Runtime Complexity proof of /tmp/tmp4tBeLR/ag01.pl

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

0 Prolog

↳1 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 61 ms)

↳2 CdtProblem

    ↳3 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳4 CdtProblem

    ↳5 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳6 CdtProblem

    ↳7 CdtKnowledgeProof (⇔, 0 ms)

    ↳8 CdtProblem

    ↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 270 ms)

    ↳10 CdtProblem

    ↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 54 ms)

    ↳12 CdtProblem

    ↳13 SIsEmptyProof (⇔, 0 ms)

    ↳14 BOUNDS(O(1), O(1))

↳15 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 77 ms)

↳16 CdtProblem

    ↳17 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳18 CdtProblem

    ↳19 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳20 CdtProblem

    ↳21 CdtKnowledgeProof (⇔, 0 ms)

    ↳22 CdtProblem

    ↳23 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 227 ms)

    ↳24 CdtProblem

    ↳25 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 55 ms)

    ↳26 CdtProblem

(0) Obligation:

Clauses:

f(c(s(X), Y)) :- f(c(X, s(Y))).
g(c(X, s(Y))) :- g(c(s(X), Y)).
h(X) :- ','(f(X), g(X)).

Query: h(g)

(1) 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: h(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: h(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(f(T6), g(T6))\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: f(T6)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: g(T6)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: f(T6), SCOPE: 2, CLAUSE: 0\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: f(c(T15, s(T16)))\n\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: g(T6), SCOPE: 3, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: g(c(s(T25), T26))\n\nT25 is ground\nT26 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\nh(X5) :- ','(f(X5), g(X5)).\nand substitutionT1 -> T6,\nX5 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 25 [arrowhead = none ];
25 -> 6;

26 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
6 -> 26 [arrowhead = none ];
26 -> 10;

27 [label="SPLIT 2\nnew knowledge:\nT6 is ground", fontsize=14, style = filled, fillcolor = violet];
6 -> 27 [arrowhead = none ];
27 -> 12;

28 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 28 [arrowhead = none ];
28 -> 14;

29 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 29 [arrowhead = none ];
29 -> 21;

30 [label="EVAL with clause\nf(c(s(X14), X15)) :- f(c(X14, s(X15))).\nand substitutionX14 -> T15,\nX15 -> T16,\nT6 -> c(s(T15), T16)", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 30 [arrowhead = none ];
30 -> 18;

31 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 31 [arrowhead = none ];
31 -> 19;

32 [label="INSTANCE with matching:\nT6 -> c(T15, s(T16))", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 32 [arrowhead = none , style=dashed];
32 -> 10[style=dashed];

33 [label="EVAL with clause\ng(c(X24, s(X25))) :- g(c(s(X24), X25)).\nand substitutionX24 -> T25,\nX25 -> T26,\nT6 -> c(T25, s(T26))", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 33 [arrowhead = none ];
33 -> 22;

34 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 34 [arrowhead = none ];
34 -> 23;

35 [label="INSTANCE with matching:\nT6 -> c(s(T25), T26)", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 35 [arrowhead = none , style=dashed];
35 -> 12[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f6_in(z0), z0)
U1(f6_out1, z0) → f2_out1
f10_in(c(s(z0), z1)) → U2(f10_in(c(z0, s(z1))), c(s(z0), z1))
U2(f10_out1, c(s(z0), z1)) → f10_out1
f12_in(c(z0, s(z1))) → U3(f12_in(c(s(z0), z1)), c(z0, s(z1)))
U3(f12_out1, c(z0, s(z1))) → f12_out1
f6_in(z0) → U4(f10_in(z0), z0)
U4(f10_out1, z0) → U5(f12_in(z0), z0)
U5(f12_out1, z0) → f6_out1

Tuples:

F2_IN(z0) → c1(U1'(f6_in(z0), z0), F6_IN(z0))
F10_IN(c(s(z0), z1)) → c3(U2'(f10_in(c(z0, s(z1))), c(s(z0), z1)), F10_IN(c(z0, s(z1))))
F12_IN(c(z0, s(z1))) → c5(U3'(f12_in(c(s(z0), z1)), c(z0, s(z1))), F12_IN(c(s(z0), z1)))
F6_IN(z0) → c7(U4'(f10_in(z0), z0), F10_IN(z0))
U4'(f10_out1, z0) → c8(U5'(f12_in(z0), z0), F12_IN(z0))

S tuples:

F2_IN(z0) → c1(U1'(f6_in(z0), z0), F6_IN(z0))
F10_IN(c(s(z0), z1)) → c3(U2'(f10_in(c(z0, s(z1))), c(s(z0), z1)), F10_IN(c(z0, s(z1))))
F12_IN(c(z0, s(z1))) → c5(U3'(f12_in(c(s(z0), z1)), c(z0, s(z1))), F12_IN(c(s(z0), z1)))
F6_IN(z0) → c7(U4'(f10_in(z0), z0), F10_IN(z0))
U4'(f10_out1, z0) → c8(U5'(f12_in(z0), z0), F12_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f10_in, U2, f12_in, U3, f6_in, U4, U5

