(Ys, N))
   U2_ga(Xs, N, Ys, len_out_ga(Ys, N)) -> len_out_ga(Xs, s(N))

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

[]  =  []

len_out_ga(x1, x2)  =  len_out_ga(x2)

U1_ga(x1, x2, x3)  =  U1_ga(x3)

tail_in_ga(x1, x2)  =  tail_in_ga(x1)

tail_out_ga(x1, x2)  =  tail_out_ga(x2)

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

U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)

s(x1)  =  s(x1)



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

(5) 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:

   LEN_IN_GA(Xs, s(N)) -> U1_GA(Xs, N, tail_in_ga(Xs, Ys))
   LEN_IN_GA(Xs, s(N)) -> TAIL_IN_GA(Xs, Ys)
   U1_GA(Xs, N, tail_out_ga(Xs, Ys)) -> U2_GA(Xs, N, Ys, len_in_ga(Ys, N))
   U1_GA(Xs, N, tail_out_ga(Xs, Ys)) -> LEN_IN_GA(Ys, N)

The TRS R consists of the following rules:

   len_in_ga([], 0) -> len_out_ga([], 0)
   len_in_ga(Xs, s(N)) -> U1_ga(Xs, N, tail_in_ga(Xs, Ys))
   tail_in_ga([], []) -> tail_out_ga([], [])
   tail_in_ga(.(X, Xs), Xs) -> tail_out_ga(.(X, Xs), Xs)
   U1_ga(Xs, N, tail_out_ga(Xs, Ys)) -> U2_ga(Xs, N, Ys, len_in_ga(Ys, N))
   U2_ga(Xs, N, Ys, len_out_ga(Ys, N)) -> len_out_ga(Xs, s(N))

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

[]  =  []

len_out_ga(x1, x2)  =  len_out_ga(x2)

U1_ga(x1, x2, x3)  =  U1_ga(x3)

tail_in_ga(x1, x2)  =  tail_in_ga(x1)

tail_out_ga(x1, x2)  =  tail_out_ga(x2)

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

U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)

s(x1)  =  s(x1)

LEN_IN_GA(x1, x2)  =  LEN_IN_GA(x1)

U1_GA(x1, x2, x3)  =  U1_GA(x3)

TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)

U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)


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

(6)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LEN_IN_GA(Xs, s(N)) -> U1_GA(Xs, N, tail_in_ga(Xs, Ys))
   LEN_IN_GA(Xs, s(N)) -> TAIL_IN_GA(Xs, Ys)
   U1_GA(Xs, N, tail_out_ga(Xs, Ys)) -> U2_GA(Xs, N, Ys, len_in_ga(Ys, N))
   U1_GA(Xs, N, tail_out_ga(Xs, Ys)) -> LEN_IN_GA(Ys, N)

The TRS R consists of the following rules:

   len_in_ga([], 0) -> len_out_ga([], 0)
   len_in_ga(Xs, s(N)) -> U1_ga(Xs, N, tail_in_ga(Xs, Ys))
   tail_in_ga([], []) -> tail_out_ga([], [])
   tail_in_ga(.(X, Xs), Xs) -> tail_out_ga(.(X, Xs), Xs)
   U1_ga(Xs, N, tail_out_ga(Xs, Ys)) -> U2_ga(Xs, N, Ys, len_in_ga(Ys, N))
   U2_ga(Xs, N, Ys, len_out_ga(Ys, N)) -> len_out_ga(Xs, s(N))

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

[]  =  []

len_out_ga(x1, x2)  =  len_out_ga(x2)

U1_ga(x1, x2, x3)  =  U1_ga(x3)

tail_in_ga(x1, x2)  =  tail_in_ga(x1)

tail_out_ga(x1, x2)  =  tail_out_ga(x2)

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

U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)

s(x1)  =  s(x1)

LEN_IN_GA(x1, x2)  =  LEN_IN_GA(x1)

U1_GA(x1, x2, x3)  =  U1_GA(x3)

TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)

U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)


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

(7) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.
----------------------------------------

(8)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U1_GA(Xs, N, tail_out_ga(Xs, Ys)) -> LEN_IN_GA(Ys, N)
   LEN_IN_GA(Xs, s(N)) -> U1_GA(Xs, N, tail_in_ga(Xs, Ys))

The TRS R consists of the following rules:

   len_in_ga([], 0) -> len_out_ga([], 0)
   len_in_ga(Xs, s(N)) -> U1_ga(Xs, N, tail_in_ga(Xs, Ys))
   tail_in_ga([], []) -> tail_out_ga([], [])
   tail_in_ga(.(X, Xs), Xs) -> tail_out_ga(.(X, Xs), Xs)
   U1_ga(Xs, N, tail_out_ga(Xs, Ys)) -> U2_ga(Xs, N, Ys, len_in_ga(Ys, N))
   U2_ga(Xs, N, Ys, len_out_ga(Ys, N)) -> len_out_ga(Xs, s(N))

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

[]  =  []

len_out_ga(x1, x2)  =  len_out_ga(x2)

U1_ga(x1, x2, x3)  =  U1_ga(x3)

tail_in_ga(x1, x2)  =  tail_in_ga(x1)

tail_out_ga(x1, x2)  =  tail_out_ga(x2)

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

U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)

s(x1)  =  s(x1)

LEN_IN_GA(x1, x2)  =  LEN_IN_GA(x1)

U1_GA(x1, x2, x3)  =  U1_GA(x3)


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

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

(10)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U1_GA(Xs, N, tail_out_ga(Xs, Ys)) -> LEN_IN_GA(Ys, N)
   LEN_IN_GA(Xs, s(N)) -> U1_GA(Xs, N, tail_in_ga(Xs, Ys))

The TRS R consists of the following rules:

   tail_in_ga([], []) -> tail_out_ga([], [])
   tail_in_ga(.(X, Xs), Xs) -> tail_out_ga(.(X, Xs), Xs)

