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

(0) Obligation:

Clauses:

p(X, Y) :- ','(q(X, Y), r(X)).
q(a, 0).
q(X, s(Y)) :- q(X, Y).
r(b) :- r(b).

Query: p(g,g)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

q(a, 0).
p(X, Y) :- ','(q(X, Y), r(X)).
q(X, s(Y)) :- q(X, Y).
r(b) :- r(b).

Query: p(g,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, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(q(T7, T8), r(T7))\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: q(T7, T8)\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: r(T7)\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: q(T7, T8), SCOPE: 2, CLAUSE: 0 | q(T7, T8), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true | q(a, 0), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: q(T7, T8), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nT8 is ground\nq(T7, T8) !=? q(a, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: q(a, 0), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: q(T13, T14)\n\nT13 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: r(T7), SCOPE: 3, CLAUSE: 3\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: r(b)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: r(b), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: r(b)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 31 [arrowhead = none ];
31 -> 4;

32 [label="ONLY EVAL with clause\np(X5, X6) :- ','(q(X5, X6), r(X5)).\nand substitutionT1 -> T7,\nX5 -> T7,\nT2 -> T8,\nX6 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 32 [arrowhead = none ];
32 -> 6;

33 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
6 -> 33 [arrowhead = none ];
33 -> 14;

34 [label="SPLIT 2\nnew knowledge:\nT7 is ground\nT8 is ground", fontsize=14, style = filled, fillcolor = violet];
6 -> 34 [arrowhead = none ];
34 -> 15;

35 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 35 [arrowhead = none ];
35 -> 17;

36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 36 [arrowhead = none ];
36 -> 26;

37 [label="EVAL with clause\nq(a, 0).\nand substitutionT7 -> a,\nT8 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 37 [arrowhead = none ];
37 -> 19;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 38 [arrowhead = none ];
38 -> 20;

39 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 39 [arrowhead = none ];
39 -> 21;

40 [label="EVAL with clause\nq(X13, s(X14)) :- q(X13, X14).\nand substitutionT7 -> T13,\nX13 -> T13,\nX14 -> T14,\nT8 -> s(T14)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 40 [arrowhead = none ];
40 -> 24;

41 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 41 [arrowhead = none ];
41 -> 25;

42 [label="BACKTRACK\nfor clause: q(X, s(Y)) :- q(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 42 [arrowhead = none ];
42 -> 22;

43 [label="INSTANCE with matching:\nT7 -> T13\nT8 -> T14", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 43 [arrowhead = none , style=dashed];
43 -> 14[style=dashed];

44 [label="EVAL with clause\nr(b) :- r(b).\nand substitutionT7 -> b", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 44 [arrowhead = none ];
44 -> 27;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 45 [arrowhead = none ];
45 -> 28;

46 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 46 [arrowhead = none ];
46 -> 29;

47 [label="ONLY EVAL with clause\nr(b) :- r(b).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 47 [arrowhead = none ];
47 -> 30;

48 [label="INSTANCE", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 48 [arrowhead = none , style=dashed];
48 -> 27[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f6_in(z0, z1), z0, z1)
U1(f6_out1, z0, z1) → f1_out1
f14_in(a, 0) → f14_out1
f14_in(z0, s(z1)) → U2(f14_in(z0, z1), z0, s(z1))
U2(f14_out1, z0, s(z1)) → f14_out1
f27_in → U3(f27_in)
U3(f27_out1) → f27_out1
f15_in(b) → U4(f27_in, b)
U4(f27_out1, b) → f15_out1
f6_in(z0, z1) → U5(f14_in(z0, z1), z0, z1)
U5(f14_out1, z0, z1) → U6(f15_in(z0), z0, z1)
U6(f15_out1, z0, z1) → f6_out1

Tuples:

F1_IN(z0, z1) → c(U1'(f6_in(z0, z1), z0, z1), F6_IN(z0, z1))
F14_IN(z0, s(z1)) → c3(U2'(f14_in(z0, z1), z0, s(z1)), F14_IN(z0, z1))
F27_IN → c5(U3'(f27_in), F27_IN)
F15_IN(b) → c7(U4'(f27_in, b), F27_IN)
F6_IN(z0, z1) → c9(U5'(f14_in(z0, z1), z0, z1), F14_IN(z0, z1))
U5'(f14_out1, z0, z1) → c10(U6'(f15_in(z0), z0, z1), F15_IN(z0))

S tuples:

F1_IN(z0, z1) → c(U1'(f6_in(z0, z1), z0, z1), F6_IN(z0, z1))
F14_IN(z0, s(z1)) → c3(U2'(f14_in(z0, z1), z0, s(z1)), F14_IN(z0, z1))
F27_IN → c5(U3'(f27_in), F27_IN)
F15_IN(b) → c7(U4'(f27_in, b), F27_IN)
F6_IN(z0, z1) → c9(U5'(f14_in(z0, z1), z0, z1), F14_IN(z0, z1))
U5'(f14_out1, z0, z1) → c10(U6'(f15_in(z0), z0, z1), F15_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f14_in, U2, f27_in, U3, f15_in, U4, f6_in, U5, U6

Defined Pair Symbols:

F1_IN, F14_IN, F27_IN, F15_IN, F6_IN, U5'

Compound Symbols:

c, c3, c5, c7, c9, c10

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

Split RHS of tuples not part of any SCC

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f6_in(z0, z1), z0, z1)
U1(f6_out1, z0, z1) → f1_out1
f14_in(a, 0) → f14_out1
f14_in(z0, s(z1)) → U2(f14_in(z0, z1), z0, s(z1))
U2(f14_out1, z0, s(z1)) → f14_out1
f27_in → U3(f27_in)
U3(f27_out1) → f27_out1
f15_in(b) → U4(f27_in, b)
U4(f27_out1, b) → f15_out1
f6_in(z0, z1) → U5(f14_in(z0, z1), z0, z1)
U5(f14_out1, z0, z1) → U6(f15_in(z0), z0, z1)
U6(f15_out1, z0, z1) → f6_out1

Tuples:

F14_IN(z0, s(z1)) → c3(U2'(f14_in(z0, z1), z0, s(z1)), F14_IN(z0, z1))
F27_IN → c5(U3'(f27_in), F27_IN)
F1_IN(z0, z1) → c1(U1'(f6_in(z0, z1), z0, z1))
F1_IN(z0, z1) → c1(F6_IN(z0, z1))
F15_IN(b) → c1(U4'(f27_in, b))
F15_IN(b) → c1(F27_IN)
F6_IN(z0, z1) → c1(U5'(f14_in(z0, z1), z0, z1))
F6_IN(z0, z1) → c1(F14_IN(z0, z1))
U5'(f14_out1, z0, z1) → c1(U6'(f15_in(z0), z0, z1))
U5'(f14_out1, z0, z1) → c1(F15_IN(z0))

S tuples:

F14_IN(z0, s(z1)) → c3(U2'(f14_in(z0, z1), z0, s(z1)), F14_IN(z0, z1))
F27_IN → c5(U3'(f27_in), F27_IN)
F1_IN(z0, z1) → c1(U1'(f6_in(z0, z1), z0, z1))
F1_IN(z0, z1) → c1(F6_IN(z0, z1))
F15_IN(b) → c1(U4'(f27_in, b))
F15_IN(b) → c1(F27_IN)
F6_IN(z0, z1) → c1(U5'(f14_in(z0, z1), z0, z1))
F6_IN(z0, z1) → c1(F14_IN(z0, z1))
U5'(f14_out1, z0, z1) → c1(U6'(f15_in(z0), z0, z1))
U5'(f14_out1, z0, z1) → c1(F15_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f14_in, U2, f27_in, U3, f15_in, U4, f6_in, U5, U6

Defined Pair Symbols:

