Left Termination of the query pattern fl_in_3(g, a, 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 PiDP
↳9 UsableRulesProof (⇔)
↳10 PiDP
↳11 PiDPToQDPProof (⇐)
↳12 QDP
↳13 QDPSizeChangeProof (⇔)
↳14 YES

(0) Obligation:

Clauses:

fl([], [], 0).
fl(.(E, X), R, s(Z)) :- ','(append(E, Y, R), fl(X, Y, Z)).
append([], X, X).
append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs).

Queries:

fl(g,a,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

p7([], T18, T18, T9, T11) :- fl1(T9, T18, T11).
p7(.(T25, T26), X43, .(T25, T28), T9, T11) :- p7(T26, X43, T28, T9, T11).
fl1([], [], 0).
fl1(.(T8, T9), T12, s(T11)) :- p7(T8, X13, T12, T9, T11).

Queries:

fl1(g,a,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:
fl1_in: (b,f,b)
p7_in: (b,f,f,b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

fl1_in_gag([], [], 0) → fl1_out_gag([], [], 0)
fl1_in_gag(.(T8, T9), T12, s(T11)) → U3_gag(T8, T9, T12, T11, p7_in_gaagg(T8, X13, T12, T9, T11))
p7_in_gaagg([], T18, T18, T9, T11) → U1_gaagg(T18, T9, T11, fl1_in_gag(T9, T18, T11))
U1_gaagg(T18, T9, T11, fl1_out_gag(T9, T18, T11)) → p7_out_gaagg([], T18, T18, T9, T11)
p7_in_gaagg(.(T25, T26), X43, .(T25, T28), T9, T11) → U2_gaagg(T25, T26, X43, T28, T9, T11, p7_in_gaagg(T26, X43, T28, T9, T11))
U2_gaagg(T25, T26, X43, T28, T9, T11, p7_out_gaagg(T26, X43, T28, T9, T11)) → p7_out_gaagg(.(T25, T26), X43, .(T25, T28), T9, T11)
U3_gag(T8, T9, T12, T11, p7_out_gaagg(T8, X13, T12, T9, T11)) → fl1_out_gag(.(T8, T9), T12, s(T11))

The argument filtering Pi contains the following mapping:
fl1_in_gag(x1, x2, x3)  =  fl1_in_gag(x1, x3)
[]  =  []
0  =  0
fl1_out_gag(x1, x2, x3)  =  fl1_out_gag(x2)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
p7_in_gaagg(x1, x2, x3, x4, x5)  =  p7_in_gaagg(x1, x4, x5)
U1_gaagg(x1, x2, x3, x4)  =  U1_gaagg(x4)
p7_out_gaagg(x1, x2, x3, x4, x5)  =  p7_out_gaagg(x2, x3)
U2_gaagg(x1, x2, x3, x4, x5, x6, x7)  =  U2_gaagg(x1, x7)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

fl1_in_gag([], [], 0) → fl1_out_gag([], [], 0)
fl1_in_gag(.(T8, T9), T12, s(T11)) → U3_gag(T8, T9, T12, T11, p7_in_gaagg(T8, X13, T12, T9, T11))
p7_in_gaagg([], T18, T18, T9, T11) → U1_gaagg(T18, T9, T11, fl1_in_gag(T9, T18, T11))
U1_gaagg(T18, T9, T11, fl1_out_gag(T9, T18, T11)) → p7_out_gaagg([], T18, T18, T9, T11)
p7_in_gaagg(.(T25, T26), X43, .(T25, T28), T9, T11) → U2_gaagg(T25, T26, X43, T28, T9, T11, p7_in_gaagg(T26, X43, T28, T9, T11))
U2_gaagg(T25, T26, X43, T28, T9, T11, p7_out_gaagg(T26, X43, T28, T9, T11)) → p7_out_gaagg(.(T25, T26), X43, .(T25, T28), T9, T11)
U3_gag(T8, T9, T12, T11, p7_out_gaagg(T8, X13, T12, T9, T11)) → fl1_out_gag(.(T8, T9), T12, s(T11))

