/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files


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MAYBE
proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Left Termination of the query pattern

sublist(g,a)

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

(0) Prolog
(1) PrologToPiTRSProof [SOUND, 0 ms]
(2) PiTRS
    (3) DependencyPairsProof [EQUIVALENT, 12 ms]
    (4) PiDP
    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
    (6) AND
        (7) PiDP
            (8) UsableRulesProof [EQUIVALENT, 0 ms]
            (9) PiDP
            (10) PiDPToQDPProof [SOUND, 0 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) PrologToPiTRSProof [SOUND, 0 ms]
(20) PiTRS
    (21) DependencyPairsProof [EQUIVALENT, 6 ms]
    (22) PiDP
    (23) DependencyGraphProof [EQUIVALENT, 0 ms]
    (24) AND
        (25) PiDP
            (26) UsableRulesProof [EQUIVALENT, 0 ms]
            (27) PiDP
            (28) PiDPToQDPProof [SOUND, 0 ms]
            (29) QDP
            (30) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (31) YES
        (32) PiDP
            (33) UsableRulesProof [EQUIVALENT, 0 ms]
            (34) PiDP
            (35) PiDPToQDPProof [SOUND, 0 ms]
            (36) QDP
(37) PrologToTRSTransformerProof [SOUND, 0 ms]
(38) QTRS
    (39) QTRSRRRProof [EQUIVALENT, 85 ms]
    (40) QTRS
    (41) QTRSRRRProof [EQUIVALENT, 0 ms]
    (42) QTRS
    (43) Overlay + Local Confluence [EQUIVALENT, 0 ms]
    (44) QTRS
    (45) DependencyPairsProof [EQUIVALENT, 0 ms]
    (46) QDP
    (47) UsableRulesProof [EQUIVALENT, 0 ms]
    (48) QDP
    (49) QReductionProof [EQUIVALENT, 0 ms]
    (50) QDP
(51) PrologToDTProblemTransformerProof [SOUND, 0 ms]
(52) TRIPLES
    (53) TriplesToPiDPProof [SOUND, 0 ms]
    (54) PiDP
    (55) DependencyGraphProof [EQUIVALENT, 0 ms]
    (56) AND
        (57) PiDP
            (58) PiDPToQDPProof [SOUND, 0 ms]
            (59) QDP
            (60) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (61) YES
        (62) PiDP
            (63) PiDPToQDPProof [SOUND, 0 ms]
            (64) QDP
(65) PrologToIRSwTTransformerProof [SOUND, 29 ms]
(66) AND
    (67) IRSwT
        (68) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
        (69) TRUE
    (70) IRSwT
        (71) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
        (72) IRSwT
        (73) IntTRSCompressionProof [EQUIVALENT, 19 ms]
        (74) IRSwT
        (75) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
        (76) IRSwT
        (77) IRSwTTerminationDigraphProof [EQUIVALENT, 6 ms]
        (78) IRSwT
        (79) FilterProof [EQUIVALENT, 0 ms]
        (80) IntTRS
        (81) IntTRSNonPeriodicNontermProof [COMPLETE, 0 ms]
        (82) NO


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

(0)
Obligation:
Clauses:

sublist(Xs, Ys) :- ','(app(X1, Zs, Ys), app(Xs, X2, Zs)).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).


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

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

sublist_in_2: (b,f)

app_in_3: (f,f,f) (b,f,f)

Transforming Prolog into the following Term Rewriting System:

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

   sublist_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aaa(X1, Zs, Ys))
   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
   U3_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
   U1_ga(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> U2_ga(Xs, Ys, app_in_gaa(Xs, X2, Zs))
   app_in_gaa([], X, X) -> app_out_gaa([], X, X)
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   U3_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U2_ga(Xs, Ys, app_out_gaa(Xs, X2, Zs)) -> sublist_out_ga(Xs, Ys)

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

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

app_in_aaa(x1, x2, x3)  =  app_in_aaa

app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)

U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)

.(x1, x2)  =  .(x2)

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

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

[]  =  []

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)

sublist_out_ga(x1, x2)  =  sublist_out_ga





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



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

(2)
Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:

   sublist_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aaa(X1, Zs, Ys))
   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
   U3_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
   U1_ga(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> U2_ga(Xs, Ys, app_in_gaa(Xs, X2, Zs))
   app_in_gaa([], X, X) -> app_out_gaa([], X, X)
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   U3_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U2_ga(Xs, Ys, app_out_gaa(Xs, X2, Zs)) -> sublist_out_ga(Xs, Ys)

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

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

app_in_aaa(x1, x2, x3)  =  app_in_aaa

app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)

U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)

.(x1, x2)  =  .(x2)

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

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

[]  =  []

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)

sublist_out_ga(x1, x2)  =  sublist_out_ga



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

(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:

   SUBLIST_IN_GA(Xs, Ys) -> U1_GA(Xs, Ys, app_in_aaa(X1, Zs, Ys))
   SUBLIST_IN_GA(Xs, Ys) -> APP_IN_AAA(X1, Zs, Ys)
   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U3_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs)
   U1_GA(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> U2_GA(Xs, Ys, app_in_gaa(Xs, X2, Zs))
   U1_GA(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> APP_IN_GAA(Xs, X2, Zs)
   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> U3_GAA(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

   sublist_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aaa(X1, Zs, Ys))
   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
   U3_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
   U1_ga(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> U2_ga(Xs, Ys, app_in_gaa(Xs, X2, Zs))
   app_in_gaa([], X, X) -> app_out_gaa([], X, X)
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   U3_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U2_ga(Xs, Ys, app_out_gaa(Xs, X2, Zs)) -> sublist_out_ga(Xs, Ys)

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

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

app_in_aaa(x1, x2, x3)  =  app_in_aaa

app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)

U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)

.(x1, x2)  =  .(x2)

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

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

[]  =  []

