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

(0) Obligation:

Clauses:

p(X) :- ','(q(f(Y)), p(Y)).
p(g(X)) :- p(X).
q(g(Y)).

Query: p(g)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

q(g(Y)).
p(X) :- ','(q(f(Y)), p(Y)).
p(g(X)) :- p(X).

Query: p(g)

(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: p(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: p(T1), SCOPE: 1, CLAUSE: 1 | p(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: p(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: p(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(q(f(X14)), p(X14))\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(q(f(X14)), p(X14)), SCOPE: 2, CLAUSE: 0\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: p(T9)\n\nT9 is ground", 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];
1 -> 18 [arrowhead = none ];
18 -> 4;

19 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 19 [arrowhead = none ];
19 -> 6;

20 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 20 [arrowhead = none ];
20 -> 7;

21 [label="ONLY EVAL with clause\np(X13) :- ','(q(f(X14)), p(X14)).\nand substitutionT1 -> T6,\nX13 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 21 [arrowhead = none ];
21 -> 13;

22 [label="EVAL with clause\np(g(X21)) :- p(X21).\nand substitutionX21 -> T9,\nT1 -> g(T9)", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 22 [arrowhead = none ];
22 -> 16;

23 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 23 [arrowhead = none ];
23 -> 17;

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

25 [label="BACKTRACK\nfor clause: q(g(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 25 [arrowhead = none ];
25 -> 15;

26 [label="INSTANCE with matching:\nT1 -> T9", 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(z0) → U1(f4_in(z0), z0)
U1(f4_out1, z0) → f1_out1
U1(f4_out2, z0) → f1_out1
f7_in(g(z0)) → U2(f1_in(z0), g(z0))
U2(f1_out1, g(z0)) → f7_out1
f4_in(z0) → U3(f6_in(z0), f7_in(z0), z0)
U3(f6_out1, z0, z1) → f4_out1
U3(z0, f7_out1, z1) → f4_out2

Tuples:

F1_IN(z0) → c(U1'(f4_in(z0), z0), F4_IN(z0))
F7_IN(g(z0)) → c3(U2'(f1_in(z0), g(z0)), F1_IN(z0))
F4_IN(z0) → c5(U3'(f6_in(z0), f7_in(z0), z0), F7_IN(z0))

S tuples:

F1_IN(z0) → c(U1'(f4_in(z0), z0), F4_IN(z0))
F7_IN(g(z0)) → c3(U2'(f1_in(z0), g(z0)), F1_IN(z0))
F4_IN(z0) → c5(U3'(f6_in(z0), f7_in(z0), z0), F7_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, f4_in, U3

Defined Pair Symbols:

F1_IN, F7_IN, F4_IN

Compound Symbols:

c, c3, c5

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

Removed 3 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f4_in(z0), z0)
U1(f4_out1, z0) → f1_out1
U1(f4_out2, z0) → f1_out1
f7_in(g(z0)) → U2(f1_in(z0), g(z0))
U2(f1_out1, g(z0)) → f7_out1
f4_in(z0) → U3(f6_in(z0), f7_in(z0), z0)
U3(f6_out1, z0, z1) → f4_out1
U3(z0, f7_out1, z1) → f4_out2

Tuples:

F1_IN(z0) → c(F4_IN(z0))
F7_IN(g(z0)) → c3(F1_IN(z0))
F4_IN(z0) → c5(F7_IN(z0))

S tuples:

F1_IN(z0) → c(F4_IN(z0))
F7_IN(g(z0)) → c3(F1_IN(z0))
F4_IN(z0) → c5(F7_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f7_in, U2, f4_in, U3

Defined Pair Symbols:

F1_IN, F7_IN, F4_IN

Compound Symbols:

c, c3, c5

(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.

