YES(O(1), O(n^1)) 3.06/1.23 YES(O(1), O(n^1)) 3.48/1.32 3.48/1.32 3.48/1.32 3.48/1.32 3.48/1.32 3.48/1.32 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 3.48/1.32 3.48/1.32 3.48/1.32
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3.48/1.32
3.48/1.32
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
3.48/1.32 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
3.48/1.32 
↳2 CdtProblem
3.48/1.32 
↳3 CdtUnreachableProof (⇔)
3.48/1.32 
↳4 CdtProblem
3.48/1.32 
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
3.48/1.32 
↳6 CdtProblem
3.48/1.32 
↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
3.48/1.32 
↳8 CdtProblem
3.48/1.32 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
3.48/1.32 
↳10 CdtProblem
3.48/1.32 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
3.48/1.32 
↳12 CdtProblem
3.48/1.32 
↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID))
3.48/1.32 
↳14 BOUNDS(O(1), O(1))
3.48/1.32
3.48/1.32 3.48/1.32
3.48/1.32
3.48/1.32

(0) Obligation:

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

i(x, x) → i(a, b) 3.48/1.32
g(x, x) → g(a, b) 3.48/1.32
h(s(f(x))) → h(f(x)) 3.48/1.32
f(s(x)) → s(s(f(h(s(x))))) 3.48/1.32
f(g(s(x), y)) → f(g(x, s(y))) 3.48/1.32
h(g(x, s(y))) → h(g(s(x), y)) 3.48/1.32
h(i(x, y)) → i(i(c, h(h(y))), x) 3.48/1.32
g(a, g(x, g(b, g(a, g(x, y))))) → g(a, g(a, g(a, g(x, g(b, g(b, y))))))

Rewrite Strategy: INNERMOST
3.48/1.32
3.48/1.32

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
3.48/1.32
3.48/1.32

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

i(z0, z0) → i(a, b) 3.48/1.32
g(z0, z0) → g(a, b) 3.48/1.32
g(a, g(z0, g(b, g(a, g(z0, z1))))) → g(a, g(a, g(a, g(z0, g(b, g(b, z1)))))) 3.48/1.32
h(s(f(z0))) → h(f(z0)) 3.48/1.32
h(g(z0, s(z1))) → h(g(s(z0), z1)) 3.48/1.32
h(i(z0, z1)) → i(i(c, h(h(z1))), z0) 3.48/1.32
f(s(z0)) → s(s(f(h(s(z0))))) 3.48/1.32
f(g(s(z0), z1)) → f(g(z0, s(z1)))
Tuples:

I(z0, z0) → c1(I(a, b)) 3.48/1.32
G(z0, z0) → c2(G(a, b)) 3.48/1.32
G(a, g(z0, g(b, g(a, g(z0, z1))))) → c3(G(a, g(a, g(a, g(z0, g(b, g(b, z1)))))), G(a, g(a, g(z0, g(b, g(b, z1))))), G(a, g(z0, g(b, g(b, z1)))), G(z0, g(b, g(b, z1))), G(b, g(b, z1)), G(b, z1)) 3.48/1.32
H(s(f(z0))) → c4(H(f(z0)), F(z0)) 3.48/1.32
H(g(z0, s(z1))) → c5(H(g(s(z0), z1)), G(s(z0), z1)) 3.48/1.32
H(i(z0, z1)) → c6(I(i(c, h(h(z1))), z0), I(c, h(h(z1))), H(h(z1)), H(z1)) 3.48/1.32
F(s(z0)) → c7(F(h(s(z0))), H(s(z0))) 3.48/1.32
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1)))
S tuples:

I(z0, z0) → c1(I(a, b)) 3.48/1.32
G(z0, z0) → c2(G(a, b)) 3.48/1.32
G(a, g(z0, g(b, g(a, g(z0, z1))))) → c3(G(a, g(a, g(a, g(z0, g(b, g(b, z1)))))), G(a, g(a, g(z0, g(b, g(b, z1))))), G(a, g(z0, g(b, g(b, z1)))), G(z0, g(b, g(b, z1))), G(b, g(b, z1)), G(b, z1)) 3.48/1.32
H(s(f(z0))) → c4(H(f(z0)), F(z0)) 3.48/1.32
H(g(z0, s(z1))) → c5(H(g(s(z0), z1)), G(s(z0), z1)) 3.48/1.32
H(i(z0, z1)) → c6(I(i(c, h(h(z1))), z0), I(c, h(h(z1))), H(h(z1)), H(z1)) 3.48/1.32
F(s(z0)) → c7(F(h(s(z0))), H(s(z0))) 3.48/1.32
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1)))
K tuples:none
Defined Rule Symbols:

i, g, h, f

Defined Pair Symbols:

