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

(0) Obligation:

Clauses:

p(val_i, val_j).
map(.(X, Xs), .(Y, Ys)) :- ','(p(X, Y), map(Xs, Ys)).
map([], []).

Query: map(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: map(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: map(T1, T2), SCOPE: 1, CLAUSE: 1 | map(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(p(T7, T11), map(T8, T12)) | map(.(T7, T8), T2), SCOPE: 1, CLAUSE: 2\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: map(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX5 is free\nX6 is free\nX7 is free\nX8 is free\nmap(T1, T2) !=? map(.(X5, X6), .(X7, X8))", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(p(T7, T11), map(T8, T12)), SCOPE: 2, CLAUSE: 0 | ?_2 | map(.(T7, T8), T2), SCOPE: 1, CLAUSE: 2\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(p(T7, T11), map(T8, T12)), SCOPE: 2, CLAUSE: 0\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ?_2 | map(.(T7, T8), T2), SCOPE: 1, CLAUSE: 2\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: map(T8, T13)\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: map(.(T7, T8), T2), SCOPE: 1, CLAUSE: 2\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 35 [arrowhead = none ];
35 -> 5;

36 [label="EVAL with clause\nmap(.(X5, X6), .(X7, X8)) :- ','(p(X5, X7), map(X6, X8)).\nand substitutionX5 -> T7,\nX6 -> T8,\nT1 -> .(T7, T8),\nX7 -> T11,\nX8 -> T12,\nT2 -> .(T11, T12),\nT9 -> T11,\nT10 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 36 [arrowhead = none ];
36 -> 14;

37 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 37 [arrowhead = none ];
37 -> 17;

38 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 38 [arrowhead = none ];
38 -> 18;

39 [label="EVAL with clause\nmap([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 39 [arrowhead = none ];
39 -> 30;

40 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 40 [arrowhead = none ];
40 -> 32;

41 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 41 [arrowhead = none ];
41 -> 19;

42 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 42 [arrowhead = none ];
42 -> 20;

43 [label="EVAL with clause\np(val_i, val_j).\nand substitutionT7 -> val_i,\nT11 -> val_j,\nT12 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 43 [arrowhead = none ];
43 -> 21;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 44 [arrowhead = none ];
44 -> 22;

45 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 45 [arrowhead = none ];
45 -> 23;

46 [label="INSTANCE with matching:\nT1 -> T8\nT2 -> T13", fontsize=14, style = filled, fillcolor = lightgrey];
21 -> 46 [arrowhead = none , style=dashed];
46 -> 3[style=dashed];

47 [label="BACKTRACK\nfor clause: map([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 47 [arrowhead = none ];
47 -> 25;

48 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 48 [arrowhead = none ];
48 -> 34;

}

(2) Obligation:

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

mapA_in_ga(.(val_i, T8), .(val_j, T13)) → U1_ga(T8, T13, mapA_in_ga(T8, T13))
mapA_in_ga([], []) → mapA_out_ga([], [])
U1_ga(T8, T13, mapA_out_ga(T8, T13)) → mapA_out_ga(.(val_i, T8), .(val_j, T13))

The argument filtering Pi contains the following mapping:
mapA_in_ga(x1, x2)  =  mapA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
val_i  =  val_i
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
[]  =  []
mapA_out_ga(x1, x2)  =  mapA_out_ga(x1, x2)
val_j  =  val_j

(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:

MAPA_IN_GA(.(val_i, T8), .(val_j, T13)) → U1_GA(T8, T13, mapA_in_ga(T8, T13))
MAPA_IN_GA(.(val_i, T8), .(val_j, T13)) → MAPA_IN_GA(T8, T13)

The TRS R consists of the following rules:

mapA_in_ga(.(val_i, T8), .(val_j, T13)) → U1_ga(T8, T13, mapA_in_ga(T8, T13))
mapA_in_ga([], []) → mapA_out_ga([], [])
U1_ga(T8, T13, mapA_out_ga(T8, T13)) → mapA_out_ga(.(val_i, T8), .(val_j, T13))

The argument filtering Pi contains the following mapping:
mapA_in_ga(x1, x2)  =  mapA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
val_i  =  val_i
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
[]  =  []
mapA_out_ga(x1, x2)  =  mapA_out_ga(x1, x2)
val_j  =  val_j
MAPA_IN_GA(x1, x2)  =  MAPA_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)

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

(4) Obligation:

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

MAPA_IN_GA(.(val_i, T8), .(val_j, T13)) → U1_GA(T8, T13, mapA_in_ga(T8, T13))
MAPA_IN_GA(.(val_i, T8), .(val_j, T13)) → MAPA_IN_GA(T8, T13)

The TRS R consists of the following rules:

mapA_in_ga(.(val_i, T8), .(val_j, T13)) → U1_ga(T8, T13, mapA_in_ga(T8, T13))
mapA_in_ga([], []) → mapA_out_ga([], [])
U1_ga(T8, T13, mapA_out_ga(T8, T13)) → mapA_out_ga(.(val_i, T8), .(val_j, T13))

The argument filtering Pi contains the following mapping:
mapA_in_ga(x1, x2)  =  mapA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
val_i  =  val_i
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
[]  =  []
mapA_out_ga(x1, x2)  =  mapA_out_ga(x1, x2)
val_j  =  val_j
MAPA_IN_GA(x1, x2)  =  MAPA_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)

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:

MAPA_IN_GA(.(val_i, T8), .(val_j, T13)) → MAPA_IN_GA(T8, T13)

The TRS R consists of the following rules:

mapA_in_ga(.(val_i, T8), .(val_j, T13)) → U1_ga(T8, T13, mapA_in_ga(T8, T13))
mapA_in_ga([], []) → mapA_out_ga([], [])
U1_ga(T8, T13, mapA_out_ga(T8, T13)) → mapA_out_ga(.(val_i, T8), .(val_j, T13))

The argument filtering Pi contains the following mapping:
mapA_in_ga(x1, x2)  =  mapA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
val_i  =  val_i
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
[]  =  []
mapA_out_ga(x1, x2)  =  mapA_out_ga(x1, x2)
val_j  =  val_j
MAPA_IN_GA(x1, x2)  =  MAPA_IN_GA(x1)

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:

MAPA_IN_GA(.(val_i, T8), .(val_j, T13)) → MAPA_IN_GA(T8, T13)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
val_i  =  val_i
val_j  =  val_j
MAPA_IN_GA(x1, x2)  =  MAPA_IN_GA(x1)

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:

MAPA_IN_GA(.(val_i, T8)) → MAPA_IN_GA(T8)

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:

• MAPA_IN_GA(.(val_i, T8)) → MAPA_IN_GA(T8)
  The graph contains the following edges 1 > 1

(12) YES