F14_IN, F27_IN, F1_IN, F15_IN, F6_IN, U5'

Compound Symbols:

c3, c5, c1

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

Removed 5 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f6_in(z0, z1), z0, z1)
U1(f6_out1, z0, z1) → f1_out1
f14_in(a, 0) → f14_out1
f14_in(z0, s(z1)) → U2(f14_in(z0, z1), z0, s(z1))
U2(f14_out1, z0, s(z1)) → f14_out1
f27_in → U3(f27_in)
U3(f27_out1) → f27_out1
f15_in(b) → U4(f27_in, b)
U4(f27_out1, b) → f15_out1
f6_in(z0, z1) → U5(f14_in(z0, z1), z0, z1)
U5(f14_out1, z0, z1) → U6(f15_in(z0), z0, z1)
U6(f15_out1, z0, z1) → f6_out1

Tuples:

F1_IN(z0, z1) → c1(F6_IN(z0, z1))
F15_IN(b) → c1(F27_IN)
F6_IN(z0, z1) → c1(U5'(f14_in(z0, z1), z0, z1))
F6_IN(z0, z1) → c1(F14_IN(z0, z1))
U5'(f14_out1, z0, z1) → c1(F15_IN(z0))
F14_IN(z0, s(z1)) → c3(F14_IN(z0, z1))
F27_IN → c5(F27_IN)
F1_IN(z0, z1) → c1
F15_IN(b) → c1
U5'(f14_out1, z0, z1) → c1

S tuples:

F1_IN(z0, z1) → c1(F6_IN(z0, z1))
F15_IN(b) → c1(F27_IN)
F6_IN(z0, z1) → c1(U5'(f14_in(z0, z1), z0, z1))
F6_IN(z0, z1) → c1(F14_IN(z0, z1))
U5'(f14_out1, z0, z1) → c1(F15_IN(z0))
F14_IN(z0, s(z1)) → c3(F14_IN(z0, z1))
F27_IN → c5(F27_IN)
F1_IN(z0, z1) → c1
F15_IN(b) → c1
U5'(f14_out1, z0, z1) → c1

K tuples:none
Defined Rule Symbols:

f1_in, U1, f14_in, U2, f27_in, U3, f15_in, U4, f6_in, U5, U6

Defined Pair Symbols:

F1_IN, F15_IN, F6_IN, U5', F14_IN, F27_IN

Compound Symbols:

c1, c3, c5, c1

(9) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F1_IN(z0, z1) → c1(F6_IN(z0, z1))
F6_IN(z0, z1) → c1(U5'(f14_in(z0, z1), z0, z1))
F6_IN(z0, z1) → c1(F14_IN(z0, z1))
U5'(f14_out1, z0, z1) → c1(F15_IN(z0))
F1_IN(z0, z1) → c1
F15_IN(b) → c1
U5'(f14_out1, z0, z1) → c1
F6_IN(z0, z1) → c1(U5'(f14_in(z0, z1), z0, z1))
F6_IN(z0, z1) → c1(F14_IN(z0, z1))
U5'(f14_out1, z0, z1) → c1(F15_IN(z0))
U5'(f14_out1, z0, z1) → c1
F15_IN(b) → c1(F27_IN)
F15_IN(b) → c1

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f6_in(z0, z1), z0, z1)
U1(f6_out1, z0, z1) → f1_out1
f14_in(a, 0) → f14_out1
f14_in(z0, s(z1)) → U2(f14_in(z0, z1), z0, s(z1))
U2(f14_out1, z0, s(z1)) → f14_out1
f27_in → U3(f27_in)
U3(f27_out1) → f27_out1
f15_in(b) → U4(f27_in, b)
U4(f27_out1, b) → f15_out1
f6_in(z0, z1) → U5(f14_in(z0, z1), z0, z1)
U5(f14_out1, z0, z1) → U6(f15_in(z0), z0, z1)
U6(f15_out1, z0, z1) → f6_out1

Tuples:

F1_IN(z0, z1) → c1(F6_IN(z0, z1))
F15_IN(b) → c1(F27_IN)
F6_IN(z0, z1) → c1(U5'(f14_in(z0, z1), z0, z1))
F6_IN(z0, z1) → c1(F14_IN(z0, z1))
U5'(f14_out1, z0, z1) → c1(F15_IN(z0))
F14_IN(z0, s(z1)) → c3(F14_IN(z0, z1))
F27_IN → c5(F27_IN)
F1_IN(z0, z1) → c1
F15_IN(b) → c1
U5'(f14_out1, z0, z1) → c1

S tuples:

F14_IN(z0, s(z1)) → c3(F14_IN(z0, z1))
F27_IN → c5(F27_IN)

K tuples:

F1_IN(z0, z1) → c1(F6_IN(z0, z1))
F6_IN(z0, z1) → c1(U5'(f14_in(z0, z1), z0, z1))
F6_IN(z0, z1) → c1(F14_IN(z0, z1))
U5'(f14_out1, z0, z1) → c1(F15_IN(z0))
F1_IN(z0, z1) → c1
F15_IN(b) → c1
U5'(f14_out1, z0, z1) → c1
F15_IN(b) → c1(F27_IN)

Defined Rule Symbols:

f1_in, U1, f14_in, U2, f27_in, U3, f15_in, U4, f6_in, U5, U6

Defined Pair Symbols:

F1_IN, F15_IN, F6_IN, U5', F14_IN, F27_IN

Compound Symbols:

c1, c3, c5, c1

(11) 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.

F14_IN(z0, s(z1)) → c3(F14_IN(z0, z1))

We considered the (Usable) Rules:

f14_in(a, 0) → f14_out1
f14_in(z0, s(z1)) → U2(f14_in(z0, z1), z0, s(z1))
U2(f14_out1, z0, s(z1)) → f14_out1

And the Tuples:

F1_IN(z0, z1) → c1(F6_IN(z0, z1))
F15_IN(b) → c1(F27_IN)
F6_IN(z0, z1) → c1(U5'(f14_in(z0, z1), z0, z1))
F6_IN(z0, z1) → c1(F14_IN(z0, z1))
U5'(f14_out1, z0, z1) → c1(F15_IN(z0))
F14_IN(z0, s(z1)) → c3(F14_IN(z0, z1))
F27_IN → c5(F27_IN)
F1_IN(z0, z1) → c1
F15_IN(b) → c1
U5'(f14_out1, z0, z1) → c1

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

POL(0) = [1]   
POL(F14_IN(x₁, x₂)) = [1] + [2]x₁ + x₂   
POL(F15_IN(x₁)) = x₁   
POL(F1_IN(x₁, x₂)) = [3] + [3]x₁ + [2]x₂   
POL(F27_IN) = 0   
POL(F6_IN(x₁, x₂)) = [2] + [2]x₁ + x₂   
POL(U2(x₁, x₂, x₃)) = 0   
POL(U5'(x₁, x₂, x₃)) = [2] + [2]x₂ + x₃   
POL(a) = [1]   
POL(b) = 0   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(f14_in(x₁, x₂)) = [1] + [3]x₁ + [2]x₂   
POL(f14_out1) = 0   
POL(s(x₁)) = [2] + x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, z1) → U1(f6_in(z0, z1), z0, z1)
U1(f6_out1, z0, z1) → f1_out1
f14_in(a, 0) → f14_out1
f14_in(z0, s(z1)) → U2(f14_in(z0, z1), z0, s(z1))
U2(f14_out1, z0, s(z1)) → f14_out1
f27_in → U3(f27_in)
U3(f27_out1) → f27_out1
f15_in(b) → U4(f27_in, b)
U4(f27_out1, b) → f15_out1
f6_in(z0, z1) → U5(f14_in(z0, z1), z0, z1)
U5(f14_out1, z0, z1) → U6(f15_in(z0), z0, z1)
U6(f15_out1, z0, z1) → f6_out1