The argument filtering Pi contains the following mapping:
[]  =  []

tail_in_ga(x1, x2)  =  tail_in_ga(x1)

tail_out_ga(x1, x2)  =  tail_out_ga(x2)

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

s(x1)  =  s(x1)

LEN_IN_GA(x1, x2)  =  LEN_IN_GA(x1)

U1_GA(x1, x2, x3)  =  U1_GA(x3)


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

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

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U1_GA(tail_out_ga(Ys)) -> LEN_IN_GA(Ys)
   LEN_IN_GA(Xs) -> U1_GA(tail_in_ga(Xs))

The TRS R consists of the following rules:

   tail_in_ga([]) -> tail_out_ga([])
   tail_in_ga(.(X, Xs)) -> tail_out_ga(Xs)

The set Q consists of the following terms:

   tail_in_ga(x0)

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

(13) UsableRulesReductionPairsProof (EQUIVALENT)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

The following rules are removed from R:

   tail_in_ga(.(X, Xs)) -> tail_out_ga(Xs)
Used ordering: POLO with Polynomial interpretation [POLO]:

   POL(.(x_1, x_2)) = x_1 + 2*x_2
   POL(LEN_IN_GA(x_1)) = x_1
   POL(U1_GA(x_1)) = x_1
   POL([]) = 0
   POL(tail_in_ga(x_1)) = x_1
   POL(tail_out_ga(x_1)) = x_1


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

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

   U1_GA(tail_out_ga(Ys)) -> LEN_IN_GA(Ys)
   LEN_IN_GA(Xs) -> U1_GA(tail_in_ga(Xs))

The TRS R consists of the following rules:

   tail_in_ga([]) -> tail_out_ga([])

The set Q consists of the following terms:

   tail_in_ga(x0)

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

(15) TransformationProof (SOUND)
By narrowing [LPAR04] the rule LEN_IN_GA(Xs) -> U1_GA(tail_in_ga(Xs)) at position [0] we obtained the following new rules [LPAR04]:

   (LEN_IN_GA([]) -> U1_GA(tail_out_ga([])),LEN_IN_GA([]) -> U1_GA(tail_out_ga([])))


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

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U1_GA(tail_out_ga(Ys)) -> LEN_IN_GA(Ys)
   LEN_IN_GA([]) -> U1_GA(tail_out_ga([]))

The TRS R consists of the following rules:

   tail_in_ga([]) -> tail_out_ga([])

The set Q consists of the following terms:

   tail_in_ga(x0)

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

(17) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U1_GA(tail_out_ga(Ys)) -> LEN_IN_GA(Ys)
   LEN_IN_GA([]) -> U1_GA(tail_out_ga([]))

R is empty.
The set Q consists of the following terms:

   tail_in_ga(x0)

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

(19) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   tail_in_ga(x0)


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

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

   U1_GA(tail_out_ga(Ys)) -> LEN_IN_GA(Ys)
   LEN_IN_GA([]) -> U1_GA(tail_out_ga([]))

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

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

len_in_2: (b,f)

Transforming Prolog into the following Term Rewriting System:

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

   len_in_ga([], 0) -> len_out_ga([], 0)
   len_in_ga(Xs, s(N)) -> U1_ga(Xs, N, tail_in_ga(Xs, Ys))
   tail_in_ga([], []) -> tail_out_ga([], [])
   tail_in_ga(.(X, Xs), Xs) -> tail_out_ga(.(X, Xs), Xs)
   U1_ga(Xs, N, tail_out_ga(Xs, Ys)) -> U2_ga(Xs, N, Ys, len_in_ga(Ys, N))
   U2_ga(Xs, N, Ys, len_out_ga(Ys, N)) -> len_out_ga(Xs, s(N))

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

[]  =  []

len_out_ga(x1, x2)  =  len_out_ga(x1, x2)

U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)

tail_in_ga(x1, x2)  =  tail_in_ga(x1)

tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)

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

U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)

s(x1)  =  s(x1)





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



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

(22)
Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:

   len_in_ga([], 0) -> len_out_ga([], 0)
   len_in_ga(Xs, s(N)) -> U1_ga(Xs, N, tail_in_ga(Xs, Ys))
   tail_in_ga([], []) -> tail_out_ga([], [])
   tail_in_ga(.(X, Xs), Xs) -> tail_out_ga(.(X, Xs), Xs)
   U1_ga(Xs, N, tail_out_ga(Xs, Ys)) -> U2_ga(Xs, N, Ys, len_in_ga(Ys, N))
   U2_ga(Xs, N, Ys, len_out_ga(Ys, N)) -> len_out_ga(Xs, s(N))

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

[]  =  []

len_out_ga(x1, x2)  =  len_out_ga(x1, x2)

U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)

tail_in_ga(x1, x2)  =  tail_in_ga(x1)

tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)

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

U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)

s(x1)  =  s(x1)



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

(23) 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:

   LEN_IN_GA(Xs, s(N)) -> U1_GA(Xs, N, tail_in_ga(Xs, Ys))
   LEN_IN_GA(Xs, s(N)) -> TAIL_IN_GA(Xs, Ys)
   U1_GA(Xs, N, tail_out_ga(Xs, Ys)) -> U2_GA(Xs, N, Ys, len_in_ga(Ys, N))
   U1_GA(Xs, N, tail_out_ga(Xs, Ys)) -> LEN_IN_GA(Ys, N)

