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

(0) Obligation:

Clauses:

p(X, Y).
map(.(X, Xs), .(Y, Ys)) :- ','(p(X, Y), map(Xs, Ys)).
map([], []).

Query: map(g,a)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

p(X, Y).
map([], []).
map(.(X, Xs), .(Y, Ys)) :- ','(p(X, Y), map(Xs, Ys)).

Query: map(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];
1 [label="1: map(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: map(T1, T2), SCOPE: 1, CLAUSE: 1 | map(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | map([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: map(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nmap(T1, T2) !=? map([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: map([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(p(T11, T15), map(T12, T16))\n\nT11 is ground\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(T11, T15), map(T12, T16)), SCOPE: 2, CLAUSE: 0\n\nT11 is ground\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: map(T12, T24)\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 17 [arrowhead = none ];
17 -> 4;

18 [label="EVAL with clause\nmap([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 18 [arrowhead = none ];
18 -> 5;

19 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 19 [arrowhead = none ];
19 -> 7;

20 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 20 [arrowhead = none ];
20 -> 10;

21 [label="EVAL with clause\nmap(.(X13, X14), .(X15, X16)) :- ','(p(X13, X15), map(X14, X16)).\nand substitutionX13 -> T11,\nX14 -> T12,\nT1 -> .(T11, T12),\nX15 -> T15,\nX16 -> T16,\nT2 -> .(T15, T16),\nT13 -> T15,\nT14 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 21 [arrowhead = none ];
21 -> 13;

22 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 22 [arrowhead = none ];
22 -> 14;

23 [label="BACKTRACK\nfor clause: map(.(X, Xs), .(Y, Ys)) :- ','(p(X, Y), map(Xs, Ys))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 23 [arrowhead = none ];
23 -> 11;

24 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 24 [arrowhead = none ];
24 -> 15;

25 [label="ONLY EVAL with clause\np(X21, X22).\nand substitutionT11 -> T22,\nX21 -> T22,\nT15 -> T23,\nX22 -> T23,\nT16 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 25 [arrowhead = none ];
25 -> 16;

26 [label="INSTANCE with matching:\nT1 -> T12\nT2 -> T24", fontsize=14, style = filled, fillcolor = lightgrey];
16 -> 26 [arrowhead = none , style=dashed];
26 -> 1[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f1_in(z1), .(z0, z1))
U1(f1_out1, .(z0, z1)) → f1_out1

Tuples:

F1_IN(.(z0, z1)) → c1(U1'(f1_in(z1), .(z0, z1)), F1_IN(z1))

S tuples:

F1_IN(.(z0, z1)) → c1(U1'(f1_in(z1), .(z0, z1)), F1_IN(z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1

(5) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f1_in(z1), .(z0, z1))
U1(f1_out1, .(z0, z1)) → f1_out1

Tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

S tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1

(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

We considered the (Usable) Rules:none
And the Tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(.(x₁, x₂)) = [1] + x₂   
POL(F1_IN(x₁)) = x₁   
POL(c1(x₁)) = x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f1_in(z1), .(z0, z1))
U1(f1_out1, .(z0, z1)) → f1_out1

Tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

S tuples:none
K tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(10) BOUNDS(O(1), O(1))

(11) 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, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: map(T1, T2), SCOPE: 1, CLAUSE: 1 | map(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | map([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: map(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nmap(T1, T2) !=? map([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: map([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(p(T7, T11), map(T8, T12))\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(p(T7, T11), map(T8, T12)), SCOPE: 2, CLAUSE: 0\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: map(T8, T20)\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 21 [arrowhead = none ];
21 -> 3;

22 [label="EVAL with clause\nmap([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 22 [arrowhead = none ];
22 -> 6;

23 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 23 [arrowhead = none ];
23 -> 8;

24 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 24 [arrowhead = none ];
24 -> 9;

25 [label="EVAL with clause\nmap(.(X9, X10), .(X11, X12)) :- ','(p(X9, X11), map(X10, X12)).\nand substitutionX9 -> T7,\nX10 -> T8,\nT1 -> .(T7, T8),\nX11 -> T11,\nX12 -> T12,\nT2 -> .(T11, T12),\nT9 -> T11,\nT10 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 25 [arrowhead = none ];
25 -> 17;

26 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 26 [arrowhead = none ];
26 -> 18;

27 [label="BACKTRACK\nfor clause: map(.(X, Xs), .(Y, Ys)) :- ','(p(X, Y), map(Xs, Ys))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 27 [arrowhead = none ];
27 -> 12;

28 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 28 [arrowhead = none ];
28 -> 19;

29 [label="ONLY EVAL with clause\np(X17, X18).\nand substitutionT7 -> T18,\nX17 -> T18,\nT11 -> T19,\nX18 -> T19,\nT12 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 29 [arrowhead = none ];
29 -> 20;

30 [label="INSTANCE with matching:\nT1 -> T8\nT2 -> T20", fontsize=14, style = filled, fillcolor = lightgrey];
20 -> 30 [arrowhead = none , style=dashed];
30 -> 2[style=dashed];

}

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, z1)) → U1(f2_in(z1), .(z0, z1))
U1(f2_out1, .(z0, z1)) → f2_out1

Tuples:

F2_IN(.(z0, z1)) → c1(U1'(f2_in(z1), .(z0, z1)), F2_IN(z1))

S tuples:

F2_IN(.(z0, z1)) → c1(U1'(f2_in(z1), .(z0, z1)), F2_IN(z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1

(13) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, z1)) → U1(f2_in(z1), .(z0, z1))
U1(f2_out1, .(z0, z1)) → f2_out1

Tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

S tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1

(15) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

We considered the (Usable) Rules:none
And the Tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(.(x₁, x₂)) = [1] + x₂   
POL(F2_IN(x₁)) = x₁   
POL(c1(x₁)) = x₁   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, z1)) → U1(f2_in(z1), .(z0, z1))
U1(f2_out1, .(z0, z1)) → f2_out1

Tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

S tuples:none
K tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1