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


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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Left Termination of the query pattern

sublist(g,g)

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

(0) Prolog
(1) PrologToPiTRSProof [SOUND, 0 ms]
(2) PiTRS
    (3) DependencyPairsProof [EQUIVALENT, 4 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, 2 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) DependencyPairsProof [EQUIVALENT, 0 ms]
    (40) QDP
    (41) DependencyGraphProof [EQUIVALENT, 0 ms]
    (42) AND
        (43) QDP
            (44) UsableRulesProof [EQUIVALENT, 0 ms]
            (45) QDP
            (46) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (47) YES
        (48) QDP
            (49) UsableRulesProof [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) UsableRulesProof [EQUIVALENT, 0 ms]
            (59) PiDP
            (60) PiDPToQDPProof [SOUND, 0 ms]
            (61) QDP
            (62) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (63) YES
        (64) PiDP
            (65) UsableRulesProof [EQUIVALENT, 0 ms]
            (66) PiDP
            (67) PiDPToQDPProof [SOUND, 0 ms]
            (68) QDP
        (69) PiDP
            (70) UsableRulesProof [EQUIVALENT, 0 ms]
            (71) PiDP
(72) PrologToIRSwTTransformerProof [SOUND, 34 ms]
(73) AND
    (74) IRSwT
        (75) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
        (76) TRUE
    (77) IRSwT
        (78) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
        (79) IRSwT
        (80) IntTRSCompressionProof [EQUIVALENT, 27 ms]
        (81) IRSwT
        (82) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
        (83) IRSwT
        (84) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
        (85) IRSwT
        (86) FilterProof [EQUIVALENT, 0 ms]
        (87) IntTRS
        (88) IntTRSNonPeriodicNontermProof [COMPLETE, 6 ms]
        (89) NO


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

(0)
Obligation:
Clauses:

append1([], Ys, Ys).
append1(.(X, Xs), Ys, .(X, Zs)) :- append1(Xs, Ys, Zs).
append2([], Ys, Ys).
append2(.(X, Xs), Ys, .(X, Zs)) :- append2(Xs, Ys, Zs).
sublist(X, Y) :- ','(append1(U, X, V), append2(V, W, Y)).


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

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

sublist_in_2: (b,b)

append1_in_3: (f,b,f)

append2_in_3: (b,f,b)

Transforming Prolog into the following Term Rewriting System:

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

   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)

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

U3_gg(x1, x2, x3)  =  U3_gg(x2, x3)

append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)

append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x3)

U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)

.(x1, x2)  =  .(x2)

U4_gg(x1, x2, x3)  =  U4_gg(x3)

append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)

[]  =  []

append2_out_gag(x1, x2, x3)  =  append2_out_gag(x2)

U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)

sublist_out_gg(x1, x2)  =  sublist_out_gg





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_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)

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

U3_gg(x1, x2, x3)  =  U3_gg(x2, x3)

append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)

append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x3)

U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)

.(x1, x2)  =  .(x2)

U4_gg(x1, x2, x3)  =  U4_gg(x3)

append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)

[]  =  []

append2_out_gag(x1, x2, x3)  =  append2_out_gag(x2)

U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)

sublist_out_gg(x1, x2)  =  sublist_out_gg



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

(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_GG(X, Y) -> U3_GG(X, Y, append1_in_aga(U, X, V))
   SUBLIST_IN_GG(X, Y) -> APPEND1_IN_AGA(U, X, V)
   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U1_AGA(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs)
   U3_GG(X, Y, append1_out_aga(U, X, V)) -> U4_GG(X, Y, append2_in_gag(V, W, Y))
   U3_GG(X, Y, append1_out_aga(U, X, V)) -> APPEND2_IN_GAG(V, W, Y)
   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> U2_GAG(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)

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

U3_gg(x1, x2, x3)  =  U3_gg(x2, x3)

append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)

append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x3)

U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)

.(x1, x2)  =  .(x2)

U4_gg(x1, x2, x3)  =  U4_gg(x3)

append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)

[]  =  []

append2_out_gag(x1, x2, x3)  =  append2_out_gag(x2)

U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)

sublist_out_gg(x1, x2)  =  sublist_out_gg

SUBLIST_IN_GG(x1, x2)  =  SUBLIST_IN_GG(x1, x2)

U3_GG(x1, x2, x3)  =  U3_GG(x2, x3)

APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)

U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x5)

U4_GG(x1, x2, x3)  =  U4_GG(x3)

APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)

U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(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_GG(X, Y) -> U3_GG(X, Y, append1_in_aga(U, X, V))
   SUBLIST_IN_GG(X, Y) -> APPEND1_IN_AGA(U, X, V)
   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U1_AGA(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs)
   U3_GG(X, Y, append1_out_aga(U, X, V)) -> U4_GG(X, Y, append2_in_gag(V, W, Y))
   U3_GG(X, Y, append1_out_aga(U, X, V)) -> APPEND2_IN_GAG(V, W, Y)
   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> U2_GAG(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)

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

U3_gg(x1, x2, x3)  =  U3_gg(x2, x3)

append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)

append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x3)

U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)

.(x1, x2)  =  .(x2)

U4_gg(x1, x2, x3)  =  U4_gg(x3)

append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)

[]  =  []

append2_out_gag(x1, x2, x3)  =  append2_out_gag(x2)

U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)

sublist_out_gg(x1, x2)  =  sublist_out_gg

SUBLIST_IN_GG(x1, x2)  =  SUBLIST_IN_GG(x1, x2)

U3_GG(x1, x2, x3)  =  U3_GG(x2, x3)

APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)

U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x5)

U4_GG(x1, x2, x3)  =  U4_GG(x3)

APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)

U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(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:

   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)

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

U3_gg(x1, x2, x3)  =  U3_gg(x2, x3)

append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)

append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x3)

U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)

.(x1, x2)  =  .(x2)

U4_gg(x1, x2, x3)  =  U4_gg(x3)

append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)

[]  =  []

append2_out_gag(x1, x2, x3)  =  append2_out_gag(x2)

U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)

sublist_out_gg(x1, x2)  =  sublist_out_gg

APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)


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:

   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs)

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

APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)


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:

   APPEND2_IN_GAG(.(Xs), .(Zs)) -> APPEND2_IN_GAG(Xs, Zs)

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:
*APPEND2_IN_GAG(.(Xs), .(Zs)) -> APPEND2_IN_GAG(Xs, Zs)
The graph contains the following edges 1 > 1, 2 > 2


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

(13)
YES

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

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

   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)

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

U3_gg(x1, x2, x3)  =  U3_gg(x2, x3)

append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)

append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x3)

U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)

.(x1, x2)  =  .(x2)

U4_gg(x1, x2, x3)  =  U4_gg(x3)

append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)

[]  =  []

append2_out_gag(x1, x2, x3)  =  append2_out_gag(x2)

U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)

sublist_out_gg(x1, x2)  =  sublist_out_gg

APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)


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:

   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs)

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

APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)


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:

   APPEND1_IN_AGA(Ys) -> APPEND1_IN_AGA(Ys)

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,b)

append1_in_3: (f,b,f)

append2_in_3: (b,f,b)

Transforming Prolog into the following Term Rewriting System:

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

   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)

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

U3_gg(x1, x2, x3)  =  U3_gg(x1, x2, x3)

append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)

append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x2, x3)

U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)