Defined Pair Symbols:

F2_IN, F10_IN, F12_IN, F6_IN, U4'

Compound Symbols:

c1, c3, c5, c7, c8

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

Split RHS of tuples not part of any SCC

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f6_in(z0), z0)
U1(f6_out1, z0) → f2_out1
f10_in(c(s(z0), z1)) → U2(f10_in(c(z0, s(z1))), c(s(z0), z1))
U2(f10_out1, c(s(z0), z1)) → f10_out1
f12_in(c(z0, s(z1))) → U3(f12_in(c(s(z0), z1)), c(z0, s(z1)))
U3(f12_out1, c(z0, s(z1))) → f12_out1
f6_in(z0) → U4(f10_in(z0), z0)
U4(f10_out1, z0) → U5(f12_in(z0), z0)
U5(f12_out1, z0) → f6_out1

Tuples:

F10_IN(c(s(z0), z1)) → c3(U2'(f10_in(c(z0, s(z1))), c(s(z0), z1)), F10_IN(c(z0, s(z1))))
F12_IN(c(z0, s(z1))) → c5(U3'(f12_in(c(s(z0), z1)), c(z0, s(z1))), F12_IN(c(s(z0), z1)))
F2_IN(z0) → c2(U1'(f6_in(z0), z0))
F2_IN(z0) → c2(F6_IN(z0))
F6_IN(z0) → c2(U4'(f10_in(z0), z0))
F6_IN(z0) → c2(F10_IN(z0))
U4'(f10_out1, z0) → c2(U5'(f12_in(z0), z0))
U4'(f10_out1, z0) → c2(F12_IN(z0))

S tuples:

F10_IN(c(s(z0), z1)) → c3(U2'(f10_in(c(z0, s(z1))), c(s(z0), z1)), F10_IN(c(z0, s(z1))))
F12_IN(c(z0, s(z1))) → c5(U3'(f12_in(c(s(z0), z1)), c(z0, s(z1))), F12_IN(c(s(z0), z1)))
F2_IN(z0) → c2(U1'(f6_in(z0), z0))
F2_IN(z0) → c2(F6_IN(z0))
F6_IN(z0) → c2(U4'(f10_in(z0), z0))
F6_IN(z0) → c2(F10_IN(z0))
U4'(f10_out1, z0) → c2(U5'(f12_in(z0), z0))
U4'(f10_out1, z0) → c2(F12_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f10_in, U2, f12_in, U3, f6_in, U4, U5

Defined Pair Symbols:

F10_IN, F12_IN, F2_IN, F6_IN, U4'

Compound Symbols:

c3, c5, c2

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

Removed 4 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f6_in(z0), z0)
U1(f6_out1, z0) → f2_out1
f10_in(c(s(z0), z1)) → U2(f10_in(c(z0, s(z1))), c(s(z0), z1))
U2(f10_out1, c(s(z0), z1)) → f10_out1
f12_in(c(z0, s(z1))) → U3(f12_in(c(s(z0), z1)), c(z0, s(z1)))
U3(f12_out1, c(z0, s(z1))) → f12_out1
f6_in(z0) → U4(f10_in(z0), z0)
U4(f10_out1, z0) → U5(f12_in(z0), z0)
U5(f12_out1, z0) → f6_out1

Tuples:

F2_IN(z0) → c2(F6_IN(z0))
F6_IN(z0) → c2(U4'(f10_in(z0), z0))
F6_IN(z0) → c2(F10_IN(z0))
U4'(f10_out1, z0) → c2(F12_IN(z0))
F10_IN(c(s(z0), z1)) → c3(F10_IN(c(z0, s(z1))))
F12_IN(c(z0, s(z1))) → c5(F12_IN(c(s(z0), z1)))
F2_IN(z0) → c2
U4'(f10_out1, z0) → c2

S tuples:

F2_IN(z0) → c2(F6_IN(z0))
F6_IN(z0) → c2(U4'(f10_in(z0), z0))
F6_IN(z0) → c2(F10_IN(z0))
U4'(f10_out1, z0) → c2(F12_IN(z0))
F10_IN(c(s(z0), z1)) → c3(F10_IN(c(z0, s(z1))))
F12_IN(c(z0, s(z1))) → c5(F12_IN(c(s(z0), z1)))
F2_IN(z0) → c2
U4'(f10_out1, z0) → c2

K tuples:none
Defined Rule Symbols:

f2_in, U1, f10_in, U2, f12_in, U3, f6_in, U4, U5

Defined Pair Symbols:

F2_IN, F6_IN, U4', F10_IN, F12_IN

Compound Symbols:

c2, c3, c5, c2

(7) CdtKnowledgeProof (EQUIVALENT transformation)