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)

sublist_out_ga(x1, x2)  =  sublist_out_ga

SUBLIST_IN_GA(x1, x2)  =  SUBLIST_IN_GA(x1)

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

APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA

U3_AAA(x1, x2, x3, x4, x5)  =  U3_AAA(x5)

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

APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)

U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)


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

(4)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   SUBLIST_IN_GA(Xs, Ys) -> U1_GA(Xs, Ys, app_in_aaa(X1, Zs, Ys))
   SUBLIST_IN_GA(Xs, Ys) -> APP_IN_AAA(X1, Zs, Ys)
   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U3_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs)
   U1_GA(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> U2_GA(Xs, Ys, app_in_gaa(Xs, X2, Zs))
   U1_GA(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> APP_IN_GAA(Xs, X2, Zs)
   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> U3_GAA(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

   sublist_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aaa(X1, Zs, Ys))
   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
   U3_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
   U1_ga(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> U2_ga(Xs, Ys, app_in_gaa(Xs, X2, Zs))
   app_in_gaa([], X, X) -> app_out_gaa([], X, X)
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   U3_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U2_ga(Xs, Ys, app_out_gaa(Xs, X2, Zs)) -> sublist_out_ga(Xs, Ys)

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

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

app_in_aaa(x1, x2, x3)  =  app_in_aaa

app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)

U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)

.(x1, x2)  =  .(x2)

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

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

[]  =  []

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)

sublist_out_ga(x1, x2)  =  sublist_out_ga

SUBLIST_IN_GA(x1, x2)  =  SUBLIST_IN_GA(x1)

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

APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA

U3_AAA(x1, x2, x3, x4, x5)  =  U3_AAA(x5)

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

APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)

U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)


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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

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

(7)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

   sublist_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aaa(X1, Zs, Ys))
   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
   U3_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
   U1_ga(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> U2_ga(Xs, Ys, app_in_gaa(Xs, X2, Zs))
   app_in_gaa([], X, X) -> app_out_gaa([], X, X)
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   U3_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U2_ga(Xs, Ys, app_out_gaa(Xs, X2, Zs)) -> sublist_out_ga(Xs, Ys)

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

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

app_in_aaa(x1, x2, x3)  =  app_in_aaa

app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)

U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)

.(x1, x2)  =  .(x2)

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

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

[]  =  []

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)

sublist_out_ga(x1, x2)  =  sublist_out_ga

APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)


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:

   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAA(Xs, Ys, Zs)

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

APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)


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:

   APP_IN_GAA(.(Xs)) -> APP_IN_GAA(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:
*APP_IN_GAA(.(Xs)) -> APP_IN_GAA(Xs)
The graph contains the following edges 1 > 1


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

(13)
YES

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

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

   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

   sublist_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aaa(X1, Zs, Ys))
   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
   U3_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
   U1_ga(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> U2_ga(Xs, Ys, app_in_gaa(Xs, X2, Zs))
   app_in_gaa([], X, X) -> app_out_gaa([], X, X)
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   U3_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U2_ga(Xs, Ys, app_out_gaa(Xs, X2, Zs)) -> sublist_out_ga(Xs, Ys)

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

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

app_in_aaa(x1, x2, x3)  =  app_in_aaa

app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)

U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)

.(x1, x2)  =  .(x2)

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

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

[]  =  []

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)

sublist_out_ga(x1, x2)  =  sublist_out_ga

APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA


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:

   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs)

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

APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA


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:

   APP_IN_AAA -> APP_IN_AAA

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

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

sublist_in_2: (b,f)

app_in_3: (f,f,f) (b,f,f)

Transforming Prolog into the following Term Rewriting System:

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

   sublist_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aaa(X1, Zs, Ys))
   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
   U3_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
   U1_ga(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> U2_ga(Xs, Ys, app_in_gaa(Xs, X2, Zs))
   app_in_gaa([], X, X) -> app_out_gaa([], X, X)
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   U3_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U2_ga(Xs, Ys, app_out_gaa(Xs, X2, Zs)) -> sublist_out_ga(Xs, Ys)

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

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

app_in_aaa(x1, x2, x3)  =  app_in_aaa

app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)

U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)

.(x1, x2)  =  .(x2)

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

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

[]  =  []

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)

sublist_out_ga(x1, x2)  =  sublist_out_ga(x1)





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



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

(20)
Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:

   sublist_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aaa(X1, Zs, Ys))
   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
   U3_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
   U1_ga(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> U2_ga(Xs, Ys, app_in_gaa(Xs, X2, Zs))
   app_in_gaa([], X, X) -> app_out_gaa([], X, X)
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   U3_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U2_ga(Xs, Ys, app_out_gaa(Xs, X2, Zs)) -> sublist_out_ga(Xs, Ys)

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

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

app_in_aaa(x1, x2, x3)  =  app_in_aaa

app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)

U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)

.(x1, x2)  =  .(x2)

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

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

[]  =  []

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)

sublist_out_ga(x1, x2)  =  sublist_out_ga(x1)



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

(21) 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:

   SUBLIST_IN_GA(Xs, Ys) -> U1_GA(Xs, Ys, app_in_aaa(X1, Zs, Ys))
   SUBLIST_IN_GA(Xs, Ys) -> APP_IN_AAA(X1, Zs, Ys)
   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U3_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs)
   U1_GA(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> U2_GA(Xs, Ys, app_in_gaa(Xs, X2, Zs))
   U1_GA(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> APP_IN_GAA(Xs, X2, Zs)
   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> U3_GAA(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

   sublist_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aaa(X1, Zs, Ys))
   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
   U3_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
   U1_ga(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> U2_ga(Xs, Ys, app_in_gaa(Xs, X2, Zs))
   app_in_gaa([], X, X) -> app_out_gaa([], X, X)
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   U3_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U2_ga(Xs, Ys, app_out_gaa(Xs, X2, Zs)) -> sublist_out_ga(Xs, Ys)

