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

(0) Obligation:

Clauses:

n :- ','(=(X, f(1, f(0, X))), g(X)).
g(f(X, f(X, X1))) :- !.
g(f(X2, X)) :- g(X).

Query: n()

(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: n\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: n, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(=(X4, f(1, f(0, X4))), g(X4))\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 9 [arrowhead = none ];
9 -> 3;

10 [label="ONLY EVAL with clause\nn :- ','(=(X4, f(1, f(0, X4))), g(X4)).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 10 [arrowhead = none ];
10 -> 6;

11 [label="UNIFY-FAIL\nbecause of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 11 [arrowhead = none ];
11 -> 8;

}

(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: n\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: n, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(=(X3, f(1, f(0, X3))), g(X3))\n\nX3 is free", 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="ONLY EVAL with clause\nn :- ','(=(X3, f(1, f(0, X3))), g(X3)).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 9 [arrowhead = none ];
9 -> 5;

10 [label="UNIFY-FAIL\nbecause of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 10 [arrowhead = none ];
10 -> 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