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

(0) Obligation:

Clauses:

p(.(A, [])) :- l(.(A, [])).
r(1).
l([]).
l(.(H, T)) :- ','(r(H), l(T)).

Query: p(a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: p(T1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: p(T1), SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: l(.(T4, []))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: l(.(T4, [])), SCOPE: 2, CLAUSE: 2 | l(.(T4, [])), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: l(.(T4, [])), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(r(T9), l([]))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(r(T9), l([])), SCOPE: 3, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: l([])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: l([]), SCOPE: 4, CLAUSE: 2 | l([]), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: l([]), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: l([]), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 46 [arrowhead = none ];
46 -> 5;

47 [label="EVAL with clause\np(.(X2, [])) :- l(.(X2, [])).\nand substitutionX2 -> T4,\nT1 -> .(T4, []),\nT3 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 47 [arrowhead = none ];
47 -> 7;

48 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 48 [arrowhead = none ];
48 -> 10;

49 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 49 [arrowhead = none ];
49 -> 13;

50 [label="BACKTRACK\nfor clause: l([])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 50 [arrowhead = none ];
50 -> 15;

51 [label="ONLY EVAL with clause\nl(.(X7, X8)) :- ','(r(X7), l(X8)).\nand substitutionT4 -> T9,\nX7 -> T9,\nX8 -> [],\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 51 [arrowhead = none ];
51 -> 17;

52 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 52 [arrowhead = none ];
52 -> 20;

53 [label="EVAL with clause\nr(1).\nand substitutionT9 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 53 [arrowhead = none ];
53 -> 23;

54 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 54 [arrowhead = none ];
54 -> 26;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 55 [arrowhead = none ];
55 -> 27;

56 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 56 [arrowhead = none ];
56 -> 32;

57 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 57 [arrowhead = none ];
57 -> 35;

58 [label="ONLY EVAL with clause\nl([]).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 58 [arrowhead = none ];
58 -> 37;

59 [label="BACKTRACK\nfor clause: l(.(H, T)) :- ','(r(H), l(T))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 59 [arrowhead = none ];
59 -> 45;

60 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 60 [arrowhead = none ];
60 -> 40;

}

(2) Obligation:

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

pA_in_a(.(1, [])) → pA_out_a(.(1, []))

The argument filtering Pi contains the following mapping:
pA_in_a(x1)  =  pA_in_a
pA_out_a(x1)  =  pA_out_a(x1)

(3) DependencyPairsProof (EQUIVALENT transformation)

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

pA_in_a(.(1, [])) → pA_out_a(.(1, []))

The argument filtering Pi contains the following mapping:
pA_in_a(x1)  =  pA_in_a
pA_out_a(x1)  =  pA_out_a(x1)

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

(4) Obligation:

Pi DP problem:
P is empty.
The TRS R consists of the following rules:

pA_in_a(.(1, [])) → pA_out_a(.(1, []))

The argument filtering Pi contains the following mapping:
pA_in_a(x1)  =  pA_in_a
pA_out_a(x1)  =  pA_out_a(x1)

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

(5) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,R,Pi) chain.

(6) YES