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

(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).

Query: fl(g,g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: fl(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: fl(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | fl(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true | fl([], [], T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: fl(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground\nfl(T1, T2, T3) !=? fl([], [], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: fl([], [], T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="B: ','(append(T8, X13, T10), fl(T9, X13, T12))\n\nT8 is ground\nT9 is ground\nT10 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(append(T8, X13, T10), fl(T9, X13, T12)), SCOPE: 2, CLAUSE: 2 | ','(append(T8, X13, T10), fl(T9, X13, T12)), SCOPE: 2, CLAUSE: 3\n\nT8 is ground\nT9 is ground\nT10 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(append(T8, X13, T10), fl(T9, X13, T12)), SCOPE: 2, CLAUSE: 2\n\nT8 is ground\nT9 is ground\nT10 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(append(T8, X13, T10), fl(T9, X13, T12)), SCOPE: 2, CLAUSE: 3\n\nT8 is ground\nT9 is ground\nT10 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: fl(T9, T17, T12)\n\nT9 is ground\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(append(T25, X38, T26), fl(T9, X38, T12))\n\nT9 is ground\nT25 is ground\nT26 is ground\nX38 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 45 [arrowhead = none ];
45 -> 6;

46 [label="EVAL with clause\nfl([], [], 0).\nand substitutionT1 -> [],\nT2 -> [],\nT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 46 [arrowhead = none ];
46 -> 11;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 47 [arrowhead = none ];
47 -> 13;

48 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 48 [arrowhead = none ];
48 -> 17;

49 [label="EVAL with clause\nfl(.(X9, X10), X11, s(X12)) :- ','(append(X9, X13, X11), fl(X10, X13, X12)).\nand substitutionX9 -> T8,\nX10 -> T9,\nT1 -> .(T8, T9),\nT2 -> T10,\nX11 -> T10,\nX12 -> T12,\nT3 -> s(T12),\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 49 [arrowhead = none ];
49 -> 21;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 50 [arrowhead = none ];
50 -> 24;

51 [label="BACKTRACK\nfor clause: fl(.(E, X), R, s(Z)) :- ','(append(E, Y, R), fl(X, Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 51 [arrowhead = none ];
51 -> 19;

52 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 52 [arrowhead = none ];
52 -> 27;

53 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 53 [arrowhead = none ];
53 -> 29;

54 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 54 [arrowhead = none ];
54 -> 31;

55 [label="EVAL with clause\nappend([], X22, X22).\nand substitutionT8 -> [],\nX13 -> T17,\nX22 -> T17,\nT10 -> T17,\nX23 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 55 [arrowhead = none ];
55 -> 33;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 56 [arrowhead = none ];
56 -> 35;

57 [label="EVAL with clause\nappend(.(X34, X35), X36, .(X34, X37)) :- append(X35, X36, X37).\nand substitutionX34 -> T24,\nX35 -> T25,\nT8 -> .(T24, T25),\nX13 -> X38,\nX36 -> X38,\nX37 -> T26,\nT10 -> .(T24, T26)", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 57 [arrowhead = none ];
57 -> 42;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 58 [arrowhead = none ];
58 -> 44;

59 [label="INSTANCE with matching:\nT1 -> T9\nT2 -> T17\nT3 -> T12", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 59 [arrowhead = none , style=dashed];
59 -> 3[style=dashed];

60 [label="INSTANCE with matching:\nT8 -> T25\nX13 -> X38\nT10 -> T26", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 60 [arrowhead = none , style=dashed];
60 -> 21[style=dashed];

}

(2) Obligation:

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

flA_in_gga([], [], 0) → flA_out_gga([], [], 0)
flA_in_gga(.(T8, T9), T10, s(T12)) → U1_gga(T8, T9, T10, T12, pB_in_gagga(T8, X13, T10, T9, T12))
pB_in_gagga([], T17, T17, T9, T12) → U2_gagga(T17, T9, T12, flA_in_gga(T9, T17, T12))
U2_gagga(T17, T9, T12, flA_out_gga(T9, T17, T12)) → pB_out_gagga([], T17, T17, T9, T12)
pB_in_gagga(.(T24, T25), X38, .(T24, T26), T9, T12) → U3_gagga(T24, T25, X38, T26, T9, T12, pB_in_gagga(T25, X38, T26, T9, T12))
U3_gagga(T24, T25, X38, T26, T9, T12, pB_out_gagga(T25, X38, T26, T9, T12)) → pB_out_gagga(.(T24, T25), X38, .(T24, T26), T9, T12)
U1_gga(T8, T9, T10, T12, pB_out_gagga(T8, X13, T10, T9, T12)) → flA_out_gga(.(T8, T9), T10, s(T12))

The argument filtering Pi contains the following mapping:
flA_in_gga(x1, x2, x3)  =  flA_in_gga(x1, x2)
[]  =  []
flA_out_gga(x1, x2, x3)  =  flA_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
pB_in_gagga(x1, x2, x3, x4, x5)  =  pB_in_gagga(x1, x3, x4)
U2_gagga(x1, x2, x3, x4)  =  U2_gagga(x1, x2, x4)
pB_out_gagga(x1, x2, x3, x4, x5)  =  pB_out_gagga(x1, x2, x3, x4, x5)
U3_gagga(x1, x2, x3, x4, x5, x6, x7)  =  U3_gagga(x1, x2, x4, x5, x7)
s(x1)  =  s(x1)

(3) 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:

FLA_IN_GGA(.(T8, T9), T10, s(T12)) → U1_GGA(T8, T9, T10, T12, pB_in_gagga(T8, X13, T10, T9, T12))
FLA_IN_GGA(.(T8, T9), T10, s(T12)) → PB_IN_GAGGA(T8, X13, T10, T9, T12)
PB_IN_GAGGA([], T17, T17, T9, T12) → U2_GAGGA(T17, T9, T12, flA_in_gga(T9, T17, T12))
PB_IN_GAGGA([], T17, T17, T9, T12) → FLA_IN_GGA(T9, T17, T12)
PB_IN_GAGGA(.(T24, T25), X38, .(T24, T26), T9, T12) → U3_GAGGA(T24, T25, X38, T26, T9, T12, pB_in_gagga(T25, X38, T26, T9, T12))
PB_IN_GAGGA(.(T24, T25), X38, .(T24, T26), T9, T12) → PB_IN_GAGGA(T25, X38, T26, T9, T12)

The TRS R consists of the following rules:

flA_in_gga([], [], 0) → flA_out_gga([], [], 0)
flA_in_gga(.(T8, T9), T10, s(T12)) → U1_gga(T8, T9, T10, T12, pB_in_gagga(T8, X13, T10, T9, T12))
pB_in_gagga([], T17, T17, T9, T12) → U2_gagga(T17, T9, T12, flA_in_gga(T9, T17, T12))
U2_gagga(T17, T9, T12, flA_out_gga(T9, T17, T12)) → pB_out_gagga([], T17, T17, T9, T12)
pB_in_gagga(.(T24, T25), X38, .(T24, T26), T9, T12) → U3_gagga(T24, T25, X38, T26, T9, T12, pB_in_gagga(T25, X38, T26, T9, T12))
U3_gagga(T24, T25, X38, T26, T9, T12, pB_out_gagga(T25, X38, T26, T9, T12)) → pB_out_gagga(.(T24, T25), X38, .(T24, T26), T9, T12)
U1_gga(T8, T9, T10, T12, pB_out_gagga(T8, X13, T10, T9, T12)) → flA_out_gga(.(T8, T9), T10, s(T12))