The TRS R consists of the following rules:

   len_in_ga([], 0) -> len_out_ga([], 0)
   len_in_ga(Xs, s(N)) -> U1_ga(Xs, N, tail_in_ga(Xs, Ys))
   tail_in_ga([], []) -> tail_out_ga([], [])
   tail_in_ga(.(X, Xs), Xs) -> tail_out_ga(.(X, Xs), Xs)
   U1_ga(Xs, N, tail_out_ga(Xs, Ys)) -> U2_ga(Xs, N, Ys, len_in_ga(Ys, N))
   U2_ga(Xs, N, Ys, len_out_ga(Ys, N)) -> len_out_ga(Xs, s(N))

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

[]  =  []

len_out_ga(x1, x2)  =  len_out_ga(x1, x2)

U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)

tail_in_ga(x1, x2)  =  tail_in_ga(x1)

tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)

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

U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)

s(x1)  =  s(x1)

LEN_IN_GA(x1, x2)  =  LEN_IN_GA(x1)

U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)

TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)

U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)


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

(24)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LEN_IN_GA(Xs, s(N)) -> U1_GA(Xs, N, tail_in_ga(Xs, Ys))
   LEN_IN_GA(Xs, s(N)) -> TAIL_IN_GA(Xs, Ys)
   U1_GA(Xs, N, tail_out_ga(Xs, Ys)) -> U2_GA(Xs, N, Ys, len_in_ga(Ys, N))
   U1_GA(Xs, N, tail_out_ga(Xs, Ys)) -> LEN_IN_GA(Ys, N)

The TRS R consists of the following rules:

   len_in_ga([], 0) -> len_out_ga([], 0)
   len_in_ga(Xs, s(N)) -> U1_ga(Xs, N, tail_in_ga(Xs, Ys))
   tail_in_ga([], []) -> tail_out_ga([], [])
   tail_in_ga(.(X, Xs), Xs) -> tail_out_ga(.(X, Xs), Xs)
   U1_ga(Xs, N, tail_out_ga(Xs, Ys)) -> U2_ga(Xs, N, Ys, len_in_ga(Ys, N))
   U2_ga(Xs, N, Ys, len_out_ga(Ys, N)) -> len_out_ga(Xs, s(N))

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

[]  =  []

len_out_ga(x1, x2)  =  len_out_ga(x1, x2)

U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)

tail_in_ga(x1, x2)  =  tail_in_ga(x1)

tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)

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

U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)

s(x1)  =  s(x1)

LEN_IN_GA(x1, x2)  =  LEN_IN_GA(x1)

U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)

TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)

U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)


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

(25) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.
----------------------------------------

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

   U1_GA(Xs, N, tail_out_ga(Xs, Ys)) -> LEN_IN_GA(Ys, N)
   LEN_IN_GA(Xs, s(N)) -> U1_GA(Xs, N, tail_in_ga(Xs, Ys))

The TRS R consists of the following rules:

   len_in_ga([], 0) -> len_out_ga([], 0)
   len_in_ga(Xs, s(N)) -> U1_ga(Xs, N, tail_in_ga(Xs, Ys))
   tail_in_ga([], []) -> tail_out_ga([], [])
   tail_in_ga(.(X, Xs), Xs) -> tail_out_ga(.(X, Xs), Xs)
   U1_ga(Xs, N, tail_out_ga(Xs, Ys)) -> U2_ga(Xs, N, Ys, len_in_ga(Ys, N))
   U2_ga(Xs, N, Ys, len_out_ga(Ys, N)) -> len_out_ga(Xs, s(N))

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

[]  =  []

len_out_ga(x1, x2)  =  len_out_ga(x1, x2)

U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)

tail_in_ga(x1, x2)  =  tail_in_ga(x1)

tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)

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

U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)

s(x1)  =  s(x1)

LEN_IN_GA(x1, x2)  =  LEN_IN_GA(x1)

U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)


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

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

(28)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U1_GA(Xs, N, tail_out_ga(Xs, Ys)) -> LEN_IN_GA(Ys, N)
   LEN_IN_GA(Xs, s(N)) -> U1_GA(Xs, N, tail_in_ga(Xs, Ys))

The TRS R consists of the following rules:

   tail_in_ga([], []) -> tail_out_ga([], [])
   tail_in_ga(.(X, Xs), Xs) -> tail_out_ga(.(X, Xs), Xs)

The argument filtering Pi contains the following mapping:
[]  =  []

tail_in_ga(x1, x2)  =  tail_in_ga(x1)

tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)

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

s(x1)  =  s(x1)

LEN_IN_GA(x1, x2)  =  LEN_IN_GA(x1)

U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)


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

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

(30)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U1_GA(Xs, tail_out_ga(Xs, Ys)) -> LEN_IN_GA(Ys)
   LEN_IN_GA(Xs) -> U1_GA(Xs, tail_in_ga(Xs))

The TRS R consists of the following rules:

   tail_in_ga([]) -> tail_out_ga([], [])
   tail_in_ga(.(X, Xs)) -> tail_out_ga(.(X, Xs), Xs)

The set Q consists of the following terms:

   tail_in_ga(x0)

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

(31) TransformationProof (SOUND)
By narrowing [LPAR04] the rule LEN_IN_GA(Xs) -> U1_GA(Xs, tail_in_ga(Xs)) at position [1] we obtained the following new rules [LPAR04]:

   (LEN_IN_GA([]) -> U1_GA([], tail_out_ga([], [])),LEN_IN_GA([]) -> U1_GA([], tail_out_ga([], [])))
   (LEN_IN_GA(.(x0, x1)) -> U1_GA(.(x0, x1), tail_out_ga(.(x0, x1), x1)),LEN_IN_GA(.(x0, x1)) -> U1_GA(.(x0, x1), tail_out_ga(.(x0, x1), x1)))


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

(32)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U1_GA(Xs, tail_out_ga(Xs, Ys)) -> LEN_IN_GA(Ys)
   LEN_IN_GA([]) -> U1_GA([], tail_out_ga([], []))
   LEN_IN_GA(.(x0, x1)) -> U1_GA(.(x0, x1), tail_out_ga(.(x0, x1), x1))

