Determinacy of the query pattern
t()
w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

t :- ','(eq(X, f(X)), !).
t :- t.
eq(X, X).

Query: t()

(1) PrologDeterminacyProcessorProof (EQUIVALENT transformation)

The root node satisfies the determinacy criterion.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: t\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: t, SCOPE: 1, CLAUSE: 0 | t, SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(eq(X1, f(X1)), !_1) | t, SCOPE: 1, CLAUSE: 1\n\nX1 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(eq(X1, f(X1)), !_1), SCOPE: 2, CLAUSE: 2 | ?_2 | t, SCOPE: 1, CLAUSE: 1\n\nX1 is free", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ?_2 | t, SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: t, SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: t\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\nt :- ','(eq(X1, f(X1)), !_1).\nand substitution", 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: eq(X, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 11 [arrowhead = none ];
11 -> 5;

12 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 12 [arrowhead = none ];
12 -> 6;

13 [label="ONLY EVAL with clause\nt :- t.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 13 [arrowhead = none ];
13 -> 7;

14 [label="INSTANCE", fontsize=14, style = filled, fillcolor = lightgrey];
7 -> 14 [arrowhead = none , style=dashed];
14 -> 1[style=dashed];

}

(2) YES