Left Termination of the query pattern goal_in_1(g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
↳3 PrologToPiTRSProof (⇐)
↳4 PiTRS
↳5 DependencyPairsProof (⇔)
↳6 PiDP
↳7 DependencyGraphProof (⇔)
↳8 AND
    ↳9 PiDP
        ↳10 UsableRulesProof (⇔)
        ↳11 PiDP
        ↳12 PiDPToQDPProof (⇐)
        ↳13 QDP
        ↳14 QDPSizeChangeProof (⇔)
        ↳15 YES
    ↳16 PiDP
        ↳17 UsableRulesProof (⇔)
        ↳18 PiDP
        ↳19 PiDPToQDPProof (⇐)
        ↳20 QDP
        ↳21 QDPSizeChangeProof (⇔)
        ↳22 YES

(0) Obligation:

Clauses:

append([], XS, XS).
append(.(X, XS), YS, .(X, ZS)) :- append(XS, YS, ZS).
s2l(s(X), .(Y, Xs)) :- s2l(X, Xs).
s2l(0, []).
goal(X) :- ','(s2l(X, XS), append(XS, YS, ZS)).

Queries:

goal(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

s2l9(s(T16), .(X67, X68)) :- s2l9(T16, X68).
s2l9(0, []).
append21([], X125, X125).
append21(.(T28, T30), X143, .(T28, X144)) :- append21(T30, X143, X144).
goal1(s(T8)) :- s2l9(T8, X30).
goal1(s(T8)) :- ','(s2l9(T8, T23), append21(T23, X110, X111)).
goal1(0).

Queries:

goal1(g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal1_in: (b)
s2l9_in: (b,f)
append21_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal1_in_g(s(T8)) → U3_g(T8, s2l9_in_ga(T8, X30))
s2l9_in_ga(s(T16), .(X67, X68)) → U1_ga(T16, X67, X68, s2l9_in_ga(T16, X68))
s2l9_in_ga(0, []) → s2l9_out_ga(0, [])
U1_ga(T16, X67, X68, s2l9_out_ga(T16, X68)) → s2l9_out_ga(s(T16), .(X67, X68))
U3_g(T8, s2l9_out_ga(T8, X30)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U4_g(T8, s2l9_in_ga(T8, T23))
U4_g(T8, s2l9_out_ga(T8, T23)) → U5_g(T8, append21_in_gaa(T23, X110, X111))
append21_in_gaa([], X125, X125) → append21_out_gaa([], X125, X125)
append21_in_gaa(.(T28, T30), X143, .(T28, X144)) → U2_gaa(T28, T30, X143, X144, append21_in_gaa(T30, X143, X144))
U2_gaa(T28, T30, X143, X144, append21_out_gaa(T30, X143, X144)) → append21_out_gaa(.(T28, T30), X143, .(T28, X144))
U5_g(T8, append21_out_gaa(T23, X110, X111)) → goal1_out_g(s(T8))
goal1_in_g(0) → goal1_out_g(0)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
s2l9_in_ga(x1, x2)  =  s2l9_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
s2l9_out_ga(x1, x2)  =  s2l9_out_ga(x2)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
append21_in_gaa(x1, x2, x3)  =  append21_in_gaa(x1)
[]  =  []
append21_out_gaa(x1, x2, x3)  =  append21_out_gaa
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

