YES(O(1), O(1)) 0.00/0.58 YES(O(1), O(1)) 0.00/0.61 0.00/0.61 0.00/0.61 0.00/0.61 0.00/0.61 0.00/0.61 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.61 0.00/0.61 0.00/0.61
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(1)).

0 CpxTRS
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↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
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↳2 CdtProblem
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↳3 CdtUnreachableProof (⇔)
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↳4 CdtProblem
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↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
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↳6 CdtProblem
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↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
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↳8 CdtProblem
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↳9 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳10 BOUNDS(O(1), O(1))
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0.00/0.61 0.00/0.61
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(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

f1 → g1 0.00/0.61
f1 → g2 0.00/0.61
f2 → g1 0.00/0.61
f2 → g2 0.00/0.61
g1 → h1 0.00/0.61
g1 → h2 0.00/0.61
g2 → h1 0.00/0.61
g2 → h2 0.00/0.61
h1 → i 0.00/0.61
h2 → i 0.00/0.61
e1(h1, h2, x, y, z) → e2(x, x, y, z, z) 0.00/0.61
e1(x1, x1, x, y, z) → e5(x1, x, y, z) 0.00/0.61
e2(f1, x, y, z, f2) → e3(x, y, x, y, y, z, y, z, x, y, z) 0.00/0.61
e2(x, x, y, z, z) → e6(x, y, z) 0.00/0.61
e2(i, x, y, z, i) → e6(x, y, z) 0.00/0.61
e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) → e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) 0.00/0.61
e3(x, y, x, y, y, z, y, z, x, y, z) → e6(x, y, z) 0.00/0.61
e4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) → e1(x1, x1, x, y, z) 0.00/0.61
e4(i, x1, i, x1, i, x1, i, x1, x, y, z) → e5(x1, x, y, z) 0.00/0.61
e4(x, x, x, x, x, x, x, x, x, x, x) → e6(x, x, x) 0.00/0.61
e5(i, x, y, z) → e6(x, y, z)

Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
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(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1 → g1 0.00/0.61
f1 → g2 0.00/0.61
f2 → g1 0.00/0.61
f2 → g2 0.00/0.61
g1 → h1 0.00/0.61
g1 → h2 0.00/0.61
g2 → h1 0.00/0.61
g2 → h2 0.00/0.61
h1 → i 0.00/0.61
h2 → i 0.00/0.61
e1(h1, h2, z0, z1, z2) → e2(z0, z0, z1, z2, z2) 0.00/0.61
e1(z0, z0, z1, z2, z3) → e5(z0, z1, z2, z3) 0.00/0.61
e2(f1, z0, z1, z2, f2) → e3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2) 0.00/0.61
e2(z0, z0, z1, z2, z2) → e6(z0, z1, z2) 0.00/0.61
e2(i, z0, z1, z2, i) → e6(z0, z1, z2) 0.00/0.61
e3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) → e4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) 0.00/0.61
e3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2) → e6(z0, z1, z2) 0.00/0.61
e4(g1, z0, g2, z0, g1, z0, g2, z0, z1, z2, z3) → e1(z0, z0, z1, z2, z3) 0.00/0.61
e4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3) → e5(z0, z1, z2, z3) 0.00/0.61
e4(z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, z0) → e6(z0, z0, z0) 0.00/0.61
e5(i, z0, z1, z2) → e6(z0, z1, z2)
Tuples:

F1 → c(G1) 0.00/0.61
F1 → c1(G2) 0.00/0.61
F2 → c2(G1) 0.00/0.61
F2 → c3(G2) 0.00/0.61
G1 → c4(H1) 0.00/0.61
G1 → c5(H2) 0.00/0.61
G2 → c6(H1) 0.00/0.61
G2 → c7(H2) 0.00/0.61
E1(h1, h2, z0, z1, z2) → c10(E2(z0, z0, z1, z2, z2)) 0.00/0.61
E1(z0, z0, z1, z2, z3) → c11(E5(z0, z1, z2, z3)) 0.00/0.61
E2(f1, z0, z1, z2, f2) → c12(E3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2)) 0.00/0.61
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) → c15(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6)) 0.00/0.61
E4(g1, z0, g2, z0, g1, z0, g2, z0, z1, z2, z3) → c17(E1(z0, z0, z1, z2, z3)) 0.00/0.61
E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3) → c18(E5(z0, z1, z2, z3))
S tuples:

F1 → c(G1) 0.00/0.61
F1 → c1(G2) 0.00/0.61
F2 → c2(G1) 0.00/0.61
F2 → c3(G2) 0.00/0.61
G1 → c4(H1) 0.00/0.61
G1 → c5(H2) 0.00/0.61
G2 → c6(H1) 0.00/0.61
G2 → c7(H2) 0.00/0.61
E1(h1, h2, z0, z1, z2) → c10(E2(z0, z0, z1, z2, z2)) 0.00/0.61
E1(z0, z0, z1, z2, z3) → c11(E5(z0, z1, z2, z3)) 0.00/0.61
E2(f1, z0, z1, z2, f2) → c12(E3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2)) 0.00/0.61
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) → c15(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6)) 0.00/0.61
E4(g1, z0, g2, z0, g1, z0, g2, z0, z1, z2, z3) → c17(E1(z0, z0, z1, z2, z3)) 0.00/0.61
E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3) → c18(E5(z0, z1, z2, z3))
K tuples:none
Defined Rule Symbols:

f1, f2, g1, g2, h1, h2, e1, e2, e3, e4, e5

Defined Pair Symbols:

F1, F2, G1, G2, E1, E2, E3, E4

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c10, c11, c12, c15, c17, c18

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(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

E1(h1, h2, z0, z1, z2) → c10(E2(z0, z0, z1, z2, z2)) 0.00/0.61
E2(f1, z0, z1, z2, f2) → c12(E3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2)) 0.00/0.61
E4(g1, z0, g2, z0, g1, z0, g2, z0, z1, z2, z3) → c17(E1(z0, z0, z1, z2, z3))
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1 → g1 0.00/0.61
f1 → g2 0.00/0.61
f2 → g1 0.00/0.61
f2 → g2 0.00/0.61
g1 → h1 0.00/0.61
g1 → h2 0.00/0.61
g2 → h1 0.00/0.61
g2 → h2 0.00/0.61
h1 → i 0.00/0.61
h2 → i 0.00/0.61
e1(h1, h2, z0, z1, z2) → e2(z0, z0, z1, z2, z2) 0.00/0.61
e1(z0, z0, z1, z2, z3) → e5(z0, z1, z2, z3) 0.00/0.61
e2(f1, z0, z1, z2, f2) → e3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2) 0.00/0.61
e2(z0, z0, z1, z2, z2) → e6(z0, z1, z2) 0.00/0.61
e2(i, z0, z1, z2, i) → e6(z0, z1, z2) 0.00/0.61
e3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) → e4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) 0.00/0.61
e3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2) → e6(z0, z1, z2) 0.00/0.61
e4(g1, z0, g2, z0, g1, z0, g2, z0, z1, z2, z3) → e1(z0, z0, z1, z2, z3) 0.00/0.61
e4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3) → e5(z0, z1, z2, z3) 0.00/0.61
e4(z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, z0) → e6(z0, z0, z0) 0.00/0.61
e5(i, z0, z1, z2) → e6(z0, z1, z2)
Tuples:

F1 → c(G1) 0.00/0.61
F1 → c1(G2) 0.00/0.61
F2 → c2(G1) 0.00/0.61
F2 → c3(G2) 0.00/0.61
G1 → c4(H1) 0.00/0.61
G1 → c5(H2) 0.00/0.61
G2 → c6(H1) 0.00/0.61
G2 → c7(H2) 0.00/0.61
E1(z0, z0, z1, z2, z3) → c11(E5(z0, z1, z2, z3)) 0.00/0.61
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) → c15(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6)) 0.00/0.61
E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3) → c18(E5(z0, z1, z2, z3))
S tuples:

F1 → c(G1) 0.00/0.61
F1 → c1(G2) 0.00/0.61
F2 → c2(G1) 0.00/0.61
F2 → c3(G2) 0.00/0.61
G1 → c4(H1) 0.00/0.61
G1 → c5(H2) 0.00/0.61
G2 → c6(H1) 0.00/0.61
G2 → c7(H2) 0.00/0.61
E1(z0, z0, z1, z2, z3) → c11(E5(z0, z1, z2, z3)) 0.00/0.61
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) → c15(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6)) 0.00/0.61
E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3) → c18(E5(z0, z1, z2, z3))
K tuples:none
Defined Rule Symbols:

f1, f2, g1, g2, h1, h2, e1, e2, e3, e4, e5

Defined Pair Symbols:

F1, F2, G1, G2, E1, E3, E4

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c11, c15, c18

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

Removed 6 trailing tuple parts
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1 → g1 0.00/0.61
f1 → g2 0.00/0.61
f2 → g1 0.00/0.61
f2 → g2 0.00/0.61
g1 → h1 0.00/0.61
g1 → h2 0.00/0.61
g2 → h1 0.00/0.61
g2 → h2 0.00/0.61
h1 → i 0.00/0.61
h2 → i 0.00/0.61
e1(h1, h2, z0, z1, z2) → e2(z0, z0, z1, z2, z2) 0.00/0.61
e1(z0, z0, z1, z2, z3) → e5(z0, z1, z2, z3) 0.00/0.61
e2(f1, z0, z1, z2, f2) → e3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2) 0.00/0.61
e2(z0, z0, z1, z2, z2) → e6(z0, z1, z2) 0.00/0.61
e2(i, z0, z1, z2, i) → e6(z0, z1, z2) 0.00/0.61
e3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) → e4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) 0.00/0.61
e3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2) → e6(z0, z1, z2) 0.00/0.61
e4(g1, z0, g2, z0, g1, z0, g2, z0, z1, z2, z3) → e1(z0, z0, z1, z2, z3) 0.00/0.61
e4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3) → e5(z0, z1, z2, z3) 0.00/0.61
e4(z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, z0) → e6(z0, z0, z0) 0.00/0.61
e5(i, z0, z1, z2) → e6(z0, z1, z2)
Tuples:

F1 → c(G1) 0.00/0.61
F1 → c1(G2) 0.00/0.61
F2 → c2(G1) 0.00/0.61
F2 → c3(G2) 0.00/0.61
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) → c15(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6)) 0.00/0.61
G1 → c4 0.00/0.61
G1 → c5 0.00/0.61
G2 → c6 0.00/0.61
G2 → c7 0.00/0.61
E1(z0, z0, z1, z2, z3) → c11 0.00/0.61
E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3) → c18
S tuples:

F1 → c(G1) 0.00/0.61
F1 → c1(G2) 0.00/0.61
F2 → c2(G1) 0.00/0.61
F2 → c3(G2) 0.00/0.61
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) → c15(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6)) 0.00/0.61
G1 → c4 0.00/0.61
G1 → c5 0.00/0.61
G2 → c6 0.00/0.61
G2 → c7 0.00/0.61
E1(z0, z0, z1, z2, z3) → c11 0.00/0.61
E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3) → c18
K tuples:none
Defined Rule Symbols:

f1, f2, g1, g2, h1, h2, e1, e2, e3, e4, e5

Defined Pair Symbols:

F1, F2, E3, G1, G2, E1, E4

Compound Symbols:

c, c1, c2, c3, c15, c4, c5, c6, c7, c11, c18

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

Removed 11 trailing nodes:

E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3) → c18 0.00/0.61
F1 → c(G1) 0.00/0.61
F2 → c3(G2) 0.00/0.61
G1 → c5 0.00/0.61
F1 → c1(G2) 0.00/0.61
E1(z0, z0, z1, z2, z3) → c11 0.00/0.61
G2 → c6 0.00/0.61
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) → c15(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6)) 0.00/0.61
G2 → c7 0.00/0.61
G1 → c4 0.00/0.61
F2 → c2(G1)
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1 → g1 0.00/0.61
f1 → g2 0.00/0.61
f2 → g1 0.00/0.61
f2 → g2 0.00/0.61
g1 → h1 0.00/0.61
g1 → h2 0.00/0.61
g2 → h1 0.00/0.61
g2 → h2 0.00/0.61
h1 → i 0.00/0.61
h2 → i 0.00/0.61
e1(h1, h2, z0, z1, z2) → e2(z0, z0, z1, z2, z2) 0.00/0.61
e1(z0, z0, z1, z2, z3) → e5(z0, z1, z2, z3) 0.00/0.61
e2(f1, z0, z1, z2, f2) → e3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2) 0.00/0.61
e2(z0, z0, z1, z2, z2) → e6(z0, z1, z2) 0.00/0.61
e2(i, z0, z1, z2, i) → e6(z0, z1, z2) 0.00/0.61
e3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) → e4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) 0.00/0.61
e3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2) → e6(z0, z1, z2) 0.00/0.61
e4(g1, z0, g2, z0, g1, z0, g2, z0, z1, z2, z3) → e1(z0, z0, z1, z2, z3) 0.00/0.61
e4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3) → e5(z0, z1, z2, z3) 0.00/0.61
e4(z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, z0) → e6(z0, z0, z0) 0.00/0.61
e5(i, z0, z1, z2) → e6(z0, z1, z2)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f1, f2, g1, g2, h1, h2, e1, e2, e3, e4, e5

Defined Pair Symbols:none

Compound Symbols:none

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(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty
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(10) BOUNDS(O(1), O(1))

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0.00/0.64 EOF