.(x1, x2)  =  .(x2)

U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)

append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)

[]  =  []

append2_out_gag(x1, x2, x3)  =  append2_out_gag(x1, x2, x3)

U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)

sublist_out_gg(x1, x2)  =  sublist_out_gg(x1, x2)





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_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)

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

U3_gg(x1, x2, x3)  =  U3_gg(x1, x2, x3)

append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)

append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x2, x3)

U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)

.(x1, x2)  =  .(x2)

U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)

append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)

[]  =  []

append2_out_gag(x1, x2, x3)  =  append2_out_gag(x1, x2, x3)

U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)

sublist_out_gg(x1, x2)  =  sublist_out_gg(x1, x2)



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

(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_GG(X, Y) -> U3_GG(X, Y, append1_in_aga(U, X, V))
   SUBLIST_IN_GG(X, Y) -> APPEND1_IN_AGA(U, X, V)
   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U1_AGA(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs)
   U3_GG(X, Y, append1_out_aga(U, X, V)) -> U4_GG(X, Y, append2_in_gag(V, W, Y))
   U3_GG(X, Y, append1_out_aga(U, X, V)) -> APPEND2_IN_GAG(V, W, Y)
   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> U2_GAG(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)

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

U3_gg(x1, x2, x3)  =  U3_gg(x1, x2, x3)

append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)

append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x2, x3)

U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)

.(x1, x2)  =  .(x2)

U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)

append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)

[]  =  []

append2_out_gag(x1, x2, x3)  =  append2_out_gag(x1, x2, x3)

U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)

sublist_out_gg(x1, x2)  =  sublist_out_gg(x1, x2)

SUBLIST_IN_GG(x1, x2)  =  SUBLIST_IN_GG(x1, x2)

U3_GG(x1, x2, x3)  =  U3_GG(x1, x2, x3)

APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)

U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, x5)

U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)

APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)

U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x2, x4, 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_GG(X, Y) -> U3_GG(X, Y, append1_in_aga(U, X, V))
   SUBLIST_IN_GG(X, Y) -> APPEND1_IN_AGA(U, X, V)
   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U1_AGA(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs)
   U3_GG(X, Y, append1_out_aga(U, X, V)) -> U4_GG(X, Y, append2_in_gag(V, W, Y))
   U3_GG(X, Y, append1_out_aga(U, X, V)) -> APPEND2_IN_GAG(V, W, Y)
   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> U2_GAG(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)

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

U3_gg(x1, x2, x3)  =  U3_gg(x1, x2, x3)

append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)

append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x2, x3)

U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)

.(x1, x2)  =  .(x2)

U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)

append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)

[]  =  []

append2_out_gag(x1, x2, x3)  =  append2_out_gag(x1, x2, x3)

U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)

sublist_out_gg(x1, x2)  =  sublist_out_gg(x1, x2)

SUBLIST_IN_GG(x1, x2)  =  SUBLIST_IN_GG(x1, x2)

U3_GG(x1, x2, x3)  =  U3_GG(x1, x2, x3)

APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)

U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, x5)

U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)

APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)

U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x2, x4, 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:

   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)

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

U3_gg(x1, x2, x3)  =  U3_gg(x1, x2, x3)

append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)

append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x2, x3)

U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)

.(x1, x2)  =  .(x2)

U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)

append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)

[]  =  []

append2_out_gag(x1, x2, x3)  =  append2_out_gag(x1, x2, x3)

U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)

sublist_out_gg(x1, x2)  =  sublist_out_gg(x1, x2)

APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)


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:

   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs)

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

APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)


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:

   APPEND2_IN_GAG(.(Xs), .(Zs)) -> APPEND2_IN_GAG(Xs, Zs)

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:
*APPEND2_IN_GAG(.(Xs), .(Zs)) -> APPEND2_IN_GAG(Xs, Zs)
The graph contains the following edges 1 > 1, 2 > 2


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

(31)
YES

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

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

   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)

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

U3_gg(x1, x2, x3)  =  U3_gg(x1, x2, x3)

append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)

append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x2, x3)

U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)

.(x1, x2)  =  .(x2)

U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)

append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)

[]  =  []

append2_out_gag(x1, x2, x3)  =  append2_out_gag(x1, x2, x3)

U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)

sublist_out_gg(x1, x2)  =  sublist_out_gg(x1, x2)

APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)


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:

   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs)

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

APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)


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:

   APPEND1_IN_AGA(Ys) -> APPEND1_IN_AGA(Ys)

