Left Termination of the query pattern
goal(g)
w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇒, 188 ms)
↳2 TRIPLES
↳3 UndefinedPredicateInTriplesTransformerProof (⇒, 0 ms)
↳4 TRIPLES
↳5 TriplesToPiDPProof (⇒, 0 ms)
↳6 PiDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 AND
    ↳9 PiDP
        ↳10 UsableRulesProof (⇔, 0 ms)
        ↳11 PiDP
        ↳12 PiDPToQDPProof (⇒, 12 ms)
        ↳13 QDP
        ↳14 QDPSizeChangeProof (⇔, 0 ms)
        ↳15 YES
    ↳16 PiDP
        ↳17 UsableRulesProof (⇔, 0 ms)
        ↳18 PiDP
        ↳19 PiDPToQDPProof (⇒, 0 ms)
        ↳20 QDP
        ↳21 QDPSizeChangeProof (⇔, 0 ms)
        ↳22 YES

(0) Obligation:

Clauses:

goal(X) :- ','(s2l(X, Xs), append(Xs, X1, X2)).
append([], Y, Z) :- ','(!, eq(Y, Z)).
append(X, Y, .(H, Z)) :- ','(head(X, H), ','(tail(X, T), append(T, Y, Z))).
s2l(0, L) :- ','(!, eq(L, [])).
s2l(X, .(X3, Xs)) :- ','(p(X, P), s2l(P, Xs)).
head([], X4).
head(.(H, X5), H).
tail([], []).
tail(.(X6, Xs), Xs).
p(0, 0).
p(s(X), X).
eq(X, X).

Query: goal(g)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

s2lA(s(X1), .(X2, X3)) :- s2lA(X1, X3).
appendC([], X1, .(X2, X3)) :- appendB(X1, X3).
appendC(.(X1, X2), X3, .(X1, X4)) :- appendC(X2, X3, X4).
goalD(0) :- appendB(X1, X2).
goalD(s(X1)) :- s2lA(X1, X2).
goalD(s(X1)) :- ','(s2lcA(X1, X2), appendC(X2, X3, X4)).

Clauses:

s2lcA(0, []).
s2lcA(s(X1), .(X2, X3)) :- s2lcA(X1, X3).
appendcB(X1, X1).
appendcC([], X1, X1).
appendcC([], X1, .(X2, X3)) :- appendcB(X1, X3).
appendcC(.(X1, X2), X3, .(X1, X4)) :- appendcC(X2, X3, X4).

Afs:

goalD(x1)  =  goalD(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [DT09].

(4) Obligation:

Triples:

s2lA(s(X1), .(X2, X3)) :- s2lA(X1, X3).
appendC(.(X1, X2), X3, .(X1, X4)) :- appendC(X2, X3, X4).
goalD(s(X1)) :- s2lA(X1, X2).
goalD(s(X1)) :- ','(s2lcA(X1, X2), appendC(X2, X3, X4)).

Clauses:

s2lcA(0, []).
s2lcA(s(X1), .(X2, X3)) :- s2lcA(X1, X3).
appendcB(X1, X1).
appendcC([], X1, X1).
appendcC([], X1, .(X2, X3)) :- appendcB(X1, X3).
appendcC(.(X1, X2), X3, .(X1, X4)) :- appendcC(X2, X3, X4).

Afs:

goalD(x1)  =  goalD(x1)

(5) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
goalD_in: (b)
s2lA_in: (b,f)
s2lcA_in: (b,f)
appendC_in: (b,f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

GOALD_IN_G(s(X1)) → U3_G(X1, s2lA_in_ga(X1, X2))
GOALD_IN_G(s(X1)) → S2LA_IN_GA(X1, X2)
S2LA_IN_GA(s(X1), .(X2, X3)) → U1_GA(X1, X2, X3, s2lA_in_ga(X1, X3))
S2LA_IN_GA(s(X1), .(X2, X3)) → S2LA_IN_GA(X1, X3)
GOALD_IN_G(s(X1)) → U4_G(X1, s2lcA_in_ga(X1, X2))
U4_G(X1, s2lcA_out_ga(X1, X2)) → U5_G(X1, appendC_in_gaa(X2, X3, X4))
U4_G(X1, s2lcA_out_ga(X1, X2)) → APPENDC_IN_GAA(X2, X3, X4)
APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) → U2_GAA(X1, X2, X3, X4, appendC_in_gaa(X2, X3, X4))
APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDC_IN_GAA(X2, X3, X4)