The TRS R consists of the following rules:

   tail_in_ga([]) -> tail_out_ga([], [])
   tail_in_ga(.(X, Xs)) -> tail_out_ga(.(X, Xs), Xs)

The set Q consists of the following terms:

   tail_in_ga(x0)

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

(33) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(34)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U1_GA(Xs, tail_out_ga(Xs, Ys)) -> LEN_IN_GA(Ys)
   LEN_IN_GA([]) -> U1_GA([], tail_out_ga([], []))
   LEN_IN_GA(.(x0, x1)) -> U1_GA(.(x0, x1), tail_out_ga(.(x0, x1), x1))

R is empty.
The set Q consists of the following terms:

   tail_in_ga(x0)

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

(35) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   tail_in_ga(x0)


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

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

   U1_GA(Xs, tail_out_ga(Xs, Ys)) -> LEN_IN_GA(Ys)
   LEN_IN_GA([]) -> U1_GA([], tail_out_ga([], []))
   LEN_IN_GA(.(x0, x1)) -> U1_GA(.(x0, x1), tail_out_ga(.(x0, x1), x1))

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

(37) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   LEN_IN_GA(.(x0, x1)) -> U1_GA(.(x0, x1), tail_out_ga(.(x0, x1), x1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U1_GA_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( tail_out_ga_2(x_1, x_2) ) = x_2 + 1
POL( LEN_IN_GA_1(x_1) ) = 2x_1
POL( [] ) = 0
POL( ._2(x_1, x_2) ) = 2x_2 + 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(38)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U1_GA(Xs, tail_out_ga(Xs, Ys)) -> LEN_IN_GA(Ys)
   LEN_IN_GA([]) -> U1_GA([], tail_out_ga([], []))

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

(39) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule U1_GA(Xs, tail_out_ga(Xs, Ys)) -> LEN_IN_GA(Ys) we obtained the following new rules [LPAR04]:

   (U1_GA([], tail_out_ga([], [])) -> LEN_IN_GA([]),U1_GA([], tail_out_ga([], [])) -> LEN_IN_GA([]))


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

(40)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LEN_IN_GA([]) -> U1_GA([], tail_out_ga([], []))
   U1_GA([], tail_out_ga([], [])) -> LEN_IN_GA([])

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

(41) 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 = U1_GA([], tail_out_ga([], [])) evaluates to  t =U1_GA([], tail_out_ga([], []))

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

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

U1_GA([], tail_out_ga([], [])) -> LEN_IN_GA([])
with rule U1_GA([], tail_out_ga([], [])) -> LEN_IN_GA([]) at position [] and matcher [ ]

LEN_IN_GA([]) -> U1_GA([], tail_out_ga([], []))
with rule LEN_IN_GA([]) -> U1_GA([], tail_out_ga([], []))

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.




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

(42)
NO

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

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

{
    "root": 11,
    "program": {
        "directives": [],
        "clauses": [
            [
                "(len ([]) (0))",
                "(!)"
            ],
            [
                "(len Xs (s N))",
                "(',' (tail Xs Ys) (len Ys N))"
            ],
            [
                "(tail ([]) ([]))",
                null
            ],
            [
                "(tail (. X Xs) Xs)",
                null
            ]
        ]
    },
    "graph": {
        "nodes": {
            "11": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(len T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "22": {
                "goal": [{
                    "clause": 1,
                    "scope": 1,
                    "term": "(len T1 T2)"
                }],
                "kb": {
                    "nonunifying": [[
                        "(len T1 T2)",
                        "(len ([]) (0))"
                    ]],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "33": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "12": {
                "goal": [
                    {
                        "clause": 0,
                        "scope": 1,
                        "term": "(len T1 T2)"
                    },
                    {
                        "clause": 1,
                        "scope": 1,
                        "term": "(len T1 T2)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "48": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "type": "Nodes",
            "124": {
                "goal": [{
                    "clause": 2,
                    "scope": 2,
                    "term": "(',' (tail T7 X8) (len X8 T9))"
                }],
                "kb": {
                    "nonunifying": [[
                        "(len T7 T2)",
                        "(len ([]) (0))"
                    ]],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T7"],
                    "free": ["X8"],
                    "exprvars": []
                }
            },
            "125": {
                "goal": [{
                    "clause": 3,
                    "scope": 2,
                    "term": "(',' (tail T7 X8) (len X8 T9))"
                }],
                "kb": {
                    "nonunifying": [[
                        "(len T7 T2)",
                        "(len ([]) (0))"
                    ]],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T7"],
                    "free": ["X8"],
                    "exprvars": []
                }
            },
            "126": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(len ([]) T9)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "127": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "128": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(len T15 T9)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T15"],
                    "free": [],
                    "exprvars": []
                }
            },
            "129": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "62": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (tail T7 X8) (len X8 T9))"
                }],
                "kb": {
                    "nonunifying": [[
                        "(len T7 T2)",
                        "(len ([]) (0))"
                    ]],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T7"],
                    "free": ["X8"],
                    "exprvars": []
                }
            },
            "63": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "64": {
                "goal": [
                    {
                        "clause": 2,
                        "scope": 2,
                        "term": "(',' (tail T7 X8) (len X8 T9))"
                    },
                    {
                        "clause": 3,
                        "scope": 2,
                        "term": "(',' (tail T7 X8) (len X8 T9))"
                    }
                ],
                "kb": {
                    "nonunifying": [[
                        "(len T7 T2)",
                        "(len ([]) (0))"
                    ]],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T7"],
                    "free": ["X8"],
                    "exprvars": []
                }
            },
            "21": {
                "goal": [
                    {
                        "clause": -1,
                        "scope": -1,
                        "term": "(!_1)"
                    },
                    {
                        "clause": 1,
                        "scope": 1,
                        "term": "(len ([]) T2)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            }
        },
        "edges": [
            {
                "from": 11,
                "to": 12,
                "label": "CASE"
            },
            {
                "from": 12,
                "to": 21,
                "label": "EVAL with clause\nlen([], 0) :- !_1.\nand substitutionT1 -> [],\nT2 -> 0"
            },
            {
                "from": 12,
                "to": 22,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 21,
                "to": 33,
                "label": "CUT"
            },
            {
                "from": 22,
                "to": 62,
                "label": "EVAL with clause\nlen(X6, s(X7)) :- ','(tail(X6, X8), len(X8, X7)).\nand substitutionT1 -> T7,\nX6 -> T7,\nX7 -> T9,\nT2 -> s(T9),\nT8 -> T9"
            },
            {
                "from": 22,
                "to": 63,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 33,
                "to": 48,
                "label": "SUCCESS"
            },
            {
                "from": 62,
                "to": 64,
                "label": "CASE"
            },
            {
                "from": 64,
                "to": 124,
                "label": "PARALLEL"
            },
            {
                "from": 64,
                "to": 125,
                "label": "PARALLEL"
            },
            {
                "from": 124,
                "to": 126,
                "label": "EVAL with clause\ntail([], []).\nand substitutionT7 -> [],\nX8 -> []"
            },
            {
                "from": 124,
                "to": 127,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 125,
                "to": 128,
                "label": "EVAL with clause\ntail(.(X13, X14), X14).\nand substitutionX13 -> T14,\nX14 -> T15,\nT7 -> .(T14, T15),\nX8 -> T15"
            },
            {
                "from": 125,
                "to": 129,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 126,
                "to": 11,
                "label": "INSTANCE with matching:\nT1 -> []\nT2 -> T9"
            },
            {
                "from": 128,
                "to": 11,
                "label": "INSTANCE with matching:\nT1 -> T15\nT2 -> T9"
            }
        ],
        "type": "Graph"
    }
}

