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

(0) Obligation:

Clauses:

p(X, g(X)).
p(X, f(X)) :- p(X, g(Y)).

Query: p(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: p(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: p(T1, T2), SCOPE: 1, CLAUSE: 0 | p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | p(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX3 is free\np(T1, T2) !=? p(X3, g(X3))", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: p(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: p(T8, g(X8))\n\nT8 is ground\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: p(T8, g(X8)), SCOPE: 2, CLAUSE: 0 | p(T8, g(X8)), SCOPE: 2, CLAUSE: 1\n\nT8 is ground\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true | p(T12, g(X8)), SCOPE: 2, CLAUSE: 1\n\nT12 is ground\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: p(T12, g(X8)), SCOPE: 2, CLAUSE: 1\n\nT12 is ground\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: p(T16, g(X18))\n\nT16 is ground\nX3 is free\nX18 is free\np(T16, T2) !=? p(X3, g(X3))", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 29 [arrowhead = none ];
29 -> 4;

30 [label="EVAL with clause\np(X3, g(X3)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> g(T5)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 30 [arrowhead = none ];
30 -> 6;

31 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 31 [arrowhead = none ];
31 -> 7;

32 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 32 [arrowhead = none ];
32 -> 10;

33 [label="EVAL with clause\np(X17, f(X17)) :- p(X17, g(X18)).\nand substitutionT1 -> T16,\nX17 -> T16,\nT2 -> f(T16)", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 33 [arrowhead = none ];
33 -> 26;

34 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 34 [arrowhead = none ];
34 -> 28;

35 [label="EVAL with clause\np(X7, f(X7)) :- p(X7, g(X8)).\nand substitutionT5 -> T8,\nX7 -> T8,\nT2 -> f(T8)", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 35 [arrowhead = none ];
35 -> 11;

36 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 36 [arrowhead = none ];
36 -> 12;

37 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 37 [arrowhead = none ];
37 -> 19;

38 [label="ONLY EVAL with clause\np(X12, g(X12)).\nand substitutionT8 -> T12,\nX12 -> T12,\nX8 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 38 [arrowhead = none ];
38 -> 21;

39 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 39 [arrowhead = none ];
39 -> 22;

40 [label="BACKTRACK\nfor clause: p(X, f(X)) :- p(X, g(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 40 [arrowhead = none ];
40 -> 24;

41 [label="INSTANCE with matching:\nT8 -> T16\nX8 -> X18", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 41 [arrowhead = none , style=dashed];
41 -> 11[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1(g(z0))
f2_in(z0) → U1(f11_in(z0), z0)
f2_in(z0) → U2(f11_in(z0), z0)
U1(f11_out1(z0), z1) → f2_out1(f(z1))
U2(f11_out1(z0), z1) → f2_out1(f(z1))
f11_in(z0) → f11_out1(z0)

Tuples:

F2_IN(z0) → c1(U1'(f11_in(z0), z0), F11_IN(z0))
F2_IN(z0) → c2(U2'(f11_in(z0), z0), F11_IN(z0))

S tuples:

F2_IN(z0) → c1(U1'(f11_in(z0), z0), F11_IN(z0))
F2_IN(z0) → c2(U2'(f11_in(z0), z0), F11_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f11_in

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1, c2

(3) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1(g(z0))
f2_in(z0) → U1(f11_in(z0), z0)
f2_in(z0) → U2(f11_in(z0), z0)
U1(f11_out1(z0), z1) → f2_out1(f(z1))
U2(f11_out1(z0), z1) → f2_out1(f(z1))
f11_in(z0) → f11_out1(z0)

Tuples:

F2_IN(z0) → c(U1'(f11_in(z0), z0))
F2_IN(z0) → c(F11_IN(z0))
F2_IN(z0) → c(U2'(f11_in(z0), z0))

S tuples:

F2_IN(z0) → c(U1'(f11_in(z0), z0))
F2_IN(z0) → c(F11_IN(z0))
F2_IN(z0) → c(U2'(f11_in(z0), z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f11_in

Defined Pair Symbols:

F2_IN

Compound Symbols:

c

(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, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: p(T1, T2), SCOPE: 1, CLAUSE: 0 | p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | p(T4, T2), SCOPE: 1, CLAUSE: 1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX2 is free\np(T1, T2) !=? p(X2, g(X2))", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: p(T4, T2), SCOPE: 1, CLAUSE: 1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: p(T6, g(X5))\n\nT6 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: p(T6, g(X5)), SCOPE: 2, CLAUSE: 0 | p(T6, g(X5)), SCOPE: 2, CLAUSE: 1\n\nT6 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true | p(T9, g(X5)), SCOPE: 2, CLAUSE: 1\n\nT9 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: p(T9, g(X5)), SCOPE: 2, CLAUSE: 1\n\nT9 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: p(T12, g(X12))\n\nT12 is ground\nX2 is free\nX12 is free\np(T12, T2) !=? p(X2, g(X2))", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: p(T12, g(X12)), SCOPE: 3, CLAUSE: 0 | p(T12, g(X12)), SCOPE: 3, CLAUSE: 1\n\nT12 is ground\nX2 is free\nX12 is free\np(T12, T2) !=? p(X2, g(X2))", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: true | p(T15, g(X12)), SCOPE: 3, CLAUSE: 1\n\nT15 is ground\nX2 is free\nX12 is free\np(T15, T2) !=? p(X2, g(X2))", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: p(T15, g(X12)), SCOPE: 3, CLAUSE: 1\n\nT15 is ground\nX2 is free\nX12 is free\np(T15, T2) !=? p(X2, g(X2))", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 31 [arrowhead = none ];
31 -> 3;

32 [label="EVAL with clause\np(X2, g(X2)).\nand substitutionT1 -> T4,\nX2 -> T4,\nT2 -> g(T4)", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 32 [arrowhead = none ];
32 -> 5;

33 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 33 [arrowhead = none ];
33 -> 8;

34 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 34 [arrowhead = none ];
34 -> 9;

35 [label="EVAL with clause\np(X11, f(X11)) :- p(X11, g(X12)).\nand substitutionT1 -> T12,\nX11 -> T12,\nT2 -> f(T12)", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 35 [arrowhead = none ];
35 -> 20;

36 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 36 [arrowhead = none ];
36 -> 23;

37 [label="EVAL with clause\np(X4, f(X4)) :- p(X4, g(X5)).\nand substitutionT4 -> T6,\nX4 -> T6,\nT2 -> f(T6)", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 37 [arrowhead = none ];
37 -> 13;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 38 [arrowhead = none ];
38 -> 14;

39 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 39 [arrowhead = none ];
39 -> 15;

40 [label="ONLY EVAL with clause\np(X8, g(X8)).\nand substitutionT6 -> T9,\nX8 -> T9,\nX5 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 40 [arrowhead = none ];
40 -> 16;

41 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 41 [arrowhead = none ];
41 -> 17;

42 [label="BACKTRACK\nfor clause: p(X, f(X)) :- p(X, g(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 42 [arrowhead = none ];
42 -> 18;

43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 43 [arrowhead = none ];
43 -> 25;

44 [label="ONLY EVAL with clause\np(X15, g(X15)).\nand substitutionT12 -> T15,\nX15 -> T15,\nX12 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 44 [arrowhead = none ];
44 -> 27;

45 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 45 [arrowhead = none ];
45 -> 29;

46 [label="BACKTRACK\nfor clause: p(X, f(X)) :- p(X, g(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 46 [arrowhead = none ];
46 -> 30;

}

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1(g(z0))
f1_in(z0) → f1_out1(f(z0))

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

f1_in

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))