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

F2_IN(z0) → c2(F6_IN(z0))
F6_IN(z0) → c2(U4'(f10_in(z0), z0))
F6_IN(z0) → c2(F10_IN(z0))
U4'(f10_out1, z0) → c2(F12_IN(z0))
F2_IN(z0) → c2
U4'(f10_out1, z0) → c2
F6_IN(z0) → c2(U4'(f10_in(z0), z0))
F6_IN(z0) → c2(F10_IN(z0))
U4'(f10_out1, z0) → c2(F12_IN(z0))
U4'(f10_out1, z0) → c2

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f6_in(z0), z0)
U1(f6_out1, z0) → f2_out1
f10_in(c(s(z0), z1)) → U2(f10_in(c(z0, s(z1))), c(s(z0), z1))
U2(f10_out1, c(s(z0), z1)) → f10_out1
f12_in(c(z0, s(z1))) → U3(f12_in(c(s(z0), z1)), c(z0, s(z1)))
U3(f12_out1, c(z0, s(z1))) → f12_out1
f6_in(z0) → U4(f10_in(z0), z0)
U4(f10_out1, z0) → U5(f12_in(z0), z0)
U5(f12_out1, z0) → f6_out1

Tuples:

F2_IN(z0) → c2(F6_IN(z0))
F6_IN(z0) → c2(U4'(f10_in(z0), z0))
F6_IN(z0) → c2(F10_IN(z0))
U4'(f10_out1, z0) → c2(F12_IN(z0))
F10_IN(c(s(z0), z1)) → c3(F10_IN(c(z0, s(z1))))
F12_IN(c(z0, s(z1))) → c5(F12_IN(c(s(z0), z1)))
F2_IN(z0) → c2
U4'(f10_out1, z0) → c2

S tuples:

F10_IN(c(s(z0), z1)) → c3(F10_IN(c(z0, s(z1))))
F12_IN(c(z0, s(z1))) → c5(F12_IN(c(s(z0), z1)))

K tuples:

F2_IN(z0) → c2(F6_IN(z0))
F6_IN(z0) → c2(U4'(f10_in(z0), z0))
F6_IN(z0) → c2(F10_IN(z0))
U4'(f10_out1, z0) → c2(F12_IN(z0))
F2_IN(z0) → c2
U4'(f10_out1, z0) → c2

Defined Rule Symbols:

f2_in, U1, f10_in, U2, f12_in, U3, f6_in, U4, U5

Defined Pair Symbols:

F2_IN, F6_IN, U4', F10_IN, F12_IN

Compound Symbols:

c2, c3, c5, c2

(9) 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.

F10_IN(c(s(z0), z1)) → c3(F10_IN(c(z0, s(z1))))

We considered the (Usable) Rules:

f10_in(c(s(z0), z1)) → U2(f10_in(c(z0, s(z1))), c(s(z0), z1))
U2(f10_out1, c(s(z0), z1)) → f10_out1

And the Tuples:

F2_IN(z0) → c2(F6_IN(z0))
F6_IN(z0) → c2(U4'(f10_in(z0), z0))
F6_IN(z0) → c2(F10_IN(z0))
U4'(f10_out1, z0) → c2(F12_IN(z0))
F10_IN(c(s(z0), z1)) → c3(F10_IN(c(z0, s(z1))))
F12_IN(c(z0, s(z1))) → c5(F12_IN(c(s(z0), z1)))
F2_IN(z0) → c2
U4'(f10_out1, z0) → c2

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

POL(F10_IN(x₁)) = [1] + x₁
POL(F12_IN(x₁)) = [1]
POL(F2_IN(x₁)) = [2] + [2]x₁
POL(F6_IN(x₁)) = [2] + [2]x₁
POL(U2(x₁, x₂)) = 0
POL(U4'(x₁, x₂)) = [2]
POL(c(x₁, x₂)) = x₁
POL(c2) = 0
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(f10_in(x₁)) = [2]x₁
POL(f10_out1) = 0
POL(s(x₁)) = [2] + x₁

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f6_in(z0), z0)
U1(f6_out1, z0) → f2_out1
f10_in(c(s(z0), z1)) → U2(f10_in(c(z0, s(z1))), c(s(z0), z1))
U2(f10_out1, c(s(z0), z1)) → f10_out1
f12_in(c(z0, s(z1))) → U3(f12_in(c(s(z0), z1)), c(z0, s(z1)))
U3(f12_out1, c(z0, s(z1))) → f12_out1
f6_in(z0) → U4(f10_in(z0), z0)
U4(f10_out1, z0) → U5(f12_in(z0), z0)
U5(f12_out1, z0) → f6_out1

Tuples:

F2_IN(z0) → c2(F6_IN(z0))
F6_IN(z0) → c2(U4'(f10_in(z0), z0))
F6_IN(z0) → c2(F10_IN(z0))
U4'(f10_out1, z0) → c2(F12_IN(z0))
F10_IN(c(s(z0), z1)) → c3(F10_IN(c(z0, s(z1))))
F12_IN(c(z0, s(z1))) → c5(F12_IN(c(s(z0), z1)))
F2_IN(z0) → c2
U4'(f10_out1, z0) → c2

S tuples:

F12_IN(c(z0, s(z1))) → c5(F12_IN(c(s(z0), z1)))

