Runtime Complexity of the query pattern
kay4(g)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(1)).

(0) Obligation:

Clauses:

kay4(I) :- ','(length(.(X1, .(X2, L)), I), ','(top(L), ','(mid([], L), bot(L)))).
top(L) :- ','(write(a), ','(mwrite('-', L), ','(write(b), nl))).
bot(L) :- ','(write(c), ','(mwrite('-', L), ','(write(d), nl))).
mid(X, []).
mid(X, .(X3, [])) :- ','(write('|'), ','(mwrite(' ', X), ','(write('X'), ','(mwrite(' ', X), ','(write('|'), nl))))).
mid(X, .(X4, .(X5, Y))) :- ','(write('|'), ','(mwrite(' ', X), ','(write('\\'), ','(mwrite(' ', Y), ','(write('/'), ','(mwrite(' ', X), ','(write('|'), ','(nl, ','(mid(.(X6, X), Y), ','(write('|'), ','(mwrite(' ', X), ','(write('/'), ','(mwrite(' ', Y), ','(write('\\'), ','(mwrite(' ', X), ','(write('|'), nl)))))))))))))))).
mwrite(Char, L) :- findall(X7, ','(member(X8, L), write(Char)), X9).
member(X, .(X, X10)).
member(X, .(X11, Xs)) :- member(X, Xs).

Query: kay4(g)

(1) 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: kay4(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: kay4(T1), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(length(.(X18, .(X19, X20)), T4), ','(top(X20), ','(mid([], X20), bot(X20))))\n\nT4 is ground\nX18 is free\nX19 is free\nX20 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 -> 4;

10 [label="ONLY EVAL with clause\nkay4(X17) :- ','(length(.(X18, .(X19, X20)), X17), ','(top(X20), ','(mid([], X20), bot(X20)))).\nand substitutionT1 -> T4,\nX17 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 10 [arrowhead = none ];
10 -> 6;

11 [label="UNDEFINED ERROR", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 11 [arrowhead = none ];
11 -> 8;

}

(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) 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: kay4(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: kay4(T1), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(length(.(X14, .(X15, X16)), T3), ','(top(X16), ','(mid([], X16), bot(X16))))\n\nT3 is ground\nX14 is free\nX15 is free\nX16 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 -> 3;

9 [label="ONLY EVAL with clause\nkay4(X13) :- ','(length(.(X14, .(X15, X16)), X13), ','(top(X16), ','(mid([], X16), bot(X16)))).\nand substitutionT1 -> T3,\nX13 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 9 [arrowhead = none ];
9 -> 5;

10 [label="UNDEFINED ERROR", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 10 [arrowhead = none ];
10 -> 7;

}

(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))