goal1_in_g(s(T8)) → U3_g(T8, s2l9_in_ga(T8, X30))
s2l9_in_ga(s(T16), .(X67, X68)) → U1_ga(T16, X67, X68, s2l9_in_ga(T16, X68))
s2l9_in_ga(0, []) → s2l9_out_ga(0, [])
U1_ga(T16, X67, X68, s2l9_out_ga(T16, X68)) → s2l9_out_ga(s(T16), .(X67, X68))
U3_g(T8, s2l9_out_ga(T8, X30)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U4_g(T8, s2l9_in_ga(T8, T23))
U4_g(T8, s2l9_out_ga(T8, T23)) → U5_g(T8, append21_in_gaa(T23, X110, X111))
append21_in_gaa([], X125, X125) → append21_out_gaa([], X125, X125)
append21_in_gaa(.(T28, T30), X143, .(T28, X144)) → U2_gaa(T28, T30, X143, X144, append21_in_gaa(T30, X143, X144))
U2_gaa(T28, T30, X143, X144, append21_out_gaa(T30, X143, X144)) → append21_out_gaa(.(T28, T30), X143, .(T28, X144))
U5_g(T8, append21_out_gaa(T23, X110, X111)) → goal1_out_g(s(T8))
goal1_in_g(0) → goal1_out_g(0)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
s2l9_in_ga(x1, x2)  =  s2l9_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
s2l9_out_ga(x1, x2)  =  s2l9_out_ga(x2)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
append21_in_gaa(x1, x2, x3)  =  append21_in_gaa(x1)
[]  =  []
append21_out_gaa(x1, x2, x3)  =  append21_out_gaa
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

GOAL1_IN_G(s(T8)) → U3_G(T8, s2l9_in_ga(T8, X30))
GOAL1_IN_G(s(T8)) → S2L9_IN_GA(T8, X30)
S2L9_IN_GA(s(T16), .(X67, X68)) → U1_GA(T16, X67, X68, s2l9_in_ga(T16, X68))
S2L9_IN_GA(s(T16), .(X67, X68)) → S2L9_IN_GA(T16, X68)
GOAL1_IN_G(s(T8)) → U4_G(T8, s2l9_in_ga(T8, T23))
U4_G(T8, s2l9_out_ga(T8, T23)) → U5_G(T8, append21_in_gaa(T23, X110, X111))
U4_G(T8, s2l9_out_ga(T8, T23)) → APPEND21_IN_GAA(T23, X110, X111)
APPEND21_IN_GAA(.(T28, T30), X143, .(T28, X144)) → U2_GAA(T28, T30, X143, X144, append21_in_gaa(T30, X143, X144))
APPEND21_IN_GAA(.(T28, T30), X143, .(T28, X144)) → APPEND21_IN_GAA(T30, X143, X144)

The TRS R consists of the following rules:

goal1_in_g(s(T8)) → U3_g(T8, s2l9_in_ga(T8, X30))
s2l9_in_ga(s(T16), .(X67, X68)) → U1_ga(T16, X67, X68, s2l9_in_ga(T16, X68))
s2l9_in_ga(0, []) → s2l9_out_ga(0, [])
U1_ga(T16, X67, X68, s2l9_out_ga(T16, X68)) → s2l9_out_ga(s(T16), .(X67, X68))
U3_g(T8, s2l9_out_ga(T8, X30)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U4_g(T8, s2l9_in_ga(T8, T23))
U4_g(T8, s2l9_out_ga(T8, T23)) → U5_g(T8, append21_in_gaa(T23, X110, X111))
append21_in_gaa([], X125, X125) → append21_out_gaa([], X125, X125)
append21_in_gaa(.(T28, T30), X143, .(T28, X144)) → U2_gaa(T28, T30, X143, X144, append21_in_gaa(T30, X143, X144))
U2_gaa(T28, T30, X143, X144, append21_out_gaa(T30, X143, X144)) → append21_out_gaa(.(T28, T30), X143, .(T28, X144))
U5_g(T8, append21_out_gaa(T23, X110, X111)) → goal1_out_g(s(T8))
goal1_in_g(0) → goal1_out_g(0)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
s2l9_in_ga(x1, x2)  =  s2l9_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
s2l9_out_ga(x1, x2)  =  s2l9_out_ga(x2)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
append21_in_gaa(x1, x2, x3)  =  append21_in_gaa(x1)
[]  =  []
append21_out_gaa(x1, x2, x3)  =  append21_out_gaa
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x2)
S2L9_IN_GA(x1, x2)  =  S2L9_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U4_G(x1, x2)  =  U4_G(x2)
U5_G(x1, x2)  =  U5_G(x2)
APPEND21_IN_GAA(x1, x2, x3)  =  APPEND21_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x5)

