YES(O(1), O(n^1)) 5.15/1.77 YES(O(1), O(n^1)) 5.57/1.84 5.57/1.84 5.57/1.84 5.57/1.84 5.57/1.85 5.57/1.85 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 5.57/1.85 5.57/1.85 5.57/1.85
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5.57/1.85
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
5.57/1.85 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
5.57/1.85 
↳2 CdtProblem
5.57/1.85 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
5.57/1.85 
↳4 CdtProblem
5.57/1.85 
↳5 CdtLeafRemovalProof (ComplexityIfPolyImplication)
5.57/1.85 
↳6 CdtProblem
5.57/1.85 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
5.57/1.85 
↳8 CdtProblem
5.57/1.85 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
5.57/1.85 
↳10 CdtProblem
5.57/1.85 
↳11 CdtKnowledgeProof (⇔)
5.57/1.85 
↳12 BOUNDS(O(1), O(1))
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5.57/1.85

(0) Obligation:

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

U11(tt, N, X, XS) → U12(splitAt(activate(N), activate(XS)), activate(X)) 5.57/1.85
U12(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS) 5.57/1.85
afterNth(N, XS) → snd(splitAt(N, XS)) 5.57/1.85
and(tt, X) → activate(X) 5.57/1.85
fst(pair(X, Y)) → X 5.57/1.85
head(cons(N, XS)) → N 5.57/1.85
natsFrom(N) → cons(N, n__natsFrom(s(N))) 5.57/1.85
sel(N, XS) → head(afterNth(N, XS)) 5.57/1.85
snd(pair(X, Y)) → Y 5.57/1.85
splitAt(0, XS) → pair(nil, XS) 5.57/1.85
splitAt(s(N), cons(X, XS)) → U11(tt, N, X, activate(XS)) 5.57/1.85
tail(cons(N, XS)) → activate(XS) 5.57/1.85
take(N, XS) → fst(splitAt(N, XS)) 5.57/1.85
natsFrom(X) → n__natsFrom(X) 5.57/1.85
activate(n__natsFrom(X)) → natsFrom(X) 5.57/1.85
activate(X) → X

Rewrite Strategy: INNERMOST
5.57/1.85
5.57/1.85

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

Converted CpxTRS to CDT
5.57/1.85
5.57/1.85

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1, z2) → U12(splitAt(activate(z0), activate(z2)), activate(z1)) 5.57/1.85
U12(pair(z0, z1), z2) → pair(cons(activate(z2), z0), z1) 5.57/1.85
afterNth(z0, z1) → snd(splitAt(z0, z1)) 5.57/1.85
and(tt, z0) → activate(z0) 5.57/1.85
fst(pair(z0, z1)) → z0 5.57/1.85
head(cons(z0, z1)) → z0 5.57/1.85
natsFrom(z0) → cons(z0, n__natsFrom(s(z0))) 5.57/1.85
natsFrom(z0) → n__natsFrom(z0) 5.57/1.85
sel(z0, z1) → head(afterNth(z0, z1)) 5.57/1.85
snd(pair(z0, z1)) → z1 5.57/1.85
splitAt(0, z0) → pair(nil, z0) 5.57/1.85
splitAt(s(z0), cons(z1, z2)) → U11(tt, z0, z1, activate(z2)) 5.57/1.85
tail(cons(z0, z1)) → activate(z1) 5.57/1.85
take(z0, z1) → fst(splitAt(z0, z1)) 5.57/1.85
activate(n__natsFrom(z0)) → natsFrom(z0) 5.57/1.85
activate(z0) → z0
Tuples:

U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1)) 5.57/1.85
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2)) 5.57/1.85
AFTERNTH(z0, z1) → c2(SND(splitAt(z0, z1)), SPLITAT(z0, z1)) 5.57/1.85
AND(tt, z0) → c3(ACTIVATE(z0)) 5.57/1.85
SEL(z0, z1) → c8(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1)) 5.57/1.85
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2)) 5.57/1.85
TAIL(cons(z0, z1)) → c12(ACTIVATE(z1)) 5.57/1.85
TAKE(z0, z1) → c13(FST(splitAt(z0, z1)), SPLITAT(z0, z1)) 5.57/1.85
ACTIVATE(n__natsFrom(z0)) → c14(NATSFROM(z0))
S tuples:

U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1)) 5.57/1.85
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2)) 5.57/1.85
AFTERNTH(z0, z1) → c2(SND(splitAt(z0, z1)), SPLITAT(z0, z1)) 5.57/1.85
AND(tt, z0) → c3(ACTIVATE(z0)) 5.57/1.85
SEL(z0, z1) → c8(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1)) 5.57/1.85
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2)) 5.57/1.85
TAIL(cons(z0, z1)) → c12(ACTIVATE(z1)) 5.57/1.85
TAKE(z0, z1) → c13(FST(splitAt(z0, z1)), SPLITAT(z0, z1)) 5.57/1.85
ACTIVATE(n__natsFrom(z0)) → c14(NATSFROM(z0))
K tuples:none
Defined Rule Symbols:

U11, U12, afterNth, and, fst, head, natsFrom, sel, snd, splitAt, tail, take, activate

Defined Pair Symbols:

U11', U12', AFTERNTH, AND, SEL, SPLITAT, TAIL, TAKE, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c8, c11, c12, c13, c14

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5.57/1.85

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

Removed 4 trailing tuple parts
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5.57/1.85

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1, z2) → U12(splitAt(activate(z0), activate(z2)), activate(z1)) 5.57/1.85
U12(pair(z0, z1), z2) → pair(cons(activate(z2), z0), z1) 5.57/1.85
afterNth(z0, z1) → snd(splitAt(z0, z1)) 5.57/1.85
and(tt, z0) → activate(z0) 5.57/1.85
fst(pair(z0, z1)) → z0 5.57/1.85
head(cons(z0, z1)) → z0 5.57/1.85
natsFrom(z0) → cons(z0, n__natsFrom(s(z0))) 5.57/1.85
natsFrom(z0) → n__natsFrom(z0) 5.57/1.85
sel(z0, z1) → head(afterNth(z0, z1)) 5.57/1.85
snd(pair(z0, z1)) → z1 5.57/1.85
splitAt(0, z0) → pair(nil, z0) 5.57/1.85
splitAt(s(z0), cons(z1, z2)) → U11(tt, z0, z1, activate(z2)) 5.57/1.85
tail(cons(z0, z1)) → activate(z1) 5.57/1.85
take(z0, z1) → fst(splitAt(z0, z1)) 5.57/1.85
activate(n__natsFrom(z0)) → natsFrom(z0) 5.57/1.85
activate(z0) → z0
Tuples:

U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1)) 5.57/1.85
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2)) 5.57/1.85
AND(tt, z0) → c3(ACTIVATE(z0)) 5.57/1.85
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2)) 5.57/1.85
TAIL(cons(z0, z1)) → c12(ACTIVATE(z1)) 5.57/1.85
AFTERNTH(z0, z1) → c2(SPLITAT(z0, z1)) 5.57/1.85
SEL(z0, z1) → c8(AFTERNTH(z0, z1)) 5.57/1.85
TAKE(z0, z1) → c13(SPLITAT(z0, z1)) 5.57/1.85
ACTIVATE(n__natsFrom(z0)) → c14
S tuples:

U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1)) 5.57/1.85
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2)) 5.57/1.85
AND(tt, z0) → c3(ACTIVATE(z0)) 5.57/1.85
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2)) 5.57/1.85
TAIL(cons(z0, z1)) → c12(ACTIVATE(z1)) 5.57/1.85
AFTERNTH(z0, z1) → c2(SPLITAT(z0, z1)) 5.57/1.85
SEL(z0, z1) → c8(AFTERNTH(z0, z1)) 5.57/1.85
TAKE(z0, z1) → c13(SPLITAT(z0, z1)) 5.57/1.85
ACTIVATE(n__natsFrom(z0)) → c14
K tuples:none
Defined Rule Symbols:

U11, U12, afterNth, and, fst, head, natsFrom, sel, snd, splitAt, tail, take, activate

Defined Pair Symbols:

U11', U12', AND, SPLITAT, TAIL, AFTERNTH, SEL, TAKE, ACTIVATE

Compound Symbols:

c, c1, c3, c11, c12, c2, c8, c13, c14

5.57/1.85
5.57/1.85

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 3 leading nodes:

SEL(z0, z1) → c8(AFTERNTH(z0, z1)) 5.57/1.85
AFTERNTH(z0, z1) → c2(SPLITAT(z0, z1)) 5.57/1.85
TAKE(z0, z1) → c13(SPLITAT(z0, z1))
Removed 4 trailing nodes:

AND(tt, z0) → c3(ACTIVATE(z0)) 5.57/1.85
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2)) 5.57/1.85
TAIL(cons(z0, z1)) → c12(ACTIVATE(z1)) 5.57/1.85
ACTIVATE(n__natsFrom(z0)) → c14
5.57/1.85
5.57/1.85

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1, z2) → U12(splitAt(activate(z0), activate(z2)), activate(z1)) 5.57/1.85
U12(pair(z0, z1), z2) → pair(cons(activate(z2), z0), z1) 5.57/1.85
afterNth(z0, z1) → snd(splitAt(z0, z1)) 5.57/1.85
and(tt, z0) → activate(z0) 5.57/1.85
fst(pair(z0, z1)) → z0 5.57/1.85
head(cons(z0, z1)) → z0 5.57/1.85
natsFrom(z0) → cons(z0, n__natsFrom(s(z0))) 5.57/1.85
natsFrom(z0) → n__natsFrom(z0) 5.57/1.85
sel(z0, z1) → head(afterNth(z0, z1)) 5.57/1.85
snd(pair(z0, z1)) → z1 5.57/1.85
splitAt(0, z0) → pair(nil, z0) 5.57/1.85
splitAt(s(z0), cons(z1, z2)) → U11(tt, z0, z1, activate(z2)) 5.57/1.85
tail(cons(z0, z1)) → activate(z1) 5.57/1.85
take(z0, z1) → fst(splitAt(z0, z1)) 5.57/1.85
activate(n__natsFrom(z0)) → natsFrom(z0) 5.57/1.85
activate(z0) → z0
Tuples:

U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1)) 5.57/1.85
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2)) 5.57/1.85
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2)) 5.57/1.85
ACTIVATE(n__natsFrom(z0)) → c14
S tuples:

U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1)) 5.57/1.85
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2)) 5.57/1.85
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2)) 5.57/1.85
ACTIVATE(n__natsFrom(z0)) → c14
K tuples:none
Defined Rule Symbols:

U11, U12, afterNth, and, fst, head, natsFrom, sel, snd, splitAt, tail, take, activate

Defined Pair Symbols:

U11', U12', SPLITAT, ACTIVATE

Compound Symbols:

c, c1, c11, c14

5.57/1.85
5.57/1.85

(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.

U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
We considered the (Usable) Rules:

activate(n__natsFrom(z0)) → natsFrom(z0) 5.57/1.85
activate(z0) → z0 5.57/1.85
natsFrom(z0) → cons(z0, n__natsFrom(s(z0))) 5.57/1.85
natsFrom(z0) → n__natsFrom(z0) 5.57/1.85
splitAt(0, z0) → pair(nil, z0) 5.57/1.85
splitAt(s(z0), cons(z1, z2)) → U11(tt, z0, z1, activate(z2)) 5.57/1.85
U11(tt, z0, z1, z2) → U12(splitAt(activate(z0), activate(z2)), activate(z1)) 5.57/1.85
U12(pair(z0, z1), z2) → pair(cons(activate(z2), z0), z1)
And the Tuples:

U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1)) 5.57/1.85
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2)) 5.57/1.85
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2)) 5.57/1.85
ACTIVATE(n__natsFrom(z0)) → c14
The order we found is given by the following interpretation:
Polynomial interpretation : 5.57/1.85