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

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

app_in_aaa(x1, x2, x3)  =  app_in_aaa

app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)

U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)

.(x1, x2)  =  .(x2)

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

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

[]  =  []

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)

sublist_out_ga(x1, x2)  =  sublist_out_ga(x1)

SUBLIST_IN_GA(x1, x2)  =  SUBLIST_IN_GA(x1)

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

APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA

U3_AAA(x1, x2, x3, x4, x5)  =  U3_AAA(x5)

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

APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)

U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x2, x5)


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

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

   SUBLIST_IN_GA(Xs, Ys) -> U1_GA(Xs, Ys, app_in_aaa(X1, Zs, Ys))
   SUBLIST_IN_GA(Xs, Ys) -> APP_IN_AAA(X1, Zs, Ys)
   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U3_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs)
   U1_GA(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> U2_GA(Xs, Ys, app_in_gaa(Xs, X2, Zs))
   U1_GA(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> APP_IN_GAA(Xs, X2, Zs)
   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> U3_GAA(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

   sublist_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aaa(X1, Zs, Ys))
   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
   U3_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
   U1_ga(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> U2_ga(Xs, Ys, app_in_gaa(Xs, X2, Zs))
   app_in_gaa([], X, X) -> app_out_gaa([], X, X)
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   U3_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U2_ga(Xs, Ys, app_out_gaa(Xs, X2, Zs)) -> sublist_out_ga(Xs, Ys)

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

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

app_in_aaa(x1, x2, x3)  =  app_in_aaa

app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)

U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)

.(x1, x2)  =  .(x2)

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

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

[]  =  []

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)

sublist_out_ga(x1, x2)  =  sublist_out_ga(x1)

SUBLIST_IN_GA(x1, x2)  =  SUBLIST_IN_GA(x1)

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

APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA

U3_AAA(x1, x2, x3, x4, x5)  =  U3_AAA(x5)

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

APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)

U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x2, x5)


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

(23) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.
----------------------------------------

(24)
Complex Obligation (AND)

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

(25)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

   sublist_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aaa(X1, Zs, Ys))
   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
   U3_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
   U1_ga(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> U2_ga(Xs, Ys, app_in_gaa(Xs, X2, Zs))
   app_in_gaa([], X, X) -> app_out_gaa([], X, X)
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   U3_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U2_ga(Xs, Ys, app_out_gaa(Xs, X2, Zs)) -> sublist_out_ga(Xs, Ys)

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

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

app_in_aaa(x1, x2, x3)  =  app_in_aaa

app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)

U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)

.(x1, x2)  =  .(x2)

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

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

[]  =  []

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)

sublist_out_ga(x1, x2)  =  sublist_out_ga(x1)

APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)


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

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

(27)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAA(Xs, Ys, Zs)

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

APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)


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

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

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

   APP_IN_GAA(.(Xs)) -> APP_IN_GAA(Xs)

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

(30) 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:
*APP_IN_GAA(.(Xs)) -> APP_IN_GAA(Xs)
The graph contains the following edges 1 > 1


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

(31)
YES

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

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

   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

   sublist_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aaa(X1, Zs, Ys))
   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U3_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
   U3_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
   U1_ga(Xs, Ys, app_out_aaa(X1, Zs, Ys)) -> U2_ga(Xs, Ys, app_in_gaa(Xs, X2, Zs))
   app_in_gaa([], X, X) -> app_out_gaa([], X, X)
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   U3_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U2_ga(Xs, Ys, app_out_gaa(Xs, X2, Zs)) -> sublist_out_ga(Xs, Ys)

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

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

app_in_aaa(x1, x2, x3)  =  app_in_aaa

app_out_aaa(x1, x2, x3)  =  app_out_aaa(x1)

U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)

.(x1, x2)  =  .(x2)

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

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

[]  =  []

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)

sublist_out_ga(x1, x2)  =  sublist_out_ga(x1)

APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA


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

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

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

   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs)