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

(6) Obligation:

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

GOAL1_IN_G(s(T8)) → U3_G(T8, s2l9_in_ga(T8, X30))
GOAL1_IN_G(s(T8)) → S2L9_IN_GA(T8, X30)
S2L9_IN_GA(s(T16), .(X67, X68)) → U1_GA(T16, X67, X68, s2l9_in_ga(T16, X68))
S2L9_IN_GA(s(T16), .(X67, X68)) → S2L9_IN_GA(T16, X68)
GOAL1_IN_G(s(T8)) → U4_G(T8, s2l9_in_ga(T8, T23))
U4_G(T8, s2l9_out_ga(T8, T23)) → U5_G(T8, append21_in_gaa(T23, X110, X111))
U4_G(T8, s2l9_out_ga(T8, T23)) → APPEND21_IN_GAA(T23, X110, X111)
APPEND21_IN_GAA(.(T28, T30), X143, .(T28, X144)) → U2_GAA(T28, T30, X143, X144, append21_in_gaa(T30, X143, X144))
APPEND21_IN_GAA(.(T28, T30), X143, .(T28, X144)) → APPEND21_IN_GAA(T30, X143, X144)

The TRS R consists of the following rules:

goal1_in_g(s(T8)) → U3_g(T8, s2l9_in_ga(T8, X30))
s2l9_in_ga(s(T16), .(X67, X68)) → U1_ga(T16, X67, X68, s2l9_in_ga(T16, X68))
s2l9_in_ga(0, []) → s2l9_out_ga(0, [])
U1_ga(T16, X67, X68, s2l9_out_ga(T16, X68)) → s2l9_out_ga(s(T16), .(X67, X68))
U3_g(T8, s2l9_out_ga(T8, X30)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U4_g(T8, s2l9_in_ga(T8, T23))
U4_g(T8, s2l9_out_ga(T8, T23)) → U5_g(T8, append21_in_gaa(T23, X110, X111))
append21_in_gaa([], X125, X125) → append21_out_gaa([], X125, X125)
append21_in_gaa(.(T28, T30), X143, .(T28, X144)) → U2_gaa(T28, T30, X143, X144, append21_in_gaa(T30, X143, X144))
U2_gaa(T28, T30, X143, X144, append21_out_gaa(T30, X143, X144)) → append21_out_gaa(.(T28, T30), X143, .(T28, X144))
U5_g(T8, append21_out_gaa(T23, X110, X111)) → goal1_out_g(s(T8))
goal1_in_g(0) → goal1_out_g(0)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
s2l9_in_ga(x1, x2)  =  s2l9_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
s2l9_out_ga(x1, x2)  =  s2l9_out_ga(x2)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
append21_in_gaa(x1, x2, x3)  =  append21_in_gaa(x1)
[]  =  []
append21_out_gaa(x1, x2, x3)  =  append21_out_gaa
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x2)
S2L9_IN_GA(x1, x2)  =  S2L9_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U4_G(x1, x2)  =  U4_G(x2)
U5_G(x1, x2)  =  U5_G(x2)
APPEND21_IN_GAA(x1, x2, x3)  =  APPEND21_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(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:

APPEND21_IN_GAA(.(T28, T30), X143, .(T28, X144)) → APPEND21_IN_GAA(T30, X143, X144)