K tuples:

F2_IN(z0) → c2(F6_IN(z0))
F6_IN(z0) → c2(U4'(f10_in(z0), z0))
F6_IN(z0) → c2(F10_IN(z0))
U4'(f10_out1, z0) → c2(F12_IN(z0))
F2_IN(z0) → c2
U4'(f10_out1, z0) → c2
F10_IN(c(s(z0), z1)) → c3(F10_IN(c(z0, s(z1))))

Defined Rule Symbols:

f2_in, U1, f10_in, U2, f12_in, U3, f6_in, U4, U5

Defined Pair Symbols:

F2_IN, F6_IN, U4', F10_IN, F12_IN

Compound Symbols:

c2, c3, c5, c2

(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.

F12_IN(c(z0, s(z1))) → c5(F12_IN(c(s(z0), z1)))

We considered the (Usable) Rules:

f10_in(c(s(z0), z1)) → U2(f10_in(c(z0, s(z1))), c(s(z0), z1))
U2(f10_out1, c(s(z0), z1)) → f10_out1

And the Tuples:

F2_IN(z0) → c2(F6_IN(z0))
F6_IN(z0) → c2(U4'(f10_in(z0), z0))
F6_IN(z0) → c2(F10_IN(z0))
U4'(f10_out1, z0) → c2(F12_IN(z0))
F10_IN(c(s(z0), z1)) → c3(F10_IN(c(z0, s(z1))))
F12_IN(c(z0, s(z1))) → c5(F12_IN(c(s(z0), z1)))
F2_IN(z0) → c2
U4'(f10_out1, z0) → c2

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

POL(F10_IN(x₁)) = 0
POL(F12_IN(x₁)) = [1] + x₁
POL(F2_IN(x₁)) = [3] + x₁
POL(F6_IN(x₁)) = [3] + x₁
POL(U2(x₁, x₂)) = [2] + [2]x₂
POL(U4'(x₁, x₂)) = [2] + x₂
POL(c(x₁, x₂)) = x₂
POL(c2) = 0
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(f10_in(x₁)) = [3] + [2]x₁
POL(f10_out1) = 0
POL(s(x₁)) = [2] + x₁

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → U1(f6_in(z0), z0)
U1(f6_out1, z0) → f2_out1
f10_in(c(s(z0), z1)) → U2(f10_in(c(z0, s(z1))), c(s(z0), z1))
U2(f10_out1, c(s(z0), z1)) → f10_out1
f12_in(c(z0, s(z1))) → U3(f12_in(c(s(z0), z1)), c(z0, s(z1)))
U3(f12_out1, c(z0, s(z1))) → f12_out1
f6_in(z0) → U4(f10_in(z0), z0)
U4(f10_out1, z0) → U5(f12_in(z0), z0)
U5(f12_out1, z0) → f6_out1

Tuples:

F2_IN(z0) → c2(F6_IN(z0))
F6_IN(z0) → c2(U4'(f10_in(z0), z0))
F6_IN(z0) → c2(F10_IN(z0))
U4'(f10_out1, z0) → c2(F12_IN(z0))
F10_IN(c(s(z0), z1)) → c3(F10_IN(c(z0, s(z1))))
F12_IN(c(z0, s(z1))) → c5(F12_IN(c(s(z0), z1)))
F2_IN(z0) → c2
U4'(f10_out1, z0) → c2

S tuples:none
K tuples:

F2_IN(z0) → c2(F6_IN(z0))
F6_IN(z0) → c2(U4'(f10_in(z0), z0))
F6_IN(z0) → c2(F10_IN(z0))
U4'(f10_out1, z0) → c2(F12_IN(z0))
F2_IN(z0) → c2
U4'(f10_out1, z0) → c2
F10_IN(c(s(z0), z1)) → c3(F10_IN(c(z0, s(z1))))
F12_IN(c(z0, s(z1))) → c5(F12_IN(c(s(z0), z1)))

Defined Rule Symbols:

f2_in, U1, f10_in, U2, f12_in, U3, f6_in, U4, U5

Defined Pair Symbols:

F2_IN, F6_IN, U4', F10_IN, F12_IN

Compound Symbols:

c2, c3, c5, c2

(13) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(14) BOUNDS(O(1), O(1))

(15) 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: h(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: h(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(f(T3), g(T3))\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(f(T3), g(T3)), SCOPE: 2, CLAUSE: 0\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(f(c(T8, s(T9))), g(c(s(T8), T9)))\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: f(c(T8, s(T9)))\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: g(c(s(T8), T9))\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: f(c(T8, s(T9))), SCOPE: 3, CLAUSE: 0\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: f(c(T18, s(s(T19))))\n\nT18 is ground\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: g(c(s(T8), T9)), SCOPE: 4, CLAUSE: 1\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: g(c(s(s(T28)), T29))\n\nT28 is ground\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 26 [arrowhead = none ];
26 -> 4;

