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

(0) Obligation:

Clauses:

p(b).
p(a) :- p1(X).
p1(b) :- !.
p1(a) :- p1(X).

Query: p(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: p(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: p(T1), SCOPE: 1, CLAUSE: 0 | p(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | p(b), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: p(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\np(T1) !=? p(b)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: p(b), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: p1(X2)\n\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: p1(X2), SCOPE: 2, CLAUSE: 2 | p1(X2), SCOPE: 2, CLAUSE: 3\n\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: !_2 | p1(X2), SCOPE: 2, CLAUSE: 3\n\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 25 [arrowhead = none ];
25 -> 3;

26 [label="EVAL with clause\np(b).\nand substitutionT1 -> b", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 26 [arrowhead = none ];
26 -> 5;

27 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 27 [arrowhead = none ];
27 -> 8;

28 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 28 [arrowhead = none ];
28 -> 9;

29 [label="EVAL with clause\np(a) :- p1(X2).\nand substitutionT1 -> a", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 29 [arrowhead = none ];
29 -> 14;

30 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 30 [arrowhead = none ];
30 -> 16;

31 [label="BACKTRACK\nfor clause: p(a) :- p1(X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 31 [arrowhead = none ];
31 -> 12;

32 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 32 [arrowhead = none ];
32 -> 21;

33 [label="ONLY EVAL with clause\np1(b) :- !_2.\nand substitutionX2 -> b", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 33 [arrowhead = none ];
33 -> 22;

34 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 34 [arrowhead = none ];
34 -> 23;

35 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 35 [arrowhead = none ];
35 -> 24;

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(b) → f2_out1
f2_in(a) → f2_out1

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f2_in

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: p(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: p(T1), SCOPE: 1, CLAUSE: 0 | p(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | p(b), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: p(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\np(T1) !=? p(b)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: p(b), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: p1(X1)\n\nX1 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: p1(X1), SCOPE: 2, CLAUSE: 2 | p1(X1), SCOPE: 2, CLAUSE: 3\n\nX1 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: !_2 | p1(X1), SCOPE: 2, CLAUSE: 3\n\nX1 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 21 [arrowhead = none ];
21 -> 4;

22 [label="EVAL with clause\np(b).\nand substitutionT1 -> b", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 22 [arrowhead = none ];
22 -> 6;

23 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 23 [arrowhead = none ];
23 -> 7;

24 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 24 [arrowhead = none ];
24 -> 10;

25 [label="EVAL with clause\np(a) :- p1(X1).\nand substitutionT1 -> a", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 25 [arrowhead = none ];
25 -> 13;

26 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 26 [arrowhead = none ];
26 -> 15;

27 [label="BACKTRACK\nfor clause: p(a) :- p1(X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 27 [arrowhead = none ];
27 -> 11;

28 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 28 [arrowhead = none ];
28 -> 17;

29 [label="ONLY EVAL with clause\np1(b) :- !_2.\nand substitutionX1 -> b", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 29 [arrowhead = none ];
29 -> 18;

30 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 30 [arrowhead = none ];
30 -> 19;

31 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 31 [arrowhead = none ];
31 -> 20;

}

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(b) → f1_out1
f1_in(a) → f1_out1

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f1_in

Defined Pair Symbols:none

Compound Symbols:none