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) Obligation:

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

terms(N) → cons(recip(sqr(N)), n__terms(s(N))) 0.00/0.61
sqr(0) → 0 0.00/0.61
sqr(s(X)) → s(n__add(sqr(activate(X)), dbl(activate(X)))) 0.00/0.61
dbl(0) → 0 0.00/0.61
dbl(s(X)) → s(n__s(n__dbl(activate(X)))) 0.00/0.61
add(0, X) → X 0.00/0.61
add(s(X), Y) → s(n__add(activate(X), Y)) 0.00/0.61
first(0, X) → nil 0.00/0.61
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z))) 0.00/0.61
terms(X) → n__terms(X) 0.00/0.61
add(X1, X2) → n__add(X1, X2) 0.00/0.61
s(X) → n__s(X) 0.00/0.61
dbl(X) → n__dbl(X) 0.00/0.61
first(X1, X2) → n__first(X1, X2) 0.00/0.61
activate(n__terms(X)) → terms(X) 0.00/0.61
activate(n__add(X1, X2)) → add(X1, X2) 0.00/0.61
activate(n__s(X)) → s(X) 0.00/0.61
activate(n__dbl(X)) → dbl(X) 0.00/0.61
activate(n__first(X1, X2)) → first(X1, X2) 0.00/0.61
activate(X) → X

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:

terms(z0) → cons(recip(sqr(z0)), n__terms(s(z0))) 0.00/0.61
terms(z0) → n__terms(z0) 0.00/0.61
sqr(0) → 0 0.00/0.61
sqr(s(z0)) → s(n__add(sqr(activate(z0)), dbl(activate(z0)))) 0.00/0.61
dbl(0) → 0 0.00/0.61
dbl(s(z0)) → s(n__s(n__dbl(activate(z0)))) 0.00/0.61
dbl(z0) → n__dbl(z0) 0.00/0.61
add(0, z0) → z0 0.00/0.61
add(s(z0), z1) → s(n__add(activate(z0), z1)) 0.00/0.61
add(z0, z1) → n__add(z0, z1) 0.00/0.61
first(0, z0) → nil 0.00/0.61
first(s(z0), cons(z1, z2)) → cons(z1, n__first(activate(z0), activate(z2))) 0.00/0.61
first(z0, z1) → n__first(z0, z1) 0.00/0.61
s(z0) → n__s(z0) 0.00/0.61
activate(n__terms(z0)) → terms(z0) 0.00/0.61
activate(n__add(z0, z1)) → add(z0, z1) 0.00/0.61
activate(n__s(z0)) → s(z0) 0.00/0.61
activate(n__dbl(z0)) → dbl(z0) 0.00/0.61
activate(n__first(z0, z1)) → first(z0, z1) 0.00/0.61
activate(z0) → z0
Tuples:

TERMS(z0) → c(SQR(z0), S(z0)) 0.00/0.61
SQR(s(z0)) → c3(S(n__add(sqr(activate(z0)), dbl(activate(z0)))), SQR(activate(z0)), ACTIVATE(z0), DBL(activate(z0)), ACTIVATE(z0)) 0.00/0.61
DBL(s(z0)) → c5(S(n__s(n__dbl(activate(z0)))), ACTIVATE(z0)) 0.00/0.61
ADD(s(z0), z1) → c8(S(n__add(activate(z0), z1)), ACTIVATE(z0)) 0.00/0.61
FIRST(s(z0), cons(z1, z2)) → c11(ACTIVATE(z0), ACTIVATE(z2)) 0.00/0.61
ACTIVATE(n__terms(z0)) → c14(TERMS(z0)) 0.00/0.61
ACTIVATE(n__add(z0, z1)) → c15(ADD(z0, z1)) 0.00/0.61
ACTIVATE(n__s(z0)) → c16(S(z0)) 0.00/0.61
ACTIVATE(n__dbl(z0)) → c17(DBL(z0)) 0.00/0.61
ACTIVATE(n__first(z0, z1)) → c18(FIRST(z0, z1))
S tuples:

TERMS(z0) → c(SQR(z0), S(z0)) 0.00/0.61
SQR(s(z0)) → c3(S(n__add(sqr(activate(z0)), dbl(activate(z0)))), SQR(activate(z0)), ACTIVATE(z0), DBL(activate(z0)), ACTIVATE(z0)) 0.00/0.61
DBL(s(z0)) → c5(S(n__s(n__dbl(activate(z0)))), ACTIVATE(z0)) 0.00/0.61
ADD(s(z0), z1) → c8(S(n__add(activate(z0), z1)), ACTIVATE(z0)) 0.00/0.61
FIRST(s(z0), cons(z1, z2)) → c11(ACTIVATE(z0), ACTIVATE(z2)) 0.00/0.61
ACTIVATE(n__terms(z0)) → c14(TERMS(z0)) 0.00/0.61
ACTIVATE(n__add(z0, z1)) → c15(ADD(z0, z1)) 0.00/0.61
ACTIVATE(n__s(z0)) → c16(S(z0)) 0.00/0.61
ACTIVATE(n__dbl(z0)) → c17(DBL(z0)) 0.00/0.61
ACTIVATE(n__first(z0, z1)) → c18(FIRST(z0, z1))
K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first, s, activate

Defined Pair Symbols:

TERMS, SQR, DBL, ADD, FIRST, ACTIVATE

Compound Symbols:

c, c3, c5, c8, c11, c14, c15, c16, 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:

SQR(s(z0)) → c3(S(n__add(sqr(activate(z0)), dbl(activate(z0)))), SQR(activate(z0)), ACTIVATE(z0), DBL(activate(z0)), ACTIVATE(z0)) 0.00/0.61
DBL(s(z0)) → c5(S(n__s(n__dbl(activate(z0)))), ACTIVATE(z0)) 0.00/0.61
ADD(s(z0), z1) → c8(S(n__add(activate(z0), z1)), ACTIVATE(z0)) 0.00/0.61
FIRST(s(z0), cons(z1, z2)) → c11(ACTIVATE(z0), ACTIVATE(z2))
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(s(z0))) 0.00/0.61
terms(z0) → n__terms(z0) 0.00/0.61
sqr(0) → 0 0.00/0.61
sqr(s(z0)) → s(n__add(sqr(activate(z0)), dbl(activate(z0)))) 0.00/0.61
dbl(0) → 0 0.00/0.61
dbl(s(z0)) → s(n__s(n__dbl(activate(z0)))) 0.00/0.61
dbl(z0) → n__dbl(z0) 0.00/0.61
add(0, z0) → z0 0.00/0.61
add(s(z0), z1) → s(n__add(activate(z0), z1)) 0.00/0.61
add(z0, z1) → n__add(z0, z1) 0.00/0.61
first(0, z0) → nil 0.00/0.61
first(s(z0), cons(z1, z2)) → cons(z1, n__first(activate(z0), activate(z2))) 0.00/0.61
first(z0, z1) → n__first(z0, z1) 0.00/0.61
s(z0) → n__s(z0) 0.00/0.61
activate(n__terms(z0)) → terms(z0) 0.00/0.61
activate(n__add(z0, z1)) → add(z0, z1) 0.00/0.61
activate(n__s(z0)) → s(z0) 0.00/0.61
activate(n__dbl(z0)) → dbl(z0) 0.00/0.61
activate(n__first(z0, z1)) → first(z0, z1) 0.00/0.61
activate(z0) → z0
Tuples:

TERMS(z0) → c(SQR(z0), S(z0)) 0.00/0.61
ACTIVATE(n__terms(z0)) → c14(TERMS(z0)) 0.00/0.61
ACTIVATE(n__add(z0, z1)) → c15(ADD(z0, z1)) 0.00/0.61
ACTIVATE(n__s(z0)) → c16(S(z0)) 0.00/0.61
ACTIVATE(n__dbl(z0)) → c17(DBL(z0)) 0.00/0.61
ACTIVATE(n__first(z0, z1)) → c18(FIRST(z0, z1))
S tuples:

TERMS(z0) → c(SQR(z0), S(z0)) 0.00/0.61
ACTIVATE(n__terms(z0)) → c14(TERMS(z0)) 0.00/0.61
ACTIVATE(n__add(z0, z1)) → c15(ADD(z0, z1)) 0.00/0.61
ACTIVATE(n__s(z0)) → c16(S(z0)) 0.00/0.61
ACTIVATE(n__dbl(z0)) → c17(DBL(z0)) 0.00/0.61
ACTIVATE(n__first(z0, z1)) → c18(FIRST(z0, z1))
K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first, s, activate

Defined Pair Symbols:

TERMS, ACTIVATE

Compound Symbols:

c, c14, c15, c16, c17, 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:

terms(z0) → cons(recip(sqr(z0)), n__terms(s(z0))) 0.00/0.61
terms(z0) → n__terms(z0) 0.00/0.61
sqr(0) → 0 0.00/0.61
sqr(s(z0)) → s(n__add(sqr(activate(z0)), dbl(activate(z0)))) 0.00/0.61
dbl(0) → 0 0.00/0.61
dbl(s(z0)) → s(n__s(n__dbl(activate(z0)))) 0.00/0.61
dbl(z0) → n__dbl(z0) 0.00/0.61
add(0, z0) → z0 0.00/0.61
add(s(z0), z1) → s(n__add(activate(z0), z1)) 0.00/0.61
add(z0, z1) → n__add(z0, z1) 0.00/0.61
first(0, z0) → nil 0.00/0.61
first(s(z0), cons(z1, z2)) → cons(z1, n__first(activate(z0), activate(z2))) 0.00/0.61
first(z0, z1) → n__first(z0, z1) 0.00/0.61
s(z0) → n__s(z0) 0.00/0.61
activate(n__terms(z0)) → terms(z0) 0.00/0.61
activate(n__add(z0, z1)) → add(z0, z1) 0.00/0.61
activate(n__s(z0)) → s(z0) 0.00/0.61
activate(n__dbl(z0)) → dbl(z0) 0.00/0.61
activate(n__first(z0, z1)) → first(z0, z1) 0.00/0.61
activate(z0) → z0
Tuples:

ACTIVATE(n__terms(z0)) → c14(TERMS(z0)) 0.00/0.61
TERMS(z0) → c 0.00/0.61
ACTIVATE(n__add(z0, z1)) → c15 0.00/0.61
ACTIVATE(n__s(z0)) → c16 0.00/0.61
ACTIVATE(n__dbl(z0)) → c17 0.00/0.61
ACTIVATE(n__first(z0, z1)) → c18
S tuples:

ACTIVATE(n__terms(z0)) → c14(TERMS(z0)) 0.00/0.61
TERMS(z0) → c 0.00/0.61
ACTIVATE(n__add(z0, z1)) → c15 0.00/0.61
ACTIVATE(n__s(z0)) → c16 0.00/0.61
ACTIVATE(n__dbl(z0)) → c17 0.00/0.61
ACTIVATE(n__first(z0, z1)) → c18
K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first, s, activate

Defined Pair Symbols:

ACTIVATE, TERMS

Compound Symbols:

c14, c, c15, c16, c17, c18

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

Removed 6 trailing nodes:

TERMS(z0) → c 0.00/0.61
ACTIVATE(n__first(z0, z1)) → c18 0.00/0.61
ACTIVATE(n__add(z0, z1)) → c15 0.00/0.61
ACTIVATE(n__terms(z0)) → c14(TERMS(z0)) 0.00/0.61
ACTIVATE(n__s(z0)) → c16 0.00/0.61
ACTIVATE(n__dbl(z0)) → c17
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(s(z0))) 0.00/0.61
terms(z0) → n__terms(z0) 0.00/0.61
sqr(0) → 0 0.00/0.61
sqr(s(z0)) → s(n__add(sqr(activate(z0)), dbl(activate(z0)))) 0.00/0.61
dbl(0) → 0 0.00/0.61
dbl(s(z0)) → s(n__s(n__dbl(activate(z0)))) 0.00/0.61
dbl(z0) → n__dbl(z0) 0.00/0.61
add(0, z0) → z0 0.00/0.61
add(s(z0), z1) → s(n__add(activate(z0), z1)) 0.00/0.61
add(z0, z1) → n__add(z0, z1) 0.00/0.61
first(0, z0) → nil 0.00/0.61
first(s(z0), cons(z1, z2)) → cons(z1, n__first(activate(z0), activate(z2))) 0.00/0.61
first(z0, z1) → n__first(z0, z1) 0.00/0.61
s(z0) → n__s(z0) 0.00/0.61
activate(n__terms(z0)) → terms(z0) 0.00/0.61
activate(n__add(z0, z1)) → add(z0, z1) 0.00/0.61
activate(n__s(z0)) → s(z0) 0.00/0.61
activate(n__dbl(z0)) → dbl(z0) 0.00/0.61
activate(n__first(z0, z1)) → first(z0, z1) 0.00/0.61
activate(z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first, s, activate

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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0.00/0.63 EOF