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

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

{
    "root": 1,
    "program": {
        "directives": [],
        "clauses": [
            [
                "(append1 ([]) Ys Ys)",
                null
            ],
            [
                "(append1 (. X Xs) Ys (. X Zs))",
                "(append1 Xs Ys Zs)"
            ],
            [
                "(append2 ([]) Ys Ys)",
                null
            ],
            [
                "(append2 (. X Xs) Ys (. X Zs))",
                "(append2 Xs Ys Zs)"
            ],
            [
                "(sublist X Y)",
                "(',' (append1 U X V) (append2 V W Y))"
            ]
        ]
    },
    "graph": {
        "nodes": {
            "44": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "45": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "56": {
                "goal": [
                    {
                        "clause": 2,
                        "scope": 3,
                        "term": "(append2 T14 X15 T10)"
                    },
                    {
                        "clause": 3,
                        "scope": 3,
                        "term": "(append2 T14 X15 T10)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T10"],
                    "free": ["X15"],
                    "exprvars": []
                }
            },
            "35": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(append1 X13 T9 X14)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T9"],
                    "free": [
                        "X13",
                        "X14"
                    ],
                    "exprvars": []
                }
            },
            "57": {
                "goal": [{
                    "clause": 2,
                    "scope": 3,
                    "term": "(append2 T14 X15 T10)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T10"],
                    "free": ["X15"],
                    "exprvars": []
                }
            },
            "14": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (append1 X13 T9 X14) (append2 X14 X15 T10))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T9",
                        "T10"
                    ],
                    "free": [
                        "X13",
                        "X14",
                        "X15"
                    ],
                    "exprvars": []
                }
            },
            "36": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(append2 T14 X15 T10)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T10"],
                    "free": ["X15"],
                    "exprvars": []
                }
            },
            "58": {
                "goal": [{
                    "clause": 3,
                    "scope": 3,
                    "term": "(append2 T14 X15 T10)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T10"],
                    "free": ["X15"],
                    "exprvars": []
                }
            },
            "type": "Nodes",
            "252": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(append2 T42 X76 T41)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T41"],
                    "free": ["X76"],
                    "exprvars": []
                }
            },
            "253": {
                "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",
                        "T2"
                    ],
                    "free": [],
                    "exprvars": []
                }
            },
            "2": {
                "goal": [{
                    "clause": 4,
                    "scope": 1,
                    "term": "(sublist T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T1",
                        "T2"
                    ],
                    "free": [],
                    "exprvars": []
                }
            },
            "50": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(append1 X45 T24 X46)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T24"],
                    "free": [
                        "X45",
                        "X46"
                    ],
                    "exprvars": []
                }
            },
            "61": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "62": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "41": {
                "goal": [
                    {
                        "clause": 0,
                        "scope": 2,
                        "term": "(append1 X13 T9 X14)"
                    },
                    {
                        "clause": 1,
                        "scope": 2,
                        "term": "(append1 X13 T9 X14)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T9"],
                    "free": [
                        "X13",
                        "X14"
                    ],
                    "exprvars": []
                }
            },
            "63": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "42": {
                "goal": [{
                    "clause": 0,
                    "scope": 2,
                    "term": "(append1 X13 T9 X14)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T9"],
                    "free": [
                        "X13",
                        "X14"
                    ],
                    "exprvars": []
                }
            },
            "43": {
                "goal": [{
                    "clause": 1,
                    "scope": 2,
                    "term": "(append1 X13 T9 X14)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T9"],
                    "free": [
                        "X13",
                        "X14"
                    ],
                    "exprvars": []
                }
            }
        },
        "edges": [
            {
                "from": 1,
                "to": 2,
                "label": "CASE"
            },
            {
                "from": 2,
                "to": 14,
                "label": "ONLY EVAL with clause\nsublist(X11, X12) :- ','(append1(X13, X11, X14), append2(X14, X15, X12)).\nand substitutionT1 -> T9,\nX11 -> T9,\nT2 -> T10,\nX12 -> T10"
            },
            {
                "from": 14,
                "to": 35,
                "label": "SPLIT 1"
            },
            {
                "from": 14,
                "to": 36,
                "label": "SPLIT 2\nnew knowledge:\nT9 is ground\nreplacements:X13 -> T13,\nX14 -> T14"
            },
            {
                "from": 35,
                "to": 41,
                "label": "CASE"
            },
            {
                "from": 36,
                "to": 56,
                "label": "CASE"
            },
            {
                "from": 41,
                "to": 42,
                "label": "PARALLEL"
            },
            {
                "from": 41,
                "to": 43,
                "label": "PARALLEL"
            },
            {
                "from": 42,
                "to": 44,
                "label": "ONLY EVAL with clause\nappend1([], X24, X24).\nand substitutionX13 -> [],\nT9 -> T20,\nX24 -> T20,\nX14 -> T20"
            },
            {
                "from": 43,
                "to": 50,
                "label": "ONLY EVAL with clause\nappend1(.(X40, X41), X42, .(X40, X43)) :- append1(X41, X42, X43).\nand substitutionX40 -> X44,\nX41 -> X45,\nX13 -> .(X44, X45),\nT9 -> T24,\nX42 -> T24,\nX43 -> X46,\nX14 -> .(X44, X46)"
            },
            {
                "from": 44,
                "to": 45,
                "label": "SUCCESS"
            },
            {
                "from": 50,
                "to": 35,
                "label": "INSTANCE with matching:\nX13 -> X45\nT9 -> T24\nX14 -> X46"
            },
            {
                "from": 56,
                "to": 57,
                "label": "PARALLEL"
            },
            {
                "from": 56,
                "to": 58,
                "label": "PARALLEL"
            },
            {
                "from": 57,
                "to": 61,
                "label": "EVAL with clause\nappend2([], X60, X60).\nand substitutionT14 -> [],\nX15 -> T32,\nX60 -> T32,\nT10 -> T32,\nX61 -> T32"
            },
            {
                "from": 57,
                "to": 62,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 58,
                "to": 252,
                "label": "EVAL with clause\nappend2(.(X72, X73), X74, .(X72, X75)) :- append2(X73, X74, X75).\nand substitutionX72 -> T39,\nX73 -> T42,\nT14 -> .(T39, T42),\nX15 -> X76,\nX74 -> X76,\nX75 -> T41,\nT10 -> .(T39, T41),\nT40 -> T42"
            },
            {
                "from": 58,
                "to": 253,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 61,
                "to": 63,
                "label": "SUCCESS"
            },
            {
                "from": 252,
                "to": 36,
                "label": "INSTANCE with matching:\nT14 -> T42\nX15 -> X76\nT10 -> T41"
            }
        ],
        "type": "Graph"
    }
}

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

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

   f1_in(T9, T10) -> U1(f14_in(T9, T10), T9, T10)
   U1(f14_out1(X14, X15), T9, T10) -> f1_out1
   f35_in(T20) -> f35_out1
   f35_in(T24) -> U2(f35_in(T24), T24)
   U2(f35_out1, T24) -> f35_out1
   f36_in(T32) -> f36_out1([], T32)
   f36_in(.(T39, T41)) -> U3(f36_in(T41), .(T39, T41))
   U3(f36_out1(T42, X76), .(T39, T41)) -> f36_out1(.(T39, T42), X76)
   f14_in(T9, T10) -> U4(f35_in(T9), T9, T10)
   U4(f35_out1, T9, T10) -> U5(f36_in(T10), T9, T10)
   U5(f36_out1(T14, X15), T9, T10) -> f14_out1(T14, X15)

Q is empty.

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

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

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

   F1_IN(T9, T10) -> U1^1(f14_in(T9, T10), T9, T10)
   F1_IN(T9, T10) -> F14_IN(T9, T10)
   F35_IN(T24) -> U2^1(f35_in(T24), T24)
   F35_IN(T24) -> F35_IN(T24)
   F36_IN(.(T39, T41)) -> U3^1(f36_in(T41), .(T39, T41))
   F36_IN(.(T39, T41)) -> F36_IN(T41)
   F14_IN(T9, T10) -> U4^1(f35_in(T9), T9, T10)
   F14_IN(T9, T10) -> F35_IN(T9)
   U4^1(f35_out1, T9, T10) -> U5^1(f36_in(T10), T9, T10)
   U4^1(f35_out1, T9, T10) -> F36_IN(T10)

The TRS R consists of the following rules:

   f1_in(T9, T10) -> U1(f14_in(T9, T10), T9, T10)
   U1(f14_out1(X14, X15), T9, T10) -> f1_out1
   f35_in(T20) -> f35_out1
   f35_in(T24) -> U2(f35_in(T24), T24)
   U2(f35_out1, T24) -> f35_out1
   f36_in(T32) -> f36_out1([], T32)
   f36_in(.(T39, T41)) -> U3(f36_in(T41), .(T39, T41))
   U3(f36_out1(T42, X76), .(T39, T41)) -> f36_out1(.(T39, T42), X76)
   f14_in(T9, T10) -> U4(f35_in(T9), T9, T10)
   U4(f35_out1, T9, T10) -> U5(f36_in(T10), T9, T10)
   U5(f36_out1(T14, X15), T9, T10) -> f14_out1(T14, X15)

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

(41) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 8 less nodes.
----------------------------------------

(42)
Complex Obligation (AND)

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

(43)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F36_IN(.(T39, T41)) -> F36_IN(T41)

The TRS R consists of the following rules:

   f1_in(T9, T10) -> U1(f14_in(T9, T10), T9, T10)
   U1(f14_out1(X14, X15), T9, T10) -> f1_out1
   f35_in(T20) -> f35_out1
   f35_in(T24) -> U2(f35_in(T24), T24)
   U2(f35_out1, T24) -> f35_out1
   f36_in(T32) -> f36_out1([], T32)
   f36_in(.(T39, T41)) -> U3(f36_in(T41), .(T39, T41))
   U3(f36_out1(T42, X76), .(T39, T41)) -> f36_out1(.(T39, T42), X76)
   f14_in(T9, T10) -> U4(f35_in(T9), T9, T10)
   U4(f35_out1, T9, T10) -> U5(f36_in(T10), T9, T10)
   U5(f36_out1(T14, X15), T9, T10) -> f14_out1(T14, X15)