27 [label="ONLY EVAL with clause\nh(X2) :- ','(f(X2), g(X2)).\nand substitutionT1 -> T3,\nX2 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 27 [arrowhead = none ];
27 -> 5;

28 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 28 [arrowhead = none ];
28 -> 7;

29 [label="EVAL with clause\nf(c(s(X7), X8)) :- f(c(X7, s(X8))).\nand substitutionX7 -> T8,\nX8 -> T9,\nT3 -> c(s(T8), T9)", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 29 [arrowhead = none ];
29 -> 8;

30 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 30 [arrowhead = none ];
30 -> 9;

31 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
8 -> 31 [arrowhead = none ];
31 -> 11;

32 [label="SPLIT 2\nnew knowledge:\nT8 is ground\nT9 is ground", fontsize=14, style = filled, fillcolor = violet];
8 -> 32 [arrowhead = none ];
32 -> 13;

33 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 33 [arrowhead = none ];
33 -> 15;

34 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 34 [arrowhead = none ];
34 -> 20;

35 [label="EVAL with clause\nf(c(s(X17), X18)) :- f(c(X17, s(X18))).\nand substitutionX17 -> T18,\nT8 -> s(T18),\nT9 -> T19,\nX18 -> s(T19)", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 35 [arrowhead = none ];
35 -> 16;

36 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 36 [arrowhead = none ];
36 -> 17;

37 [label="INSTANCE with matching:\nT8 -> T18\nT9 -> s(T19)", fontsize=14, style = filled, fillcolor = lightgrey];
16 -> 37 [arrowhead = none , style=dashed];
37 -> 11[style=dashed];

38 [label="EVAL with clause\ng(c(X27, s(X28))) :- g(c(s(X27), X28)).\nand substitutionT8 -> T28,\nX27 -> s(T28),\nX28 -> T29,\nT9 -> s(T29)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 38 [arrowhead = none ];
38 -> 24;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 39 [arrowhead = none ];
39 -> 25;

40 [label="INSTANCE with matching:\nT8 -> s(T28)\nT9 -> T29", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 40 [arrowhead = none , style=dashed];
40 -> 13[style=dashed];

}

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(c(s(z0), z1)) → U1(f8_in(z0, z1), c(s(z0), z1))
U1(f8_out1, c(s(z0), z1)) → f1_out1
f11_in(s(z0), z1) → U2(f11_in(z0, s(z1)), s(z0), z1)
U2(f11_out1, s(z0), z1) → f11_out1
f13_in(z0, s(z1)) → U3(f13_in(s(z0), z1), z0, s(z1))
U3(f13_out1, z0, s(z1)) → f13_out1
f8_in(z0, z1) → U4(f11_in(z0, z1), z0, z1)
U4(f11_out1, z0, z1) → U5(f13_in(z0, z1), z0, z1)
U5(f13_out1, z0, z1) → f8_out1

Tuples:

F1_IN(c(s(z0), z1)) → c1(U1'(f8_in(z0, z1), c(s(z0), z1)), F8_IN(z0, z1))
F11_IN(s(z0), z1) → c3(U2'(f11_in(z0, s(z1)), s(z0), z1), F11_IN(z0, s(z1)))
F13_IN(z0, s(z1)) → c5(U3'(f13_in(s(z0), z1), z0, s(z1)), F13_IN(s(z0), z1))
F8_IN(z0, z1) → c7(U4'(f11_in(z0, z1), z0, z1), F11_IN(z0, z1))
U4'(f11_out1, z0, z1) → c8(U5'(f13_in(z0, z1), z0, z1), F13_IN(z0, z1))

S tuples:

F1_IN(c(s(z0), z1)) → c1(U1'(f8_in(z0, z1), c(s(z0), z1)), F8_IN(z0, z1))
F11_IN(s(z0), z1) → c3(U2'(f11_in(z0, s(z1)), s(z0), z1), F11_IN(z0, s(z1)))
F13_IN(z0, s(z1)) → c5(U3'(f13_in(s(z0), z1), z0, s(z1)), F13_IN(s(z0), z1))
F8_IN(z0, z1) → c7(U4'(f11_in(z0, z1), z0, z1), F11_IN(z0, z1))
U4'(f11_out1, z0, z1) → c8(U5'(f13_in(z0, z1), z0, z1), F13_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f11_in, U2, f13_in, U3, f8_in, U4, U5

Defined Pair Symbols:

F1_IN, F11_IN, F13_IN, F8_IN, U4'

Compound Symbols:

c1, c3, c5, c7, c8

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

Split RHS of tuples not part of any SCC

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(c(s(z0), z1)) → U1(f8_in(z0, z1), c(s(z0), z1))
U1(f8_out1, c(s(z0), z1)) → f1_out1
f11_in(s(z0), z1) → U2(f11_in(z0, s(z1)), s(z0), z1)
U2(f11_out1, s(z0), z1) → f11_out1
f13_in(z0, s(z1)) → U3(f13_in(s(z0), z1), z0, s(z1))
U3(f13_out1, z0, s(z1)) → f13_out1
f8_in(z0, z1) → U4(f11_in(z0, z1), z0, z1)
U4(f11_out1, z0, z1) → U5(f13_in(z0, z1), z0, z1)
U5(f13_out1, z0, z1) → f8_out1

