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

(0) Obligation:

Clauses:

even(X) :- ','(eq(Y, f(e, f(o, Y))), c(Y, X)).
c(f(e, X1), 0).
c(f(X2, X), s(Y)) :- c(X, Y).
eq(X, X).

Query: even(g)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

c(f(e, X1), 0).
eq(X, X).
even(X) :- ','(eq(Y, f(e, f(o, Y))), c(Y, X)).
c(f(X2, X), s(Y)) :- c(X, Y).

Query: even(g)

(3) 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: even(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: even(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(eq(X11, f(e, f(o, X11))), c(X11, T4))\n\nT4 is ground\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(eq(X11, f(e, f(o, X11))), c(X11, T4)), SCOPE: 2, CLAUSE: 1\n\nT4 is ground\nX11 is free", 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 -> 4;

12 [label="ONLY EVAL with clause\neven(X10) :- ','(eq(X11, f(e, f(o, X11))), c(X11, X10)).\nand substitutionT1 -> T4,\nX10 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 12 [arrowhead = none ];
12 -> 7;

13 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 13 [arrowhead = none ];
13 -> 9;

14 [label="BACKTRACK\nfor clause: eq(X, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 14 [arrowhead = none ];
14 -> 10;

}

(4) 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

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))

(7) 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: even(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: even(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(eq(X5, f(e, f(o, X5))), c(X5, T3))\n\nT3 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(eq(X5, f(e, f(o, X5))), c(X5, T3)), SCOPE: 2, CLAUSE: 1\n\nT3 is ground\nX5 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];
1 -> 9 [arrowhead = none ];
9 -> 3;

10 [label="ONLY EVAL with clause\neven(X4) :- ','(eq(X5, f(e, f(o, X5))), c(X5, X4)).\nand substitutionT1 -> T3,\nX4 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 10 [arrowhead = none ];
10 -> 5;

11 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 11 [arrowhead = none ];
11 -> 6;

12 [label="BACKTRACK\nfor clause: eq(X, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 12 [arrowhead = none ];
12 -> 8;

}

(8) 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