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

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

   f11_in([]) -> f11_out1(0)
   f11_in([]) -> U1(f11_in([]), [])
   U1(f11_out1(T9), []) -> f11_out1(s(T9))
   f11_in(.(T14, T15)) -> U2(f11_in(T15), .(T14, T15))
   U2(f11_out1(T9), .(T14, T15)) -> f11_out1(s(T9))

Q is empty.

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

(45) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(.(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(0) = 0
   POL(U1(x_1, x_2)) = x_1 + 2*x_2
   POL(U2(x_1, x_2)) = x_1 + x_2
   POL([]) = 0
   POL(f11_in(x_1)) = 1 + 2*x_1
   POL(f11_out1(x_1)) = 2*x_1
   POL(s(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f11_in([]) -> f11_out1(0)




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

(46)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f11_in([]) -> U1(f11_in([]), [])
   U1(f11_out1(T9), []) -> f11_out1(s(T9))
   f11_in(.(T14, T15)) -> U2(f11_in(T15), .(T14, T15))
   U2(f11_out1(T9), .(T14, T15)) -> f11_out1(s(T9))

Q is empty.

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

(47) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(.(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(U1(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(U2(x_1, x_2)) = x_1 + x_2
   POL([]) = 0
   POL(f11_in(x_1)) = 2*x_1
   POL(f11_out1(x_1)) = 2 + x_1
   POL(s(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U1(f11_out1(T9), []) -> f11_out1(s(T9))




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

(48)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f11_in([]) -> U1(f11_in([]), [])
   f11_in(.(T14, T15)) -> U2(f11_in(T15), .(T14, T15))
   U2(f11_out1(T9), .(T14, T15)) -> f11_out1(s(T9))

Q is empty.

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

(49) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(.(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(U1(x_1, x_2)) = x_1 + x_2
   POL(U2(x_1, x_2)) = 1 + x_1 + x_2
   POL([]) = 0
   POL(f11_in(x_1)) = 2*x_1
   POL(f11_out1(x_1)) = 2*x_1
   POL(s(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U2(f11_out1(T9), .(T14, T15)) -> f11_out1(s(T9))




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

(50)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f11_in([]) -> U1(f11_in([]), [])
   f11_in(.(T14, T15)) -> U2(f11_in(T15), .(T14, T15))

Q is empty.

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

(51) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(.(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(U1(x_1, x_2)) = x_1 + x_2
   POL(U2(x_1, x_2)) = 1 + x_1 + x_2
   POL([]) = 0
   POL(f11_in(x_1)) = 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f11_in(.(T14, T15)) -> U2(f11_in(T15), .(T14, T15))




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

(52)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f11_in([]) -> U1(f11_in([]), [])

Q is empty.

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

(53) Overlay + Local Confluence (EQUIVALENT)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
----------------------------------------

(54)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f11_in([]) -> U1(f11_in([]), [])

The set Q consists of the following terms:

   f11_in([])


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

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

(56)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F11_IN([]) -> F11_IN([])

The TRS R consists of the following rules:

   f11_in([]) -> U1(f11_in([]), [])

The set Q consists of the following terms:

   f11_in([])

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

(57) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(58)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F11_IN([]) -> F11_IN([])

R is empty.
The set Q consists of the following terms:

   f11_in([])

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

(59) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   f11_in([])


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

(60)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F11_IN([]) -> F11_IN([])

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

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

s = F11_IN([]) evaluates to  t =F11_IN([])

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

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

The DP semiunifies directly so there is only one rewrite step from F11_IN([]) to F11_IN([]).