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

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

(45)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F36_IN(.(T39, T41)) -> F36_IN(T41)

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

(46) 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:
*F36_IN(.(T39, T41)) -> F36_IN(T41)
The graph contains the following edges 1 > 1


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

(47)
YES

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

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

   F35_IN(T24) -> F35_IN(T24)

The TRS R consists of the following rules:

   f1_in(T9, T10) -> U1(f14_in(T9, T10), T9, T10)
   U1(f14_out1(X14, X15), T9, T10) -> f1_out1
   f35_in(T20) -> f35_out1
   f35_in(T24) -> U2(f35_in(T24), T24)
   U2(f35_out1, T24) -> f35_out1
   f36_in(T32) -> f36_out1([], T32)
   f36_in(.(T39, T41)) -> U3(f36_in(T41), .(T39, T41))
   U3(f36_out1(T42, X76), .(T39, T41)) -> f36_out1(.(T39, T42), X76)
   f14_in(T9, T10) -> U4(f35_in(T9), T9, T10)
   U4(f35_out1, T9, T10) -> U5(f36_in(T10), T9, T10)
   U5(f36_out1(T14, X15), T9, T10) -> f14_out1(T14, X15)

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

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

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

   F35_IN(T24) -> F35_IN(T24)

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": 3,
    "program": {
        "directives": [],
        "clauses": [
            [
                "(append1 ([]) Ys Ys)",
                null
            ],
            [
                "(append1 (. X Xs) Ys (. X Zs))",
                "(append1 Xs Ys Zs)"
            ],
            [
                "(append2 ([]) Ys Ys)",
                null
            ],
            [
                "(append2 (. X Xs) Ys (. X Zs))",
                "(append2 Xs Ys Zs)"
            ],
            [
                "(sublist X Y)",
                "(',' (append1 U X V) (append2 V W Y))"
            ]
        ]
    },
    "graph": {
        "nodes": {
            "66": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (append1 X76 T37 X77) (append2 (. X75 X77) X7 T6))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T6",
                        "T37"
                    ],
                    "free": [
                        "X7",
                        "X75",
                        "X76",
                        "X77"
                    ],
                    "exprvars": []
                }
            },
            "type": "Nodes",
            "250": {
                "goal": [
                    {
                        "clause": 2,
                        "scope": 5,
                        "term": "(append2 (. X75 T40) X7 T6)"
                    },
                    {
                        "clause": 3,
                        "scope": 5,
                        "term": "(append2 (. X75 T40) X7 T6)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T6"],
                    "free": [
                        "X7",
                        "X75"
                    ],
                    "exprvars": []
                }
            },
            "251": {
                "goal": [{
                    "clause": 3,
                    "scope": 5,
                    "term": "(append2 (. X75 T40) X7 T6)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T6"],
                    "free": [
                        "X7",
                        "X75"
                    ],
                    "exprvars": []
                }
            },
            "254": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(append2 T61 X130 T60)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T60"],
                    "free": ["X130"],
                    "exprvars": []
                }
            },
            "255": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "256": {
                "goal": [
                    {
                        "clause": 2,
                        "scope": 6,
                        "term": "(append2 T61 X130 T60)"
                    },
                    {
                        "clause": 3,
                        "scope": 6,
                        "term": "(append2 T61 X130 T60)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T60"],
                    "free": ["X130"],
                    "exprvars": []
                }
            },
            "257": {
                "goal": [{
                    "clause": 2,
                    "scope": 6,
                    "term": "(append2 T61 X130 T60)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T60"],
                    "free": ["X130"],
                    "exprvars": []
                }
            },
            "258": {
                "goal": [{
                    "clause": 3,
                    "scope": 6,
                    "term": "(append2 T61 X130 T60)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T60"],
                    "free": ["X130"],
                    "exprvars": []
                }
            },
            "259": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "139": {
                "goal": [
                    {
                        "clause": 0,
                        "scope": 4,
                        "term": "(append1 X76 T37 X77)"
                    },
                    {
                        "clause": 1,
                        "scope": 4,
                        "term": "(append1 X76 T37 X77)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T37"],
                    "free": [
                        "X76",
                        "X77"
                    ],
                    "exprvars": []
                }
            },
            "53": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "54": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
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                    "exprvars": []
                }
            },
            "55": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "12": {
                "goal": [{
                    "clause": 0,
                    "scope": 2,
                    "term": "(',' (append1 X5 T5 X6) (append2 X6 X7 T6))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T5",
                        "T6"
                    ],
                    "free": [
                        "X5",
                        "X6",
                        "X7"
                    ],
                    "exprvars": []
                }
            },
            "13": {
                "goal": [{
                    "clause": 1,
                    "scope": 2,
                    "term": "(',' (append1 X5 T5 X6) (append2 X6 X7 T6))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T5",
                        "T6"
                    ],
                    "free": [
                        "X5",
                        "X6",
                        "X7"
                    ],
                    "exprvars": []
                }
            },
            "15": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(append2 T15 X7 T6)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T6",
                        "T15"
                    ],
                    "free": ["X7"],
                    "exprvars": []
                }
            },
            "59": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(append2 T30 X49 T31)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T30",
                        "T31"
                    ],
                    "free": ["X49"],
                    "exprvars": []
                }
            },
            "38": {
                "goal": [
                    {
                        "clause": 2,
                        "scope": 3,
                        "term": "(append2 T15 X7 T6)"
                    },
                    {
                        "clause": 3,
                        "scope": 3,
                        "term": "(append2 T15 X7 T6)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T6",
                        "T15"
                    ],
                    "free": ["X7"],
                    "exprvars": []
                }
            },
            "39": {
                "goal": [{
                    "clause": 2,
                    "scope": 3,
                    "term": "(append2 T15 X7 T6)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T6",
                        "T15"
                    ],
                    "free": ["X7"],
                    "exprvars": []
                }
            },
            "180": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "181": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "260": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "261": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "262": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(append2 T78 X159 T77)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T77"],
                    "free": ["X159"],
                    "exprvars": []
                }
            },
            "263": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "165": {
                "goal": [{
                    "clause": 0,
                    "scope": 4,
                    "term": "(append1 X76 T37 X77)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T37"],
                    "free": [
                        "X76",
                        "X77"
                    ],
                    "exprvars": []
                }
            },
            "3": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(sublist T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T1",
                        "T2"
                    ],
                    "free": [],
                    "exprvars": []
                }
            },
            "102": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(append1 X76 T37 X77)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T37"],
