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

(0) Obligation:

Clauses:

t :- ','(=(X, f(0, f(1, f(0, X)))), ','(!, g(X))).
t :- t.
g(f(X, f(X, X1))) :- !.
g(f(X2, X)) :- g(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: ','(=(X3, f(0, f(1, f(0, X3)))), ','(!_1, g(X3))) | t, SCOPE: 1, CLAUSE: 1\n\nX3 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: t, SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: t\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: t, SCOPE: 2, CLAUSE: 0 | t, SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: t, SCOPE: 2, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: t, SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(=(X7, f(0, f(1, f(0, X7)))), ','(!_2, g(X7)))\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: t\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 12 [arrowhead = none ];
12 -> 2;

13 [label="ONLY EVAL with clause\nt :- ','(=(X3, f(0, f(1, f(0, X3)))), ','(!_1, g(X3))).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 13 [arrowhead = none ];
13 -> 3;

14 [label="UNIFY-FAIL\nbecause of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 14 [arrowhead = none ];
14 -> 4;

15 [label="ONLY EVAL with clause\nt :- t.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 15 [arrowhead = none ];
15 -> 5;

16 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 16 [arrowhead = none ];
16 -> 6;

17 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 17 [arrowhead = none ];
17 -> 7;

18 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 18 [arrowhead = none ];
18 -> 8;

19 [label="ONLY EVAL with clause\nt :- ','(=(X7, f(0, f(1, f(0, X7)))), ','(!_2, g(X7))).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 19 [arrowhead = none ];
19 -> 9;

20 [label="ONLY EVAL with clause\nt :- t.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 20 [arrowhead = none ];
20 -> 11;

21 [label="UNIFY-FAIL\nbecause of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 21 [arrowhead = none ];
21 -> 10;

22 [label="INSTANCE", fontsize=14, style = filled, fillcolor = lightgrey];
11 -> 22 [arrowhead = none , style=dashed];
22 -> 1[style=dashed];

}

(2) YES