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

APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA


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

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

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

   APP_IN_AAA -> APP_IN_AAA

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

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

{
    "root": 2,
    "program": {
        "directives": [],
        "clauses": [
            [
                "(sublist Xs Ys)",
                "(',' (app X1 Zs Ys) (app Xs X2 Zs))"
            ],
            [
                "(app ([]) X X)",
                null
            ],
            [
                "(app (. X Xs) Ys (. X Zs))",
                "(app Xs Ys Zs)"
            ]
        ]
    },
    "graph": {
        "nodes": {
            "88": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(app T9 X20 T14)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T9"],
                    "free": ["X20"],
                    "exprvars": []
                }
            },
            "190": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "191": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "192": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "193": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(app T43 X90 T45)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T43"],
                    "free": ["X90"],
                    "exprvars": []
                }
            },
            "type": "Nodes",
            "194": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "152": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "186": {
                "goal": [
                    {
                        "clause": 1,
                        "scope": 3,
                        "term": "(app T9 X20 T14)"
                    },
                    {
                        "clause": 2,
                        "scope": 3,
                        "term": "(app T9 X20 T14)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T9"],
                    "free": ["X20"],
                    "exprvars": []
                }
            },
            "143": {
                "goal": [{
                    "clause": 1,
                    "scope": 2,
                    "term": "(app X18 X19 T11)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X18",
                        "X19"
                    ],
                    "exprvars": []
                }
            },
            "154": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "144": {
                "goal": [{
                    "clause": 2,
                    "scope": 2,
                    "term": "(app X18 X19 T11)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X18",
                        "X19"
                    ],
                    "exprvars": []
                }
            },
            "177": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(app X58 X59 T27)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X58",
                        "X59"
                    ],
                    "exprvars": []
                }
            },
            "188": {
                "goal": [{
                    "clause": 1,
                    "scope": 3,
                    "term": "(app T9 X20 T14)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T9"],
                    "free": ["X20"],
                    "exprvars": []
                }
            },
            "2": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(sublist T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "178": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
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                    "exprvars": []
                }
            },
            "189": {
                "goal": [{
                    "clause": 2,
                    "scope": 3,
                    "term": "(app T9 X20 T14)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T9"],
                    "free": ["X20"],
                    "exprvars": []
                }
            },
            "6": {
                "goal": [{
                    "clause": 0,
                    "scope": 1,
                    "term": "(sublist T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "139": {
                "goal": [
                    {
                        "clause": 1,
                        "scope": 2,
                        "term": "(app X18 X19 T11)"
                    },
                    {
                        "clause": 2,
                        "scope": 2,
                        "term": "(app X18 X19 T11)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X18",
                        "X19"
                    ],
                    "exprvars": []
                }
            },
            "83": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (app X18 X19 T11) (app T9 X20 X19))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T9"],
                    "free": [
                        "X18",
                        "X19",
                        "X20"
                    ],
                    "exprvars": []
                }
            },
            "86": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(app X18 X19 T11)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X18",
                        "X19"
                    ],
                    "exprvars": []
                }
            }
        },
        "edges": [
            {
                "from": 2,
                "to": 6,
                "label": "CASE"
            },
            {
                "from": 6,
                "to": 83,
                "label": "ONLY EVAL with clause\nsublist(X16, X17) :- ','(app(X18, X19, X17), app(X16, X20, X19)).\nand substitutionT1 -> T9,\nX16 -> T9,\nT2 -> T11,\nX17 -> T11,\nT10 -> T11"
            },
            {
                "from": 83,
                "to": 86,
                "label": "SPLIT 1"
            },
            {
                "from": 83,
                "to": 88,
                "label": "SPLIT 2\nreplacements:X18 -> T13,\nX19 -> T14"
            },
            {
                "from": 86,
                "to": 139,
                "label": "CASE"
            },
            {
                "from": 88,
                "to": 186,
                "label": "CASE"
            },
            {
                "from": 139,
                "to": 143,
                "label": "PARALLEL"
            },
            {
                "from": 139,
                "to": 144,
                "label": "PARALLEL"
            },
            {
                "from": 143,
                "to": 152,
                "label": "ONLY EVAL with clause\napp([], X37, X37).\nand substitutionX18 -> [],\nX19 -> T20,\nX37 -> T20,\nT11 -> T20,\nX38 -> T20"
            },
            {
                "from": 144,
                "to": 177,
                "label": "EVAL with clause\napp(.(X53, X54), X55, .(X53, X56)) :- app(X54, X55, X56).\nand substitutionX53 -> T25,\nX54 -> X58,\nX18 -> .(T25, X58),\nX19 -> X59,\nX55 -> X59,\nX57 -> T25,\nX56 -> T27,\nT11 -> .(T25, T27),\nT26 -> T27"
            },
            {
                "from": 144,
                "to": 178,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 152,
                "to": 154,
                "label": "SUCCESS"
            },
            {
                "from": 177,
                "to": 86,
                "label": "INSTANCE with matching:\nX18 -> X58\nX19 -> X59\nT11 -> T27"
            },
            {
                "from": 186,
                "to": 188,
                "label": "PARALLEL"
            },
            {
                "from": 186,
                "to": 189,
                "label": "PARALLEL"
            },
            {
                "from": 188,
                "to": 190,
                "label": "EVAL with clause\napp([], X74, X74).\nand substitutionT9 -> [],\nX20 -> T35,\nX74 -> T35,\nT14 -> T35,\nX75 -> T35"
            },
            {
                "from": 188,
                "to": 191,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 189,
                "to": 193,
                "label": "EVAL with clause\napp(.(X86, X87), X88, .(X86, X89)) :- app(X87, X88, X89).\nand substitutionX86 -> T42,\nX87 -> T43,\nT9 -> .(T42, T43),\nX20 -> X90,\nX88 -> X90,\nX89 -> T45,\nT14 -> .(T42, T45),\nT44 -> T45"
            },
            {
                "from": 189,
                "to": 194,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 190,
                "to": 192,
                "label": "SUCCESS"
            },
            {
                "from": 193,
                "to": 88,
                "label": "INSTANCE with matching:\nT9 -> T43\nX20 -> X90\nT14 -> T45"
            }
        ],
        "type": "Graph"
    }
}

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

(38)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f2_in(T9) -> U1(f83_in(T9), T9)
   U1(f83_out1, T9) -> f2_out1
   f86_in -> f86_out1
   f86_in -> U2(f86_in)
   U2(f86_out1) -> f86_out1
   f88_in([]) -> f88_out1
   f88_in(.(T42, T43)) -> U3(f88_in(T43), .(T42, T43))
   U3(f88_out1, .(T42, T43)) -> f88_out1
   f83_in(T9) -> U4(f86_in, T9)
   U4(f86_out1, T9) -> U5(f88_in(T9), T9)
   U5(f88_out1, T9) -> f83_out1

Q is empty.