                    "free": [
                        "X76",
                        "X77"
                    ],
                    "exprvars": []
                }
            },
            "168": {
                "goal": [{
                    "clause": 1,
                    "scope": 4,
                    "term": "(append1 X76 T37 X77)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T37"],
                    "free": [
                        "X76",
                        "X77"
                    ],
                    "exprvars": []
                }
            },
            "6": {
                "goal": [{
                    "clause": 4,
                    "scope": 1,
                    "term": "(sublist T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T1",
                        "T2"
                    ],
                    "free": [],
                    "exprvars": []
                }
            },
            "105": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(append2 (. X75 T40) X7 T6)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T6"],
                    "free": [
                        "X7",
                        "X75"
                    ],
                    "exprvars": []
                }
            },
            "249": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(append1 X107 T50 X108)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T50"],
                    "free": [
                        "X107",
                        "X108"
                    ],
                    "exprvars": []
                }
            },
            "8": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (append1 X5 T5 X6) (append2 X6 X7 T6))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T5",
                        "T6"
                    ],
                    "free": [
                        "X5",
                        "X6",
                        "X7"
                    ],
                    "exprvars": []
                }
            },
            "9": {
                "goal": [
                    {
                        "clause": 0,
                        "scope": 2,
                        "term": "(',' (append1 X5 T5 X6) (append2 X6 X7 T6))"
                    },
                    {
                        "clause": 1,
                        "scope": 2,
                        "term": "(',' (append1 X5 T5 X6) (append2 X6 X7 T6))"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T5",
                        "T6"
                    ],
                    "free": [
                        "X5",
                        "X6",
                        "X7"
                    ],
                    "exprvars": []
                }
            },
            "60": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "40": {
                "goal": [{
                    "clause": 3,
                    "scope": 3,
                    "term": "(append2 T15 X7 T6)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T6",
                        "T15"
                    ],
                    "free": ["X7"],
                    "exprvars": []
                }
            }
        },
        "edges": [
            {
                "from": 3,
                "to": 6,
                "label": "CASE"
            },
            {
                "from": 6,
                "to": 8,
                "label": "ONLY EVAL with clause\nsublist(X3, X4) :- ','(append1(X5, X3, X6), append2(X6, X7, X4)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T6,\nX4 -> T6"
            },
            {
                "from": 8,
                "to": 9,
                "label": "CASE"
            },
            {
                "from": 9,
                "to": 12,
                "label": "PARALLEL"
            },
            {
                "from": 9,
                "to": 13,
                "label": "PARALLEL"
            },
            {
                "from": 12,
                "to": 15,
                "label": "ONLY EVAL with clause\nappend1([], X20, X20).\nand substitutionX5 -> [],\nT5 -> T15,\nX20 -> T15,\nX6 -> T15"
            },
            {
                "from": 13,
                "to": 66,
                "label": "ONLY EVAL with clause\nappend1(.(X71, X72), X73, .(X71, X74)) :- append1(X72, X73, X74).\nand substitutionX71 -> X75,\nX72 -> X76,\nX5 -> .(X75, X76),\nT5 -> T37,\nX73 -> T37,\nX74 -> X77,\nX6 -> .(X75, X77)"
            },
            {
                "from": 15,
                "to": 38,
                "label": "CASE"
            },
            {
                "from": 38,
                "to": 39,
                "label": "PARALLEL"
            },
            {
                "from": 38,
                "to": 40,
                "label": "PARALLEL"
            },
            {
                "from": 39,
                "to": 53,
                "label": "EVAL with clause\nappend2([], X33, X33).\nand substitutionT15 -> [],\nX7 -> T22,\nX33 -> T22,\nT6 -> T22,\nX34 -> T22"
            },
            {
                "from": 39,
                "to": 54,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 40,
                "to": 59,
                "label": "EVAL with clause\nappend2(.(X45, X46), X47, .(X45, X48)) :- append2(X46, X47, X48).\nand substitutionX45 -> T29,\nX46 -> T30,\nT15 -> .(T29, T30),\nX7 -> X49,\nX47 -> X49,\nX48 -> T31,\nT6 -> .(T29, T31)"
            },
            {
                "from": 40,
                "to": 60,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 53,
                "to": 55,
                "label": "SUCCESS"
            },
            {
                "from": 59,
                "to": 15,
                "label": "INSTANCE with matching:\nT15 -> T30\nX7 -> X49\nT6 -> T31"
            },
            {
                "from": 66,
                "to": 102,
                "label": "SPLIT 1"
            },
            {
                "from": 66,
                "to": 105,
                "label": "SPLIT 2\nnew knowledge:\nT37 is ground\nreplacements:X76 -> T39,\nX77 -> T40"
            },
            {
                "from": 102,
                "to": 139,
                "label": "CASE"
            },
            {
                "from": 105,
                "to": 250,
                "label": "CASE"
            },
            {
                "from": 139,
                "to": 165,
                "label": "PARALLEL"
            },
            {
                "from": 139,
                "to": 168,
                "label": "PARALLEL"
            },
            {
                "from": 165,
                "to": 180,
                "label": "ONLY EVAL with clause\nappend1([], X86, X86).\nand substitutionX76 -> [],\nT37 -> T46,\nX86 -> T46,\nX77 -> T46"
            },
            {
                "from": 168,
                "to": 249,
                "label": "ONLY EVAL with clause\nappend1(.(X102, X103), X104, .(X102, X105)) :- append1(X103, X104, X105).\nand substitutionX102 -> X106,\nX103 -> X107,\nX76 -> .(X106, X107),\nT37 -> T50,\nX104 -> T50,\nX105 -> X108,\nX77 -> .(X106, X108)"
            },
            {
                "from": 180,
                "to": 181,
                "label": "SUCCESS"
            },
            {
                "from": 249,
                "to": 102,
                "label": "INSTANCE with matching:\nX76 -> X107\nT37 -> T50\nX77 -> X108"
            },
            {
                "from": 250,
                "to": 251,
                "label": "BACKTRACK\nfor clause: append2([], Ys, Ys)because of non-unification"
            },
            {
                "from": 251,
                "to": 254,
                "label": "EVAL with clause\nappend2(.(X125, X126), X127, .(X125, X128)) :- append2(X126, X127, X128).\nand substitutionX75 -> T59,\nX125 -> T59,\nT40 -> T61,\nX126 -> T61,\nX7 -> X130,\nX127 -> X130,\nX129 -> T59,\nX128 -> T60,\nT6 -> .(T59, T60),\nT58 -> T61"
            },
            {
                "from": 251,
                "to": 255,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 254,
                "to": 256,
                "label": "CASE"
            },
            {
                "from": 256,
                "to": 257,
                "label": "PARALLEL"
            },
            {
                "from": 256,
                "to": 258,
                "label": "PARALLEL"
            },
            {
                "from": 257,
                "to": 259,
                "label": "EVAL with clause\nappend2([], X143, X143).\nand substitutionT61 -> [],\nX130 -> T68,\nX143 -> T68,\nT60 -> T68,\nX144 -> T68"
            },
            {
                "from": 257,
                "to": 260,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 258,
                "to": 262,
                "label": "EVAL with clause\nappend2(.(X155, X156), X157, .(X155, X158)) :- append2(X156, X157, X158).\nand substitutionX155 -> T75,\nX156 -> T78,\nT61 -> .(T75, T78),\nX130 -> X159,\nX157 -> X159,\nX158 -> T77,\nT60 -> .(T75, T77),\nT76 -> T78"
            },
            {
                "from": 258,
                "to": 263,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 259,
                "to": 261,
                "label": "SUCCESS"
            },
            {
                "from": 262,
                "to": 254,
                "label": "INSTANCE with matching:\nT61 -> T78\nX130 -> X159\nT60 -> T77"
            }
        ],
        "type": "Graph"
    }
}

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

(52)
Obligation:
Triples:

append2A(.(X1, X2), X3, .(X1, X4)) :- append2A(X2, X3, X4).
append1B(.(X1, X2), X3, .(X1, X4)) :- append1B(X2, X3, X4).
append2C(.(X1, X2), X3, .(X1, X4)) :- append2C(X2, X3, X4).
sublistD(X1, X2) :- append2A(X1, X3, X2).
sublistD(X1, X2) :- append1B(X3, X1, X4).
sublistD(X1, .(X2, X3)) :- ','(append1cB(X4, X1, X5), append2C(X5, X6, X3)).

