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

(0) Obligation:

Clauses:

q(X) :- ','(grammar(G), ','(-(terminal, string(X)), produces(G, X))).
grammar(S) :- ','(equal(S, or(N, and(a, and(S, b)))), equal(N, or(c, and(c, N)))).
equal(X, X).
-(terminal, string(nil)).
-(terminal, string(and(A, X))) :- ','(terminal(A), -(terminal, string(X))).
terminal(a).
terminal(b).
terminal(c).
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: q(g)

(1) PrologToTRSTransformerProof (SOUND transformation)

Transformed Prolog program to TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: q(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: q(T1), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(grammar(X3), ','(-(terminal, string(T3)), produces(X3, T3)))\n\nT3 is ground\nX3 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(grammar(X3), ','(-(terminal, string(T3)), produces(X3, T3))), SCOPE: 2, CLAUSE: 1\n\nT3 is ground\nX3 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(','(equal(X11, or(X10, and(a, and(X11, b)))), equal(X10, or(c, and(c, X10)))), ','(-(terminal, string(T3)), produces(X11, T3)))\n\nT3 is ground\nX11 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(','(equal(X11, or(X10, and(a, and(X11, b)))), equal(X10, or(c, and(c, X10)))), ','(-(terminal, string(T3)), produces(X11, T3))), SCOPE: 3, CLAUSE: 2\n\nT3 is ground\nX11 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 17 [arrowhead = none ];
17 -> 5;

18 [label="ONLY EVAL with clause\nq(X2) :- ','(grammar(X3), ','(-(terminal, string(X2)), produces(X3, X2))).\nand substitutionT1 -> T3,\nX2 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 18 [arrowhead = none ];
18 -> 7;

19 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 19 [arrowhead = none ];
19 -> 11;

20 [label="ONLY EVAL with clause\ngrammar(X9) :- ','(equal(X9, or(X10, and(a, and(X9, b)))), equal(X10, or(c, and(c, X10)))).\nand substitutionX3 -> X11,\nX9 -> X11", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 20 [arrowhead = none ];
20 -> 13;

21 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 21 [arrowhead = none ];
21 -> 14;

22 [label="BACKTRACK\nfor clause: equal(X, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 22 [arrowhead = none ];
22 -> 16;

}

(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