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

(62)
NO

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

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

{
    "root": 2,
    "program": {
        "directives": [],
        "clauses": [
            [
                "(len ([]) (0))",
                "(!)"
            ],
            [
                "(len Xs (s N))",
                "(',' (tail Xs Ys) (len Ys N))"
            ],
            [
                "(tail ([]) ([]))",
                null
            ],
            [
                "(tail (. X Xs) Xs)",
                null
            ]
        ]
    },
    "graph": {
        "nodes": {
            "68": {
                "goal": [
                    {
                        "clause": -1,
                        "scope": -1,
                        "term": "(!_1)"
                    },
                    {
                        "clause": 1,
                        "scope": 1,
                        "term": "(len ([]) T2)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "69": {
                "goal": [{
                    "clause": 1,
                    "scope": 1,
                    "term": "(len T1 T2)"
                }],
                "kb": {
                    "nonunifying": [[
                        "(len T1 T2)",
                        "(len ([]) (0))"
                    ]],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "type": "Nodes",
            "130": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "141": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(len T13 T7)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T13"],
                    "free": [],
                    "exprvars": []
                }
            },
            "131": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (tail T5 X5) (len X5 T7))"
                }],
                "kb": {
                    "nonunifying": [[
                        "(len T5 T2)",
                        "(len ([]) (0))"
                    ]],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T5"],
                    "free": ["X5"],
                    "exprvars": []
                }
            },
            "142": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "132": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "133": {
                "goal": [
                    {
                        "clause": 2,
                        "scope": 2,
                        "term": "(',' (tail T5 X5) (len X5 T7))"
                    },
                    {
                        "clause": 3,
                        "scope": 2,
                        "term": "(',' (tail T5 X5) (len X5 T7))"
                    }
                ],
                "kb": {
                    "nonunifying": [[
                        "(len T5 T2)",
                        "(len ([]) (0))"
                    ]],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T5"],
                    "free": ["X5"],
                    "exprvars": []
                }
            },
            "2": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(len T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "134": {
                "goal": [{
                    "clause": 2,
                    "scope": 2,
                    "term": "(',' (tail T5 X5) (len X5 T7))"
                }],
                "kb": {
                    "nonunifying": [[
                        "(len T5 T2)",
                        "(len ([]) (0))"
                    ]],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T5"],
                    "free": ["X5"],
                    "exprvars": []
                }
            },
            "135": {
                "goal": [{
                    "clause": 3,
                    "scope": 2,
                    "term": "(',' (tail T5 X5) (len X5 T7))"
                }],
                "kb": {
                    "nonunifying": [[
                        "(len T5 T2)",
                        "(len ([]) (0))"
                    ]],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T5"],
                    "free": ["X5"],
                    "exprvars": []
                }
            },
            "136": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(len ([]) T7)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "5": {
                "goal": [
                    {
                        "clause": 0,
                        "scope": 1,
                        "term": "(len T1 T2)"
                    },
                    {
                        "clause": 1,
                        "scope": 1,
                        "term": "(len T1 T2)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "139": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "70": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            }
        },
        "edges": [
            {
                "from": 2,
                "to": 5,
                "label": "CASE"
            },
            {
                "from": 5,
                "to": 68,
                "label": "EVAL with clause\nlen([], 0) :- !_1.\nand substitutionT1 -> [],\nT2 -> 0"
            },
            {
                "from": 5,
                "to": 69,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 68,
                "to": 70,
                "label": "CUT"
            },
            {
                "from": 69,
                "to": 131,
                "label": "EVAL with clause\nlen(X3, s(X4)) :- ','(tail(X3, X5), len(X5, X4)).\nand substitutionT1 -> T5,\nX3 -> T5,\nX4 -> T7,\nT2 -> s(T7),\nT6 -> T7"
            },
            {
                "from": 69,
                "to": 132,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 70,
                "to": 130,
                "label": "SUCCESS"
            },
            {
                "from": 131,
                "to": 133,
                "label": "CASE"
            },
            {
                "from": 133,
                "to": 134,
                "label": "PARALLEL"
            },
            {
                "from": 133,
                "to": 135,
                "label": "PARALLEL"
            },
            {
                "from": 134,
                "to": 136,
                "label": "EVAL with clause\ntail([], []).\nand substitutionT5 -> [],\nX5 -> []"
            },
            {
                "from": 134,
                "to": 139,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 135,
                "to": 141,
                "label": "EVAL with clause\ntail(.(X10, X11), X11).\nand substitutionX10 -> T12,\nX11 -> T13,\nT5 -> .(T12, T13),\nX5 -> T13"
            },
            {
                "from": 135,
                "to": 142,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 136,
                "to": 2,
                "label": "INSTANCE with matching:\nT1 -> []\nT2 -> T7"
            },
            {
                "from": 141,
                "to": 2,
                "label": "INSTANCE with matching:\nT1 -> T13\nT2 -> T7"
            }
        ],
        "type": "Graph"
    }
}

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

(64)
Obligation:
Triples:

lenA([], s(X1)) :- lenA([], X1).
lenA(.(X1, X2), s(X3)) :- lenA(X2, X3).

Clauses:

lencA([], 0).
lencA([], s(X1)) :- lencA([], X1).
lencA(.(X1, X2), s(X3)) :- lencA(X2, X3).

Afs:

lenA(x1, x2)  =  lenA(x1)


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

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

lenA_in_2: (b,f)

Transforming TRIPLES into the following Term Rewriting System:

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

   LENA_IN_GA([], s(X1)) -> U1_GA(X1, lenA_in_ga([], X1))
   LENA_IN_GA([], s(X1)) -> LENA_IN_GA([], X1)
   LENA_IN_GA(.(X1, X2), s(X3)) -> U2_GA(X1, X2, X3, lenA_in_ga(X2, X3))
   LENA_IN_GA(.(X1, X2), s(X3)) -> LENA_IN_GA(X2, X3)

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

[]  =  []

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

s(x1)  =  s(x1)

LENA_IN_GA(x1, x2)  =  LENA_IN_GA(x1)

U1_GA(x1, x2)  =  U1_GA(x2)

U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)