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

(39) QTRSRRRProof (EQUIVALENT)
Used ordering:
f2_in/1(YES)
U1/2(YES,YES)
f83_in/1(YES)
f83_out1/0)
f2_out1/0)
f86_in/0)
f86_out1/0)
U2/1)YES(
f88_in/1(YES)
[]/0)
f88_out1/0)
./2(YES,YES)
U3/2(YES,YES)
U4/2(YES,YES)
U5/2(YES,YES)

Quasi precedence:
f2_in_1 > U1_2 > f2_out1
f2_in_1 > [f83_in_1, f86_in] > U4_2 > [f83_out1, f86_out1, f88_in_1, [], f88_out1, ._2] > U3_2
f2_in_1 > [f83_in_1, f86_in] > U4_2 > [f83_out1, f86_out1, f88_in_1, [], f88_out1, ._2] > U5_2


Status:
f2_in_1: multiset status
U1_2: [1,2]
f83_in_1: [1]
f83_out1: multiset status
f2_out1: multiset status
f86_in: multiset status
f86_out1: multiset status
f88_in_1: multiset status
[]: multiset status
f88_out1: multiset status
._2: multiset status
U3_2: multiset status
U4_2: multiset status
U5_2: multiset status

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f2_in(T9) -> U1(f83_in(T9), T9)
   U1(f83_out1, T9) -> f2_out1
   f86_in -> f86_out1
   f88_in([]) -> f88_out1
   f88_in(.(T42, T43)) -> U3(f88_in(T43), .(T42, T43))
   U3(f88_out1, .(T42, T43)) -> f88_out1
   f83_in(T9) -> U4(f86_in, T9)
   U4(f86_out1, T9) -> U5(f88_in(T9), T9)
   U5(f88_out1, T9) -> f83_out1




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

(40)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f86_in -> U2(f86_in)
   U2(f86_out1) -> f86_out1

Q is empty.

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

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

   POL(U2(x_1)) = 2*x_1
   POL(f86_in) = 0
   POL(f86_out1) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U2(f86_out1) -> f86_out1




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

(42)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f86_in -> U2(f86_in)

Q is empty.

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

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

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

   f86_in -> U2(f86_in)

The set Q consists of the following terms:

   f86_in


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

(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:

   F86_IN -> F86_IN

The TRS R consists of the following rules:

   f86_in -> U2(f86_in)

The set Q consists of the following terms:

   f86_in

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

(47) 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.
----------------------------------------

(48)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F86_IN -> F86_IN

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

   f86_in

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

(49) 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].

   f86_in


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

(50)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F86_IN -> F86_IN

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

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

{
    "root": 1,
    "program": {
        "directives": [],
        "clauses": [
            [
                "(sublist Xs Ys)",
                "(',' (app X1 Zs Ys) (app Xs X2 Zs))"
            ],
            [
                "(app ([]) X X)",
                null
            ],
            [
                "(app (. X Xs) Ys (. X Zs))",
                "(app Xs Ys Zs)"
            ]
        ]
    },
    "graph": {
        "nodes": {
            "22": {
                "goal": [
                    {
                        "clause": 1,
                        "scope": 2,
                        "term": "(',' (app X7 X8 T7) (app T5 X9 X8))"
                    },
                    {
                        "clause": 2,
                        "scope": 2,
                        "term": "(',' (app X7 X8 T7) (app T5 X9 X8))"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T5"],
                    "free": [
                        "X7",
                        "X8",
                        "X9"
                    ],
                    "exprvars": []
                }
            },
            "23": {
                "goal": [{
                    "clause": 1,
                    "scope": 2,
                    "term": "(',' (app X7 X8 T7) (app T5 X9 X8))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T5"],
                    "free": [
                        "X7",
                        "X8",
                        "X9"
                    ],
                    "exprvars": []
                }
            },
            "24": {
                "goal": [{
                    "clause": 2,
                    "scope": 2,
                    "term": "(',' (app X7 X8 T7) (app T5 X9 X8))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T5"],
                    "free": [
                        "X7",
                        "X8",
                        "X9"
                    ],
                    "exprvars": []
                }
            },
            "25": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(app T5 X9 T19)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T5"],
                    "free": ["X9"],
                    "exprvars": []
                }
            },
            "type": "Nodes",
            "195": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (app X80 X81 T45) (app T5 X9 X81))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T5"],
                    "free": [
                        "X9",
                        "X80",
                        "X81"
                    ],
                    "exprvars": []
                }
            },
            "163": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "196": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(sublist T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "168": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(app T34 X56 T36)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T34"],
                    "free": ["X56"],
                    "exprvars": []
                }
            },
            "169": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "80": {
                "goal": [
                    {
                        "clause": 1,
                        "scope": 3,
                        "term": "(app T5 X9 T19)"
                    },
                    {
                        "clause": 2,
                        "scope": 3,
                        "term": "(app T5 X9 T19)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T5"],
                    "free": ["X9"],
                    "exprvars": []
                }
            },
            "81": {
                "goal": [{
                    "clause": 1,
                    "scope": 3,
                    "term": "(app T5 X9 T19)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T5"],
                    "free": ["X9"],
                    "exprvars": []
                }
            },
            "82": {
                "goal": [{
                    "clause": 2,
                    "scope": 3,
                    "term": "(app T5 X9 T19)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T5"],
                    "free": ["X9"],
                    "exprvars": []
                }
            },
            "84": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "85": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "20": {
                "goal": [{
                    "clause": 0,
                    "scope": 1,
                    "term": "(sublist T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "21": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (app X7 X8 T7) (app T5 X9 X8))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T5"],
                    "free": [
                        "X7",
                        "X8",
                        "X9"
                    ],
                    "exprvars": []
                }
            }
        },
        "edges": [
            {
                "from": 1,
                "to": 20,
                "label": "CASE"
            },
            {
                "from": 20,
                "to": 21,
                "label": "ONLY EVAL with clause\nsublist(X5, X6) :- ','(app(X7, X8, X6), app(X5, X9, X8)).\nand substitutionT1 -> T5,\nX5 -> T5,\nT2 -> T7,\nX6 -> T7,\nT6 -> T7"
            },
            {
                "from": 21,
                "to": 22,
                "label": "CASE"
            },
            {
                "from": 22,
                "to": 23,
                "label": "PARALLEL"
            },
            {
                "from": 22,
                "to": 24,
                "label": "PARALLEL"
            },
            {
                "from": 23,
                "to": 25,
                "label": "ONLY EVAL with clause\napp([], X26, X26).\nand substitutionX7 -> [],\nX8 -> T19,\nX26 -> T19,\nT7 -> T19,\nX27 -> T19,\nT18 -> T19"
            },
            {
                "from": 24,
                "to": 195,
                "label": "EVAL with clause\napp(.(X75, X76), X77, .(X75, X78)) :- app(X76, X77, X78).\nand substitutionX75 -> T43,\nX76 -> X80,\nX7 -> .(T43, X80),\nX8 -> X81,\nX77 -> X81,\nX79 -> T43,\nX78 -> T45,\nT7 -> .(T43, T45),\nT44 -> T45"
            },
            {
                "from": 24,
                "to": 196,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 25,
                "to": 80,
                "label": "CASE"
            },
            {
                "from": 80,
                "to": 81,
                "label": "PARALLEL"
            },
            {
                "from": 80,
                "to": 82,
                "label": "PARALLEL"
            },
            {
                "from": 81,
                "to": 84,
                "label": "EVAL with clause\napp([], X40, X40).\nand substitutionT5 -> [],\nX9 -> T26,\nX40 -> T26,\nT19 -> T26,\nX41 -> T26"
            },
            {
                "from": 81,
                "to": 85,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 82,
                "to": 168,
                "label": "EVAL with clause\napp(.(X52, X53), X54, .(X52, X55)) :- app(X53, X54, X55).\nand substitutionX52 -> T33,\nX53 -> T34,\nT5 -> .(T33, T34),\nX9 -> X56,\nX54 -> X56,\nX55 -> T36,\nT19 -> .(T33, T36),\nT35 -> T36"
            },
            {
                "from": 82,
                "to": 169,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 84,
                "to": 163,
                "label": "SUCCESS"
            },
            {
                "from": 168,
                "to": 25,
                "label": "INSTANCE with matching:\nT5 -> T34\nX9 -> X56\nT19 -> T36"
            },
            {
                "from": 195,
                "to": 21,
                "label": "INSTANCE with matching:\nX7 -> X80\nX8 -> X81\nT7 -> T45"
            }
        ],
        "type": "Graph"
    }
}