The TRS R consists of the following rules:

s2lcA_in_ga(0, []) → s2lcA_out_ga(0, [])
s2lcA_in_ga(s(X1), .(X2, X3)) → U7_ga(X1, X2, X3, s2lcA_in_ga(X1, X3))
U7_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) → s2lcA_out_ga(s(X1), .(X2, X3))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
s2lA_in_ga(x1, x2)  =  s2lA_in_ga(x1)
.(x1, x2)  =  .(x2)
s2lcA_in_ga(x1, x2)  =  s2lcA_in_ga(x1)
0  =  0
s2lcA_out_ga(x1, x2)  =  s2lcA_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
appendC_in_gaa(x1, x2, x3)  =  appendC_in_gaa(x1)
GOALD_IN_G(x1)  =  GOALD_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)
S2LA_IN_GA(x1, x2)  =  S2LA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U4_G(x1, x2)  =  U4_G(x1, x2)
U5_G(x1, x2)  =  U5_G(x1, x2)
APPENDC_IN_GAA(x1, x2, x3)  =  APPENDC_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x2, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(6) Obligation:

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

GOALD_IN_G(s(X1)) → U3_G(X1, s2lA_in_ga(X1, X2))
GOALD_IN_G(s(X1)) → S2LA_IN_GA(X1, X2)
S2LA_IN_GA(s(X1), .(X2, X3)) → U1_GA(X1, X2, X3, s2lA_in_ga(X1, X3))
S2LA_IN_GA(s(X1), .(X2, X3)) → S2LA_IN_GA(X1, X3)
GOALD_IN_G(s(X1)) → U4_G(X1, s2lcA_in_ga(X1, X2))
U4_G(X1, s2lcA_out_ga(X1, X2)) → U5_G(X1, appendC_in_gaa(X2, X3, X4))
U4_G(X1, s2lcA_out_ga(X1, X2)) → APPENDC_IN_GAA(X2, X3, X4)
APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) → U2_GAA(X1, X2, X3, X4, appendC_in_gaa(X2, X3, X4))
APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDC_IN_GAA(X2, X3, X4)

The TRS R consists of the following rules:

s2lcA_in_ga(0, []) → s2lcA_out_ga(0, [])
s2lcA_in_ga(s(X1), .(X2, X3)) → U7_ga(X1, X2, X3, s2lcA_in_ga(X1, X3))
U7_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) → s2lcA_out_ga(s(X1), .(X2, X3))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
s2lA_in_ga(x1, x2)  =  s2lA_in_ga(x1)
.(x1, x2)  =  .(x2)
s2lcA_in_ga(x1, x2)  =  s2lcA_in_ga(x1)
0  =  0
s2lcA_out_ga(x1, x2)  =  s2lcA_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
appendC_in_gaa(x1, x2, x3)  =  appendC_in_gaa(x1)
GOALD_IN_G(x1)  =  GOALD_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)
S2LA_IN_GA(x1, x2)  =  S2LA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U4_G(x1, x2)  =  U4_G(x1, x2)
U5_G(x1, x2)  =  U5_G(x1, x2)
APPENDC_IN_GAA(x1, x2, x3)  =  APPENDC_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x2, x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 7 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDC_IN_GAA(X2, X3, X4)

The TRS R consists of the following rules:

s2lcA_in_ga(0, []) → s2lcA_out_ga(0, [])
s2lcA_in_ga(s(X1), .(X2, X3)) → U7_ga(X1, X2, X3, s2lcA_in_ga(X1, X3))
U7_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) → s2lcA_out_ga(s(X1), .(X2, X3))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
s2lcA_in_ga(x1, x2)  =  s2lcA_in_ga(x1)
0  =  0
s2lcA_out_ga(x1, x2)  =  s2lcA_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
APPENDC_IN_GAA(x1, x2, x3)  =  APPENDC_IN_GAA(x1)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDC_IN_GAA(X2, X3, X4)

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

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

APPENDC_IN_GAA(.(X2)) → APPENDC_IN_GAA(X2)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • APPENDC_IN_GAA(.(X2)) → APPENDC_IN_GAA(X2)
    The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

S2LA_IN_GA(s(X1), .(X2, X3)) → S2LA_IN_GA(X1, X3)

The TRS R consists of the following rules:

s2lcA_in_ga(0, []) → s2lcA_out_ga(0, [])
s2lcA_in_ga(s(X1), .(X2, X3)) → U7_ga(X1, X2, X3, s2lcA_in_ga(X1, X3))
U7_ga(X1, X2, X3, s2lcA_out_ga(X1, X3)) → s2lcA_out_ga(s(X1), .(X2, X3))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
s2lcA_in_ga(x1, x2)  =  s2lcA_in_ga(x1)
0  =  0
s2lcA_out_ga(x1, x2)  =  s2lcA_out_ga(x1, x2)
U7_ga(x1, x2, x3, x4)  =  U7_ga(x1, x4)
S2LA_IN_GA(x1, x2)  =  S2LA_IN_GA(x1)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

S2LA_IN_GA(s(X1), .(X2, X3)) → S2LA_IN_GA(X1, X3)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2LA_IN_GA(x1, x2)  =  S2LA_IN_GA(x1)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

S2LA_IN_GA(s(X1)) → S2LA_IN_GA(X1)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • S2LA_IN_GA(s(X1)) → S2LA_IN_GA(X1)
    The graph contains the following edges 1 > 1

(22) YES