The TRS R consists of the following rules:

goal1_in_g(s(T8)) → U3_g(T8, s2l9_in_ga(T8, X30))
s2l9_in_ga(s(T16), .(X67, X68)) → U1_ga(T16, X67, X68, s2l9_in_ga(T16, X68))
s2l9_in_ga(0, []) → s2l9_out_ga(0, [])
U1_ga(T16, X67, X68, s2l9_out_ga(T16, X68)) → s2l9_out_ga(s(T16), .(X67, X68))
U3_g(T8, s2l9_out_ga(T8, X30)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U4_g(T8, s2l9_in_ga(T8, T23))
U4_g(T8, s2l9_out_ga(T8, T23)) → U5_g(T8, append21_in_gaa(T23, X110, X111))
append21_in_gaa([], X125, X125) → append21_out_gaa([], X125, X125)
append21_in_gaa(.(T28, T30), X143, .(T28, X144)) → U2_gaa(T28, T30, X143, X144, append21_in_gaa(T30, X143, X144))
U2_gaa(T28, T30, X143, X144, append21_out_gaa(T30, X143, X144)) → append21_out_gaa(.(T28, T30), X143, .(T28, X144))
U5_g(T8, append21_out_gaa(T23, X110, X111)) → goal1_out_g(s(T8))
goal1_in_g(0) → goal1_out_g(0)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
s2l9_in_ga(x1, x2)  =  s2l9_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
s2l9_out_ga(x1, x2)  =  s2l9_out_ga(x2)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
append21_in_gaa(x1, x2, x3)  =  append21_in_gaa(x1)
[]  =  []
append21_out_gaa(x1, x2, x3)  =  append21_out_gaa
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
APPEND21_IN_GAA(x1, x2, x3)  =  APPEND21_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:

APPEND21_IN_GAA(.(T28, T30), X143, .(T28, X144)) → APPEND21_IN_GAA(T30, X143, X144)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND21_IN_GAA(x1, x2, x3)  =  APPEND21_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:

APPEND21_IN_GAA(.(T30)) → APPEND21_IN_GAA(T30)

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:

  • APPEND21_IN_GAA(.(T30)) → APPEND21_IN_GAA(T30)
    The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

S2L9_IN_GA(s(T16), .(X67, X68)) → S2L9_IN_GA(T16, X68)

The TRS R consists of the following rules:

goal1_in_g(s(T8)) → U3_g(T8, s2l9_in_ga(T8, X30))
s2l9_in_ga(s(T16), .(X67, X68)) → U1_ga(T16, X67, X68, s2l9_in_ga(T16, X68))
s2l9_in_ga(0, []) → s2l9_out_ga(0, [])
U1_ga(T16, X67, X68, s2l9_out_ga(T16, X68)) → s2l9_out_ga(s(T16), .(X67, X68))
U3_g(T8, s2l9_out_ga(T8, X30)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U4_g(T8, s2l9_in_ga(T8, T23))
U4_g(T8, s2l9_out_ga(T8, T23)) → U5_g(T8, append21_in_gaa(T23, X110, X111))
append21_in_gaa([], X125, X125) → append21_out_gaa([], X125, X125)
append21_in_gaa(.(T28, T30), X143, .(T28, X144)) → U2_gaa(T28, T30, X143, X144, append21_in_gaa(T30, X143, X144))
U2_gaa(T28, T30, X143, X144, append21_out_gaa(T30, X143, X144)) → append21_out_gaa(.(T28, T30), X143, .(T28, X144))
U5_g(T8, append21_out_gaa(T23, X110, X111)) → goal1_out_g(s(T8))
goal1_in_g(0) → goal1_out_g(0)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
s2l9_in_ga(x1, x2)  =  s2l9_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
s2l9_out_ga(x1, x2)  =  s2l9_out_ga(x2)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
append21_in_gaa(x1, x2, x3)  =  append21_in_gaa(x1)
[]  =  []
append21_out_gaa(x1, x2, x3)  =  append21_out_gaa
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
S2L9_IN_GA(x1, x2)  =  S2L9_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:

S2L9_IN_GA(s(T16), .(X67, X68)) → S2L9_IN_GA(T16, X68)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2L9_IN_GA(x1, x2)  =  S2L9_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:

S2L9_IN_GA(s(T16)) → S2L9_IN_GA(T16)

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:

  • S2L9_IN_GA(s(T16)) → S2L9_IN_GA(T16)
    The graph contains the following edges 1 > 1

(22) YES