Clauses:

append2cA([], X1, X1).
append2cA(.(X1, X2), X3, .(X1, X4)) :- append2cA(X2, X3, X4).
append1cB([], X1, X1).
append1cB(.(X1, X2), X3, .(X1, X4)) :- append1cB(X2, X3, X4).
append2cC([], X1, X1).
append2cC(.(X1, X2), X3, .(X1, X4)) :- append2cC(X2, X3, X4).

Afs:

sublistD(x1, x2)  =  sublistD(x1, x2)


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

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

sublistD_in_2: (b,b)

append2A_in_3: (b,f,b)

append1B_in_3: (f,b,f)

append1cB_in_3: (f,b,f)

append2C_in_3: (b,f,b)

Transforming TRIPLES into the following Term Rewriting System:

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

   SUBLISTD_IN_GG(X1, X2) -> U4_GG(X1, X2, append2A_in_gag(X1, X3, X2))
   SUBLISTD_IN_GG(X1, X2) -> APPEND2A_IN_GAG(X1, X3, X2)
   APPEND2A_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> U1_GAG(X1, X2, X3, X4, append2A_in_gag(X2, X3, X4))
   APPEND2A_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2A_IN_GAG(X2, X3, X4)
   SUBLISTD_IN_GG(X1, X2) -> U5_GG(X1, X2, append1B_in_aga(X3, X1, X4))
   SUBLISTD_IN_GG(X1, X2) -> APPEND1B_IN_AGA(X3, X1, X4)
   APPEND1B_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> U2_AGA(X1, X2, X3, X4, append1B_in_aga(X2, X3, X4))
   APPEND1B_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPEND1B_IN_AGA(X2, X3, X4)
   SUBLISTD_IN_GG(X1, .(X2, X3)) -> U6_GG(X1, X2, X3, append1cB_in_aga(X4, X1, X5))
   U6_GG(X1, X2, X3, append1cB_out_aga(X4, X1, X5)) -> U7_GG(X1, X2, X3, append2C_in_gag(X5, X6, X3))
   U6_GG(X1, X2, X3, append1cB_out_aga(X4, X1, X5)) -> APPEND2C_IN_GAG(X5, X6, X3)
   APPEND2C_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> U3_GAG(X1, X2, X3, X4, append2C_in_gag(X2, X3, X4))
   APPEND2C_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2C_IN_GAG(X2, X3, X4)

The TRS R consists of the following rules:

   append1cB_in_aga([], X1, X1) -> append1cB_out_aga([], X1, X1)
   append1cB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, append1cB_in_aga(X2, X3, X4))
   U10_aga(X1, X2, X3, X4, append1cB_out_aga(X2, X3, X4)) -> append1cB_out_aga(.(X1, X2), X3, .(X1, X4))

The argument filtering Pi contains the following mapping:
append2A_in_gag(x1, x2, x3)  =  append2A_in_gag(x1, x3)

.(x1, x2)  =  .(x2)

append1B_in_aga(x1, x2, x3)  =  append1B_in_aga(x2)

append1cB_in_aga(x1, x2, x3)  =  append1cB_in_aga(x2)

append1cB_out_aga(x1, x2, x3)  =  append1cB_out_aga(x1, x2, x3)

U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)

append2C_in_gag(x1, x2, x3)  =  append2C_in_gag(x1, x3)

SUBLISTD_IN_GG(x1, x2)  =  SUBLISTD_IN_GG(x1, x2)

U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)

APPEND2A_IN_GAG(x1, x2, x3)  =  APPEND2A_IN_GAG(x1, x3)

U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x2, x4, x5)

U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)

APPEND1B_IN_AGA(x1, x2, x3)  =  APPEND1B_IN_AGA(x2)

U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)

U6_GG(x1, x2, x3, x4)  =  U6_GG(x1, x3, x4)

U7_GG(x1, x2, x3, x4)  =  U7_GG(x1, x3, x4)

APPEND2C_IN_GAG(x1, x2, x3)  =  APPEND2C_IN_GAG(x1, x3)

U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x2, x4, x5)


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:

   SUBLISTD_IN_GG(X1, X2) -> U4_GG(X1, X2, append2A_in_gag(X1, X3, X2))
   SUBLISTD_IN_GG(X1, X2) -> APPEND2A_IN_GAG(X1, X3, X2)
   APPEND2A_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> U1_GAG(X1, X2, X3, X4, append2A_in_gag(X2, X3, X4))
   APPEND2A_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2A_IN_GAG(X2, X3, X4)
   SUBLISTD_IN_GG(X1, X2) -> U5_GG(X1, X2, append1B_in_aga(X3, X1, X4))
   SUBLISTD_IN_GG(X1, X2) -> APPEND1B_IN_AGA(X3, X1, X4)
   APPEND1B_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> U2_AGA(X1, X2, X3, X4, append1B_in_aga(X2, X3, X4))
   APPEND1B_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPEND1B_IN_AGA(X2, X3, X4)
   SUBLISTD_IN_GG(X1, .(X2, X3)) -> U6_GG(X1, X2, X3, append1cB_in_aga(X4, X1, X5))
   U6_GG(X1, X2, X3, append1cB_out_aga(X4, X1, X5)) -> U7_GG(X1, X2, X3, append2C_in_gag(X5, X6, X3))
   U6_GG(X1, X2, X3, append1cB_out_aga(X4, X1, X5)) -> APPEND2C_IN_GAG(X5, X6, X3)
   APPEND2C_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> U3_GAG(X1, X2, X3, X4, append2C_in_gag(X2, X3, X4))
   APPEND2C_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2C_IN_GAG(X2, X3, X4)

The TRS R consists of the following rules:

   append1cB_in_aga([], X1, X1) -> append1cB_out_aga([], X1, X1)
   append1cB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, append1cB_in_aga(X2, X3, X4))
   U10_aga(X1, X2, X3, X4, append1cB_out_aga(X2, X3, X4)) -> append1cB_out_aga(.(X1, X2), X3, .(X1, X4))

The argument filtering Pi contains the following mapping:
append2A_in_gag(x1, x2, x3)  =  append2A_in_gag(x1, x3)

.(x1, x2)  =  .(x2)

append1B_in_aga(x1, x2, x3)  =  append1B_in_aga(x2)

append1cB_in_aga(x1, x2, x3)  =  append1cB_in_aga(x2)

append1cB_out_aga(x1, x2, x3)  =  append1cB_out_aga(x1, x2, x3)

U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)

append2C_in_gag(x1, x2, x3)  =  append2C_in_gag(x1, x3)

SUBLISTD_IN_GG(x1, x2)  =  SUBLISTD_IN_GG(x1, x2)

U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)

APPEND2A_IN_GAG(x1, x2, x3)  =  APPEND2A_IN_GAG(x1, x3)

U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x2, x4, x5)

U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)

APPEND1B_IN_AGA(x1, x2, x3)  =  APPEND1B_IN_AGA(x2)

U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)

U6_GG(x1, x2, x3, x4)  =  U6_GG(x1, x3, x4)

U7_GG(x1, x2, x3, x4)  =  U7_GG(x1, x3, x4)

APPEND2C_IN_GAG(x1, x2, x3)  =  APPEND2C_IN_GAG(x1, x3)

U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x2, x4, x5)


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

(55) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 10 less nodes.
----------------------------------------

