YES(O(1), O(n^1)) 0.00/0.81 YES(O(1), O(n^1)) 0.00/0.84 0.00/0.84 0.00/0.84 0.00/0.84 0.00/0.84 0.00/0.84 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.84 0.00/0.84 0.00/0.84
0.00/0.84
0.00/0.84
0.00/0.84
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
0.00/0.84 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
0.00/0.84 
↳2 CdtProblem
0.00/0.84 
↳3 CdtUnreachableProof (⇔)
0.00/0.84 
↳4 CdtProblem
0.00/0.84 
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
0.00/0.84 
↳6 CdtProblem
0.00/0.84 
↳7 CdtLeafRemovalProof (ComplexityIfPolyImplication)
0.00/0.84 
↳8 CdtProblem
0.00/0.84 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/0.84 
↳10 CdtProblem
0.00/0.84 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/0.84 
↳12 CdtProblem
0.00/0.84 
↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID))
0.00/0.84 
↳14 BOUNDS(O(1), O(1))
0.00/0.84
0.00/0.84 0.00/0.84
0.00/0.84
0.00/0.84

(0) Obligation:

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

natsFrom(N) → cons(N, n__natsFrom(n__s(N))) 0.00/0.84
fst(pair(XS, YS)) → XS 0.00/0.84
snd(pair(XS, YS)) → YS 0.00/0.84
splitAt(0, XS) → pair(nil, XS) 0.00/0.84
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS)) 0.00/0.84
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS) 0.00/0.84
head(cons(N, XS)) → N 0.00/0.84
tail(cons(N, XS)) → activate(XS) 0.00/0.84
sel(N, XS) → head(afterNth(N, XS)) 0.00/0.84
take(N, XS) → fst(splitAt(N, XS)) 0.00/0.84
afterNth(N, XS) → snd(splitAt(N, XS)) 0.00/0.84
natsFrom(X) → n__natsFrom(X) 0.00/0.84
s(X) → n__s(X) 0.00/0.84
activate(n__natsFrom(X)) → natsFrom(activate(X)) 0.00/0.84
activate(n__s(X)) → s(activate(X)) 0.00/0.84
activate(X) → X

Rewrite Strategy: INNERMOST
0.00/0.84
0.00/0.84

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

Converted CpxTRS to CDT
0.00/0.84
0.00/0.84

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0))) 0.00/0.84
natsFrom(z0) → n__natsFrom(z0) 0.00/0.84
fst(pair(z0, z1)) → z0 0.00/0.84
snd(pair(z0, z1)) → z1 0.00/0.84
splitAt(0, z0) → pair(nil, z0) 0.00/0.84
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2)) 0.00/0.84
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1) 0.00/0.84
head(cons(z0, z1)) → z0 0.00/0.84
tail(cons(z0, z1)) → activate(z1) 0.00/0.84
sel(z0, z1) → head(afterNth(z0, z1)) 0.00/0.84
take(z0, z1) → fst(splitAt(z0, z1)) 0.00/0.84
afterNth(z0, z1) → snd(splitAt(z0, z1)) 0.00/0.84
s(z0) → n__s(z0) 0.00/0.84
activate(n__natsFrom(z0)) → natsFrom(activate(z0)) 0.00/0.84
activate(n__s(z0)) → s(activate(z0)) 0.00/0.84
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)) 0.00/0.84
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 0.00/0.84
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1)) 0.00/0.84
SEL(z0, z1) → c9(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1)) 0.00/0.84
TAKE(z0, z1) → c10(FST(splitAt(z0, z1)), SPLITAT(z0, z1)) 0.00/0.84
AFTERNTH(z0, z1) → c11(SND(splitAt(z0, z1)), SPLITAT(z0, z1)) 0.00/0.84
ACTIVATE(n__natsFrom(z0)) → c13(NATSFROM(activate(z0)), ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(S(activate(z0)), ACTIVATE(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)) 0.00/0.84
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 0.00/0.84
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1)) 0.00/0.84
SEL(z0, z1) → c9(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1)) 0.00/0.84
TAKE(z0, z1) → c10(FST(splitAt(z0, z1)), SPLITAT(z0, z1)) 0.00/0.84
AFTERNTH(z0, z1) → c11(SND(splitAt(z0, z1)), SPLITAT(z0, z1)) 0.00/0.84
ACTIVATE(n__natsFrom(z0)) → c13(NATSFROM(activate(z0)), ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c5, c6, c8, c9, c10, c11, c13, c14

