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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: max_valued(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: max_valued(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: max_Valued(T7, T6, T9)\n\nT6 is ground\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 6 [arrowhead = none ];
6 -> 2;

7 [label="EVAL with clause\nmax_valued(.(X5, X6), X7) :- max_Valued(X6, X5, X7).\nand substitutionX5 -> T6,\nX6 -> T7,\nT1 -> .(T6, T7),\nT2 -> T9,\nX7 -> T9,\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 7 [arrowhead = none ];
7 -> 3;

8 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 8 [arrowhead = none ];
8 -> 4;

9 [label="UNDEFINED ERROR", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 9 [arrowhead = none ];
9 -> 5;

}

(2) TRUE