POL(0) = [1]    5.57/1.85
POL(ACTIVATE(x1)) = 0    5.57/1.85
POL(SPLITAT(x1, x2)) = [4]x1    5.57/1.85
POL(U11(x1, x2, x3, x4)) = [1]    5.57/1.85
POL(U11'(x1, x2, x3, x4)) = [4]x1 + [4]x2    5.57/1.85
POL(U12(x1, x2)) = [1]    5.57/1.85
POL(U12'(x1, x2)) = [4]x1    5.57/1.85
POL(activate(x1)) = x1    5.57/1.85
POL(c(x1, x2, x3, x4, x5)) = x1 + x2 + x3 + x4 + x5    5.57/1.85
POL(c1(x1)) = x1    5.57/1.85
POL(c11(x1, x2)) = x1 + x2    5.57/1.85
POL(c14) = 0    5.57/1.85
POL(cons(x1, x2)) = 0    5.57/1.85
POL(n__natsFrom(x1)) = [1]    5.57/1.85
POL(natsFrom(x1)) = [1]    5.57/1.85
POL(nil) = [3]    5.57/1.85
POL(pair(x1, x2)) = [1]    5.57/1.85
POL(s(x1)) = [4] + x1    5.57/1.85
POL(splitAt(x1, x2)) = [4]    5.57/1.85
POL(tt) = [4]   
5.57/1.85
5.57/1.85

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1, z2) → U12(splitAt(activate(z0), activate(z2)), activate(z1)) 5.57/1.85
U12(pair(z0, z1), z2) → pair(cons(activate(z2), z0), z1) 5.57/1.85
afterNth(z0, z1) → snd(splitAt(z0, z1)) 5.57/1.85
and(tt, z0) → activate(z0) 5.57/1.85
fst(pair(z0, z1)) → z0 5.57/1.85
head(cons(z0, z1)) → z0 5.57/1.85
natsFrom(z0) → cons(z0, n__natsFrom(s(z0))) 5.57/1.85
natsFrom(z0) → n__natsFrom(z0) 5.57/1.85
sel(z0, z1) → head(afterNth(z0, z1)) 5.57/1.85
snd(pair(z0, z1)) → z1 5.57/1.85
splitAt(0, z0) → pair(nil, z0) 5.57/1.85
splitAt(s(z0), cons(z1, z2)) → U11(tt, z0, z1, activate(z2)) 5.57/1.85
tail(cons(z0, z1)) → activate(z1) 5.57/1.85
take(z0, z1) → fst(splitAt(z0, z1)) 5.57/1.85
activate(n__natsFrom(z0)) → natsFrom(z0) 5.57/1.85
activate(z0) → z0
Tuples:

U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1)) 5.57/1.85
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2)) 5.57/1.85
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2)) 5.57/1.85
ACTIVATE(n__natsFrom(z0)) → c14
S tuples:

U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1)) 5.57/1.85
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2)) 5.57/1.85
ACTIVATE(n__natsFrom(z0)) → c14
K tuples:

U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
Defined Rule Symbols:

U11, U12, afterNth, and, fst, head, natsFrom, sel, snd, splitAt, tail, take, activate

Defined Pair Symbols:

U11', U12', SPLITAT, ACTIVATE

Compound Symbols:

c, c1, c11, c14

5.57/1.85
5.57/1.85

(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.

SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2))
We considered the (Usable) Rules:

activate(n__natsFrom(z0)) → natsFrom(z0) 5.57/1.85
activate(z0) → z0 5.57/1.85
natsFrom(z0) → cons(z0, n__natsFrom(s(z0))) 5.57/1.85
natsFrom(z0) → n__natsFrom(z0) 5.57/1.85
splitAt(0, z0) → pair(nil, z0) 5.57/1.85
splitAt(s(z0), cons(z1, z2)) → U11(tt, z0, z1, activate(z2)) 5.57/1.85
U11(tt, z0, z1, z2) → U12(splitAt(activate(z0), activate(z2)), activate(z1)) 5.57/1.85
U12(pair(z0, z1), z2) → pair(cons(activate(z2), z0), z1)
And the Tuples:

U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1)) 5.57/1.85
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2)) 5.57/1.85
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2)) 5.57/1.85
ACTIVATE(n__natsFrom(z0)) → c14
The order we found is given by the following interpretation:
Polynomial interpretation : 5.57/1.85

POL(0) = [1]    5.57/1.85
POL(ACTIVATE(x1)) = 0    5.57/1.85
POL(SPLITAT(x1, x2)) = [2]x1    5.57/1.85
POL(U11(x1, x2, x3, x4)) = [3] + [3]x1 + [3]x2 + [3]x3    5.57/1.85
POL(U11'(x1, x2, x3, x4)) = x1 + [2]x2    5.57/1.85
POL(U12(x1, x2)) = [3]    5.57/1.85
POL(U12'(x1, x2)) = 0    5.57/1.85
POL(activate(x1)) = x1    5.57/1.85
POL(c(x1, x2, x3, x4, x5)) = x1 + x2 + x3 + x4 + x5    5.57/1.85
POL(c1(x1)) = x1    5.57/1.85
POL(c11(x1, x2)) = x1 + x2    5.57/1.85
POL(c14) = 0    5.57/1.85
POL(cons(x1, x2)) = 0    5.57/1.85
POL(n__natsFrom(x1)) = 0    5.57/1.85
POL(natsFrom(x1)) = 0    5.57/1.85
POL(nil) = [3]    5.57/1.85
POL(pair(x1, x2)) = x2    5.57/1.85
POL(s(x1)) = [4] + x1    5.57/1.85
POL(splitAt(x1, x2)) = 0    5.57/1.85
POL(tt) = 0   
5.57/1.85
5.57/1.85

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1, z2) → U12(splitAt(activate(z0), activate(z2)), activate(z1)) 5.57/1.85
U12(pair(z0, z1), z2) → pair(cons(activate(z2), z0), z1) 5.57/1.85
afterNth(z0, z1) → snd(splitAt(z0, z1)) 5.57/1.85
and(tt, z0) → activate(z0) 5.57/1.85
fst(pair(z0, z1)) → z0 5.57/1.85
head(cons(z0, z1)) → z0 5.57/1.85
natsFrom(z0) → cons(z0, n__natsFrom(s(z0))) 5.57/1.85
natsFrom(z0) → n__natsFrom(z0) 5.57/1.85
sel(z0, z1) → head(afterNth(z0, z1)) 5.57/1.85
snd(pair(z0, z1)) → z1 5.57/1.85
splitAt(0, z0) → pair(nil, z0) 5.57/1.85
splitAt(s(z0), cons(z1, z2)) → U11(tt, z0, z1, activate(z2)) 5.57/1.85
tail(cons(z0, z1)) → activate(z1) 5.57/1.85
take(z0, z1) → fst(splitAt(z0, z1)) 5.57/1.85
activate(n__natsFrom(z0)) → natsFrom(z0) 5.57/1.85
activate(z0) → z0
Tuples:

U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1)) 5.57/1.85
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2)) 5.57/1.85
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2)) 5.57/1.85
ACTIVATE(n__natsFrom(z0)) → c14
S tuples:

U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1)) 5.57/1.85
ACTIVATE(n__natsFrom(z0)) → c14
K tuples:

U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2)) 5.57/1.85
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2))
Defined Rule Symbols:

U11, U12, afterNth, and, fst, head, natsFrom, sel, snd, splitAt, tail, take, activate

Defined Pair Symbols:

U11', U12', SPLITAT, ACTIVATE

Compound Symbols:

c, c1, c11, c14

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(11) CdtKnowledgeProof (EQUIVALENT transformation)

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

U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1)) 5.57/1.85
ACTIVATE(n__natsFrom(z0)) → c14 5.57/1.85
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2)) 5.57/1.85
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2)) 5.57/1.85
ACTIVATE(n__natsFrom(z0)) → c14
Now S is empty
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(12) BOUNDS(O(1), O(1))

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5.90/1.91 EOF