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


Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES



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

(66)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LENA_IN_GA([], s(X1)) -> U1_GA(X1, lenA_in_ga([], X1))
   LENA_IN_GA([], s(X1)) -> LENA_IN_GA([], X1)
   LENA_IN_GA(.(X1, X2), s(X3)) -> U2_GA(X1, X2, X3, lenA_in_ga(X2, X3))
   LENA_IN_GA(.(X1, X2), s(X3)) -> LENA_IN_GA(X2, X3)

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

[]  =  []

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

s(x1)  =  s(x1)

LENA_IN_GA(x1, x2)  =  LENA_IN_GA(x1)

U1_GA(x1, x2)  =  U1_GA(x2)

U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)


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

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

(68)
Complex Obligation (AND)

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

(69)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LENA_IN_GA([], s(X1)) -> LENA_IN_GA([], X1)

R is empty.
The argument filtering Pi contains the following mapping:
[]  =  []

s(x1)  =  s(x1)

LENA_IN_GA(x1, x2)  =  LENA_IN_GA(x1)


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

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

(71)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LENA_IN_GA([]) -> LENA_IN_GA([])

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

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

s = LENA_IN_GA([]) evaluates to  t =LENA_IN_GA([])

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

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

The DP semiunifies directly so there is only one rewrite step from LENA_IN_GA([]) to LENA_IN_GA([]).




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

(73)
NO

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

(74)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LENA_IN_GA(.(X1, X2), s(X3)) -> LENA_IN_GA(X2, X3)

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

s(x1)  =  s(x1)

LENA_IN_GA(x1, x2)  =  LENA_IN_GA(x1)


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

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

(76)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LENA_IN_GA(.(X1, X2)) -> LENA_IN_GA(X2)

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

(77) 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:
*LENA_IN_GA(.(X1, X2)) -> LENA_IN_GA(X2)
The graph contains the following edges 1 > 1


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

(78)
YES

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

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

{
    "root": 1,
    "program": {
        "directives": [],
        "clauses": [
            [
                "(len ([]) (0))",
                "(!)"
            ],
            [
                "(len Xs (s N))",
                "(',' (tail Xs Ys) (len Ys N))"
            ],
            [
                "(tail ([]) ([]))",
                null
            ],
            [
                "(tail (. X Xs) Xs)",
                null
            ]
        ]
    },
    "graph": {
        "nodes": {
            "67": {
                "goal": [
                    {
                        "clause": -1,
                        "scope": -1,
                        "term": "(!_1)"
                    },
                    {
                        "clause": 1,
                        "scope": 1,
                        "term": "(len ([]) T2)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "13": {
                "goal": [
                    {
                        "clause": 0,
                        "scope": 1,
                        "term": "(len T1 T2)"
                    },
                    {
                        "clause": 1,
                        "scope": 1,
                        "term": "(len T1 T2)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "type": "Nodes",
            "150": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(len T15 T9)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T15"],
                    "free": [],
                    "exprvars": []
                }
            },
            "140": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "151": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "143": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (tail T7 X8) (len X8 T9))"
                }],
                "kb": {
                    "nonunifying": [[
                        "(len T7 T2)",
                        "(len ([]) (0))"
                    ]],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T7"],
                    "free": ["X8"],
                    "exprvars": []
                }
            },
            "1": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(len T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "144": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "145": {
                "goal": [
                    {
                        "clause": 2,
                        "scope": 2,
                        "term": "(',' (tail T7 X8) (len X8 T9))"
                    },
                    {
                        "clause": 3,
                        "scope": 2,
                        "term": "(',' (tail T7 X8) (len X8 T9))"
                    }
                ],
                "kb": {
                    "nonunifying": [[
                        "(len T7 T2)",
                        "(len ([]) (0))"
                    ]],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T7"],
                    "free": ["X8"],
                    "exprvars": []
                }
            },
            "146": {
                "goal": [{
                    "clause": 2,
                    "scope": 2,
                    "term": "(',' (tail T7 X8) (len X8 T9))"
                }],
                "kb": {
                    "nonunifying": [[
                        "(len T7 T2)",
                        "(len ([]) (0))"
                    ]],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T7"],
                    "free": ["X8"],
                    "exprvars": []
                }
            },
            "147": {
                "goal": [{
                    "clause": 3,
                    "scope": 2,
                    "term": "(',' (tail T7 X8) (len X8 T9))"
                }],
                "kb": {
                    "nonunifying": [[
                        "(len T7 T2)",
                        "(len ([]) (0))"
                    ]],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T7"],
                    "free": ["X8"],
                    "exprvars": []
                }
            },
            "137": {
                "goal": [{
                    "clause": 1,
                    "scope": 1,
                    "term": "(len T1 T2)"
                }],
                "kb": {
                    "nonunifying": [[
                        "(len T1 T2)",
                        "(len ([]) (0))"
                    ]],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "148": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(len ([]) T9)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "138": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "149": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            }
        },
        "edges": [
            {
                "from": 1,
                "to": 13,
                "label": "CASE"
            },
            {
                "from": 13,
                "to": 67,
                "label": "EVAL with clause\nlen([], 0) :- !_1.\nand substitutionT1 -> [],\nT2 -> 0"
            },
            {
                "from": 13,
                "to": 137,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 67,
                "to": 138,
                "label": "CUT"
            },
            {
                "from": 137,
                "to": 143,
                "label": "EVAL with clause\nlen(X6, s(X7)) :- ','(tail(X6, X8), len(X8, X7)).\nand substitutionT1 -> T7,\nX6 -> T7,\nX7 -> T9,\nT2 -> s(T9),\nT8 -> T9"
            },
            {
                "from": 137,
                "to": 144,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 138,
                "to": 140,
                "label": "SUCCESS"
            },
            {
                "from": 143,
                "to": 145,
                "label": "CASE"
            },
            {
                "from": 145,
                "to": 146,
                "label": "PARALLEL"
            },
            {
                "from": 145,
                "to": 147,
                "label": "PARALLEL"
            },
            {
                "from": 146,
                "to": 148,
                "label": "EVAL with clause\ntail([], []).\nand substitutionT7 -> [],\nX8 -> []"
            },
            {
                "from": 146,
                "to": 149,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 147,
                "to": 150,
                "label": "EVAL with clause\ntail(.(X13, X14), X14).\nand substitutionX13 -> T14,\nX14 -> T15,\nT7 -> .(T14, T15),\nX8 -> T15"
            },
            {
                "from": 147,
                "to": 151,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 148,
                "to": 1,
                "label": "INSTANCE with matching:\nT1 -> []\nT2 -> T9"
            },
            {
                "from": 150,
                "to": 1,
                "label": "INSTANCE with matching:\nT1 -> T15\nT2 -> T9"
            }
        ],
        "type": "Graph"
    }
}

