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

(0) Obligation:

Clauses:

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

Query: e(g)

(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: e(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: e(T1), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(=(X7, .(e, .(o, X7))), c(X7, T4))\n\nT4 is ground\nX7 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];
2 -> 8 [arrowhead = none ];
8 -> 4;

9 [label="ONLY EVAL with clause\ne(X6) :- ','(=(X7, .(e, .(o, X7))), c(X7, X6)).\nand substitutionT1 -> T4,\nX6 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 9 [arrowhead = none ];
9 -> 6;

10 [label="UNIFY-FAIL\nbecause of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 10 [arrowhead = none ];
10 -> 7;

}

(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: e(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: e(T1), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(=(X5, .(e, .(o, X5))), c(X5, T3))\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\ne(X4) :- ','(=(X5, .(e, .(o, X5))), c(X5, X4)).\nand substitutionT1 -> T3,\nX4 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 10 [arrowhead = none ];
10 -> 5;

11 [label="UNIFY-FAIL\nbecause of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 11 [arrowhead = none ];
11 -> 8;

}

(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