0.00/0.84
0.00/0.84

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

SPLITAT(s(z0), cons(z1, z2)) → c5(U(splitAt(z0, activate(z2)), z0, z1, activate(z2)), SPLITAT(z0, activate(z2)), ACTIVATE(z2), ACTIVATE(z2))
0.00/0.84
0.00/0.84

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0))) 0.00/0.84
natsFrom(z0) → n__natsFrom(z0) 0.00/0.84
fst(pair(z0, z1)) → z0 0.00/0.84
snd(pair(z0, z1)) → z1 0.00/0.84
splitAt(0, z0) → pair(nil, z0) 0.00/0.84
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2)) 0.00/0.84
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1) 0.00/0.84
head(cons(z0, z1)) → z0 0.00/0.84
tail(cons(z0, z1)) → activate(z1) 0.00/0.84
sel(z0, z1) → head(afterNth(z0, z1)) 0.00/0.84
take(z0, z1) → fst(splitAt(z0, z1)) 0.00/0.84
afterNth(z0, z1) → snd(splitAt(z0, z1)) 0.00/0.84
s(z0) → n__s(z0) 0.00/0.84
activate(n__natsFrom(z0)) → natsFrom(activate(z0)) 0.00/0.84
activate(n__s(z0)) → s(activate(z0)) 0.00/0.84
activate(z0) → z0
Tuples:

U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 0.00/0.84
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1)) 0.00/0.84
SEL(z0, z1) → c9(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1)) 0.00/0.84
TAKE(z0, z1) → c10(FST(splitAt(z0, z1)), SPLITAT(z0, z1)) 0.00/0.84
AFTERNTH(z0, z1) → c11(SND(splitAt(z0, z1)), SPLITAT(z0, z1)) 0.00/0.84
ACTIVATE(n__natsFrom(z0)) → c13(NATSFROM(activate(z0)), ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(S(activate(z0)), ACTIVATE(z0))
S tuples:

U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 0.00/0.84
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1)) 0.00/0.84
SEL(z0, z1) → c9(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1)) 0.00/0.84
TAKE(z0, z1) → c10(FST(splitAt(z0, z1)), SPLITAT(z0, z1)) 0.00/0.84
AFTERNTH(z0, z1) → c11(SND(splitAt(z0, z1)), SPLITAT(z0, z1)) 0.00/0.84
ACTIVATE(n__natsFrom(z0)) → c13(NATSFROM(activate(z0)), ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

U, TAIL, SEL, TAKE, AFTERNTH, ACTIVATE

Compound Symbols:

c6, c8, c9, c10, c11, c13, c14

0.00/0.84
0.00/0.84

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

Removed 7 trailing tuple parts
0.00/0.84
0.00/0.84

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0))) 0.00/0.84
natsFrom(z0) → n__natsFrom(z0) 0.00/0.84
fst(pair(z0, z1)) → z0 0.00/0.84
snd(pair(z0, z1)) → z1 0.00/0.84
splitAt(0, z0) → pair(nil, z0) 0.00/0.84
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2)) 0.00/0.84
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1) 0.00/0.84
head(cons(z0, z1)) → z0 0.00/0.84
tail(cons(z0, z1)) → activate(z1) 0.00/0.84
sel(z0, z1) → head(afterNth(z0, z1)) 0.00/0.84
take(z0, z1) → fst(splitAt(z0, z1)) 0.00/0.84
afterNth(z0, z1) → snd(splitAt(z0, z1)) 0.00/0.84
s(z0) → n__s(z0) 0.00/0.84
activate(n__natsFrom(z0)) → natsFrom(activate(z0)) 0.00/0.84
activate(n__s(z0)) → s(activate(z0)) 0.00/0.84
activate(z0) → z0
Tuples:

U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 0.00/0.84
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1)) 0.00/0.84
SEL(z0, z1) → c9(AFTERNTH(z0, z1)) 0.00/0.84
TAKE(z0, z1) → c10 0.00/0.84
AFTERNTH(z0, z1) → c11 0.00/0.84
ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
S tuples:

U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 0.00/0.84
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1)) 0.00/0.84
SEL(z0, z1) → c9(AFTERNTH(z0, z1)) 0.00/0.84
TAKE(z0, z1) → c10 0.00/0.84
AFTERNTH(z0, z1) → c11 0.00/0.84
ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

U, TAIL, SEL, TAKE, AFTERNTH, ACTIVATE

Compound Symbols:

c6, c8, c9, c10, c11, c13, c14

0.00/0.84
0.00/0.84

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 0.00/0.84
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1))
Removed 3 trailing nodes:

AFTERNTH(z0, z1) → c11 0.00/0.84
TAKE(z0, z1) → c10 0.00/0.84
SEL(z0, z1) → c9(AFTERNTH(z0, z1))
0.00/0.84
0.00/0.84

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0))) 0.00/0.84
natsFrom(z0) → n__natsFrom(z0) 0.00/0.84
fst(pair(z0, z1)) → z0 0.00/0.84
snd(pair(z0, z1)) → z1 0.00/0.84
splitAt(0, z0) → pair(nil, z0) 0.00/0.84
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2)) 0.00/0.84
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1) 0.00/0.84
head(cons(z0, z1)) → z0 0.00/0.84
tail(cons(z0, z1)) → activate(z1) 0.00/0.84
sel(z0, z1) → head(afterNth(z0, z1)) 0.00/0.84
take(z0, z1) → fst(splitAt(z0, z1)) 0.00/0.84
afterNth(z0, z1) → snd(splitAt(z0, z1)) 0.00/0.84
s(z0) → n__s(z0) 0.00/0.84
activate(n__natsFrom(z0)) → natsFrom(activate(z0)) 0.00/0.84
activate(n__s(z0)) → s(activate(z0)) 0.00/0.84
activate(z0) → z0
Tuples:

ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
S tuples:

ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c13, c14

0.00/0.84
0.00/0.84

(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.

ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.84

POL(ACTIVATE(x1)) = [2]x1    0.00/0.84
POL(c13(x1)) = x1    0.00/0.84
POL(c14(x1)) = x1    0.00/0.84
POL(n__natsFrom(x1)) = [1] + x1    0.00/0.84
POL(n__s(x1)) = x1   
0.00/0.84
0.00/0.84

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0))) 0.00/0.84
natsFrom(z0) → n__natsFrom(z0) 0.00/0.84
fst(pair(z0, z1)) → z0 0.00/0.84
snd(pair(z0, z1)) → z1 0.00/0.84
splitAt(0, z0) → pair(nil, z0) 0.00/0.84
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2)) 0.00/0.84
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1) 0.00/0.84
head(cons(z0, z1)) → z0 0.00/0.84
tail(cons(z0, z1)) → activate(z1) 0.00/0.84
sel(z0, z1) → head(afterNth(z0, z1)) 0.00/0.84
take(z0, z1) → fst(splitAt(z0, z1)) 0.00/0.84
afterNth(z0, z1) → snd(splitAt(z0, z1)) 0.00/0.84
s(z0) → n__s(z0) 0.00/0.84
activate(n__natsFrom(z0)) → natsFrom(activate(z0)) 0.00/0.84
activate(n__s(z0)) → s(activate(z0)) 0.00/0.84
activate(z0) → z0
Tuples:

ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
S tuples:

ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
K tuples:

ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c13, c14

0.00/0.84
0.00/0.84

(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.

ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.84

POL(ACTIVATE(x1)) = [3]x1    0.00/0.84
POL(c13(x1)) = x1    0.00/0.84
POL(c14(x1)) = x1    0.00/0.84
POL(n__natsFrom(x1)) = x1    0.00/0.84
POL(n__s(x1)) = [1] + x1   
0.00/0.84
0.00/0.84

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0))) 0.00/0.84
natsFrom(z0) → n__natsFrom(z0) 0.00/0.84
fst(pair(z0, z1)) → z0 0.00/0.84
snd(pair(z0, z1)) → z1 0.00/0.84
splitAt(0, z0) → pair(nil, z0) 0.00/0.84
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2)) 0.00/0.84
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1) 0.00/0.84
head(cons(z0, z1)) → z0 0.00/0.84
tail(cons(z0, z1)) → activate(z1) 0.00/0.84
sel(z0, z1) → head(afterNth(z0, z1)) 0.00/0.84
take(z0, z1) → fst(splitAt(z0, z1)) 0.00/0.84
afterNth(z0, z1) → snd(splitAt(z0, z1)) 0.00/0.84
s(z0) → n__s(z0) 0.00/0.84
activate(n__natsFrom(z0)) → natsFrom(activate(z0)) 0.00/0.84
activate(n__s(z0)) → s(activate(z0)) 0.00/0.84
activate(z0) → z0
Tuples:

ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c13, c14

0.00/0.84
0.00/0.84

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

The set S is empty
0.00/0.84
0.00/0.84

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

0.00/0.84
0.00/0.84
0.00/0.90 EOF