Runtime Complexity of the query pattern
max_valued(g,a)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(1)).

(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) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: max_valued(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: max_valued(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: max_Valued(T10, T9, T12)\n\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 11 [arrowhead = none ];
11 -> 3;

12 [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];
3 -> 12 [arrowhead = none ];
12 -> 8;

13 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 13 [arrowhead = none ];
13 -> 9;

14 [label="UNDEFINED ERROR", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 14 [arrowhead = none ];
14 -> 10;

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:none

Compound Symbols:none

(3) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(4) BOUNDS(O(1), O(1))

(5) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

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];
4 [label="4: max_valued(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: max_Valued(T7, T6, T9)\n\nT6 is ground\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 8 [arrowhead = none ];
8 -> 4;

9 [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];
4 -> 9 [arrowhead = none ];
9 -> 5;

10 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 10 [arrowhead = none ];
10 -> 6;

11 [label="UNDEFINED ERROR", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 11 [arrowhead = none ];
11 -> 7;

}

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:none

Compound Symbols:none