Tuples:

F11_IN(s(z0), z1) → c3(U2'(f11_in(z0, s(z1)), s(z0), z1), F11_IN(z0, s(z1)))
F13_IN(z0, s(z1)) → c5(U3'(f13_in(s(z0), z1), z0, s(z1)), F13_IN(s(z0), z1))
F1_IN(c(s(z0), z1)) → c2(U1'(f8_in(z0, z1), c(s(z0), z1)))
F1_IN(c(s(z0), z1)) → c2(F8_IN(z0, z1))
F8_IN(z0, z1) → c2(U4'(f11_in(z0, z1), z0, z1))
F8_IN(z0, z1) → c2(F11_IN(z0, z1))
U4'(f11_out1, z0, z1) → c2(U5'(f13_in(z0, z1), z0, z1))
U4'(f11_out1, z0, z1) → c2(F13_IN(z0, z1))

S tuples:

F11_IN(s(z0), z1) → c3(U2'(f11_in(z0, s(z1)), s(z0), z1), F11_IN(z0, s(z1)))
F13_IN(z0, s(z1)) → c5(U3'(f13_in(s(z0), z1), z0, s(z1)), F13_IN(s(z0), z1))
F1_IN(c(s(z0), z1)) → c2(U1'(f8_in(z0, z1), c(s(z0), z1)))
F1_IN(c(s(z0), z1)) → c2(F8_IN(z0, z1))
F8_IN(z0, z1) → c2(U4'(f11_in(z0, z1), z0, z1))
F8_IN(z0, z1) → c2(F11_IN(z0, z1))
U4'(f11_out1, z0, z1) → c2(U5'(f13_in(z0, z1), z0, z1))
U4'(f11_out1, z0, z1) → c2(F13_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f11_in, U2, f13_in, U3, f8_in, U4, U5

Defined Pair Symbols:

F11_IN, F13_IN, F1_IN, F8_IN, U4'

Compound Symbols:

c3, c5, c2

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

Removed 4 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(c(s(z0), z1)) → U1(f8_in(z0, z1), c(s(z0), z1))
U1(f8_out1, c(s(z0), z1)) → f1_out1
f11_in(s(z0), z1) → U2(f11_in(z0, s(z1)), s(z0), z1)
U2(f11_out1, s(z0), z1) → f11_out1
f13_in(z0, s(z1)) → U3(f13_in(s(z0), z1), z0, s(z1))
U3(f13_out1, z0, s(z1)) → f13_out1
f8_in(z0, z1) → U4(f11_in(z0, z1), z0, z1)
U4(f11_out1, z0, z1) → U5(f13_in(z0, z1), z0, z1)
U5(f13_out1, z0, z1) → f8_out1

Tuples:

F1_IN(c(s(z0), z1)) → c2(F8_IN(z0, z1))
F8_IN(z0, z1) → c2(U4'(f11_in(z0, z1), z0, z1))
F8_IN(z0, z1) → c2(F11_IN(z0, z1))
U4'(f11_out1, z0, z1) → c2(F13_IN(z0, z1))
F11_IN(s(z0), z1) → c3(F11_IN(z0, s(z1)))
F13_IN(z0, s(z1)) → c5(F13_IN(s(z0), z1))
F1_IN(c(s(z0), z1)) → c2
U4'(f11_out1, z0, z1) → c2

S tuples:

F1_IN(c(s(z0), z1)) → c2(F8_IN(z0, z1))
F8_IN(z0, z1) → c2(U4'(f11_in(z0, z1), z0, z1))
F8_IN(z0, z1) → c2(F11_IN(z0, z1))
U4'(f11_out1, z0, z1) → c2(F13_IN(z0, z1))
F11_IN(s(z0), z1) → c3(F11_IN(z0, s(z1)))
F13_IN(z0, s(z1)) → c5(F13_IN(s(z0), z1))
F1_IN(c(s(z0), z1)) → c2
U4'(f11_out1, z0, z1) → c2

K tuples:none
Defined Rule Symbols:

f1_in, U1, f11_in, U2, f13_in, U3, f8_in, U4, U5

Defined Pair Symbols:

F1_IN, F8_IN, U4', F11_IN, F13_IN

Compound Symbols:

c2, c3, c5, c2

(21) CdtKnowledgeProof (EQUIVALENT transformation)

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

F1_IN(c(s(z0), z1)) → c2(F8_IN(z0, z1))
F8_IN(z0, z1) → c2(U4'(f11_in(z0, z1), z0, z1))
F8_IN(z0, z1) → c2(F11_IN(z0, z1))
U4'(f11_out1, z0, z1) → c2(F13_IN(z0, z1))
F1_IN(c(s(z0), z1)) → c2
U4'(f11_out1, z0, z1) → c2
F8_IN(z0, z1) → c2(U4'(f11_in(z0, z1), z0, z1))
F8_IN(z0, z1) → c2(F11_IN(z0, z1))
U4'(f11_out1, z0, z1) → c2(F13_IN(z0, z1))
U4'(f11_out1, z0, z1) → c2