Tuples:

F1_IN(z0, z1) → c1(F6_IN(z0, z1))
F15_IN(b) → c1(F27_IN)
F6_IN(z0, z1) → c1(U5'(f14_in(z0, z1), z0, z1))
F6_IN(z0, z1) → c1(F14_IN(z0, z1))
U5'(f14_out1, z0, z1) → c1(F15_IN(z0))
F14_IN(z0, s(z1)) → c3(F14_IN(z0, z1))
F27_IN → c5(F27_IN)
F1_IN(z0, z1) → c1
F15_IN(b) → c1
U5'(f14_out1, z0, z1) → c1

S tuples:

F27_IN → c5(F27_IN)

K tuples:

F1_IN(z0, z1) → c1(F6_IN(z0, z1))
F6_IN(z0, z1) → c1(U5'(f14_in(z0, z1), z0, z1))
F6_IN(z0, z1) → c1(F14_IN(z0, z1))
U5'(f14_out1, z0, z1) → c1(F15_IN(z0))
F1_IN(z0, z1) → c1
F15_IN(b) → c1
U5'(f14_out1, z0, z1) → c1
F15_IN(b) → c1(F27_IN)
F14_IN(z0, s(z1)) → c3(F14_IN(z0, z1))

Defined Rule Symbols:

f1_in, U1, f14_in, U2, f27_in, U3, f15_in, U4, f6_in, U5, U6

Defined Pair Symbols:

F1_IN, F15_IN, F6_IN, U5', F14_IN, F27_IN

Compound Symbols:

c1, c3, c5, c1

(13) 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, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(q(T5, T6), r(T5))\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(q(T5, T6), r(T5)), SCOPE: 2, CLAUSE: 0 | ','(q(T5, T6), r(T5)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: r(a) | ','(q(a, 0), r(a)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(q(T5, T6), r(T5)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nT6 is ground\nq(T5, T6) !=? q(a, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: r(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(q(a, 0), r(a)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: r(a), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(q(T11, T12), r(T11))\n\nT11 is ground\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 24 [arrowhead = none ];
24 -> 3;

25 [label="ONLY EVAL with clause\np(X3, X4) :- ','(q(X3, X4), r(X3)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T6,\nX4 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 25 [arrowhead = none ];
25 -> 5;

26 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 26 [arrowhead = none ];
26 -> 7;

27 [label="EVAL with clause\nq(a, 0).\nand substitutionT5 -> a,\nT6 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 27 [arrowhead = none ];
27 -> 8;

28 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 28 [arrowhead = none ];
28 -> 9;

29 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 29 [arrowhead = none ];
29 -> 10;

30 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 30 [arrowhead = none ];
30 -> 11;

31 [label="EVAL with clause\nq(X11, s(X12)) :- q(X11, X12).\nand substitutionT5 -> T11,\nX11 -> T11,\nX12 -> T12,\nT6 -> s(T12)", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 31 [arrowhead = none ];
31 -> 18;

32 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 32 [arrowhead = none ];
32 -> 23;

33 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 33 [arrowhead = none ];
33 -> 12;

34 [label="BACKTRACK\nfor clause: q(X, s(Y)) :- q(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 34 [arrowhead = none ];
34 -> 16;

35 [label="BACKTRACK\nfor clause: r(b) :- r(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 35 [arrowhead = none ];
35 -> 13;

36 [label="INSTANCE with matching:\nT5 -> T11\nT6 -> T12", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 36 [arrowhead = none , style=dashed];
36 -> 5[style=dashed];

}

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1, z0, z1) → f2_out1
f5_in(a, 0) → U2(f8_in, a, 0)
f5_in(z0, s(z1)) → U3(f5_in(z0, z1), z0, s(z1))
U2(f8_out1, a, 0) → f5_out1
U2(f8_out2, a, 0) → f5_out1
U3(f5_out1, z0, s(z1)) → f5_out1
f8_in → U4(f10_in, f11_in)
U4(f10_out1, z0) → f8_out1
U4(z0, f11_out1) → f8_out2