F7_IN(g(z0)) → c3(F1_IN(z0))
F4_IN(z0) → c5(F7_IN(z0))

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

F1_IN(z0) → c(F4_IN(z0))
F7_IN(g(z0)) → c3(F1_IN(z0))
F4_IN(z0) → c5(F7_IN(z0))

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

POL(F1_IN(x₁)) = [1] + x₁   
POL(F4_IN(x₁)) = [1] + x₁   
POL(F7_IN(x₁)) = x₁   
POL(c(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(g(x₁)) = [2] + x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → U1(f4_in(z0), z0)
U1(f4_out1, z0) → f1_out1
U1(f4_out2, z0) → f1_out1
f7_in(g(z0)) → U2(f1_in(z0), g(z0))
U2(f1_out1, g(z0)) → f7_out1
f4_in(z0) → U3(f6_in(z0), f7_in(z0), z0)
U3(f6_out1, z0, z1) → f4_out1
U3(z0, f7_out1, z1) → f4_out2

Tuples:

F1_IN(z0) → c(F4_IN(z0))
F7_IN(g(z0)) → c3(F1_IN(z0))
F4_IN(z0) → c5(F7_IN(z0))

S tuples:

F1_IN(z0) → c(F4_IN(z0))

K tuples:

F7_IN(g(z0)) → c3(F1_IN(z0))
F4_IN(z0) → c5(F7_IN(z0))

Defined Rule Symbols:

f1_in, U1, f7_in, U2, f4_in, U3

Defined Pair Symbols:

F1_IN, F7_IN, F4_IN

Compound Symbols:

c, c3, c5

(9) 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)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: p(T1), SCOPE: 1, CLAUSE: 1 | p(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(q(f(X3)), p(X3)) | p(T3), SCOPE: 1, CLAUSE: 2\n\nT3 is ground\nX3 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(q(f(X3)), p(X3)), SCOPE: 2, CLAUSE: 0 | ?_2 | p(T3), SCOPE: 1, CLAUSE: 2\n\nT3 is ground\nX3 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ?_2 | p(T3), SCOPE: 1, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: p(T3), SCOPE: 1, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: p(T6)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 13 [arrowhead = none ];
13 -> 3;

14 [label="ONLY EVAL with clause\np(X2) :- ','(q(f(X3)), p(X3)).\nand substitutionT1 -> T3,\nX2 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 14 [arrowhead = none ];
14 -> 5;

15 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 15 [arrowhead = none ];
15 -> 8;

16 [label="BACKTRACK\nfor clause: q(g(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 16 [arrowhead = none ];
16 -> 9;

17 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 17 [arrowhead = none ];
17 -> 10;

18 [label="EVAL with clause\np(g(X7)) :- p(X7).\nand substitutionX7 -> T6,\nT3 -> g(T6)", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 18 [arrowhead = none ];
18 -> 11;

19 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 19 [arrowhead = none ];
19 -> 12;

20 [label="INSTANCE with matching:\nT1 -> T6", fontsize=14, style = filled, fillcolor = lightgrey];
11 -> 20 [arrowhead = none , style=dashed];
20 -> 2[style=dashed];

}

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(g(z0)) → U1(f2_in(z0), g(z0))
U1(f2_out1, g(z0)) → f2_out1

Tuples:

F2_IN(g(z0)) → c(U1'(f2_in(z0), g(z0)), F2_IN(z0))

S tuples:

F2_IN(g(z0)) → c(U1'(f2_in(z0), g(z0)), F2_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c

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

Removed 1 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(g(z0)) → U1(f2_in(z0), g(z0))
U1(f2_out1, g(z0)) → f2_out1

Tuples:

F2_IN(g(z0)) → c(F2_IN(z0))

S tuples:

F2_IN(g(z0)) → c(F2_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c

(13) 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(g(z0)) → c(F2_IN(z0))

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

F2_IN(g(z0)) → c(F2_IN(z0))

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

POL(F2_IN(x₁)) = x₁   
POL(c(x₁)) = x₁   
POL(g(x₁)) = [1] + x₁   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(g(z0)) → U1(f2_in(z0), g(z0))
U1(f2_out1, g(z0)) → f2_out1

Tuples:

F2_IN(g(z0)) → c(F2_IN(z0))

S tuples:none
K tuples:

F2_IN(g(z0)) → c(F2_IN(z0))

Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c

(15) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(16) BOUNDS(O(1), O(1))