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

(0) Obligation:

Clauses:

dyck(X) :- ','(grammar(G), ','(-(terminal, string(X)), produces(G, X))).
grammar(S) :- =(S, or(e, or(and(S, S), and(o, and(S, c))))).
equal(X, X).
-(terminal, string(nil)).
-(terminal, string(and(A, X))) :- ','(terminal(A), -(terminal, string(X))).
terminal(o).
terminal(c).
terminal(e).
produces(G, X) :- gives(G, X, nil).
gives(or(P, Q), X, Y) :- gives(P, X, Y).
gives(or(P, Q), X, Y) :- gives(Q, X, Y).
gives(and(P, Q), X, Z) :- ','(gives(P, X, Y), gives(Q, Y, Z)).
gives(A, and(A, X), X) :- terminal(A).

Query: dyck(g)

(1) PrologToTRSTransformerProof (SOUND transformation)

Transformed Prolog program to TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: dyck(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: dyck(T1), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(grammar(X5), ','(-(terminal, string(T4)), produces(X5, T4)))\n\nT4 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(grammar(X5), ','(-(terminal, string(T4)), produces(X5, T4))), SCOPE: 2, CLAUSE: 1\n\nT4 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(=(X11, or(e, or(and(X11, X11), and(o, and(X11, c))))), ','(-(terminal, string(T4)), produces(X11, T4)))\n\nT4 is ground\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 19 [arrowhead = none ];
19 -> 6;

20 [label="ONLY EVAL with clause\ndyck(X4) :- ','(grammar(X5), ','(-(terminal, string(X4)), produces(X5, X4))).\nand substitutionT1 -> T4,\nX4 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 20 [arrowhead = none ];
20 -> 8;

21 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 21 [arrowhead = none ];
21 -> 14;

22 [label="ONLY EVAL with clause\ngrammar(X10) :- =(X10, or(e, or(and(X10, X10), and(o, and(X10, c))))).\nand substitutionX5 -> X11,\nX10 -> X11", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 22 [arrowhead = none ];
22 -> 17;

23 [label="UNIFY-FAIL\nbecause of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 23 [arrowhead = none ];
23 -> 18;

}

(2) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(3) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(4) YES