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

(0) Obligation:

Clauses:

concatenate([], L, L).
concatenate(.(X, L1), L2, .(X, L3)) :- concatenate(L1, L2, L3).
member(X, .(X, L)).
member(X, .(Y, L)) :- member(X, L).
reverse(L, L1) :- reverse_concatenate(L, [], L1).
reverse_concatenate([], L, L).
reverse_concatenate(.(X, L1), L2, L3) :- reverse_concatenate(L1, .(X, L2), L3).

Query: reverse_concatenate(g,g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: reverse_concatenate(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: reverse_concatenate(T1, T2, T3), SCOPE: 1, CLAUSE: 5 | reverse_concatenate(T1, T2, T3), SCOPE: 1, CLAUSE: 6\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true | reverse_concatenate([], T5, T3), SCOPE: 1, CLAUSE: 6\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: reverse_concatenate(T1, T2, T3), SCOPE: 1, CLAUSE: 6\n\nT1 is ground\nT2 is ground\nX2 is free\nreverse_concatenate(T1, T2, T3) !=? reverse_concatenate([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: reverse_concatenate([], T5, T3), SCOPE: 1, CLAUSE: 6\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: reverse_concatenate(T11, .(T10, T12), T14)\n\nT10 is ground\nT11 is ground\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: reverse_concatenate(T11, .(T10, T12), T14), SCOPE: 2, CLAUSE: 5 | reverse_concatenate(T11, .(T10, T12), T14), SCOPE: 2, CLAUSE: 6\n\nT10 is ground\nT11 is ground\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: reverse_concatenate(T11, .(T10, T12), T14), SCOPE: 2, CLAUSE: 5\n\nT10 is ground\nT11 is ground\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: reverse_concatenate(T11, .(T10, T12), T14), SCOPE: 2, CLAUSE: 6\n\nT10 is ground\nT11 is ground\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: reverse_concatenate(T36, .(T35, .(T37, T38)), T40)\n\nT35 is ground\nT36 is ground\nT37 is ground\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 34 [arrowhead = none ];
34 -> 6;

35 [label="EVAL with clause\nreverse_concatenate([], X2, X2).\nand substitutionT1 -> [],\nT2 -> T5,\nX2 -> T5,\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 35 [arrowhead = none ];
35 -> 10;

36 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 36 [arrowhead = none ];
36 -> 12;

37 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 37 [arrowhead = none ];
37 -> 15;

38 [label="EVAL with clause\nreverse_concatenate(.(X11, X12), X13, X14) :- reverse_concatenate(X12, .(X11, X13), X14).\nand substitutionX11 -> T10,\nX12 -> T11,\nT1 -> .(T10, T11),\nT2 -> T12,\nX13 -> T12,\nT3 -> T14,\nX14 -> T14,\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 38 [arrowhead = none ];
38 -> 19;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 39 [arrowhead = none ];
39 -> 21;

40 [label="BACKTRACK\nfor clause: reverse_concatenate(.(X, L1), L2, L3) :- reverse_concatenate(L1, .(X, L2), L3)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 40 [arrowhead = none ];
40 -> 16;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 41 [arrowhead = none ];
41 -> 22;

42 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 42 [arrowhead = none ];
42 -> 24;

43 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 43 [arrowhead = none ];
43 -> 25;

44 [label="EVAL with clause\nreverse_concatenate([], X19, X19).\nand substitutionT11 -> [],\nT10 -> T23,\nT12 -> T24,\nX19 -> .(T23, T24),\nT14 -> .(T23, T24)", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 44 [arrowhead = none ];
44 -> 29;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 45 [arrowhead = none ];
45 -> 30;

46 [label="EVAL with clause\nreverse_concatenate(.(X28, X29), X30, X31) :- reverse_concatenate(X29, .(X28, X30), X31).\nand substitutionX28 -> T35,\nX29 -> T36,\nT11 -> .(T35, T36),\nT10 -> T37,\nT12 -> T38,\nX30 -> .(T37, T38),\nT14 -> T40,\nX31 -> T40,\nT39 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 46 [arrowhead = none ];
46 -> 32;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 47 [arrowhead = none ];
47 -> 33;

48 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 48 [arrowhead = none ];
48 -> 31;

49 [label="INSTANCE with matching:\nT1 -> T36\nT2 -> .(T35, .(T37, T38))\nT3 -> T40", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 49 [arrowhead = none , style=dashed];
49 -> 1[style=dashed];

}

(2) Obligation:

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

reverse_concatenateA_in_gga([], T5, T5) → reverse_concatenateA_out_gga([], T5, T5)
reverse_concatenateA_in_gga(.(T23, []), T24, .(T23, T24)) → reverse_concatenateA_out_gga(.(T23, []), T24, .(T23, T24))
reverse_concatenateA_in_gga(.(T37, .(T35, T36)), T38, T40) → U1_gga(T37, T35, T36, T38, T40, reverse_concatenateA_in_gga(T36, .(T35, .(T37, T38)), T40))
U1_gga(T37, T35, T36, T38, T40, reverse_concatenateA_out_gga(T36, .(T35, .(T37, T38)), T40)) → reverse_concatenateA_out_gga(.(T37, .(T35, T36)), T38, T40)

The argument filtering Pi contains the following mapping:
reverse_concatenateA_in_gga(x1, x2, x3)  =  reverse_concatenateA_in_gga(x1, x2)
[]  =  []
reverse_concatenateA_out_gga(x1, x2, x3)  =  reverse_concatenateA_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x4, x6)

(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:

REVERSE_CONCATENATEA_IN_GGA(.(T37, .(T35, T36)), T38, T40) → U1_GGA(T37, T35, T36, T38, T40, reverse_concatenateA_in_gga(T36, .(T35, .(T37, T38)), T40))
REVERSE_CONCATENATEA_IN_GGA(.(T37, .(T35, T36)), T38, T40) → REVERSE_CONCATENATEA_IN_GGA(T36, .(T35, .(T37, T38)), T40)

The TRS R consists of the following rules:

reverse_concatenateA_in_gga([], T5, T5) → reverse_concatenateA_out_gga([], T5, T5)
reverse_concatenateA_in_gga(.(T23, []), T24, .(T23, T24)) → reverse_concatenateA_out_gga(.(T23, []), T24, .(T23, T24))
reverse_concatenateA_in_gga(.(T37, .(T35, T36)), T38, T40) → U1_gga(T37, T35, T36, T38, T40, reverse_concatenateA_in_gga(T36, .(T35, .(T37, T38)), T40))
U1_gga(T37, T35, T36, T38, T40, reverse_concatenateA_out_gga(T36, .(T35, .(T37, T38)), T40)) → reverse_concatenateA_out_gga(.(T37, .(T35, T36)), T38, T40)

The argument filtering Pi contains the following mapping:
reverse_concatenateA_in_gga(x1, x2, x3)  =  reverse_concatenateA_in_gga(x1, x2)
[]  =  []
reverse_concatenateA_out_gga(x1, x2, x3)  =  reverse_concatenateA_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x4, x6)
REVERSE_CONCATENATEA_IN_GGA(x1, x2, x3)  =  REVERSE_CONCATENATEA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x1, x2, x3, x4, x6)

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

(4) Obligation:

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

REVERSE_CONCATENATEA_IN_GGA(.(T37, .(T35, T36)), T38, T40) → U1_GGA(T37, T35, T36, T38, T40, reverse_concatenateA_in_gga(T36, .(T35, .(T37, T38)), T40))
REVERSE_CONCATENATEA_IN_GGA(.(T37, .(T35, T36)), T38, T40) → REVERSE_CONCATENATEA_IN_GGA(T36, .(T35, .(T37, T38)), T40)

The TRS R consists of the following rules:

reverse_concatenateA_in_gga([], T5, T5) → reverse_concatenateA_out_gga([], T5, T5)
reverse_concatenateA_in_gga(.(T23, []), T24, .(T23, T24)) → reverse_concatenateA_out_gga(.(T23, []), T24, .(T23, T24))
reverse_concatenateA_in_gga(.(T37, .(T35, T36)), T38, T40) → U1_gga(T37, T35, T36, T38, T40, reverse_concatenateA_in_gga(T36, .(T35, .(T37, T38)), T40))
U1_gga(T37, T35, T36, T38, T40, reverse_concatenateA_out_gga(T36, .(T35, .(T37, T38)), T40)) → reverse_concatenateA_out_gga(.(T37, .(T35, T36)), T38, T40)

The argument filtering Pi contains the following mapping:
reverse_concatenateA_in_gga(x1, x2, x3)  =  reverse_concatenateA_in_gga(x1, x2)
[]  =  []
reverse_concatenateA_out_gga(x1, x2, x3)  =  reverse_concatenateA_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x4, x6)
REVERSE_CONCATENATEA_IN_GGA(x1, x2, x3)  =  REVERSE_CONCATENATEA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x1, x2, x3, x4, x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

REVERSE_CONCATENATEA_IN_GGA(.(T37, .(T35, T36)), T38, T40) → REVERSE_CONCATENATEA_IN_GGA(T36, .(T35, .(T37, T38)), T40)

The TRS R consists of the following rules:

reverse_concatenateA_in_gga([], T5, T5) → reverse_concatenateA_out_gga([], T5, T5)
reverse_concatenateA_in_gga(.(T23, []), T24, .(T23, T24)) → reverse_concatenateA_out_gga(.(T23, []), T24, .(T23, T24))
reverse_concatenateA_in_gga(.(T37, .(T35, T36)), T38, T40) → U1_gga(T37, T35, T36, T38, T40, reverse_concatenateA_in_gga(T36, .(T35, .(T37, T38)), T40))
U1_gga(T37, T35, T36, T38, T40, reverse_concatenateA_out_gga(T36, .(T35, .(T37, T38)), T40)) → reverse_concatenateA_out_gga(.(T37, .(T35, T36)), T38, T40)

The argument filtering Pi contains the following mapping:
reverse_concatenateA_in_gga(x1, x2, x3)  =  reverse_concatenateA_in_gga(x1, x2)
[]  =  []
reverse_concatenateA_out_gga(x1, x2, x3)  =  reverse_concatenateA_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x4, x6)
REVERSE_CONCATENATEA_IN_GGA(x1, x2, x3)  =  REVERSE_CONCATENATEA_IN_GGA(x1, x2)

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:

REVERSE_CONCATENATEA_IN_GGA(.(T37, .(T35, T36)), T38, T40) → REVERSE_CONCATENATEA_IN_GGA(T36, .(T35, .(T37, T38)), T40)

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

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:

REVERSE_CONCATENATEA_IN_GGA(.(T37, .(T35, T36)), T38) → REVERSE_CONCATENATEA_IN_GGA(T36, .(T35, .(T37, T38)))

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:

• REVERSE_CONCATENATEA_IN_GGA(.(T37, .(T35, T36)), T38) → REVERSE_CONCATENATEA_IN_GGA(T36, .(T35, .(T37, T38)))
  The graph contains the following edges 1 > 1

(12) YES