The argument filtering Pi contains the following mapping:
fl1_in_gag(x1, x2, x3)  =  fl1_in_gag(x1, x3)
[]  =  []
0  =  0
fl1_out_gag(x1, x2, x3)  =  fl1_out_gag(x2)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
p7_in_gaagg(x1, x2, x3, x4, x5)  =  p7_in_gaagg(x1, x4, x5)
U1_gaagg(x1, x2, x3, x4)  =  U1_gaagg(x4)
p7_out_gaagg(x1, x2, x3, x4, x5)  =  p7_out_gaagg(x2, x3)
U2_gaagg(x1, x2, x3, x4, x5, x6, x7)  =  U2_gaagg(x1, x7)

(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:

FL1_IN_GAG(.(T8, T9), T12, s(T11)) → U3_GAG(T8, T9, T12, T11, p7_in_gaagg(T8, X13, T12, T9, T11))
FL1_IN_GAG(.(T8, T9), T12, s(T11)) → P7_IN_GAAGG(T8, X13, T12, T9, T11)
P7_IN_GAAGG([], T18, T18, T9, T11) → U1_GAAGG(T18, T9, T11, fl1_in_gag(T9, T18, T11))
P7_IN_GAAGG([], T18, T18, T9, T11) → FL1_IN_GAG(T9, T18, T11)
P7_IN_GAAGG(.(T25, T26), X43, .(T25, T28), T9, T11) → U2_GAAGG(T25, T26, X43, T28, T9, T11, p7_in_gaagg(T26, X43, T28, T9, T11))
P7_IN_GAAGG(.(T25, T26), X43, .(T25, T28), T9, T11) → P7_IN_GAAGG(T26, X43, T28, T9, T11)

The TRS R consists of the following rules:

fl1_in_gag([], [], 0) → fl1_out_gag([], [], 0)
fl1_in_gag(.(T8, T9), T12, s(T11)) → U3_gag(T8, T9, T12, T11, p7_in_gaagg(T8, X13, T12, T9, T11))
p7_in_gaagg([], T18, T18, T9, T11) → U1_gaagg(T18, T9, T11, fl1_in_gag(T9, T18, T11))
U1_gaagg(T18, T9, T11, fl1_out_gag(T9, T18, T11)) → p7_out_gaagg([], T18, T18, T9, T11)
p7_in_gaagg(.(T25, T26), X43, .(T25, T28), T9, T11) → U2_gaagg(T25, T26, X43, T28, T9, T11, p7_in_gaagg(T26, X43, T28, T9, T11))
U2_gaagg(T25, T26, X43, T28, T9, T11, p7_out_gaagg(T26, X43, T28, T9, T11)) → p7_out_gaagg(.(T25, T26), X43, .(T25, T28), T9, T11)
U3_gag(T8, T9, T12, T11, p7_out_gaagg(T8, X13, T12, T9, T11)) → fl1_out_gag(.(T8, T9), T12, s(T11))

The argument filtering Pi contains the following mapping:
fl1_in_gag(x1, x2, x3)  =  fl1_in_gag(x1, x3)
[]  =  []
0  =  0
fl1_out_gag(x1, x2, x3)  =  fl1_out_gag(x2)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
p7_in_gaagg(x1, x2, x3, x4, x5)  =  p7_in_gaagg(x1, x4, x5)
U1_gaagg(x1, x2, x3, x4)  =  U1_gaagg(x4)
p7_out_gaagg(x1, x2, x3, x4, x5)  =  p7_out_gaagg(x2, x3)
U2_gaagg(x1, x2, x3, x4, x5, x6, x7)  =  U2_gaagg(x1, x7)
FL1_IN_GAG(x1, x2, x3)  =  FL1_IN_GAG(x1, x3)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x5)
P7_IN_GAAGG(x1, x2, x3, x4, x5)  =  P7_IN_GAAGG(x1, x4, x5)
U1_GAAGG(x1, x2, x3, x4)  =  U1_GAAGG(x4)
U2_GAAGG(x1, x2, x3, x4, x5, x6, x7)  =  U2_GAAGG(x1, x7)