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(c(s(z0), z1)) → U1(f8_in(z0, z1), c(s(z0), z1))
U1(f8_out1, c(s(z0), z1)) → f1_out1
f11_in(s(z0), z1) → U2(f11_in(z0, s(z1)), s(z0), z1)
U2(f11_out1, s(z0), z1) → f11_out1
f13_in(z0, s(z1)) → U3(f13_in(s(z0), z1), z0, s(z1))
U3(f13_out1, z0, s(z1)) → f13_out1
f8_in(z0, z1) → U4(f11_in(z0, z1), z0, z1)
U4(f11_out1, z0, z1) → U5(f13_in(z0, z1), z0, z1)
U5(f13_out1, z0, z1) → f8_out1

Tuples:

F1_IN(c(s(z0), z1)) → c2(F8_IN(z0, z1))
F8_IN(z0, z1) → c2(U4'(f11_in(z0, z1), z0, z1))
F8_IN(z0, z1) → c2(F11_IN(z0, z1))
U4'(f11_out1, z0, z1) → c2(F13_IN(z0, z1))
F11_IN(s(z0), z1) → c3(F11_IN(z0, s(z1)))
F13_IN(z0, s(z1)) → c5(F13_IN(s(z0), z1))
F1_IN(c(s(z0), z1)) → c2
U4'(f11_out1, z0, z1) → c2

S tuples:

F11_IN(s(z0), z1) → c3(F11_IN(z0, s(z1)))
F13_IN(z0, s(z1)) → c5(F13_IN(s(z0), z1))

K tuples:

F1_IN(c(s(z0), z1)) → c2(F8_IN(z0, z1))
F8_IN(z0, z1) → c2(U4'(f11_in(z0, z1), z0, z1))
F8_IN(z0, z1) → c2(F11_IN(z0, z1))
U4'(f11_out1, z0, z1) → c2(F13_IN(z0, z1))
F1_IN(c(s(z0), z1)) → c2
U4'(f11_out1, z0, z1) → c2

Defined Rule Symbols:

f1_in, U1, f11_in, U2, f13_in, U3, f8_in, U4, U5

Defined Pair Symbols:

F1_IN, F8_IN, U4', F11_IN, F13_IN

Compound Symbols:

c2, c3, c5, c2

(23) 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.

F13_IN(z0, s(z1)) → c5(F13_IN(s(z0), z1))

We considered the (Usable) Rules:

f11_in(s(z0), z1) → U2(f11_in(z0, s(z1)), s(z0), z1)
U2(f11_out1, s(z0), z1) → f11_out1

And the Tuples:

F1_IN(c(s(z0), z1)) → c2(F8_IN(z0, z1))
F8_IN(z0, z1) → c2(U4'(f11_in(z0, z1), z0, z1))
F8_IN(z0, z1) → c2(F11_IN(z0, z1))
U4'(f11_out1, z0, z1) → c2(F13_IN(z0, z1))
F11_IN(s(z0), z1) → c3(F11_IN(z0, s(z1)))
F13_IN(z0, s(z1)) → c5(F13_IN(s(z0), z1))
F1_IN(c(s(z0), z1)) → c2
U4'(f11_out1, z0, z1) → c2

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

POL(F11_IN(x₁, x₂)) = [2]
POL(F13_IN(x₁, x₂)) = [2]x₂
POL(F1_IN(x₁)) = [3] + [3]x₁
POL(F8_IN(x₁, x₂)) = [3] + [3]x₂
POL(U2(x₁, x₂, x₃)) = 0
POL(U4'(x₁, x₂, x₃)) = [2]x₃
POL(c(x₁, x₂)) = x₂
POL(c2) = 0
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(f11_in(x₁, x₂)) = 0
POL(f11_out1) = 0
POL(s(x₁)) = [3] + x₁

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(c(s(z0), z1)) → U1(f8_in(z0, z1), c(s(z0), z1))
U1(f8_out1, c(s(z0), z1)) → f1_out1
f11_in(s(z0), z1) → U2(f11_in(z0, s(z1)), s(z0), z1)
U2(f11_out1, s(z0), z1) → f11_out1
f13_in(z0, s(z1)) → U3(f13_in(s(z0), z1), z0, s(z1))
U3(f13_out1, z0, s(z1)) → f13_out1
f8_in(z0, z1) → U4(f11_in(z0, z1), z0, z1)
U4(f11_out1, z0, z1) → U5(f13_in(z0, z1), z0, z1)
U5(f13_out1, z0, z1) → f8_out1

Tuples:

F1_IN(c(s(z0), z1)) → c2(F8_IN(z0, z1))
F8_IN(z0, z1) → c2(U4'(f11_in(z0, z1), z0, z1))
F8_IN(z0, z1) → c2(F11_IN(z0, z1))
U4'(f11_out1, z0, z1) → c2(F13_IN(z0, z1))
F11_IN(s(z0), z1) → c3(F11_IN(z0, s(z1)))
F13_IN(z0, s(z1)) → c5(F13_IN(s(z0), z1))
F1_IN(c(s(z0), z1)) → c2
U4'(f11_out1, z0, z1) → c2

S tuples:

F11_IN(s(z0), z1) → c3(F11_IN(z0, s(z1)))

K tuples:

F1_IN(c(s(z0), z1)) → c2(F8_IN(z0, z1))
F8_IN(z0, z1) → c2(U4'(f11_in(z0, z1), z0, z1))
F8_IN(z0, z1) → c2(F11_IN(z0, z1))
U4'(f11_out1, z0, z1) → c2(F13_IN(z0, z1))
F1_IN(c(s(z0), z1)) → c2
U4'(f11_out1, z0, z1) → c2
F13_IN(z0, s(z1)) → c5(F13_IN(s(z0), z1))

Defined Rule Symbols:

f1_in, U1, f11_in, U2, f13_in, U3, f8_in, U4, U5

Defined Pair Symbols:

F1_IN, F8_IN, U4', F11_IN, F13_IN

Compound Symbols:

c2, c3, c5, c2

(25) 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.

F11_IN(s(z0), z1) → c3(F11_IN(z0, s(z1)))

We considered the (Usable) Rules:

f11_in(s(z0), z1) → U2(f11_in(z0, s(z1)), s(z0), z1)
U2(f11_out1, s(z0), z1) → f11_out1

And the Tuples:

F1_IN(c(s(z0), z1)) → c2(F8_IN(z0, z1))
F8_IN(z0, z1) → c2(U4'(f11_in(z0, z1), z0, z1))
F8_IN(z0, z1) → c2(F11_IN(z0, z1))
U4'(f11_out1, z0, z1) → c2(F13_IN(z0, z1))
F11_IN(s(z0), z1) → c3(F11_IN(z0, s(z1)))
F13_IN(z0, s(z1)) → c5(F13_IN(s(z0), z1))
F1_IN(c(s(z0), z1)) → c2
U4'(f11_out1, z0, z1) → c2

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

POL(F11_IN(x₁, x₂)) = x₁
POL(F13_IN(x₁, x₂)) = 0
POL(F1_IN(x₁)) = [1] + [2]x₁
POL(F8_IN(x₁, x₂)) = x₁ + x₂
POL(U2(x₁, x₂, x₃)) = 0
POL(U4'(x₁, x₂, x₃)) = x₂ + x₃
POL(c(x₁, x₂)) = x₁ + x₂
POL(c2) = 0
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(f11_in(x₁, x₂)) = 0
POL(f11_out1) = 0
POL(s(x₁)) = [1] + x₁

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(c(s(z0), z1)) → U1(f8_in(z0, z1), c(s(z0), z1))
U1(f8_out1, c(s(z0), z1)) → f1_out1
f11_in(s(z0), z1) → U2(f11_in(z0, s(z1)), s(z0), z1)
U2(f11_out1, s(z0), z1) → f11_out1
f13_in(z0, s(z1)) → U3(f13_in(s(z0), z1), z0, s(z1))
U3(f13_out1, z0, s(z1)) → f13_out1
f8_in(z0, z1) → U4(f11_in(z0, z1), z0, z1)
U4(f11_out1, z0, z1) → U5(f13_in(z0, z1), z0, z1)
U5(f13_out1, z0, z1) → f8_out1

Tuples:

F1_IN(c(s(z0), z1)) → c2(F8_IN(z0, z1))
F8_IN(z0, z1) → c2(U4'(f11_in(z0, z1), z0, z1))
F8_IN(z0, z1) → c2(F11_IN(z0, z1))
U4'(f11_out1, z0, z1) → c2(F13_IN(z0, z1))
F11_IN(s(z0), z1) → c3(F11_IN(z0, s(z1)))
F13_IN(z0, s(z1)) → c5(F13_IN(s(z0), z1))
F1_IN(c(s(z0), z1)) → c2
U4'(f11_out1, z0, z1) → c2

S tuples:none
K tuples:

F1_IN(c(s(z0), z1)) → c2(F8_IN(z0, z1))
F8_IN(z0, z1) → c2(U4'(f11_in(z0, z1), z0, z1))
F8_IN(z0, z1) → c2(F11_IN(z0, z1))
U4'(f11_out1, z0, z1) → c2(F13_IN(z0, z1))
F1_IN(c(s(z0), z1)) → c2
U4'(f11_out1, z0, z1) → c2
F13_IN(z0, s(z1)) → c5(F13_IN(s(z0), z1))
F11_IN(s(z0), z1) → c3(F11_IN(z0, s(z1)))

Defined Rule Symbols:

f1_in, U1, f11_in, U2, f13_in, U3, f8_in, U4, U5

Defined Pair Symbols:

F1_IN, F8_IN, U4', F11_IN, F13_IN

Compound Symbols:

c2, c3, c5, c2