I, G, H, F

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8

3.48/1.32
3.48/1.32

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

G(a, g(z0, g(b, g(a, g(z0, z1))))) → c3(G(a, g(a, g(a, g(z0, g(b, g(b, z1)))))), G(a, g(a, g(z0, g(b, g(b, z1))))), G(a, g(z0, g(b, g(b, z1)))), G(z0, g(b, g(b, z1))), G(b, g(b, z1)), G(b, z1)) 3.48/1.32
H(g(z0, s(z1))) → c5(H(g(s(z0), z1)), G(s(z0), z1)) 3.48/1.32
H(i(z0, z1)) → c6(I(i(c, h(h(z1))), z0), I(c, h(h(z1))), H(h(z1)), H(z1))
3.48/1.32
3.48/1.32

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

i(z0, z0) → i(a, b) 3.48/1.32
g(z0, z0) → g(a, b) 3.48/1.32
g(a, g(z0, g(b, g(a, g(z0, z1))))) → g(a, g(a, g(a, g(z0, g(b, g(b, z1)))))) 3.48/1.32
h(s(f(z0))) → h(f(z0)) 3.48/1.32
h(g(z0, s(z1))) → h(g(s(z0), z1)) 3.48/1.32
h(i(z0, z1)) → i(i(c, h(h(z1))), z0) 3.48/1.32
f(s(z0)) → s(s(f(h(s(z0))))) 3.48/1.32
f(g(s(z0), z1)) → f(g(z0, s(z1)))
Tuples:

I(z0, z0) → c1(I(a, b)) 3.48/1.32
G(z0, z0) → c2(G(a, b)) 3.48/1.32
F(s(z0)) → c7(F(h(s(z0))), H(s(z0))) 3.48/1.32
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1))) 3.48/1.32
H(s(f(z0))) → c4(H(f(z0)), F(z0))
S tuples:

I(z0, z0) → c1(I(a, b)) 3.48/1.32
G(z0, z0) → c2(G(a, b)) 3.48/1.32
H(s(f(z0))) → c4(H(f(z0)), F(z0)) 3.48/1.32
F(s(z0)) → c7(F(h(s(z0))), H(s(z0))) 3.48/1.32
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1)))
K tuples:none
Defined Rule Symbols:

i, g, h, f

Defined Pair Symbols:

I, G, F, H

Compound Symbols:

c1, c2, c7, c8, c4

3.48/1.32
3.48/1.32

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

Removed 3 trailing tuple parts
3.48/1.32
3.48/1.32

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

i(z0, z0) → i(a, b) 3.48/1.32
g(z0, z0) → g(a, b) 3.48/1.32
g(a, g(z0, g(b, g(a, g(z0, z1))))) → g(a, g(a, g(a, g(z0, g(b, g(b, z1)))))) 3.48/1.32
h(s(f(z0))) → h(f(z0)) 3.48/1.32
h(g(z0, s(z1))) → h(g(s(z0), z1)) 3.48/1.32
h(i(z0, z1)) → i(i(c, h(h(z1))), z0) 3.48/1.32
f(s(z0)) → s(s(f(h(s(z0))))) 3.48/1.32
f(g(s(z0), z1)) → f(g(z0, s(z1)))
Tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0))) 3.48/1.32
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1))) 3.48/1.32
I(z0, z0) → c1 3.48/1.32
G(z0, z0) → c2 3.48/1.32
H(s(f(z0))) → c4(F(z0))
S tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0))) 3.48/1.32
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1))) 3.48/1.32
I(z0, z0) → c1 3.48/1.32
G(z0, z0) → c2 3.48/1.32
H(s(f(z0))) → c4(F(z0))
K tuples:none
Defined Rule Symbols:

i, g, h, f

Defined Pair Symbols:

F, I, G, H

Compound Symbols:

c7, c8, c1, c2, c4

3.48/1.32
3.48/1.32

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

Removed 2 trailing nodes:

I(z0, z0) → c1 3.48/1.32
G(z0, z0) → c2
3.48/1.32
3.48/1.32

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

i(z0, z0) → i(a, b) 3.48/1.32
g(z0, z0) → g(a, b) 3.48/1.32
g(a, g(z0, g(b, g(a, g(z0, z1))))) → g(a, g(a, g(a, g(z0, g(b, g(b, z1)))))) 3.48/1.32
h(s(f(z0))) → h(f(z0)) 3.48/1.32
h(g(z0, s(z1))) → h(g(s(z0), z1)) 3.48/1.32
h(i(z0, z1)) → i(i(c, h(h(z1))), z0) 3.48/1.32
f(s(z0)) → s(s(f(h(s(z0))))) 3.48/1.32
f(g(s(z0), z1)) → f(g(z0, s(z1)))
Tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0))) 3.48/1.32
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1))) 3.48/1.32
G(z0, z0) → c2 3.48/1.32
H(s(f(z0))) → c4(F(z0))
S tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0))) 3.48/1.32
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1))) 3.48/1.32
G(z0, z0) → c2 3.48/1.32
H(s(f(z0))) → c4(F(z0))
K tuples:none
Defined Rule Symbols:

i, g, h, f

Defined Pair Symbols:

F, G, H

Compound Symbols:

c7, c8, c2, c4

3.48/1.32
3.48/1.32

(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.

