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

(0) Obligation:

Clauses:

color_map([], X1).
color_map(.(Region, Regions), Colors) :- ','(color_region(Region, Colors), color_map(Regions, Colors)).
color_region(region(X2, Color, Neighbors), Colors) :- ','(select(Color, Colors, Colors1), subset(Neighbors, Colors1)).
select(X, .(X, Xs), Xs).
select(X, .(Y, Xs), .(Y, Zs)) :- select(X, Xs, Zs).
subset([], X3).
subset(.(X, Xs), Ys) :- ','(member(X, Ys), subset(Xs, Ys)).
member(X, .(X, X4)).
member(X, .(X5, Xs)) :- member(X, Xs).
map(.(region(belize, Belize, .(Guatemala, [])), .(region(guatemala, Guatemala, .(Belize, .(El_Salvador, .(Honduras, [])))), .(region(el_salvador, El_Salvador, .(Guatemala, .(Honduras, []))), .(region(honduras, Honduras, .(Guatemala, .(El_Salvador, .(Nicaragua, [])))), .(region(nicaragua, Nicaragua, .(Honduras, .(Costa_rica, []))), .(region(costa_rica, Costa_rica, .(Nicaragua, .(Panama, []))), .(region(panama, Panama, .(Costa_rica, [])), [])))))))).
query :- ','(map(X), color_map(X, .(red, .(blue, .(green, []))))).

Query: map(a)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

color_map([], X1).
select(X, .(X, Xs), Xs).
subset([], X3).
member(X, .(X, X4)).
map(.(region(belize, Belize, .(Guatemala, [])), .(region(guatemala, Guatemala, .(Belize, .(El_Salvador, .(Honduras, [])))), .(region(el_salvador, El_Salvador, .(Guatemala, .(Honduras, []))), .(region(honduras, Honduras, .(Guatemala, .(El_Salvador, .(Nicaragua, [])))), .(region(nicaragua, Nicaragua, .(Honduras, .(Costa_rica, []))), .(region(costa_rica, Costa_rica, .(Nicaragua, .(Panama, []))), .(region(panama, Panama, .(Costa_rica, [])), [])))))))).
color_map(.(Region, Regions), Colors) :- ','(color_region(Region, Colors), color_map(Regions, Colors)).
color_region(region(X2, Color, Neighbors), Colors) :- ','(select(Color, Colors, Colors1), subset(Neighbors, Colors1)).
select(X, .(Y, Xs), .(Y, Zs)) :- select(X, Xs, Zs).
subset(.(X, Xs), Ys) :- ','(member(X, Ys), subset(Xs, Ys)).
member(X, .(X5, Xs)) :- member(X, Xs).
query :- ','(map(X), color_map(X, .(red, .(blue, .(green, []))))).

Query: map(a)

(3) 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: map(T1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: map(T1), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 11 [arrowhead = none ];
11 -> 3;

12 [label="EVAL with clause\nmap(.(region(belize, X20, .(X21, [])), .(region(guatemala, X21, .(X20, .(X22, .(X23, [])))), .(region(el_salvador, X22, .(X21, .(X23, []))), .(region(honduras, X23, .(X21, .(X22, .(X24, [])))), .(region(nicaragua, X24, .(X23, .(X25, []))), .(region(costa_rica, X25, .(X24, .(X26, []))), .(region(panama, X26, .(X25, [])), [])))))))).\nand substitutionX20 -> T16,\nX21 -> T17,\nX22 -> T18,\nX23 -> T19,\nX24 -> T20,\nX25 -> T21,\nX26 -> T22,\nT1 -> .(region(belize, T16, .(T17, [])), .(region(guatemala, T17, .(T16, .(T18, .(T19, [])))), .(region(el_salvador, T18, .(T17, .(T19, []))), .(region(honduras, T19, .(T17, .(T18, .(T20, [])))), .(region(nicaragua, T20, .(T19, .(T21, []))), .(region(costa_rica, T21, .(T20, .(T22, []))), .(region(panama, T22, .(T21, [])), [])))))))", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 12 [arrowhead = none ];
12 -> 6;

13 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 13 [arrowhead = none ];
13 -> 9;

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

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → f1_out1

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

f1_in

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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

(7) 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: map(T1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: map(T1), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", 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];
2 -> 9 [arrowhead = none ];
9 -> 4;

10 [label="EVAL with clause\nmap(.(region(belize, X13, .(X14, [])), .(region(guatemala, X14, .(X13, .(X15, .(X16, [])))), .(region(el_salvador, X15, .(X14, .(X16, []))), .(region(honduras, X16, .(X14, .(X15, .(X17, [])))), .(region(nicaragua, X17, .(X16, .(X18, []))), .(region(costa_rica, X18, .(X17, .(X19, []))), .(region(panama, X19, .(X18, [])), [])))))))).\nand substitutionX13 -> T9,\nX14 -> T10,\nX15 -> T11,\nX16 -> T12,\nX17 -> T13,\nX18 -> T14,\nX19 -> T15,\nT1 -> .(region(belize, T9, .(T10, [])), .(region(guatemala, T10, .(T9, .(T11, .(T12, [])))), .(region(el_salvador, T11, .(T10, .(T12, []))), .(region(honduras, T12, .(T10, .(T11, .(T13, [])))), .(region(nicaragua, T13, .(T12, .(T14, []))), .(region(costa_rica, T14, .(T13, .(T15, []))), .(region(panama, T15, .(T14, [])), [])))))))", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 10 [arrowhead = none ];
10 -> 5;

11 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 11 [arrowhead = none ];
11 -> 7;

12 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 12 [arrowhead = none ];
12 -> 8;

}

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → f2_out1

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

f2_in

Defined Pair Symbols:none

Compound Symbols:none