YES(O(1), O(n^2)) 10.71/3.12 YES(O(1), O(n^2)) 10.71/3.17 10.71/3.17 10.71/3.17 10.71/3.17 10.71/3.17 10.71/3.17 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 10.71/3.17 10.71/3.17 10.71/3.17
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
10.71/3.17 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
10.71/3.17 
↳2 CdtProblem
10.71/3.17 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
10.71/3.17 
↳4 CdtProblem
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↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
10.71/3.17 
↳6 CdtProblem
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↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
10.71/3.17 
↳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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10.71/3.17

(0) Obligation:

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

f(j(x, y), y) → g(f(x, k(y))) 10.71/3.17
f(x, h1(y, z)) → h2(0, x, h1(y, z)) 10.71/3.17
g(h2(x, y, h1(z, u))) → h2(s(x), y, h1(z, u)) 10.71/3.17
h2(x, j(y, h1(z, u)), h1(z, u)) → h2(s(x), y, h1(s(z), u)) 10.71/3.17
i(f(x, h(y))) → y 10.71/3.17
i(h2(s(x), y, h1(x, z))) → z 10.71/3.17
k(h(x)) → h1(0, x) 10.71/3.17
k(h1(x, y)) → h1(s(x), y)

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

Converted CpxTRS to CDT
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10.71/3.17

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(j(z0, z1), z1) → g(f(z0, k(z1))) 10.71/3.17
f(z0, h1(z1, z2)) → h2(0, z0, h1(z1, z2)) 10.71/3.17
g(h2(z0, z1, h1(z2, z3))) → h2(s(z0), z1, h1(z2, z3)) 10.71/3.17
h2(z0, j(z1, h1(z2, z3)), h1(z2, z3)) → h2(s(z0), z1, h1(s(z2), z3)) 10.71/3.17
i(f(z0, h(z1))) → z1 10.71/3.17
i(h2(s(z0), z1, h1(z0, z2))) → z2 10.71/3.17
k(h(z0)) → h1(0, z0) 10.71/3.17
k(h1(z0, z1)) → h1(s(z0), z1)
Tuples:

F(j(z0, z1), z1) → c(G(f(z0, k(z1))), F(z0, k(z1)), K(z1)) 10.71/3.17
F(z0, h1(z1, z2)) → c1(H2(0, z0, h1(z1, z2))) 10.71/3.17
G(h2(z0, z1, h1(z2, z3))) → c2(H2(s(z0), z1, h1(z2, z3))) 10.71/3.17
H2(z0, j(z1, h1(z2, z3)), h1(z2, z3)) → c3(H2(s(z0), z1, h1(s(z2), z3)))
S tuples:

F(j(z0, z1), z1) → c(G(f(z0, k(z1))), F(z0, k(z1)), K(z1)) 10.71/3.17
F(z0, h1(z1, z2)) → c1(H2(0, z0, h1(z1, z2))) 10.71/3.17
G(h2(z0, z1, h1(z2, z3))) → c2(H2(s(z0), z1, h1(z2, z3))) 10.71/3.17
H2(z0, j(z1, h1(z2, z3)), h1(z2, z3)) → c3(H2(s(z0), z1, h1(s(z2), z3)))
K tuples:none
Defined Rule Symbols:

f, g, h2, i, k

Defined Pair Symbols:

F, G, H2

Compound Symbols:

c, c1, c2, c3

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

Removed 1 trailing tuple parts
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10.71/3.17

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(j(z0, z1), z1) → g(f(z0, k(z1))) 10.71/3.17
f(z0, h1(z1, z2)) → h2(0, z0, h1(z1, z2)) 10.71/3.17
g(h2(z0, z1, h1(z2, z3))) → h2(s(z0), z1, h1(z2, z3)) 10.71/3.17
h2(z0, j(z1, h1(z2, z3)), h1(z2, z3)) → h2(s(z0), z1, h1(s(z2), z3)) 10.71/3.17
i(f(z0, h(z1))) → z1 10.71/3.17
i(h2(s(z0), z1, h1(z0, z2))) → z2 10.71/3.17
k(h(z0)) → h1(0, z0) 10.71/3.17
k(h1(z0, z1)) → h1(s(z0), z1)
Tuples:

F(z0, h1(z1, z2)) → c1(H2(0, z0, h1(z1, z2))) 10.71/3.17
G(h2(z0, z1, h1(z2, z3))) → c2(H2(s(z0), z1, h1(z2, z3))) 10.71/3.17
H2(z0, j(z1, h1(z2, z3)), h1(z2, z3)) → c3(H2(s(z0), z1, h1(s(z2), z3))) 10.71/3.17
F(j(z0, z1), z1) → c(G(f(z0, k(z1))), F(z0, k(z1)))
S tuples:

F(z0, h1(z1, z2)) → c1(H2(0, z0, h1(z1, z2))) 10.71/3.17
G(h2(z0, z1, h1(z2, z3))) → c2(H2(s(z0), z1, h1(z2, z3))) 10.71/3.17
H2(z0, j(z1, h1(z2, z3)), h1(z2, z3)) → c3(H2(s(z0), z1, h1(s(z2), z3))) 10.71/3.17
F(j(z0, z1), z1) → c(G(f(z0, k(z1))), F(z0, k(z1)))
K tuples:none
Defined Rule Symbols:

f, g, h2, i, k

Defined Pair Symbols:

F, G, H2

Compound Symbols:

c1, c2, c3, c

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10.71/3.17

(5) 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.