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

(6) Obligation:

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

FL1_IN_GAG(.(T8, T9), T12, s(T11)) → U3_GAG(T8, T9, T12, T11, p7_in_gaagg(T8, X13, T12, T9, T11))
FL1_IN_GAG(.(T8, T9), T12, s(T11)) → P7_IN_GAAGG(T8, X13, T12, T9, T11)
P7_IN_GAAGG([], T18, T18, T9, T11) → U1_GAAGG(T18, T9, T11, fl1_in_gag(T9, T18, T11))
P7_IN_GAAGG([], T18, T18, T9, T11) → FL1_IN_GAG(T9, T18, T11)
P7_IN_GAAGG(.(T25, T26), X43, .(T25, T28), T9, T11) → U2_GAAGG(T25, T26, X43, T28, T9, T11, p7_in_gaagg(T26, X43, T28, T9, T11))
P7_IN_GAAGG(.(T25, T26), X43, .(T25, T28), T9, T11) → P7_IN_GAAGG(T26, X43, T28, T9, T11)

The TRS R consists of the following rules:

fl1_in_gag([], [], 0) → fl1_out_gag([], [], 0)
fl1_in_gag(.(T8, T9), T12, s(T11)) → U3_gag(T8, T9, T12, T11, p7_in_gaagg(T8, X13, T12, T9, T11))
p7_in_gaagg([], T18, T18, T9, T11) → U1_gaagg(T18, T9, T11, fl1_in_gag(T9, T18, T11))
U1_gaagg(T18, T9, T11, fl1_out_gag(T9, T18, T11)) → p7_out_gaagg([], T18, T18, T9, T11)
p7_in_gaagg(.(T25, T26), X43, .(T25, T28), T9, T11) → U2_gaagg(T25, T26, X43, T28, T9, T11, p7_in_gaagg(T26, X43, T28, T9, T11))
U2_gaagg(T25, T26, X43, T28, T9, T11, p7_out_gaagg(T26, X43, T28, T9, T11)) → p7_out_gaagg(.(T25, T26), X43, .(T25, T28), T9, T11)
U3_gag(T8, T9, T12, T11, p7_out_gaagg(T8, X13, T12, T9, T11)) → fl1_out_gag(.(T8, T9), T12, s(T11))

The argument filtering Pi contains the following mapping:
fl1_in_gag(x1, x2, x3)  =  fl1_in_gag(x1, x3)
[]  =  []
0  =  0
fl1_out_gag(x1, x2, x3)  =  fl1_out_gag(x2)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
p7_in_gaagg(x1, x2, x3, x4, x5)  =  p7_in_gaagg(x1, x4, x5)
U1_gaagg(x1, x2, x3, x4)  =  U1_gaagg(x4)
p7_out_gaagg(x1, x2, x3, x4, x5)  =  p7_out_gaagg(x2, x3)
U2_gaagg(x1, x2, x3, x4, x5, x6, x7)  =  U2_gaagg(x1, x7)
FL1_IN_GAG(x1, x2, x3)  =  FL1_IN_GAG(x1, x3)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x5)
P7_IN_GAAGG(x1, x2, x3, x4, x5)  =  P7_IN_GAAGG(x1, x4, x5)
U1_GAAGG(x1, x2, x3, x4)  =  U1_GAAGG(x4)
U2_GAAGG(x1, x2, x3, x4, x5, x6, x7)  =  U2_GAAGG(x1, x7)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(8) Obligation:

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

FL1_IN_GAG(.(T8, T9), T12, s(T11)) → P7_IN_GAAGG(T8, X13, T12, T9, T11)
P7_IN_GAAGG([], T18, T18, T9, T11) → FL1_IN_GAG(T9, T18, T11)
P7_IN_GAAGG(.(T25, T26), X43, .(T25, T28), T9, T11) → P7_IN_GAAGG(T26, X43, T28, T9, T11)