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

(52)
Obligation:
Triples:

appA(.(X1, X2), X3, .(X1, X4)) :- appA(X2, X3, X4).
pB([], X1, X1, X2, X3) :- appA(X2, X3, X1).
pB(.(X1, X2), X3, .(X1, X4), X5, X6) :- pB(X2, X3, X4, X5, X6).
sublistC(X1, X2) :- pB(X3, X4, X2, X1, X5).

Clauses:

appcA([], X1, X1).
appcA(.(X1, X2), X3, .(X1, X4)) :- appcA(X2, X3, X4).
qcB([], X1, X1, X2, X3) :- appcA(X2, X3, X1).
qcB(.(X1, X2), X3, .(X1, X4), X5, X6) :- qcB(X2, X3, X4, X5, X6).

Afs:

sublistC(x1, x2)  =  sublistC(x1)


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

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

sublistC_in_2: (b,f)

pB_in_5: (f,f,f,b,f)

appA_in_3: (b,f,f)

Transforming TRIPLES into the following Term Rewriting System:

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

   SUBLISTC_IN_GA(X1, X2) -> U4_GA(X1, X2, pB_in_aaaga(X3, X4, X2, X1, X5))
   SUBLISTC_IN_GA(X1, X2) -> PB_IN_AAAGA(X3, X4, X2, X1, X5)
   PB_IN_AAAGA([], X1, X1, X2, X3) -> U2_AAAGA(X1, X2, X3, appA_in_gaa(X2, X3, X1))
   PB_IN_AAAGA([], X1, X1, X2, X3) -> APPA_IN_GAA(X2, X3, X1)
   APPA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> U1_GAA(X1, X2, X3, X4, appA_in_gaa(X2, X3, X4))
   APPA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPA_IN_GAA(X2, X3, X4)
   PB_IN_AAAGA(.(X1, X2), X3, .(X1, X4), X5, X6) -> U3_AAAGA(X1, X2, X3, X4, X5, X6, pB_in_aaaga(X2, X3, X4, X5, X6))
   PB_IN_AAAGA(.(X1, X2), X3, .(X1, X4), X5, X6) -> PB_IN_AAAGA(X2, X3, X4, X5, X6)

R is empty.
The argument filtering Pi contains the following mapping:
pB_in_aaaga(x1, x2, x3, x4, x5)  =  pB_in_aaaga(x4)

appA_in_gaa(x1, x2, x3)  =  appA_in_gaa(x1)

.(x1, x2)  =  .(x2)

[]  =  []

SUBLISTC_IN_GA(x1, x2)  =  SUBLISTC_IN_GA(x1)

U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)

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

U2_AAAGA(x1, x2, x3, x4)  =  U2_AAAGA(x2, x4)

APPA_IN_GAA(x1, x2, x3)  =  APPA_IN_GAA(x1)

U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x2, x5)

U3_AAAGA(x1, x2, x3, x4, x5, x6, x7)  =  U3_AAAGA(x5, x7)


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


Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES



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

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

   SUBLISTC_IN_GA(X1, X2) -> U4_GA(X1, X2, pB_in_aaaga(X3, X4, X2, X1, X5))
   SUBLISTC_IN_GA(X1, X2) -> PB_IN_AAAGA(X3, X4, X2, X1, X5)
   PB_IN_AAAGA([], X1, X1, X2, X3) -> U2_AAAGA(X1, X2, X3, appA_in_gaa(X2, X3, X1))
   PB_IN_AAAGA([], X1, X1, X2, X3) -> APPA_IN_GAA(X2, X3, X1)
   APPA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> U1_GAA(X1, X2, X3, X4, appA_in_gaa(X2, X3, X4))
   APPA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPA_IN_GAA(X2, X3, X4)
   PB_IN_AAAGA(.(X1, X2), X3, .(X1, X4), X5, X6) -> U3_AAAGA(X1, X2, X3, X4, X5, X6, pB_in_aaaga(X2, X3, X4, X5, X6))
   PB_IN_AAAGA(.(X1, X2), X3, .(X1, X4), X5, X6) -> PB_IN_AAAGA(X2, X3, X4, X5, X6)