(56)
Complex Obligation (AND)

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

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

   APPEND2C_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2C_IN_GAG(X2, X3, X4)

The TRS R consists of the following rules:

   append1cB_in_aga([], X1, X1) -> append1cB_out_aga([], X1, X1)
   append1cB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, append1cB_in_aga(X2, X3, X4))
   U10_aga(X1, X2, X3, X4, append1cB_out_aga(X2, X3, X4)) -> append1cB_out_aga(.(X1, X2), X3, .(X1, X4))

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

append1cB_in_aga(x1, x2, x3)  =  append1cB_in_aga(x2)

append1cB_out_aga(x1, x2, x3)  =  append1cB_out_aga(x1, x2, x3)

U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)

APPEND2C_IN_GAG(x1, x2, x3)  =  APPEND2C_IN_GAG(x1, x3)


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

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

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

   APPEND2C_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2C_IN_GAG(X2, X3, X4)

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

APPEND2C_IN_GAG(x1, x2, x3)  =  APPEND2C_IN_GAG(x1, x3)


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

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

(61)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APPEND2C_IN_GAG(.(X2), .(X4)) -> APPEND2C_IN_GAG(X2, X4)

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

(62) 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:
*APPEND2C_IN_GAG(.(X2), .(X4)) -> APPEND2C_IN_GAG(X2, X4)
The graph contains the following edges 1 > 1, 2 > 2


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

(63)
YES

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

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

   APPEND1B_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPEND1B_IN_AGA(X2, X3, X4)

The TRS R consists of the following rules:

   append1cB_in_aga([], X1, X1) -> append1cB_out_aga([], X1, X1)
   append1cB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, append1cB_in_aga(X2, X3, X4))
   U10_aga(X1, X2, X3, X4, append1cB_out_aga(X2, X3, X4)) -> append1cB_out_aga(.(X1, X2), X3, .(X1, X4))

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

append1cB_in_aga(x1, x2, x3)  =  append1cB_in_aga(x2)

append1cB_out_aga(x1, x2, x3)  =  append1cB_out_aga(x1, x2, x3)

U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)

APPEND1B_IN_AGA(x1, x2, x3)  =  APPEND1B_IN_AGA(x2)


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

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

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

   APPEND1B_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPEND1B_IN_AGA(X2, X3, X4)

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

APPEND1B_IN_AGA(x1, x2, x3)  =  APPEND1B_IN_AGA(x2)


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

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

(68)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APPEND1B_IN_AGA(X3) -> APPEND1B_IN_AGA(X3)

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

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

   APPEND2A_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2A_IN_GAG(X2, X3, X4)

The TRS R consists of the following rules:

   append1cB_in_aga([], X1, X1) -> append1cB_out_aga([], X1, X1)
   append1cB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, append1cB_in_aga(X2, X3, X4))
   U10_aga(X1, X2, X3, X4, append1cB_out_aga(X2, X3, X4)) -> append1cB_out_aga(.(X1, X2), X3, .(X1, X4))

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

append1cB_in_aga(x1, x2, x3)  =  append1cB_in_aga(x2)

append1cB_out_aga(x1, x2, x3)  =  append1cB_out_aga(x1, x2, x3)

U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)

APPEND2A_IN_GAG(x1, x2, x3)  =  APPEND2A_IN_GAG(x1, x3)


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

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

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

   APPEND2A_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2A_IN_GAG(X2, X3, X4)

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

APPEND2A_IN_GAG(x1, x2, x3)  =  APPEND2A_IN_GAG(x1, x3)


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

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

{
    "root": 4,
    "program": {
        "directives": [],
        "clauses": [
            [
                "(append1 ([]) Ys Ys)",
                null
            ],
            [
                "(append1 (. X Xs) Ys (. X Zs))",
                "(append1 Xs Ys Zs)"
            ],
            [
                "(append2 ([]) Ys Ys)",
                null
            ],
            [
                "(append2 (. X Xs) Ys (. X Zs))",
                "(append2 Xs Ys Zs)"
            ],
            [
                "(sublist X Y)",
                "(',' (append1 U X V) (append2 V W Y))"
            ]
        ]
    },
    "graph": {
        "nodes": {
            "34": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(append1 X13 T9 X14)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T9"],
                    "free": [
                        "X13",
                        "X14"
                    ],
                    "exprvars": []
                }
            },
            "46": {
                "goal": [
                    {
                        "clause": 0,
                        "scope": 2,
                        "term": "(append1 X13 T9 X14)"
                    },
                    {
                        "clause": 1,
                        "scope": 2,
                        "term": "(append1 X13 T9 X14)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T9"],
                    "free": [
                        "X13",
                        "X14"
                    ],
                    "exprvars": []
                }
            },
            "47": {
                "goal": [{
                    "clause": 0,
                    "scope": 2,
                    "term": "(append1 X13 T9 X14)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T9"],
                    "free": [
                        "X13",
                        "X14"
                    ],
                    "exprvars": []
                }
            },
            "37": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(append2 T14 X15 T10)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T10"],
                    "free": ["X15"],
                    "exprvars": []
                }
            },
            "48": {
                "goal": [{
                    "clause": 1,
                    "scope": 2,
                    "term": "(append1 X13 T9 X14)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T9"],
                    "free": [
                        "X13",
                        "X14"
                    ],
                    "exprvars": []
                }
            },
            "49": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "192": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "type": "Nodes",
            "194": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
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                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "196": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "186": {
                "goal": [
                    {
                        "clause": 2,
                        "scope": 3,
                        "term": "(append2 T14 X15 T10)"
                    },
                    {
                        "clause": 3,
                        "scope": 3,
                        "term": "(append2 T14 X15 T10)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T10"],
                    "free": ["X15"],
                    "exprvars": []
                }
            },
            "187": {
                "goal": [{
                    "clause": 2,
                    "scope": 3,
                    "term": "(append2 T14 X15 T10)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T10"],
                    "free": ["X15"],
                    "exprvars": []
                }
            },
            "188": {
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                    "clause": 3,
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                    "term": "(append2 T14 X15 T10)"
                }],
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                    "nonunifying": [],
                    "intvars": {},
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                        "relations": []
                    },
                    "ground": ["T10"],
                    "free": ["X15"],
                    "exprvars": []
                }
            },
            "4": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(sublist T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
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                        "T2"
                    ],
                    "free": [],
                    "exprvars": []
                }
            },
            "5": {
                "goal": [{
                    "clause": 4,
                    "scope": 1,
                    "term": "(sublist T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
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                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T1",
                        "T2"
                    ],
                    "free": [],
                    "exprvars": []
                }
            },
            "7": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (append1 X13 T9 X14) (append2 X14 X15 T10))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
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                        "T10"
                    ],
                    "free": [
                        "X13",
                        "X14",
                        "X15"
                    ],
                    "exprvars": []
                }
            },
            "206": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(append2 T42 X76 T41)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T41"],
                    "free": ["X76"],
                    "exprvars": []
                }
            },
            "207": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "51": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "52": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(append1 X45 T24 X46)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T24"],
                    "free": [
                        "X45",
                        "X46"
                    ],
                    "exprvars": []
                }
            }
        },
        "edges": [
            {
                "from": 4,
                "to": 5,
                "label": "CASE"
            },
            {
                "from": 5,
                "to": 7,
                "label": "ONLY EVAL with clause\nsublist(X11, X12) :- ','(append1(X13, X11, X14), append2(X14, X15, X12)).\nand substitutionT1 -> T9,\nX11 -> T9,\nT2 -> T10,\nX12 -> T10"
            },
            {
                "from": 7,
                "to": 34,
                "label": "SPLIT 1"
            },
            {
                "from": 7,
                "to": 37,
                "label": "SPLIT 2\nnew knowledge:\nT9 is ground\nreplacements:X13 -> T13,\nX14 -> T14"
            },
            {
                "from": 34,
                "to": 46,
                "label": "CASE"
            },
            {
                "from": 37,
                "to": 186,
                "label": "CASE"
            },
            {
                "from": 46,
                "to": 47,
                "label": "PARALLEL"
            },
            {
                "from": 46,
                "to": 48,
                "label": "PARALLEL"
            },
            {
                "from": 47,
                "to": 49,
                "label": "ONLY EVAL with clause\nappend1([], X24, X24).\nand substitutionX13 -> [],\nT9 -> T20,\nX24 -> T20,\nX14 -> T20"
            },
            {
                "from": 48,
                "to": 52,
                "label": "ONLY EVAL with clause\nappend1(.(X40, X41), X42, .(X40, X43)) :- append1(X41, X42, X43).\nand substitutionX40 -> X44,\nX41 -> X45,\nX13 -> .(X44, X45),\nT9 -> T24,\nX42 -> T24,\nX43 -> X46,\nX14 -> .(X44, X46)"
            },
            {
                "from": 49,
                "to": 51,
                "label": "SUCCESS"
            },
            {
                "from": 52,
                "to": 34,
                "label": "INSTANCE with matching:\nX13 -> X45\nT9 -> T24\nX14 -> X46"
            },
            {
                "from": 186,
                "to": 187,
                "label": "PARALLEL"
            },
            {
                "from": 186,
                "to": 188,
                "label": "PARALLEL"
            },
            {
                "from": 187,
                "to": 192,
                "label": "EVAL with clause\nappend2([], X60, X60).\nand substitutionT14 -> [],\nX15 -> T32,\nX60 -> T32,\nT10 -> T32,\nX61 -> T32"
            },
            {
                "from": 187,
                "to": 194,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 188,
                "to": 206,
                "label": "EVAL with clause\nappend2(.(X72, X73), X74, .(X72, X75)) :- append2(X73, X74, X75).\nand substitutionX72 -> T39,\nX73 -> T42,\nT14 -> .(T39, T42),\nX15 -> X76,\nX74 -> X76,\nX75 -> T41,\nT10 -> .(T39, T41),\nT40 -> T42"
            },
            {
                "from": 188,
                "to": 207,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 192,
                "to": 196,
                "label": "SUCCESS"
            },
            {
                "from": 206,
                "to": 37,
                "label": "INSTANCE with matching:\nT14 -> T42\nX15 -> X76\nT10 -> T41"
            }
        ],
        "type": "Graph"
    }
}