The argument filtering Pi contains the following mapping:
flA_in_gga(x1, x2, x3)  =  flA_in_gga(x1, x2)
[]  =  []
flA_out_gga(x1, x2, x3)  =  flA_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
pB_in_gagga(x1, x2, x3, x4, x5)  =  pB_in_gagga(x1, x3, x4)
U2_gagga(x1, x2, x3, x4)  =  U2_gagga(x1, x2, x4)
pB_out_gagga(x1, x2, x3, x4, x5)  =  pB_out_gagga(x1, x2, x3, x4, x5)
U3_gagga(x1, x2, x3, x4, x5, x6, x7)  =  U3_gagga(x1, x2, x4, x5, x7)
s(x1)  =  s(x1)
FLA_IN_GGA(x1, x2, x3)  =  FLA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)
PB_IN_GAGGA(x1, x2, x3, x4, x5)  =  PB_IN_GAGGA(x1, x3, x4)
U2_GAGGA(x1, x2, x3, x4)  =  U2_GAGGA(x1, x2, x4)
U3_GAGGA(x1, x2, x3, x4, x5, x6, x7)  =  U3_GAGGA(x1, x2, x4, x5, x7)

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

(4) Obligation:

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

FLA_IN_GGA(.(T8, T9), T10, s(T12)) → U1_GGA(T8, T9, T10, T12, pB_in_gagga(T8, X13, T10, T9, T12))
FLA_IN_GGA(.(T8, T9), T10, s(T12)) → PB_IN_GAGGA(T8, X13, T10, T9, T12)
PB_IN_GAGGA([], T17, T17, T9, T12) → U2_GAGGA(T17, T9, T12, flA_in_gga(T9, T17, T12))
PB_IN_GAGGA([], T17, T17, T9, T12) → FLA_IN_GGA(T9, T17, T12)
PB_IN_GAGGA(.(T24, T25), X38, .(T24, T26), T9, T12) → U3_GAGGA(T24, T25, X38, T26, T9, T12, pB_in_gagga(T25, X38, T26, T9, T12))
PB_IN_GAGGA(.(T24, T25), X38, .(T24, T26), T9, T12) → PB_IN_GAGGA(T25, X38, T26, T9, T12)

The TRS R consists of the following rules:

flA_in_gga([], [], 0) → flA_out_gga([], [], 0)
flA_in_gga(.(T8, T9), T10, s(T12)) → U1_gga(T8, T9, T10, T12, pB_in_gagga(T8, X13, T10, T9, T12))
pB_in_gagga([], T17, T17, T9, T12) → U2_gagga(T17, T9, T12, flA_in_gga(T9, T17, T12))
U2_gagga(T17, T9, T12, flA_out_gga(T9, T17, T12)) → pB_out_gagga([], T17, T17, T9, T12)
pB_in_gagga(.(T24, T25), X38, .(T24, T26), T9, T12) → U3_gagga(T24, T25, X38, T26, T9, T12, pB_in_gagga(T25, X38, T26, T9, T12))
U3_gagga(T24, T25, X38, T26, T9, T12, pB_out_gagga(T25, X38, T26, T9, T12)) → pB_out_gagga(.(T24, T25), X38, .(T24, T26), T9, T12)
U1_gga(T8, T9, T10, T12, pB_out_gagga(T8, X13, T10, T9, T12)) → flA_out_gga(.(T8, T9), T10, s(T12))

The argument filtering Pi contains the following mapping:
flA_in_gga(x1, x2, x3)  =  flA_in_gga(x1, x2)
[]  =  []
flA_out_gga(x1, x2, x3)  =  flA_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
pB_in_gagga(x1, x2, x3, x4, x5)  =  pB_in_gagga(x1, x3, x4)
U2_gagga(x1, x2, x3, x4)  =  U2_gagga(x1, x2, x4)
pB_out_gagga(x1, x2, x3, x4, x5)  =  pB_out_gagga(x1, x2, x3, x4, x5)
U3_gagga(x1, x2, x3, x4, x5, x6, x7)  =  U3_gagga(x1, x2, x4, x5, x7)
s(x1)  =  s(x1)
FLA_IN_GGA(x1, x2, x3)  =  FLA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)
PB_IN_GAGGA(x1, x2, x3, x4, x5)  =  PB_IN_GAGGA(x1, x3, x4)
U2_GAGGA(x1, x2, x3, x4)  =  U2_GAGGA(x1, x2, x4)
U3_GAGGA(x1, x2, x3, x4, x5, x6, x7)  =  U3_GAGGA(x1, x2, x4, x5, x7)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

