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, Y, Z) :- append(.(X, Y), Z, Y).
append([], Y, Y).
append(.(H, Xs), Ys, .(H, Zs)) :- append(Xs, Ys, Zs).

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\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: append(.(T12, T11), T10, T11)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: append(.(T12, T11), T10, T11), SCOPE: 2, CLAUSE: 1 | append(.(T12, T11), T10, T11), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: append(.(T12, T11), T10, T11), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: append(.(T27, T26), T25, T26)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 8 [arrowhead = none ];
8 -> 2;

9 [label="ONLY EVAL with clause\np(X4, X5, X6) :- append(.(X4, X5), X6, X5).\nand substitutionT1 -> T12,\nX4 -> T12,\nT2 -> T11,\nX5 -> T11,\nT3 -> T10,\nX6 -> T10,\nT9 -> T10,\nT8 -> T11,\nT7 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 9 [arrowhead = none ];
9 -> 3;

10 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 10 [arrowhead = none ];
10 -> 4;

11 [label="BACKTRACK\nfor clause: append([], Y, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 11 [arrowhead = none ];
11 -> 5;

12 [label="EVAL with clause\nappend(.(X16, X17), X18, .(X16, X19)) :- append(X17, X18, X19).\nand substitutionT12 -> T27,\nX16 -> T27,\nT11 -> .(T27, T26),\nX17 -> .(T27, T26),\nT10 -> T25,\nX18 -> T25,\nX19 -> T26,\nT22 -> .(T27, T26),\nT23 -> T25,\nT24 -> T26,\nT21 -> T27", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 12 [arrowhead = none ];
12 -> 6;

13 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 13 [arrowhead = none ];
13 -> 7;

14 [label="INSTANCE with matching:\nT12 -> T27\nT11 -> T26\nT10 -> T25", fontsize=14, style = filled, fillcolor = lightgrey];
6 -> 14 [arrowhead = none , style=dashed];
14 -> 3[style=dashed];

}

(2) YES