Tuples:

F2_IN(z0, z1) → c(U1'(f5_in(z0, z1), z0, z1), F5_IN(z0, z1))
F5_IN(a, 0) → c2(U2'(f8_in, a, 0), F8_IN)
F5_IN(z0, s(z1)) → c3(U3'(f5_in(z0, z1), z0, s(z1)), F5_IN(z0, z1))
F8_IN → c7(U4'(f10_in, f11_in))

S tuples:

F2_IN(z0, z1) → c(U1'(f5_in(z0, z1), z0, z1), F5_IN(z0, z1))
F5_IN(a, 0) → c2(U2'(f8_in, a, 0), F8_IN)
F5_IN(z0, s(z1)) → c3(U3'(f5_in(z0, z1), z0, s(z1)), F5_IN(z0, z1))
F8_IN → c7(U4'(f10_in, f11_in))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f5_in, U2, U3, f8_in, U4

Defined Pair Symbols:

F2_IN, F5_IN, F8_IN

Compound Symbols:

c, c2, c3, c7

(15) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1, z0, z1) → f2_out1
f5_in(a, 0) → U2(f8_in, a, 0)
f5_in(z0, s(z1)) → U3(f5_in(z0, z1), z0, s(z1))
U2(f8_out1, a, 0) → f5_out1
U2(f8_out2, a, 0) → f5_out1
U3(f5_out1, z0, s(z1)) → f5_out1
f8_in → U4(f10_in, f11_in)
U4(f10_out1, z0) → f8_out1
U4(z0, f11_out1) → f8_out2

Tuples:

F5_IN(z0, s(z1)) → c3(U3'(f5_in(z0, z1), z0, s(z1)), F5_IN(z0, z1))
F8_IN → c7(U4'(f10_in, f11_in))
F2_IN(z0, z1) → c1(U1'(f5_in(z0, z1), z0, z1))
F2_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(a, 0) → c1(U2'(f8_in, a, 0))
F5_IN(a, 0) → c1(F8_IN)

S tuples:

F5_IN(z0, s(z1)) → c3(U3'(f5_in(z0, z1), z0, s(z1)), F5_IN(z0, z1))
F8_IN → c7(U4'(f10_in, f11_in))
F2_IN(z0, z1) → c1(U1'(f5_in(z0, z1), z0, z1))
F2_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(a, 0) → c1(U2'(f8_in, a, 0))
F5_IN(a, 0) → c1(F8_IN)

K tuples:none
Defined Rule Symbols:

f2_in, U1, f5_in, U2, U3, f8_in, U4

Defined Pair Symbols:

F5_IN, F8_IN, F2_IN

Compound Symbols:

c3, c7, c1

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

Removed 4 trailing tuple parts

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1, z0, z1) → f2_out1
f5_in(a, 0) → U2(f8_in, a, 0)
f5_in(z0, s(z1)) → U3(f5_in(z0, z1), z0, s(z1))
U2(f8_out1, a, 0) → f5_out1
U2(f8_out2, a, 0) → f5_out1
U3(f5_out1, z0, s(z1)) → f5_out1
f8_in → U4(f10_in, f11_in)
U4(f10_out1, z0) → f8_out1
U4(z0, f11_out1) → f8_out2

Tuples:

F2_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(a, 0) → c1(F8_IN)
F5_IN(z0, s(z1)) → c3(F5_IN(z0, z1))
F8_IN → c7
F2_IN(z0, z1) → c1
F5_IN(a, 0) → c1

S tuples:

F2_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(a, 0) → c1(F8_IN)
F5_IN(z0, s(z1)) → c3(F5_IN(z0, z1))
F8_IN → c7
F2_IN(z0, z1) → c1
F5_IN(a, 0) → c1

K tuples:none
Defined Rule Symbols:

f2_in, U1, f5_in, U2, U3, f8_in, U4

Defined Pair Symbols:

F2_IN, F5_IN, F8_IN

Compound Symbols:

c1, c3, c7, c1

(19) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F2_IN(z0, z1) → c1(F5_IN(z0, z1))
F2_IN(z0, z1) → c1

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1, z0, z1) → f2_out1
f5_in(a, 0) → U2(f8_in, a, 0)
f5_in(z0, s(z1)) → U3(f5_in(z0, z1), z0, s(z1))
U2(f8_out1, a, 0) → f5_out1
U2(f8_out2, a, 0) → f5_out1
U3(f5_out1, z0, s(z1)) → f5_out1
f8_in → U4(f10_in, f11_in)
U4(f10_out1, z0) → f8_out1
U4(z0, f11_out1) → f8_out2

Tuples:

F2_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(a, 0) → c1(F8_IN)
F5_IN(z0, s(z1)) → c3(F5_IN(z0, z1))
F8_IN → c7
F2_IN(z0, z1) → c1
F5_IN(a, 0) → c1

S tuples:

F5_IN(a, 0) → c1(F8_IN)
F5_IN(z0, s(z1)) → c3(F5_IN(z0, z1))
F8_IN → c7
F5_IN(a, 0) → c1

K tuples:

F2_IN(z0, z1) → c1(F5_IN(z0, z1))
F2_IN(z0, z1) → c1

Defined Rule Symbols:

f2_in, U1, f5_in, U2, U3, f8_in, U4

Defined Pair Symbols:

F2_IN, F5_IN, F8_IN

Compound Symbols:

c1, c3, c7, c1

(21) 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.

F5_IN(z0, s(z1)) → c3(F5_IN(z0, z1))

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

F2_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(a, 0) → c1(F8_IN)
F5_IN(z0, s(z1)) → c3(F5_IN(z0, z1))
F8_IN → c7
F2_IN(z0, z1) → c1
F5_IN(a, 0) → c1

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

POL(0) = 0   
POL(F2_IN(x₁, x₂)) = x₁ + [2]x₂   
POL(F5_IN(x₁, x₂)) = x₁ + x₂   
POL(F8_IN) = 0   
POL(a) = 0   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c7) = 0   
POL(s(x₁)) = [2] + x₁   

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, z1) → U1(f5_in(z0, z1), z0, z1)
U1(f5_out1, z0, z1) → f2_out1
f5_in(a, 0) → U2(f8_in, a, 0)
f5_in(z0, s(z1)) → U3(f5_in(z0, z1), z0, s(z1))
U2(f8_out1, a, 0) → f5_out1
U2(f8_out2, a, 0) → f5_out1
U3(f5_out1, z0, s(z1)) → f5_out1
f8_in → U4(f10_in, f11_in)
U4(f10_out1, z0) → f8_out1
U4(z0, f11_out1) → f8_out2

Tuples:

F2_IN(z0, z1) → c1(F5_IN(z0, z1))
F5_IN(a, 0) → c1(F8_IN)
F5_IN(z0, s(z1)) → c3(F5_IN(z0, z1))
F8_IN → c7
F2_IN(z0, z1) → c1
F5_IN(a, 0) → c1

S tuples:

F5_IN(a, 0) → c1(F8_IN)
F8_IN → c7
F5_IN(a, 0) → c1

K tuples:

F2_IN(z0, z1) → c1(F5_IN(z0, z1))
F2_IN(z0, z1) → c1
F5_IN(z0, s(z1)) → c3(F5_IN(z0, z1))

Defined Rule Symbols:

f2_in, U1, f5_in, U2, U3, f8_in, U4

Defined Pair Symbols:

F2_IN, F5_IN, F8_IN

Compound Symbols:

c1, c3, c7, c1

(23) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F5_IN(a, 0) → c1(F8_IN)
F8_IN → c7
F5_IN(a, 0) → c1
F8_IN → c7

Now S is empty

(24) BOUNDS(O(1), O(1))