YES(O(1), O(1)) 0.00/0.59 YES(O(1), O(1)) 0.00/0.64 0.00/0.64 0.00/0.64 0.00/0.64 0.00/0.64 0.00/0.64 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.64 0.00/0.64 0.00/0.64
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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) Obligation:

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

f1(a, x) → g1(x, x) 0.00/0.64
f1(x, a) → g2(x, x) 0.00/0.64
f2(a, x) → g1(x, x) 0.00/0.64
f2(x, a) → g2(x, x) 0.00/0.64
g1(a, x) → h1(x) 0.00/0.64
g1(x, a) → h2(x) 0.00/0.64
g2(a, x) → h1(x) 0.00/0.64
g2(x, a) → h2(x) 0.00/0.64
h1(a) → i 0.00/0.64
h2(a) → i 0.00/0.64
e1(h1(w), h2(w), x, y, z, w) → e2(x, x, y, z, z, w) 0.00/0.64
e1(x1, x1, x, y, z, a) → e5(x1, x, y, z) 0.00/0.64
e2(f1(w, w), x, y, z, f2(w, w), w) → e3(x, y, x, y, y, z, y, z, x, y, z, w) 0.00/0.64
e2(x, x, y, z, z, a) → e6(x, y, z) 0.00/0.64
e2(i, x, y, z, i, a) → e6(x, y, z) 0.00/0.64
e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z, w) → e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z, w) 0.00/0.64
e3(x, y, x, y, y, z, y, z, x, y, z, a) → e6(x, y, z) 0.00/0.64
e4(g1(w, w), x1, g2(w, w), x1, g1(w, w), x1, g2(w, w), x1, x, y, z, w) → e1(x1, x1, x, y, z, w) 0.00/0.64
e4(i, x1, i, x1, i, x1, i, x1, x, y, z, a) → e5(x1, x, y, z) 0.00/0.64
e4(x, x, x, x, x, x, x, x, x, x, x, a) → e6(x, x, x) 0.00/0.64
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(a, z0) → g1(z0, z0) 0.00/0.64
f1(z0, a) → g2(z0, z0) 0.00/0.64
f2(a, z0) → g1(z0, z0) 0.00/0.64
f2(z0, a) → g2(z0, z0) 0.00/0.64
g1(a, z0) → h1(z0) 0.00/0.64
g1(z0, a) → h2(z0) 0.00/0.64
g2(a, z0) → h1(z0) 0.00/0.64
g2(z0, a) → h2(z0) 0.00/0.64
h1(a) → i 0.00/0.64
h2(a) → i 0.00/0.64
e1(h1(z0), h2(z0), z1, z2, z3, z0) → e2(z1, z1, z2, z3, z3, z0) 0.00/0.64
e1(z0, z0, z1, z2, z3, a) → e5(z0, z1, z2, z3) 0.00/0.64
e2(f1(z0, z0), z1, z2, z3, f2(z0, z0), z0) → e3(z1, z2, z1, z2, z2, z3, z2, z3, z1, z2, z3, z0) 0.00/0.64
e2(z0, z0, z1, z2, z2, a) → e6(z0, z1, z2) 0.00/0.64
e2(i, z0, z1, z2, i, a) → e6(z0, z1, z2) 0.00/0.64
e3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) → e4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) 0.00/0.64
e3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2, a) → e6(z0, z1, z2) 0.00/0.64
e4(g1(z0, z0), z1, g2(z0, z0), z1, g1(z0, z0), z1, g2(z0, z0), z1, z2, z3, z4, z0) → e1(z1, z1, z2, z3, z4, z0) 0.00/0.64
e4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3, a) → e5(z0, z1, z2, z3) 0.00/0.64
e4(z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, a) → e6(z0, z0, z0) 0.00/0.64
e5(i, z0, z1, z2) → e6(z0, z1, z2)
Tuples:

F1(a, z0) → c(G1(z0, z0)) 0.00/0.64
F1(z0, a) → c1(G2(z0, z0)) 0.00/0.64
F2(a, z0) → c2(G1(z0, z0)) 0.00/0.64
F2(z0, a) → c3(G2(z0, z0)) 0.00/0.64
G1(a, z0) → c4(H1(z0)) 0.00/0.64
G1(z0, a) → c5(H2(z0)) 0.00/0.64
G2(a, z0) → c6(H1(z0)) 0.00/0.64
G2(z0, a) → c7(H2(z0)) 0.00/0.64
E1(h1(z0), h2(z0), z1, z2, z3, z0) → c10(E2(z1, z1, z2, z3, z3, z0)) 0.00/0.64
E1(z0, z0, z1, z2, z3, a) → c11(E5(z0, z1, z2, z3)) 0.00/0.64
E2(f1(z0, z0), z1, z2, z3, f2(z0, z0), z0) → c12(E3(z1, z2, z1, z2, z2, z3, z2, z3, z1, z2, z3, z0)) 0.00/0.64
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) → c15(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7)) 0.00/0.64
E4(g1(z0, z0), z1, g2(z0, z0), z1, g1(z0, z0), z1, g2(z0, z0), z1, z2, z3, z4, z0) → c17(E1(z1, z1, z2, z3, z4, z0)) 0.00/0.64
E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3, a) → c18(E5(z0, z1, z2, z3))
S tuples:

F1(a, z0) → c(G1(z0, z0)) 0.00/0.64
F1(z0, a) → c1(G2(z0, z0)) 0.00/0.64
F2(a, z0) → c2(G1(z0, z0)) 0.00/0.64
F2(z0, a) → c3(G2(z0, z0)) 0.00/0.64
G1(a, z0) → c4(H1(z0)) 0.00/0.64
G1(z0, a) → c5(H2(z0)) 0.00/0.64
G2(a, z0) → c6(H1(z0)) 0.00/0.64
G2(z0, a) → c7(H2(z0)) 0.00/0.64
E1(h1(z0), h2(z0), z1, z2, z3, z0) → c10(E2(z1, z1, z2, z3, z3, z0)) 0.00/0.64
E1(z0, z0, z1, z2, z3, a) → c11(E5(z0, z1, z2, z3)) 0.00/0.64
E2(f1(z0, z0), z1, z2, z3, f2(z0, z0), z0) → c12(E3(z1, z2, z1, z2, z2, z3, z2, z3, z1, z2, z3, z0)) 0.00/0.64
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) → c15(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7)) 0.00/0.64
E4(g1(z0, z0), z1, g2(z0, z0), z1, g1(z0, z0), z1, g2(z0, z0), z1, z2, z3, z4, z0) → c17(E1(z1, z1, z2, z3, z4, z0)) 0.00/0.64
E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3, a) → 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(z0), h2(z0), z1, z2, z3, z0) → c10(E2(z1, z1, z2, z3, z3, z0)) 0.00/0.64
E2(f1(z0, z0), z1, z2, z3, f2(z0, z0), z0) → c12(E3(z1, z2, z1, z2, z2, z3, z2, z3, z1, z2, z3, z0)) 0.00/0.64
E4(g1(z0, z0), z1, g2(z0, z0), z1, g1(z0, z0), z1, g2(z0, z0), z1, z2, z3, z4, z0) → c17(E1(z1, z1, z2, z3, z4, z0))
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1(a, z0) → g1(z0, z0) 0.00/0.64
f1(z0, a) → g2(z0, z0) 0.00/0.64
f2(a, z0) → g1(z0, z0) 0.00/0.64
f2(z0, a) → g2(z0, z0) 0.00/0.64
g1(a, z0) → h1(z0) 0.00/0.64
g1(z0, a) → h2(z0) 0.00/0.64
g2(a, z0) → h1(z0) 0.00/0.64
g2(z0, a) → h2(z0) 0.00/0.64
h1(a) → i 0.00/0.64
h2(a) → i 0.00/0.64
e1(h1(z0), h2(z0), z1, z2, z3, z0) → e2(z1, z1, z2, z3, z3, z0) 0.00/0.64
e1(z0, z0, z1, z2, z3, a) → e5(z0, z1, z2, z3) 0.00/0.64
e2(f1(z0, z0), z1, z2, z3, f2(z0, z0), z0) → e3(z1, z2, z1, z2, z2, z3, z2, z3, z1, z2, z3, z0) 0.00/0.64
e2(z0, z0, z1, z2, z2, a) → e6(z0, z1, z2) 0.00/0.64
e2(i, z0, z1, z2, i, a) → e6(z0, z1, z2) 0.00/0.64
e3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) → e4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) 0.00/0.64
e3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2, a) → e6(z0, z1, z2) 0.00/0.64
e4(g1(z0, z0), z1, g2(z0, z0), z1, g1(z0, z0), z1, g2(z0, z0), z1, z2, z3, z4, z0) → e1(z1, z1, z2, z3, z4, z0) 0.00/0.64
e4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3, a) → e5(z0, z1, z2, z3) 0.00/0.64
e4(z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, a) → e6(z0, z0, z0) 0.00/0.64
e5(i, z0, z1, z2) → e6(z0, z1, z2)
Tuples:

F1(a, z0) → c(G1(z0, z0)) 0.00/0.64
F1(z0, a) → c1(G2(z0, z0)) 0.00/0.64
F2(a, z0) → c2(G1(z0, z0)) 0.00/0.64
F2(z0, a) → c3(G2(z0, z0)) 0.00/0.64
G1(a, z0) → c4(H1(z0)) 0.00/0.64
G1(z0, a) → c5(H2(z0)) 0.00/0.64
G2(a, z0) → c6(H1(z0)) 0.00/0.64
G2(z0, a) → c7(H2(z0)) 0.00/0.64
E1(z0, z0, z1, z2, z3, a) → c11(E5(z0, z1, z2, z3)) 0.00/0.64
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) → c15(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7)) 0.00/0.64
E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3, a) → c18(E5(z0, z1, z2, z3))
S tuples:

F1(a, z0) → c(G1(z0, z0)) 0.00/0.64
F1(z0, a) → c1(G2(z0, z0)) 0.00/0.64
F2(a, z0) → c2(G1(z0, z0)) 0.00/0.64
F2(z0, a) → c3(G2(z0, z0)) 0.00/0.64
G1(a, z0) → c4(H1(z0)) 0.00/0.64
G1(z0, a) → c5(H2(z0)) 0.00/0.64
G2(a, z0) → c6(H1(z0)) 0.00/0.64
G2(z0, a) → c7(H2(z0)) 0.00/0.64
E1(z0, z0, z1, z2, z3, a) → c11(E5(z0, z1, z2, z3)) 0.00/0.64
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) → c15(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7)) 0.00/0.64
E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3, a) → 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(a, z0) → g1(z0, z0) 0.00/0.64
f1(z0, a) → g2(z0, z0) 0.00/0.64
f2(a, z0) → g1(z0, z0) 0.00/0.64
f2(z0, a) → g2(z0, z0) 0.00/0.64
g1(a, z0) → h1(z0) 0.00/0.64
g1(z0, a) → h2(z0) 0.00/0.64
g2(a, z0) → h1(z0) 0.00/0.64
g2(z0, a) → h2(z0) 0.00/0.64
h1(a) → i 0.00/0.64
h2(a) → i 0.00/0.64
e1(h1(z0), h2(z0), z1, z2, z3, z0) → e2(z1, z1, z2, z3, z3, z0) 0.00/0.64
e1(z0, z0, z1, z2, z3, a) → e5(z0, z1, z2, z3) 0.00/0.64
e2(f1(z0, z0), z1, z2, z3, f2(z0, z0), z0) → e3(z1, z2, z1, z2, z2, z3, z2, z3, z1, z2, z3, z0) 0.00/0.64
e2(z0, z0, z1, z2, z2, a) → e6(z0, z1, z2) 0.00/0.64
e2(i, z0, z1, z2, i, a) → e6(z0, z1, z2) 0.00/0.64
e3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) → e4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) 0.00/0.64
e3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2, a) → e6(z0, z1, z2) 0.00/0.64
e4(g1(z0, z0), z1, g2(z0, z0), z1, g1(z0, z0), z1, g2(z0, z0), z1, z2, z3, z4, z0) → e1(z1, z1, z2, z3, z4, z0) 0.00/0.64
e4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3, a) → e5(z0, z1, z2, z3) 0.00/0.64
e4(z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, a) → e6(z0, z0, z0) 0.00/0.64
e5(i, z0, z1, z2) → e6(z0, z1, z2)
Tuples:

F1(a, z0) → c(G1(z0, z0)) 0.00/0.64
F1(z0, a) → c1(G2(z0, z0)) 0.00/0.64
F2(a, z0) → c2(G1(z0, z0)) 0.00/0.64
F2(z0, a) → c3(G2(z0, z0)) 0.00/0.64
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) → c15(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7)) 0.00/0.64
G1(a, z0) → c4 0.00/0.64
G1(z0, a) → c5 0.00/0.64
G2(a, z0) → c6 0.00/0.64
G2(z0, a) → c7 0.00/0.64
E1(z0, z0, z1, z2, z3, a) → c11 0.00/0.64
E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3, a) → c18
S tuples:

F1(a, z0) → c(G1(z0, z0)) 0.00/0.64
F1(z0, a) → c1(G2(z0, z0)) 0.00/0.64
F2(a, z0) → c2(G1(z0, z0)) 0.00/0.64
F2(z0, a) → c3(G2(z0, z0)) 0.00/0.64
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) → c15(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7)) 0.00/0.64
G1(a, z0) → c4 0.00/0.64
G1(z0, a) → c5 0.00/0.64
G2(a, z0) → c6 0.00/0.64
G2(z0, a) → c7 0.00/0.64
E1(z0, z0, z1, z2, z3, a) → c11 0.00/0.64
E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3, a) → 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, a) → c18 0.00/0.64
F1(a, z0) → c(G1(z0, z0)) 0.00/0.64
F2(z0, a) → c3(G2(z0, z0)) 0.00/0.64
G1(z0, a) → c5 0.00/0.64
F1(z0, a) → c1(G2(z0, z0)) 0.00/0.64
E1(z0, z0, z1, z2, z3, a) → c11 0.00/0.64
G2(a, z0) → c6 0.00/0.64
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) → c15(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7)) 0.00/0.64
G2(z0, a) → c7 0.00/0.64
G1(a, z0) → c4 0.00/0.64
F2(a, z0) → c2(G1(z0, z0))
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1(a, z0) → g1(z0, z0) 0.00/0.64
f1(z0, a) → g2(z0, z0) 0.00/0.64
f2(a, z0) → g1(z0, z0) 0.00/0.64
f2(z0, a) → g2(z0, z0) 0.00/0.64
g1(a, z0) → h1(z0) 0.00/0.64
g1(z0, a) → h2(z0) 0.00/0.64
g2(a, z0) → h1(z0) 0.00/0.64
g2(z0, a) → h2(z0) 0.00/0.64
h1(a) → i 0.00/0.64
h2(a) → i 0.00/0.64
e1(h1(z0), h2(z0), z1, z2, z3, z0) → e2(z1, z1, z2, z3, z3, z0) 0.00/0.64
e1(z0, z0, z1, z2, z3, a) → e5(z0, z1, z2, z3) 0.00/0.64
e2(f1(z0, z0), z1, z2, z3, f2(z0, z0), z0) → e3(z1, z2, z1, z2, z2, z3, z2, z3, z1, z2, z3, z0) 0.00/0.64
e2(z0, z0, z1, z2, z2, a) → e6(z0, z1, z2) 0.00/0.64
e2(i, z0, z1, z2, i, a) → e6(z0, z1, z2) 0.00/0.64
e3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) → e4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) 0.00/0.64
e3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2, a) → e6(z0, z1, z2) 0.00/0.64
e4(g1(z0, z0), z1, g2(z0, z0), z1, g1(z0, z0), z1, g2(z0, z0), z1, z2, z3, z4, z0) → e1(z1, z1, z2, z3, z4, z0) 0.00/0.64
e4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3, a) → e5(z0, z1, z2, z3) 0.00/0.64
e4(z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, a) → e6(z0, z0, z0) 0.00/0.64
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
0.00/0.68 EOF