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

(73)
Complex Obligation (AND)

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

(74)
Obligation:
Rules:
f206_in(T41) -> f37_in(T41) :|: TRUE
f37_out(x) -> f206_out(x) :|: TRUE
f186_in(T10) -> f188_in(T10) :|: TRUE
f187_out(x1) -> f186_out(x1) :|: TRUE
f186_in(x2) -> f187_in(x2) :|: TRUE
f188_out(x3) -> f186_out(x3) :|: TRUE
f207_out -> f188_out(x4) :|: TRUE
f188_in(x5) -> f207_in :|: TRUE
f188_in(.(x6, x7)) -> f206_in(x7) :|: TRUE
f206_out(x8) -> f188_out(.(x9, x8)) :|: TRUE
f186_out(x10) -> f37_out(x10) :|: TRUE
f37_in(x11) -> f186_in(x11) :|: TRUE
f4_in(T1, T2) -> f5_in(T1, T2) :|: TRUE
f5_out(x12, x13) -> f4_out(x12, x13) :|: TRUE
f5_in(x14, x15) -> f7_in(x14, x15) :|: TRUE
f7_out(x16, x17) -> f5_out(x16, x17) :|: TRUE
f7_in(x18, x19) -> f34_in(x18) :|: TRUE
f34_out(x20) -> f37_in(x21) :|: TRUE
f37_out(x22) -> f7_out(x23, x22) :|: TRUE
Start term: f4_in(T1, T2)

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

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

Simplified to the following IRSwTs:


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

(76)
TRUE

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

(77)
Obligation:
Rules:
f52_in(T24) -> f34_in(T24) :|: TRUE
f34_out(x) -> f52_out(x) :|: TRUE
f34_in(T9) -> f46_in(T9) :|: TRUE
f46_out(x1) -> f34_out(x1) :|: TRUE
f48_in(x2) -> f52_in(x2) :|: TRUE
f52_out(x3) -> f48_out(x3) :|: TRUE
f46_in(x4) -> f48_in(x4) :|: TRUE
f46_in(x5) -> f47_in(x5) :|: TRUE
f48_out(x6) -> f46_out(x6) :|: TRUE
f47_out(x7) -> f46_out(x7) :|: TRUE
f4_in(T1, T2) -> f5_in(T1, T2) :|: TRUE
f5_out(x8, x9) -> f4_out(x8, x9) :|: TRUE
f5_in(x10, x11) -> f7_in(x10, x11) :|: TRUE
f7_out(x12, x13) -> f5_out(x12, x13) :|: TRUE
f7_in(x14, x15) -> f34_in(x14) :|: TRUE
f34_out(x16) -> f37_in(x17) :|: TRUE
f37_out(x18) -> f7_out(x19, x18) :|: TRUE
Start term: f4_in(T1, T2)

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

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

Simplified to the following IRSwTs:

intTRSProblem:
f52_in(T24) -> f34_in(T24) :|: TRUE
f34_in(T9) -> f46_in(T9) :|: TRUE
f48_in(x2) -> f52_in(x2) :|: TRUE
f46_in(x4) -> f48_in(x4) :|: TRUE


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

(79)
Obligation:
Rules:
f52_in(T24) -> f34_in(T24) :|: TRUE
f34_in(T9) -> f46_in(T9) :|: TRUE
f48_in(x2) -> f52_in(x2) :|: TRUE
f46_in(x4) -> f48_in(x4) :|: TRUE

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

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

(81)
Obligation:
Rules:
f48_in(x2:0) -> f48_in(x2:0) :|: TRUE

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

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

(83)
Obligation:
Rules:
f48_in(x2:0) -> f48_in(x2:0) :|: TRUE

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

(84) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f48_in(x2:0) -> f48_in(x2:0) :|: TRUE

Arcs:
(1) -> (1)

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

(85)
Obligation:

Termination digraph:
Nodes:
(1) f48_in(x2:0) -> f48_in(x2:0) :|: TRUE

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(86) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f48_in(VARIABLE)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(87)
Obligation:
Rules:
f48_in(x2:0) -> f48_in(x2:0) :|: TRUE

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

(88) IntTRSNonPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x2:0) -> f(1, x2:0) :|: pc = 1 && TRUE
Proved unsatisfiability of the following formula, indicating that the system is never left after entering:
(((run2_0 = ((1 * 1)) and run2_1 = ((run1_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 run2_1 = ((run1_1 * 1))) and (((run1_0 * 1)) = ((1 * 1)) and T))

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

(89)
NO