R is empty.
The argument filtering Pi contains the following mapping:
pB_in_aaaga(x1, x2, x3, x4, x5)  =  pB_in_aaaga(x4)

appA_in_gaa(x1, x2, x3)  =  appA_in_gaa(x1)

.(x1, x2)  =  .(x2)

[]  =  []

SUBLISTC_IN_GA(x1, x2)  =  SUBLISTC_IN_GA(x1)

U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)

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

U2_AAAGA(x1, x2, x3, x4)  =  U2_AAAGA(x2, x4)

APPA_IN_GAA(x1, x2, x3)  =  APPA_IN_GAA(x1)

U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x2, x5)

U3_AAAGA(x1, x2, x3, x4, x5, x6, x7)  =  U3_AAAGA(x5, x7)


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

(55) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.
----------------------------------------

(56)
Complex Obligation (AND)

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

(57)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APPA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPA_IN_GAA(X2, X3, X4)

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

APPA_IN_GAA(x1, x2, x3)  =  APPA_IN_GAA(x1)


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

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

(59)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APPA_IN_GAA(.(X2)) -> APPA_IN_GAA(X2)

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

(60) 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:
*APPA_IN_GAA(.(X2)) -> APPA_IN_GAA(X2)
The graph contains the following edges 1 > 1


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

(61)
YES

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

(62)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   PB_IN_AAAGA(.(X1, X2), X3, .(X1, X4), X5, X6) -> PB_IN_AAAGA(X2, X3, X4, X5, X6)

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

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


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

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

(64)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PB_IN_AAAGA(X5) -> PB_IN_AAAGA(X5)

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

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

{
    "root": 3,
    "program": {
        "directives": [],
        "clauses": [
            [
                "(sublist Xs Ys)",
                "(',' (app X1 Zs Ys) (app Xs X2 Zs))"
            ],
            [
                "(app ([]) X X)",
                null
            ],
            [
                "(app (. X Xs) Ys (. X Zs))",
                "(app Xs Ys Zs)"
            ]
        ]
    },
    "graph": {
        "nodes": {
            "89": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(app T9 X20 T14)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T9"],
                    "free": ["X20"],
                    "exprvars": []
                }
            },
            "36": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (app X18 X19 T11) (app T9 X20 X19))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T9"],
                    "free": [
                        "X18",
                        "X19",
                        "X20"
                    ],
                    "exprvars": []
                }
            },
            "180": {
                "goal": [{
                    "clause": 1,
                    "scope": 3,
                    "term": "(app T9 X20 T14)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T9"],
                    "free": ["X20"],
                    "exprvars": []
                }
            },
            "181": {
                "goal": [{
                    "clause": 2,
                    "scope": 3,
                    "term": "(app T9 X20 T14)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T9"],
                    "free": ["X20"],
                    "exprvars": []
                }
            },
            "160": {
                "goal": [{
                    "clause": 2,
                    "scope": 2,
                    "term": "(app X18 X19 T11)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X18",
                        "X19"
                    ],
                    "exprvars": []
                }
            },
            "182": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "type": "Nodes",
            "161": {
                "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": []
                }
            },
            "162": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "184": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "185": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(app T43 X90 T45)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T43"],
                    "free": ["X90"],
                    "exprvars": []
                }
            },
            "175": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(app X58 X59 T27)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X58",
                        "X59"
                    ],
                    "exprvars": []
                }
            },
            "176": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "187": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "3": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(sublist T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "179": {
                "goal": [
                    {
                        "clause": 1,
                        "scope": 3,
                        "term": "(app T9 X20 T14)"
                    },
                    {
                        "clause": 2,
                        "scope": 3,
                        "term": "(app T9 X20 T14)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T9"],
                    "free": ["X20"],
                    "exprvars": []
                }
            },
            "158": {
                "goal": [
                    {
                        "clause": 1,
                        "scope": 2,
                        "term": "(app X18 X19 T11)"
                    },
                    {
                        "clause": 2,
                        "scope": 2,
                        "term": "(app X18 X19 T11)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X18",
                        "X19"
                    ],
                    "exprvars": []
                }
            },
            "5": {
                "goal": [{
                    "clause": 0,
                    "scope": 1,
                    "term": "(sublist T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T1"],
                    "free": [],
                    "exprvars": []
                }
            },
            "159": {
                "goal": [{
                    "clause": 1,
                    "scope": 2,
                    "term": "(app X18 X19 T11)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X18",
                        "X19"
                    ],
                    "exprvars": []
                }
            },
            "87": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(app X18 X19 T11)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X18",
                        "X19"
                    ],
                    "exprvars": []
                }
            }
        },
        "edges": [
            {
                "from": 3,
                "to": 5,
                "label": "CASE"
            },
            {
                "from": 5,
                "to": 36,
                "label": "ONLY EVAL with clause\nsublist(X16, X17) :- ','(app(X18, X19, X17), app(X16, X20, X19)).\nand substitutionT1 -> T9,\nX16 -> T9,\nT2 -> T11,\nX17 -> T11,\nT10 -> T11"
            },
            {
                "from": 36,
                "to": 87,
                "label": "SPLIT 1"
            },
            {
                "from": 36,
                "to": 89,
                "label": "SPLIT 2\nreplacements:X18 -> T13,\nX19 -> T14"
            },
            {
                "from": 87,
                "to": 158,
                "label": "CASE"
            },
            {
                "from": 89,
                "to": 179,
                "label": "CASE"
            },
            {
                "from": 158,
                "to": 159,
                "label": "PARALLEL"
            },
            {
                "from": 158,
                "to": 160,
                "label": "PARALLEL"
            },
            {
                "from": 159,
                "to": 161,
                "label": "ONLY EVAL with clause\napp([], X37, X37).\nand substitutionX18 -> [],\nX19 -> T20,\nX37 -> T20,\nT11 -> T20,\nX38 -> T20"
            },
            {
                "from": 160,
                "to": 175,
                "label": "EVAL with clause\napp(.(X53, X54), X55, .(X53, X56)) :- app(X54, X55, X56).\nand substitutionX53 -> T25,\nX54 -> X58,\nX18 -> .(T25, X58),\nX19 -> X59,\nX55 -> X59,\nX57 -> T25,\nX56 -> T27,\nT11 -> .(T25, T27),\nT26 -> T27"
            },
            {
                "from": 160,
                "to": 176,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 161,
                "to": 162,
                "label": "SUCCESS"
            },
            {
                "from": 175,
                "to": 87,
                "label": "INSTANCE with matching:\nX18 -> X58\nX19 -> X59\nT11 -> T27"
            },
            {
                "from": 179,
                "to": 180,
                "label": "PARALLEL"
            },
            {
                "from": 179,
                "to": 181,
                "label": "PARALLEL"
            },
            {
                "from": 180,
                "to": 182,
                "label": "EVAL with clause\napp([], X74, X74).\nand substitutionT9 -> [],\nX20 -> T35,\nX74 -> T35,\nT14 -> T35,\nX75 -> T35"
            },
            {
                "from": 180,
                "to": 183,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 181,
                "to": 185,
                "label": "EVAL with clause\napp(.(X86, X87), X88, .(X86, X89)) :- app(X87, X88, X89).\nand substitutionX86 -> T42,\nX87 -> T43,\nT9 -> .(T42, T43),\nX20 -> X90,\nX88 -> X90,\nX89 -> T45,\nT14 -> .(T42, T45),\nT44 -> T45"
            },
            {
                "from": 181,
                "to": 187,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 182,
                "to": 184,
                "label": "SUCCESS"
            },
            {
                "from": 185,
                "to": 89,
                "label": "INSTANCE with matching:\nT9 -> T43\nX20 -> X90\nT14 -> T45"
            }
        ],
        "type": "Graph"
    }
}