F(s(z0)) → c7(F(h(s(z0))), H(s(z0))) 3.48/1.32
H(s(f(z0))) → c4(F(z0))
We considered the (Usable) Rules:

g(z0, z0) → g(a, b) 3.48/1.32
h(s(f(z0))) → h(f(z0))
And the Tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0))) 3.48/1.32
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1))) 3.48/1.32
G(z0, z0) → c2 3.48/1.32
H(s(f(z0))) → c4(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 3.48/1.32

POL(F(x1)) = [5] + [4]x1    3.48/1.32
POL(G(x1, x2)) = 0    3.48/1.32
POL(H(x1)) = [5] + [2]x1    3.48/1.32
POL(a) = [1]    3.48/1.32
POL(b) = [1]    3.48/1.32
POL(c2) = 0    3.48/1.32
POL(c4(x1)) = x1    3.48/1.32
POL(c7(x1, x2)) = x1 + x2    3.48/1.32
POL(c8(x1, x2)) = x1 + x2    3.48/1.32
POL(f(x1)) = [4] + [4]x1    3.48/1.32
POL(g(x1, x2)) = 0    3.48/1.32
POL(h(x1)) = [1]    3.48/1.32
POL(s(x1)) = [5] + x1   
3.48/1.32
3.48/1.32

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

i(z0, z0) → i(a, b) 3.48/1.32
g(z0, z0) → g(a, b) 3.48/1.32
g(a, g(z0, g(b, g(a, g(z0, z1))))) → g(a, g(a, g(a, g(z0, g(b, g(b, z1)))))) 3.48/1.32
h(s(f(z0))) → h(f(z0)) 3.48/1.32
h(g(z0, s(z1))) → h(g(s(z0), z1)) 3.48/1.32
h(i(z0, z1)) → i(i(c, h(h(z1))), z0) 3.48/1.32
f(s(z0)) → s(s(f(h(s(z0))))) 3.48/1.32
f(g(s(z0), z1)) → f(g(z0, s(z1)))
Tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0))) 3.48/1.32
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1))) 3.48/1.32
G(z0, z0) → c2 3.48/1.32
H(s(f(z0))) → c4(F(z0))
S tuples:

F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1))) 3.48/1.32
G(z0, z0) → c2
K tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0))) 3.48/1.32
H(s(f(z0))) → c4(F(z0))
Defined Rule Symbols:

i, g, h, f

Defined Pair Symbols:

F, G, H

Compound Symbols:

c7, c8, c2, c4

3.48/1.32
3.48/1.32

(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.

F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1))) 3.48/1.32
G(z0, z0) → c2
We considered the (Usable) Rules:

g(z0, z0) → g(a, b) 3.48/1.32
h(s(f(z0))) → h(f(z0))
And the Tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0))) 3.48/1.32
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1))) 3.48/1.32
G(z0, z0) → c2 3.48/1.32
H(s(f(z0))) → c4(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 3.48/1.32

POL(F(x1)) = [2] + [2]x1    3.48/1.32
POL(G(x1, x2)) = [1]    3.48/1.32
POL(H(x1)) = [2]x1    3.48/1.32
POL(a) = 0    3.48/1.32
POL(b) = [1]    3.48/1.32
POL(c2) = 0    3.48/1.32
POL(c4(x1)) = x1    3.48/1.32
POL(c7(x1, x2)) = x1 + x2    3.48/1.32
POL(c8(x1, x2)) = x1 + x2    3.48/1.32
POL(f(x1)) = x1    3.48/1.32
POL(g(x1, x2)) = [2]x1    3.48/1.32
POL(h(x1)) = 0    3.48/1.32
POL(s(x1)) = [1] + x1   
3.48/1.32
3.48/1.32

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

i(z0, z0) → i(a, b) 3.48/1.32
g(z0, z0) → g(a, b) 3.48/1.32
g(a, g(z0, g(b, g(a, g(z0, z1))))) → g(a, g(a, g(a, g(z0, g(b, g(b, z1)))))) 3.48/1.32
h(s(f(z0))) → h(f(z0)) 3.48/1.32
h(g(z0, s(z1))) → h(g(s(z0), z1)) 3.48/1.32
h(i(z0, z1)) → i(i(c, h(h(z1))), z0) 3.48/1.32
f(s(z0)) → s(s(f(h(s(z0))))) 3.48/1.32
f(g(s(z0), z1)) → f(g(z0, s(z1)))
Tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0))) 3.48/1.32
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1))) 3.48/1.32
G(z0, z0) → c2 3.48/1.32
H(s(f(z0))) → c4(F(z0))
S tuples:none
K tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0))) 3.48/1.32
H(s(f(z0))) → c4(F(z0)) 3.48/1.32
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1))) 3.48/1.32
G(z0, z0) → c2
Defined Rule Symbols:

i, g, h, f

Defined Pair Symbols:

F, G, H

Compound Symbols:

c7, c8, c2, c4

3.48/1.32
3.48/1.32

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

The set S is empty
3.48/1.32
3.48/1.32

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

3.48/1.32
3.48/1.32
3.81/1.40 EOF