Determinacy 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) PrologDeterminacyProcessorProof (EQUIVALENT transformation)

The root node satisfies the determinacy criterion.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: p(T1, T2, T3)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: p(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: !_1 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: 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];
5 [label="5: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(eq(T12, a), ','(eq(T13, T14), p(T13, T14, T12)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(eq(T12, a), ','(eq(T13, T14), p(T13, T14, T12))), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(eq(T18, T19), p(T18, T19, a))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(eq(T18, T19), p(T18, T19, a)), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: p(T24, T24, a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: p(T24, T24, a), SCOPE: 4, CLAUSE: 0 | p(T24, T24, a), SCOPE: 4, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: !_4 | p(T24, T24, a), SCOPE: 4, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 18 [arrowhead = none ];
18 -> 2;

19 [label="EVAL with clause\np(X2, X2, a) :- !_1.\nand substitutionT1 -> T5,\nX2 -> T5,\nT2 -> T5,\nT3 -> a", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 19 [arrowhead = none ];
19 -> 3;

20 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 20 [arrowhead = none ];
20 -> 4;

21 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 21 [arrowhead = none ];
21 -> 5;

22 [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];
4 -> 22 [arrowhead = none ];
22 -> 7;

23 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 23 [arrowhead = none ];
23 -> 6;

24 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 24 [arrowhead = none ];
24 -> 8;

25 [label="EVAL with clause\neq(X11, X11).\nand substitutionT12 -> a,\nX11 -> a,\nT17 -> a,\nT13 -> T18,\nT14 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 25 [arrowhead = none ];
25 -> 9;

26 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 26 [arrowhead = none ];
26 -> 10;

27 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 27 [arrowhead = none ];
27 -> 11;

28 [label="EVAL with clause\neq(X15, X15).\nand substitutionT18 -> T24,\nX15 -> T24,\nT19 -> T24,\nT23 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 28 [arrowhead = none ];
28 -> 12;

29 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 29 [arrowhead = none ];
29 -> 13;

30 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 30 [arrowhead = none ];
30 -> 14;

31 [label="ONLY EVAL with clause\np(X19, X19, a) :- !_4.\nand substitutionT24 -> T28,\nX19 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 31 [arrowhead = none ];
31 -> 15;

32 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 32 [arrowhead = none ];
32 -> 16;

33 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 33 [arrowhead = none ];
33 -> 17;

}

(2) YES