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

(0) Obligation:

Clauses:

minus(X, Y, Z) :- ','(f(X, 0), ','(!, =(Z, 0))).
f(X, Y) :- ','(!, =(X, Y)).
f(X, Y) :- f(X, Y).
=(X, X).

Query: minus(g,g,a)

(1) BuiltinConflictTransformerProof (BOTH BOUNDS(ID, ID) transformation)

Renamed defined predicates conflicting with built-in predicates [PROLOG].

(2) Obligation:

Clauses:

minus(X, Y, Z) :- ','(f(X, 0), ','(!, user_defined_=(Z, 0))).
f(X, Y) :- ','(!, user_defined_=(X, Y)).
f(X, Y) :- f(X, Y).
user_defined_=(X, X).

Query: minus(g,g,a)

(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: minus(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: minus(T1, T2, T3), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(f(T7, 0), ','(!_1, user_defined_=(T10, 0)))\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(f(T7, 0), ','(!_1, user_defined_=(T10, 0))), SCOPE: 2, CLAUSE: 1 | ','(f(T7, 0), ','(!_1, user_defined_=(T10, 0))), SCOPE: 2, CLAUSE: 2\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(','(!_2, user_defined_=(T13, 0)), ','(!_1, user_defined_=(T10, 0))) | ','(f(T13, 0), ','(!_1, user_defined_=(T10, 0))), SCOPE: 2, CLAUSE: 2\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(user_defined_=(T13, 0), ','(!_1, user_defined_=(T10, 0)))\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(user_defined_=(T13, 0), ','(!_1, user_defined_=(T10, 0))), SCOPE: 3, CLAUSE: 3\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(!_1, user_defined_=(T10, 0))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: user_defined_=(T10, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: user_defined_=(T10, 0), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 18 [arrowhead = none ];
18 -> 3;

19 [label="ONLY EVAL with clause\nminus(X4, X5, X6) :- ','(f(X4, 0), ','(!_1, user_defined_=(X6, 0))).\nand substitutionT1 -> T7,\nX4 -> T7,\nT2 -> T8,\nX5 -> T8,\nT3 -> T10,\nX6 -> T10,\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 19 [arrowhead = none ];
19 -> 5;

20 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 20 [arrowhead = none ];
20 -> 7;

21 [label="ONLY EVAL with clause\nf(X11, X12) :- ','(!_2, user_defined_=(X11, X12)).\nand substitutionT7 -> T13,\nX11 -> T13,\nX12 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 21 [arrowhead = none ];
21 -> 8;

22 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 22 [arrowhead = none ];
22 -> 9;

23 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 23 [arrowhead = none ];
23 -> 10;

24 [label="EVAL with clause\nuser_defined_=(X15, X15).\nand substitutionT13 -> 0,\nX15 -> 0,\nT16 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 24 [arrowhead = none ];
24 -> 11;

25 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 25 [arrowhead = none ];
25 -> 12;

26 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 26 [arrowhead = none ];
26 -> 13;

27 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 27 [arrowhead = none ];
27 -> 14;

28 [label="EVAL with clause\nuser_defined_=(X18, X18).\nand substitutionT10 -> 0,\nX18 -> 0,\nT19 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 28 [arrowhead = none ];
28 -> 15;

29 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 29 [arrowhead = none ];
29 -> 16;

30 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 30 [arrowhead = none ];
30 -> 17;

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0, z0) → f2_out1(0)

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

f2_in

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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