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

(66)
Complex Obligation (AND)

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

(67)
Obligation:
Rules:
f185_in(T43) -> f89_in(T43) :|: TRUE
f89_out(x) -> f185_out(x) :|: TRUE
f179_in(T9) -> f181_in(T9) :|: TRUE
f181_out(x1) -> f179_out(x1) :|: TRUE
f180_out(x2) -> f179_out(x2) :|: TRUE
f179_in(x3) -> f180_in(x3) :|: TRUE
f185_out(x4) -> f181_out(.(x5, x4)) :|: TRUE
f187_out -> f181_out(x6) :|: TRUE
f181_in(.(x7, x8)) -> f185_in(x8) :|: TRUE
f181_in(x9) -> f187_in :|: TRUE
f89_in(x10) -> f179_in(x10) :|: TRUE
f179_out(x11) -> f89_out(x11) :|: TRUE
f5_out(T1) -> f3_out(T1) :|: TRUE
f3_in(x12) -> f5_in(x12) :|: TRUE
f5_in(x13) -> f36_in(x13) :|: TRUE
f36_out(x14) -> f5_out(x14) :|: TRUE
f36_in(x15) -> f87_in :|: TRUE
f89_out(x16) -> f36_out(x16) :|: TRUE
f87_out -> f89_in(x17) :|: TRUE
Start term: f3_in(T1)

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

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

Simplified to the following IRSwTs:


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

(69)
TRUE

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

(70)
Obligation:
Rules:
f160_in -> f176_in :|: TRUE
f175_out -> f160_out :|: TRUE
f176_out -> f160_out :|: TRUE
f160_in -> f175_in :|: TRUE
f87_in -> f158_in :|: TRUE
f158_out -> f87_out :|: TRUE
f158_in -> f160_in :|: TRUE
f160_out -> f158_out :|: TRUE
f159_out -> f158_out :|: TRUE
f158_in -> f159_in :|: TRUE
f87_out -> f175_out :|: TRUE
f175_in -> f87_in :|: TRUE
f5_out(T1) -> f3_out(T1) :|: TRUE
f3_in(x) -> f5_in(x) :|: TRUE
f5_in(T9) -> f36_in(T9) :|: TRUE
f36_out(x1) -> f5_out(x1) :|: TRUE
f36_in(x2) -> f87_in :|: TRUE
f89_out(x3) -> f36_out(x3) :|: TRUE
f87_out -> f89_in(x4) :|: TRUE
Start term: f3_in(T1)

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

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

Simplified to the following IRSwTs:

intTRSProblem:
f160_in -> f175_in :|: TRUE
f87_in -> f158_in :|: TRUE
f158_in -> f160_in :|: TRUE
f175_in -> f87_in :|: TRUE


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

(72)
Obligation:
Rules:
f160_in -> f175_in :|: TRUE
f87_in -> f158_in :|: TRUE
f158_in -> f160_in :|: TRUE
f175_in -> f87_in :|: TRUE

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

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

(74)
Obligation:
Rules:
f87_in -> f87_in :|: TRUE

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

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

(76)
Obligation:
Rules:
f87_in -> f87_in :|: TRUE

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

(77) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f87_in -> f87_in :|: TRUE

Arcs:
(1) -> (1)

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

(78)
Obligation:

Termination digraph:
Nodes:
(1) f87_in -> f87_in :|: TRUE

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(79) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f87_in()
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(80)
Obligation:
Rules:
f87_in -> f87_in :|: TRUE

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

(81) IntTRSNonPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc) -> f(1) :|: pc = 1 && TRUE
Proved unsatisfiability of the following formula, indicating that the system is never left after entering:
((run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) and !(((run2_0 * 1)) = ((1 * 1)) and T))
Proved satisfiability of the following formula, indicating that the system is entered at least once:
(run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T))

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

(82)
NO