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

(0) Obligation:

Clauses:

p(X, X, a) :- !.
p(X, Y, Z) :- ','(eq(Z, a), ','(eq(X, Y), p(X, Y, Z))).
eq(X, X).

Query: p(a,a,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: p(T1, T2, T3)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: p(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: !_1 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nX2 is free\np(T1, T2, T3) !=? p(X2, X2, a)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(eq(T12, a), ','(eq(T13, T14), p(T13, T14, T12)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(eq(T12, a), ','(eq(T13, T14), p(T13, T14, T12))), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(eq(T18, T19), p(T18, T19, a))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(eq(T18, T19), p(T18, T19, a)), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: p(T24, T24, a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: p(T24, T24, a), SCOPE: 4, CLAUSE: 0 | p(T24, T24, a), SCOPE: 4, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: !_4 | p(T24, T24, a), SCOPE: 4, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 51 [arrowhead = none ];
51 -> 4;

52 [label="EVAL with clause\np(X2, X2, a) :- !_1.\nand substitutionT1 -> T5,\nX2 -> T5,\nT2 -> T5,\nT3 -> a", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 52 [arrowhead = none ];
52 -> 7;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 53 [arrowhead = none ];
53 -> 10;

54 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 54 [arrowhead = none ];
54 -> 13;

55 [label="ONLY EVAL with clause\np(X6, X7, X8) :- ','(eq(X8, a), ','(eq(X6, X7), p(X6, X7, X8))).\nand substitutionT1 -> T13,\nX6 -> T13,\nT2 -> T14,\nX7 -> T14,\nT3 -> T12,\nX8 -> T12,\nT11 -> T12,\nT9 -> T13,\nT10 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 55 [arrowhead = none ];
55 -> 20;

56 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 56 [arrowhead = none ];
56 -> 18;

57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 57 [arrowhead = none ];
57 -> 22;

58 [label="EVAL with clause\neq(X11, X11).\nand substitutionT12 -> a,\nX11 -> a,\nT17 -> a,\nT13 -> T18,\nT14 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 58 [arrowhead = none ];
58 -> 25;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 59 [arrowhead = none ];
59 -> 27;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 60 [arrowhead = none ];
60 -> 29;

61 [label="EVAL with clause\neq(X15, X15).\nand substitutionT18 -> T24,\nX15 -> T24,\nT19 -> T24,\nT23 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 61 [arrowhead = none ];
61 -> 31;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 62 [arrowhead = none ];
62 -> 33;

63 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 63 [arrowhead = none ];
63 -> 42;

64 [label="ONLY EVAL with clause\np(X19, X19, a) :- !_4.\nand substitutionT24 -> T28,\nX19 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 64 [arrowhead = none ];
64 -> 46;

65 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 65 [arrowhead = none ];
65 -> 48;

66 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 66 [arrowhead = none ];
66 -> 50;

}

(2) Obligation:

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

pA_in_aaa(T5, T5, a) → pA_out_aaa(T5, T5, a)

The argument filtering Pi contains the following mapping:
pA_in_aaa(x1, x2, x3)  =  pA_in_aaa
pA_out_aaa(x1, x2, x3)  =  pA_out_aaa(x3)

(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_aaa(T5, T5, a) → pA_out_aaa(T5, T5, a)

The argument filtering Pi contains the following mapping:
pA_in_aaa(x1, x2, x3)  =  pA_in_aaa
pA_out_aaa(x1, x2, x3)  =  pA_out_aaa(x3)

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_aaa(T5, T5, a) → pA_out_aaa(T5, T5, a)

The argument filtering Pi contains the following mapping:
pA_in_aaa(x1, x2, x3)  =  pA_in_aaa
pA_out_aaa(x1, x2, x3)  =  pA_out_aaa(x3)

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