The TRS R consists of the following rules:

fl1_in_gag([], [], 0) → fl1_out_gag([], [], 0)
fl1_in_gag(.(T8, T9), T12, s(T11)) → U3_gag(T8, T9, T12, T11, p7_in_gaagg(T8, X13, T12, T9, T11))
p7_in_gaagg([], T18, T18, T9, T11) → U1_gaagg(T18, T9, T11, fl1_in_gag(T9, T18, T11))
U1_gaagg(T18, T9, T11, fl1_out_gag(T9, T18, T11)) → p7_out_gaagg([], T18, T18, T9, T11)
p7_in_gaagg(.(T25, T26), X43, .(T25, T28), T9, T11) → U2_gaagg(T25, T26, X43, T28, T9, T11, p7_in_gaagg(T26, X43, T28, T9, T11))
U2_gaagg(T25, T26, X43, T28, T9, T11, p7_out_gaagg(T26, X43, T28, T9, T11)) → p7_out_gaagg(.(T25, T26), X43, .(T25, T28), T9, T11)
U3_gag(T8, T9, T12, T11, p7_out_gaagg(T8, X13, T12, T9, T11)) → fl1_out_gag(.(T8, T9), T12, s(T11))

The argument filtering Pi contains the following mapping:
fl1_in_gag(x1, x2, x3)  =  fl1_in_gag(x1, x3)
[]  =  []
0  =  0
fl1_out_gag(x1, x2, x3)  =  fl1_out_gag(x2)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
U3_gag(x1, x2, x3, x4, x5)  =  U3_gag(x5)
p7_in_gaagg(x1, x2, x3, x4, x5)  =  p7_in_gaagg(x1, x4, x5)
U1_gaagg(x1, x2, x3, x4)  =  U1_gaagg(x4)
p7_out_gaagg(x1, x2, x3, x4, x5)  =  p7_out_gaagg(x2, x3)
U2_gaagg(x1, x2, x3, x4, x5, x6, x7)  =  U2_gaagg(x1, x7)
FL1_IN_GAG(x1, x2, x3)  =  FL1_IN_GAG(x1, x3)
P7_IN_GAAGG(x1, x2, x3, x4, x5)  =  P7_IN_GAAGG(x1, x4, x5)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

FL1_IN_GAG(.(T8, T9), T12, s(T11)) → P7_IN_GAAGG(T8, X13, T12, T9, T11)
P7_IN_GAAGG([], T18, T18, T9, T11) → FL1_IN_GAG(T9, T18, T11)
P7_IN_GAAGG(.(T25, T26), X43, .(T25, T28), T9, T11) → P7_IN_GAAGG(T26, X43, T28, T9, T11)

R is empty.
The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
FL1_IN_GAG(x1, x2, x3)  =  FL1_IN_GAG(x1, x3)
P7_IN_GAAGG(x1, x2, x3, x4, x5)  =  P7_IN_GAAGG(x1, x4, x5)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

FL1_IN_GAG(.(T8, T9), s(T11)) → P7_IN_GAAGG(T8, T9, T11)
P7_IN_GAAGG([], T9, T11) → FL1_IN_GAG(T9, T11)
P7_IN_GAAGG(.(T25, T26), T9, T11) → P7_IN_GAAGG(T26, T9, T11)

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

(13) 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:

  • P7_IN_GAAGG([], T9, T11) → FL1_IN_GAG(T9, T11)
    The graph contains the following edges 2 >= 1, 3 >= 2

  • P7_IN_GAAGG(.(T25, T26), T9, T11) → P7_IN_GAAGG(T26, T9, T11)
    The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

  • FL1_IN_GAG(.(T8, T9), s(T11)) → P7_IN_GAAGG(T8, T9, T11)
    The graph contains the following edges 1 > 1, 1 > 2, 2 > 3

(14) YES