FLA_IN_GGA(.(T8, T9), T10, s(T12)) → PB_IN_GAGGA(T8, X13, T10, T9, T12)
PB_IN_GAGGA([], T17, T17, T9, T12) → FLA_IN_GGA(T9, T17, T12)
PB_IN_GAGGA(.(T24, T25), X38, .(T24, T26), T9, T12) → PB_IN_GAGGA(T25, X38, T26, T9, T12)

The TRS R consists of the following rules:

flA_in_gga([], [], 0) → flA_out_gga([], [], 0)
flA_in_gga(.(T8, T9), T10, s(T12)) → U1_gga(T8, T9, T10, T12, pB_in_gagga(T8, X13, T10, T9, T12))
pB_in_gagga([], T17, T17, T9, T12) → U2_gagga(T17, T9, T12, flA_in_gga(T9, T17, T12))
U2_gagga(T17, T9, T12, flA_out_gga(T9, T17, T12)) → pB_out_gagga([], T17, T17, T9, T12)
pB_in_gagga(.(T24, T25), X38, .(T24, T26), T9, T12) → U3_gagga(T24, T25, X38, T26, T9, T12, pB_in_gagga(T25, X38, T26, T9, T12))
U3_gagga(T24, T25, X38, T26, T9, T12, pB_out_gagga(T25, X38, T26, T9, T12)) → pB_out_gagga(.(T24, T25), X38, .(T24, T26), T9, T12)
U1_gga(T8, T9, T10, T12, pB_out_gagga(T8, X13, T10, T9, T12)) → flA_out_gga(.(T8, T9), T10, s(T12))

The argument filtering Pi contains the following mapping:
flA_in_gga(x1, x2, x3)  =  flA_in_gga(x1, x2)
[]  =  []
flA_out_gga(x1, x2, x3)  =  flA_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
pB_in_gagga(x1, x2, x3, x4, x5)  =  pB_in_gagga(x1, x3, x4)
U2_gagga(x1, x2, x3, x4)  =  U2_gagga(x1, x2, x4)
pB_out_gagga(x1, x2, x3, x4, x5)  =  pB_out_gagga(x1, x2, x3, x4, x5)
U3_gagga(x1, x2, x3, x4, x5, x6, x7)  =  U3_gagga(x1, x2, x4, x5, x7)
s(x1)  =  s(x1)
FLA_IN_GGA(x1, x2, x3)  =  FLA_IN_GGA(x1, x2)
PB_IN_GAGGA(x1, x2, x3, x4, x5)  =  PB_IN_GAGGA(x1, x3, x4)

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

FLA_IN_GGA(.(T8, T9), T10, s(T12)) → PB_IN_GAGGA(T8, X13, T10, T9, T12)
PB_IN_GAGGA([], T17, T17, T9, T12) → FLA_IN_GGA(T9, T17, T12)
PB_IN_GAGGA(.(T24, T25), X38, .(T24, T26), T9, T12) → PB_IN_GAGGA(T25, X38, T26, T9, T12)

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

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

FLA_IN_GGA(.(T8, T9), T10) → PB_IN_GAGGA(T8, T10, T9)
PB_IN_GAGGA([], T17, T9) → FLA_IN_GGA(T9, T17)
PB_IN_GAGGA(.(T24, T25), .(T24, T26), T9) → PB_IN_GAGGA(T25, T26, T9)

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

(11) 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:

• PB_IN_GAGGA([], T17, T9) → FLA_IN_GGA(T9, T17)
  The graph contains the following edges 3 >= 1, 2 >= 2

• PB_IN_GAGGA(.(T24, T25), .(T24, T26), T9) → PB_IN_GAGGA(T25, T26, T9)
  The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3

• FLA_IN_GGA(.(T8, T9), T10) → PB_IN_GAGGA(T8, T10, T9)
  The graph contains the following edges 1 > 1, 2 >= 2, 1 > 3

(12) YES