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

(80)
Obligation:
Rules:
f67_out -> f13_out([]) :|: TRUE
f13_in([]) -> f67_in :|: TRUE
f13_in(T1) -> f137_in(T1) :|: TRUE
f137_out(x) -> f13_out(x) :|: TRUE
f1_out(T15) -> f150_out(T15) :|: TRUE
f150_in(x1) -> f1_in(x1) :|: TRUE
f145_in(T7) -> f147_in(T7) :|: TRUE
f145_in(x2) -> f146_in(x2) :|: TRUE
f146_out(x3) -> f145_out(x3) :|: TRUE
f147_out(x4) -> f145_out(x4) :|: TRUE
f1_out([]) -> f148_out :|: TRUE
f148_in -> f1_in([]) :|: TRUE
f150_out(x5) -> f147_out(.(x6, x5)) :|: TRUE
f147_in(.(x7, x8)) -> f150_in(x8) :|: TRUE
f147_in(x9) -> f151_in :|: TRUE
f151_out -> f147_out(x10) :|: TRUE
f146_in([]) -> f148_in :|: TRUE
f148_out -> f146_out([]) :|: TRUE
f149_out -> f146_out(x11) :|: TRUE
f146_in(x12) -> f149_in :|: TRUE
f137_in(x13) -> f144_in :|: TRUE
f137_in(x14) -> f143_in(x14) :|: TRUE
f143_out(x15) -> f137_out(x15) :|: TRUE
f144_out -> f137_out(x16) :|: TRUE
f145_out(x17) -> f143_out(x17) :|: TRUE
f143_in(x18) -> f145_in(x18) :|: TRUE
f13_out(x19) -> f1_out(x19) :|: TRUE
f1_in(x20) -> f13_in(x20) :|: TRUE
Start term: f1_in(T1)

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

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

Simplified to the following IRSwTs:

intTRSProblem:
f13_in(T1) -> f137_in(T1) :|: TRUE
f150_in(x1) -> f1_in(x1) :|: TRUE
f145_in(T7) -> f147_in(T7) :|: TRUE
f145_in(x2) -> f146_in(x2) :|: TRUE
f148_in -> f1_in([]) :|: TRUE
f147_in(.(x7, x8)) -> f150_in(x8) :|: TRUE
f146_in([]) -> f148_in :|: TRUE
f137_in(x14) -> f143_in(x14) :|: TRUE
f143_in(x18) -> f145_in(x18) :|: TRUE
f1_in(x20) -> f13_in(x20) :|: TRUE


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

(82)
Obligation:
Rules:
f13_in(T1) -> f137_in(T1) :|: TRUE
f150_in(x1) -> f1_in(x1) :|: TRUE
f145_in(T7) -> f147_in(T7) :|: TRUE
f145_in(x2) -> f146_in(x2) :|: TRUE
f148_in -> f1_in([]) :|: TRUE
f147_in(.(x7, x8)) -> f150_in(x8) :|: TRUE
f146_in([]) -> f148_in :|: TRUE
f137_in(x14) -> f143_in(x14) :|: TRUE
f143_in(x18) -> f145_in(x18) :|: TRUE
f1_in(x20) -> f13_in(x20) :|: TRUE

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

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

(84)
Obligation:
Rules:
f13_in(.(x7:0, x8:0)) -> f13_in(x8:0) :|: TRUE
f13_in([]) -> f13_in([]) :|: TRUE

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

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

(86)
Obligation:
Rules:
f13_in(.(x7:0, x8:0)) -> f13_in(x8:0) :|: TRUE
f13_in([]) -> f13_in([]) :|: TRUE

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

(87) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f13_in(.(x7:0, x8:0)) -> f13_in(x8:0) :|: TRUE
(2) f13_in([]) -> f13_in([]) :|: TRUE

Arcs:
(1) -> (1), (2)
(2) -> (2)

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

(88)
Complex Obligation (AND)

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

(89)
Obligation:

Termination digraph:
Nodes:
(1) f13_in(.(x7:0, x8:0)) -> f13_in(x8:0) :|: TRUE

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

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

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

(91)
Obligation:
Rules:
f13_in(.(x8:0)) -> f13_in(x8:0) :|: TRUE

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

(92) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f13_in(VARIABLE)
.(VARIABLE)
Removed predefined arithmetic.
----------------------------------------

(93)
Obligation:
Rules:
f13_in(.(x8:0)) -> f13_in(x8:0)

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

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

(95)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   f13_in(.(x8:0)) -> f13_in(x8:0)

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

(96) 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:
*f13_in(.(x8:0)) -> f13_in(x8:0)
The graph contains the following edges 1 > 1


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

(97)
YES

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

(98)
Obligation:

Termination digraph:
Nodes:
(1) f13_in([]) -> f13_in([]) :|: TRUE

Arcs:
(1) -> (1)

This digraph is fully evaluated!