YES(O(1), O(n^1)) 2.44/1.02 YES(O(1), O(n^1)) 2.44/1.06 2.44/1.06 2.44/1.06 2.44/1.06 2.44/1.06 2.44/1.06 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 2.44/1.06 2.44/1.06 2.44/1.06
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
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↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
2.44/1.06 
↳2 CdtProblem
2.44/1.06 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
2.44/1.06 
↳4 CdtProblem
2.44/1.06 
↳5 CdtLeafRemovalProof (ComplexityIfPolyImplication)
2.44/1.06 
↳6 CdtProblem
2.44/1.06 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
2.44/1.06 
↳8 CdtProblem
2.44/1.06 
↳9 CdtKnowledgeProof (⇔)
2.44/1.06 
↳10 BOUNDS(O(1), O(1))
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2.44/1.06

(0) Obligation:

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

natsFrom(N) → cons(N, n__natsFrom(s(N))) 2.44/1.06
fst(pair(XS, YS)) → XS 2.44/1.06
snd(pair(XS, YS)) → YS 2.44/1.06
splitAt(0, XS) → pair(nil, XS) 2.44/1.06
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS)) 2.44/1.06
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS) 2.44/1.06
head(cons(N, XS)) → N 2.44/1.06
tail(cons(N, XS)) → activate(XS) 2.44/1.06
sel(N, XS) → head(afterNth(N, XS)) 2.44/1.06
take(N, XS) → fst(splitAt(N, XS)) 2.44/1.06
afterNth(N, XS) → snd(splitAt(N, XS)) 2.44/1.06
natsFrom(X) → n__natsFrom(X) 2.44/1.06
activate(n__natsFrom(X)) → natsFrom(X) 2.44/1.06
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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2.44/1.06

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

natsFrom(z0) → cons(z0, n__natsFrom(s(z0))) 2.44/1.06
natsFrom(z0) → n__natsFrom(z0) 2.44/1.06
fst(pair(z0, z1)) → z0 2.44/1.06
snd(pair(z0, z1)) → z1 2.44/1.06
splitAt(0, z0) → pair(nil, z0) 2.44/1.06
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2)) 2.44/1.06
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1) 2.44/1.07
head(cons(z0, z1)) → z0 2.44/1.07
tail(cons(z0, z1)) → activate(z1) 2.44/1.07
sel(z0, z1) → head(afterNth(z0, z1)) 2.44/1.07
take(z0, z1) → fst(splitAt(z0, z1)) 2.44/1.07
afterNth(z0, z1) → snd(splitAt(z0, z1)) 2.44/1.07
activate(n__natsFrom(z0)) → natsFrom(z0) 2.44/1.07
activate(z0) → z0
Tuples:

SPLITAT(s(z0), cons(z1, z2)) → c5(U(splitAt(z0, activate(z2)), z0, z1, activate(z2)), SPLITAT(z0, activate(z2)), ACTIVATE(z2), ACTIVATE(z2)) 2.44/1.07
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 2.44/1.07
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1)) 2.44/1.07
SEL(z0, z1) → c9(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1)) 2.44/1.07
TAKE(z0, z1) → c10(FST(splitAt(z0, z1)), SPLITAT(z0, z1)) 2.44/1.07
AFTERNTH(z0, z1) → c11(SND(splitAt(z0, z1)), SPLITAT(z0, z1)) 2.44/1.07
ACTIVATE(n__natsFrom(z0)) → c12(NATSFROM(z0))
S tuples:

SPLITAT(s(z0), cons(z1, z2)) → c5(U(splitAt(z0, activate(z2)), z0, z1, activate(z2)), SPLITAT(z0, activate(z2)), ACTIVATE(z2), ACTIVATE(z2)) 2.44/1.07
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 2.44/1.07
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1)) 2.44/1.07
SEL(z0, z1) → c9(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1)) 2.44/1.07
TAKE(z0, z1) → c10(FST(splitAt(z0, z1)), SPLITAT(z0, z1)) 2.44/1.07
AFTERNTH(z0, z1) → c11(SND(splitAt(z0, z1)), SPLITAT(z0, z1)) 2.44/1.07
ACTIVATE(n__natsFrom(z0)) → c12(NATSFROM(z0))
K tuples:none
Defined Rule Symbols:

natsFrom, fst, snd, splitAt, u, head, tail, sel, take, afterNth, activate

Defined Pair Symbols:

SPLITAT, U, TAIL, SEL, TAKE, AFTERNTH, ACTIVATE

Compound Symbols:

c5, c6, c8, c9, c10, c11, c12

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2.44/1.07

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

Removed 4 trailing tuple parts
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

natsFrom(z0) → cons(z0, n__natsFrom(s(z0))) 2.44/1.07
natsFrom(z0) → n__natsFrom(z0) 2.44/1.07
fst(pair(z0, z1)) → z0 2.44/1.07
snd(pair(z0, z1)) → z1 2.44/1.07
splitAt(0, z0) → pair(nil, z0) 2.44/1.07
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2)) 2.44/1.07
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1) 2.44/1.07
head(cons(z0, z1)) → z0 2.44/1.07
tail(cons(z0, z1)) → activate(z1) 2.44/1.07
sel(z0, z1) → head(afterNth(z0, z1)) 2.44/1.07
take(z0, z1) → fst(splitAt(z0, z1)) 2.44/1.07
afterNth(z0, z1) → snd(splitAt(z0, z1)) 2.44/1.07
activate(n__natsFrom(z0)) → natsFrom(z0) 2.44/1.07
activate(z0) → z0
Tuples:

SPLITAT(s(z0), cons(z1, z2)) → c5(U(splitAt(z0, activate(z2)), z0, z1, activate(z2)), SPLITAT(z0, activate(z2)), ACTIVATE(z2), ACTIVATE(z2)) 2.44/1.07
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 2.44/1.07
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1)) 2.44/1.07
SEL(z0, z1) → c9(AFTERNTH(z0, z1)) 2.44/1.07
TAKE(z0, z1) → c10(SPLITAT(z0, z1)) 2.44/1.07
AFTERNTH(z0, z1) → c11(SPLITAT(z0, z1)) 2.44/1.07
ACTIVATE(n__natsFrom(z0)) → c12
S tuples:

SPLITAT(s(z0), cons(z1, z2)) → c5(U(splitAt(z0, activate(z2)), z0, z1, activate(z2)), SPLITAT(z0, activate(z2)), ACTIVATE(z2), ACTIVATE(z2)) 2.44/1.07
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 2.44/1.07
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1)) 2.44/1.07
SEL(z0, z1) → c9(AFTERNTH(z0, z1)) 2.44/1.07
TAKE(z0, z1) → c10(SPLITAT(z0, z1)) 2.44/1.07
AFTERNTH(z0, z1) → c11(SPLITAT(z0, z1)) 2.44/1.07
ACTIVATE(n__natsFrom(z0)) → c12
K tuples:none
Defined Rule Symbols:

natsFrom, fst, snd, splitAt, u, head, tail, sel, take, afterNth, activate

Defined Pair Symbols:

SPLITAT, U, TAIL, SEL, TAKE, AFTERNTH, ACTIVATE

Compound Symbols:

c5, c6, c8, c9, c10, c11, c12

2.44/1.07
2.44/1.07

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 3 leading nodes:

TAKE(z0, z1) → c10(SPLITAT(z0, z1)) 2.44/1.07
SEL(z0, z1) → c9(AFTERNTH(z0, z1)) 2.44/1.07
AFTERNTH(z0, z1) → c11(SPLITAT(z0, z1))
Removed 3 trailing nodes:

TAIL(cons(z0, z1)) → c8(ACTIVATE(z1)) 2.44/1.07
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 2.44/1.07
ACTIVATE(n__natsFrom(z0)) → c12
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2.44/1.07

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

natsFrom(z0) → cons(z0, n__natsFrom(s(z0))) 2.44/1.07
natsFrom(z0) → n__natsFrom(z0) 2.44/1.07
fst(pair(z0, z1)) → z0 2.44/1.07
snd(pair(z0, z1)) → z1 2.44/1.07
splitAt(0, z0) → pair(nil, z0) 2.44/1.07
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2)) 2.44/1.07
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1) 2.44/1.07
head(cons(z0, z1)) → z0 2.44/1.07
tail(cons(z0, z1)) → activate(z1) 2.44/1.07
sel(z0, z1) → head(afterNth(z0, z1)) 2.44/1.07
take(z0, z1) → fst(splitAt(z0, z1)) 2.44/1.07
afterNth(z0, z1) → snd(splitAt(z0, z1)) 2.44/1.07
activate(n__natsFrom(z0)) → natsFrom(z0) 2.44/1.07
activate(z0) → z0
Tuples:

SPLITAT(s(z0), cons(z1, z2)) → c5(U(splitAt(z0, activate(z2)), z0, z1, activate(z2)), SPLITAT(z0, activate(z2)), ACTIVATE(z2), ACTIVATE(z2)) 2.44/1.07
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 2.44/1.07
ACTIVATE(n__natsFrom(z0)) → c12
S tuples:

SPLITAT(s(z0), cons(z1, z2)) → c5(U(splitAt(z0, activate(z2)), z0, z1, activate(z2)), SPLITAT(z0, activate(z2)), ACTIVATE(z2), ACTIVATE(z2)) 2.44/1.07
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 2.44/1.07
ACTIVATE(n__natsFrom(z0)) → c12
K tuples:none
Defined Rule Symbols:

natsFrom, fst, snd, splitAt, u, head, tail, sel, take, afterNth, activate

Defined Pair Symbols:

SPLITAT, U, ACTIVATE

Compound Symbols:

c5, c6, c12

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(7) 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.