F(z0, h1(z1, z2)) → c1(H2(0, z0, h1(z1, z2))) 10.71/3.17
G(h2(z0, z1, h1(z2, z3))) → c2(H2(s(z0), z1, h1(z2, z3))) 10.71/3.17
F(j(z0, z1), z1) → c(G(f(z0, k(z1))), F(z0, k(z1)))
We considered the (Usable) Rules:

k(h(z0)) → h1(0, z0) 10.71/3.17
k(h1(z0, z1)) → h1(s(z0), z1) 10.71/3.17
f(j(z0, z1), z1) → g(f(z0, k(z1))) 10.71/3.17
f(z0, h1(z1, z2)) → h2(0, z0, h1(z1, z2)) 10.71/3.17
h2(z0, j(z1, h1(z2, z3)), h1(z2, z3)) → h2(s(z0), z1, h1(s(z2), z3)) 10.71/3.17
g(h2(z0, z1, h1(z2, z3))) → h2(s(z0), z1, h1(z2, z3))
And the Tuples:

F(z0, h1(z1, z2)) → c1(H2(0, z0, h1(z1, z2))) 10.71/3.17
G(h2(z0, z1, h1(z2, z3))) → c2(H2(s(z0), z1, h1(z2, z3))) 10.71/3.17
H2(z0, j(z1, h1(z2, z3)), h1(z2, z3)) → c3(H2(s(z0), z1, h1(s(z2), z3))) 10.71/3.17
F(j(z0, z1), z1) → c(G(f(z0, k(z1))), F(z0, k(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation : 10.71/3.17

POL(0) = [4]    10.71/3.17
POL(F(x1, x2)) = [1] + [4]x1    10.71/3.17
POL(G(x1)) = [1]    10.71/3.17
POL(H2(x1, x2, x3)) = 0    10.71/3.17
POL(c(x1, x2)) = x1 + x2    10.71/3.17
POL(c1(x1)) = x1    10.71/3.17
POL(c2(x1)) = x1    10.71/3.17
POL(c3(x1)) = x1    10.71/3.17
POL(f(x1, x2)) = [5]x2    10.71/3.17
POL(g(x1)) = [3]x1    10.71/3.17
POL(h(x1)) = [1] + x1    10.71/3.17
POL(h1(x1, x2)) = [5]    10.71/3.17
POL(h2(x1, x2, x3)) = [1] + [3]x1 + [2]x3    10.71/3.17
POL(j(x1, x2)) = [1] + x1    10.71/3.17
POL(k(x1)) = 0    10.71/3.17
POL(s(x1)) = x1   
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10.71/3.17

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(j(z0, z1), z1) → g(f(z0, k(z1))) 10.71/3.17
f(z0, h1(z1, z2)) → h2(0, z0, h1(z1, z2)) 10.71/3.17
g(h2(z0, z1, h1(z2, z3))) → h2(s(z0), z1, h1(z2, z3)) 10.71/3.17
h2(z0, j(z1, h1(z2, z3)), h1(z2, z3)) → h2(s(z0), z1, h1(s(z2), z3)) 10.71/3.17
i(f(z0, h(z1))) → z1 10.71/3.17
i(h2(s(z0), z1, h1(z0, z2))) → z2 10.71/3.17
k(h(z0)) → h1(0, z0) 10.71/3.17
k(h1(z0, z1)) → h1(s(z0), z1)
Tuples:

F(z0, h1(z1, z2)) → c1(H2(0, z0, h1(z1, z2))) 10.71/3.17
G(h2(z0, z1, h1(z2, z3))) → c2(H2(s(z0), z1, h1(z2, z3))) 10.71/3.17
H2(z0, j(z1, h1(z2, z3)), h1(z2, z3)) → c3(H2(s(z0), z1, h1(s(z2), z3))) 10.71/3.17
F(j(z0, z1), z1) → c(G(f(z0, k(z1))), F(z0, k(z1)))
S tuples:

H2(z0, j(z1, h1(z2, z3)), h1(z2, z3)) → c3(H2(s(z0), z1, h1(s(z2), z3)))
K tuples:

F(z0, h1(z1, z2)) → c1(H2(0, z0, h1(z1, z2))) 10.71/3.17
G(h2(z0, z1, h1(z2, z3))) → c2(H2(s(z0), z1, h1(z2, z3))) 10.71/3.17
F(j(z0, z1), z1) → c(G(f(z0, k(z1))), F(z0, k(z1)))
Defined Rule Symbols:

f, g, h2, i, k

Defined Pair Symbols:

F, G, H2

Compound Symbols:

c1, c2, c3, c

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(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

H2(z0, j(z1, h1(z2, z3)), h1(z2, z3)) → c3(H2(s(z0), z1, h1(s(z2), z3)))
We considered the (Usable) Rules:

k(h(z0)) → h1(0, z0) 10.71/3.17
k(h1(z0, z1)) → h1(s(z0), z1) 10.71/3.17
f(j(z0, z1), z1) → g(f(z0, k(z1))) 10.71/3.17
f(z0, h1(z1, z2)) → h2(0, z0, h1(z1, z2)) 10.71/3.17
h2(z0, j(z1, h1(z2, z3)), h1(z2, z3)) → h2(s(z0), z1, h1(s(z2), z3)) 10.71/3.17
g(h2(z0, z1, h1(z2, z3))) → h2(s(z0), z1, h1(z2, z3))
And the Tuples:

F(z0, h1(z1, z2)) → c1(H2(0, z0, h1(z1, z2))) 10.71/3.17
G(h2(z0, z1, h1(z2, z3))) → c2(H2(s(z0), z1, h1(z2, z3))) 10.71/3.17
H2(z0, j(z1, h1(z2, z3)), h1(z2, z3)) → c3(H2(s(z0), z1, h1(s(z2), z3))) 10.71/3.17
F(j(z0, z1), z1) → c(G(f(z0, k(z1))), F(z0, k(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation : 10.71/3.17

POL(0) = 0    10.71/3.17
POL(F(x1, x2)) = x1 + x12    10.71/3.17
POL(G(x1)) = x1    10.71/3.17
POL(H2(x1, x2, x3)) = x2    10.71/3.17
POL(c(x1, x2)) = x1 + x2    10.71/3.17
POL(c1(x1)) = x1    10.71/3.17
POL(c2(x1)) = x1    10.71/3.17
POL(c3(x1)) = x1    10.71/3.17
POL(f(x1, x2)) = x1    10.71/3.17
POL(g(x1)) = x1    10.71/3.17
POL(h(x1)) = x1    10.71/3.17
POL(h1(x1, x2)) = 0    10.71/3.17
POL(h2(x1, x2, x3)) = x2    10.71/3.17
POL(j(x1, x2)) = [2] + x1    10.71/3.17
POL(k(x1)) = 0    10.71/3.17
POL(s(x1)) = 0   
10.71/3.17
10.71/3.17

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(j(z0, z1), z1) → g(f(z0, k(z1))) 10.71/3.17
f(z0, h1(z1, z2)) → h2(0, z0, h1(z1, z2)) 10.71/3.17
g(h2(z0, z1, h1(z2, z3))) → h2(s(z0), z1, h1(z2, z3)) 10.71/3.17
h2(z0, j(z1, h1(z2, z3)), h1(z2, z3)) → h2(s(z0), z1, h1(s(z2), z3)) 10.71/3.17
i(f(z0, h(z1))) → z1 10.71/3.17
i(h2(s(z0), z1, h1(z0, z2))) → z2 10.71/3.17
k(h(z0)) → h1(0, z0) 10.71/3.17
k(h1(z0, z1)) → h1(s(z0), z1)
Tuples:

F(z0, h1(z1, z2)) → c1(H2(0, z0, h1(z1, z2))) 10.71/3.17
G(h2(z0, z1, h1(z2, z3))) → c2(H2(s(z0), z1, h1(z2, z3))) 10.71/3.17
H2(z0, j(z1, h1(z2, z3)), h1(z2, z3)) → c3(H2(s(z0), z1, h1(s(z2), z3))) 10.71/3.17
F(j(z0, z1), z1) → c(G(f(z0, k(z1))), F(z0, k(z1)))
S tuples:none
K tuples:

F(z0, h1(z1, z2)) → c1(H2(0, z0, h1(z1, z2))) 10.71/3.17
G(h2(z0, z1, h1(z2, z3))) → c2(H2(s(z0), z1, h1(z2, z3))) 10.71/3.17
F(j(z0, z1), z1) → c(G(f(z0, k(z1))), F(z0, k(z1))) 10.71/3.17
H2(z0, j(z1, h1(z2, z3)), h1(z2, z3)) → c3(H2(s(z0), z1, h1(s(z2), z3)))
Defined Rule Symbols:

f, g, h2, i, k

Defined Pair Symbols:

F, G, H2

Compound Symbols:

c1, c2, c3, c

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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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11.09/3.24 EOF