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

(0) Obligation:

Clauses:

max_valued(.(Head, Tail), Max) :- max_Valued(Tail, Head, Max).
max_valued([], Term, Term).
max_valued(.(Head, Tail), Term, Max) :- ','(higher_valued(Head, Term), ','(!, max_valued(Tail, Head, Max))).
max_valued(.(Head, Tail), Term, Max) :- ','(higher_valued(Term, Head), max_valued(Tail, Term, Max)).
higher_valued(X, Y) :- greater(s(X), Y).
greater(s(X1), 0).
greater(s(X), s(Y)) :- greater(X, Y).

Query: max_valued(g,a)

(1) PrologToTRSTransformerProof (SOUND transformation)

Transformed Prolog program to TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="3: max_valued(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: max_valued(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: max_Valued(T10, T9, T12)\n\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 16 [arrowhead = none ];
16 -> 5;

17 [label="EVAL with clause\nmax_valued(.(X8, X9), X10) :- max_Valued(X9, X8, X10).\nand substitutionX8 -> T9,\nX9 -> T10,\nT1 -> .(T9, T10),\nT2 -> T12,\nX10 -> T12,\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 17 [arrowhead = none ];
17 -> 7;

18 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 18 [arrowhead = none ];
18 -> 11;

19 [label="UNDEFINED ERROR", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 19 [arrowhead = none ];
19 -> 15;

}

(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