SPLITAT(s(z0), cons(z1, z2)) → c5(U(splitAt(z0, activate(z2)), z0, z1, activate(z2)), SPLITAT(z0, activate(z2)), ACTIVATE(z2), ACTIVATE(z2))
We considered the (Usable) Rules:

activate(n__natsFrom(z0)) → natsFrom(z0) 2.44/1.07
activate(z0) → z0 2.44/1.07
splitAt(0, z0) → pair(nil, z0) 2.44/1.07
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2)) 2.44/1.07
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1) 2.44/1.07
natsFrom(z0) → cons(z0, n__natsFrom(s(z0))) 2.44/1.07
natsFrom(z0) → n__natsFrom(z0)
And the Tuples:

SPLITAT(s(z0), cons(z1, z2)) → c5(U(splitAt(z0, activate(z2)), z0, z1, activate(z2)), SPLITAT(z0, activate(z2)), ACTIVATE(z2), ACTIVATE(z2)) 2.44/1.07
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 2.44/1.07
ACTIVATE(n__natsFrom(z0)) → c12
The order we found is given by the following interpretation:
Polynomial interpretation : 2.44/1.07

POL(0) = 0    2.44/1.07
POL(ACTIVATE(x1)) = 0    2.44/1.07
POL(SPLITAT(x1, x2)) = [2]x1    2.44/1.07
POL(U(x1, x2, x3, x4)) = 0    2.44/1.07
POL(activate(x1)) = 0    2.44/1.07
POL(c12) = 0    2.44/1.07
POL(c5(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    2.44/1.07
POL(c6(x1)) = x1    2.44/1.07
POL(cons(x1, x2)) = [4]    2.44/1.07
POL(n__natsFrom(x1)) = x1    2.44/1.07
POL(natsFrom(x1)) = [3] + [3]x1    2.44/1.07
POL(nil) = 0    2.44/1.07
POL(pair(x1, x2)) = [4] + x1 + x2    2.44/1.07
POL(s(x1)) = [1] + x1    2.44/1.07
POL(splitAt(x1, x2)) = [4] + [5]x2    2.44/1.07
POL(u(x1, x2, x3, x4)) = [2] + [5]x1   
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

natsFrom(z0) → cons(z0, n__natsFrom(s(z0))) 2.44/1.07
natsFrom(z0) → n__natsFrom(z0) 2.44/1.07
fst(pair(z0, z1)) → z0 2.44/1.07
snd(pair(z0, z1)) → z1 2.44/1.07
splitAt(0, z0) → pair(nil, z0) 2.44/1.07
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2)) 2.44/1.07
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1) 2.44/1.07
head(cons(z0, z1)) → z0 2.44/1.07
tail(cons(z0, z1)) → activate(z1) 2.44/1.07
sel(z0, z1) → head(afterNth(z0, z1)) 2.44/1.07
take(z0, z1) → fst(splitAt(z0, z1)) 2.44/1.07
afterNth(z0, z1) → snd(splitAt(z0, z1)) 2.44/1.07
activate(n__natsFrom(z0)) → natsFrom(z0) 2.44/1.07
activate(z0) → z0
Tuples:

SPLITAT(s(z0), cons(z1, z2)) → c5(U(splitAt(z0, activate(z2)), z0, z1, activate(z2)), SPLITAT(z0, activate(z2)), ACTIVATE(z2), ACTIVATE(z2)) 2.44/1.07
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 2.44/1.07
ACTIVATE(n__natsFrom(z0)) → c12
S tuples:

U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 2.44/1.07
ACTIVATE(n__natsFrom(z0)) → c12
K tuples:

SPLITAT(s(z0), cons(z1, z2)) → c5(U(splitAt(z0, activate(z2)), z0, z1, activate(z2)), SPLITAT(z0, activate(z2)), ACTIVATE(z2), ACTIVATE(z2))
Defined Rule Symbols:

natsFrom, fst, snd, splitAt, u, head, tail, sel, take, afterNth, activate

Defined Pair Symbols:

SPLITAT, U, ACTIVATE

Compound Symbols:

c5, c6, c12

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2.44/1.07

(9) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 2.44/1.07
ACTIVATE(n__natsFrom(z0)) → c12 2.44/1.07
ACTIVATE(n__natsFrom(z0)) → c12
Now S is empty
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(10) BOUNDS(O(1), O(1))

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2.75/1.13 EOF