The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
↳3 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳4 CdtProblem
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtUsableRulesProof (⇔, 0 ms)
↳8 CdtProblem
↳9 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 82 ms)
↳10 CdtProblem
↳11 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 BOUNDS(O(1), O(1))

(0) Obligation:

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

pairNs → cons(0, n__incr(n__oddNs))
oddNs → incr(pairNs)
incr(cons(X, XS)) → cons(s(X), n__incr(activate(XS)))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
zip(nil, XS) → nil
zip(X, nil) → nil
zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), n__zip(activate(XS), activate(YS)))
tail(cons(X, XS)) → activate(XS)
repItems(nil) → nil
repItems(cons(X, XS)) → cons(X, n__cons(X, n__repItems(activate(XS))))
incr(X) → n__incr(X)
oddNs → n__oddNs
take(X1, X2) → n__take(X1, X2)
zip(X1, X2) → n__zip(X1, X2)
cons(X1, X2) → n__cons(X1, X2)
repItems(X) → n__repItems(X)
activate(n__incr(X)) → incr(activate(X))
activate(n__oddNs) → oddNs
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zip(X1, X2)) → zip(activate(X1), activate(X2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__repItems(X)) → repItems(activate(X))
activate(X) → X

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

pairNs → cons(0, n__incr(n__oddNs))
oddNs → incr(pairNs)
oddNs → n__oddNs
incr(cons(z0, z1)) → cons(s(z0), n__incr(activate(z1)))
incr(z0) → n__incr(z0)
take(0, z0) → nil
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2)))
take(z0, z1) → n__take(z0, z1)
zip(nil, z0) → nil
zip(z0, nil) → nil
zip(cons(z0, z1), cons(z2, z3)) → cons(pair(z0, z2), n__zip(activate(z1), activate(z3)))
zip(z0, z1) → n__zip(z0, z1)
tail(cons(z0, z1)) → activate(z1)
repItems(nil) → nil
repItems(cons(z0, z1)) → cons(z0, n__cons(z0, n__repItems(activate(z1))))
repItems(z0) → n__repItems(z0)
cons(z0, z1) → n__cons(z0, z1)
activate(n__incr(z0)) → incr(activate(z0))
activate(n__oddNs) → oddNs
activate(n__take(z0, z1)) → take(activate(z0), activate(z1))
activate(n__zip(z0, z1)) → zip(activate(z0), activate(z1))
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
activate(n__repItems(z0)) → repItems(activate(z0))
activate(z0) → z0
Tuples:

PAIRNS → c(CONS(0, n__incr(n__oddNs)))
ODDNS → c1(INCR(pairNs), PAIRNS)
ODDNS → c2
INCR(cons(z0, z1)) → c3(CONS(s(z0), n__incr(activate(z1))), ACTIVATE(z1))
INCR(z0) → c4
TAKE(0, z0) → c5
TAKE(s(z0), cons(z1, z2)) → c6(CONS(z1, n__take(z0, activate(z2))), ACTIVATE(z2))
TAKE(z0, z1) → c7
ZIP(nil, z0) → c8
ZIP(z0, nil) → c9
ZIP(cons(z0, z1), cons(z2, z3)) → c10(CONS(pair(z0, z2), n__zip(activate(z1), activate(z3))), ACTIVATE(z1), ACTIVATE(z3))
ZIP(z0, z1) → c11
TAIL(cons(z0, z1)) → c12(ACTIVATE(z1))
REPITEMS(nil) → c13
REPITEMS(cons(z0, z1)) → c14(CONS(z0, n__cons(z0, n__repItems(activate(z1)))), ACTIVATE(z1))
REPITEMS(z0) → c15
CONS(z0, z1) → c16
ACTIVATE(n__incr(z0)) → c17(INCR(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__oddNs) → c18(ODDNS)
ACTIVATE(n__take(z0, z1)) → c19(TAKE(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__zip(z0, z1)) → c20(ZIP(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c21(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__repItems(z0)) → c22(REPITEMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c23
S tuples:

PAIRNS → c(CONS(0, n__incr(n__oddNs)))
ODDNS → c1(INCR(pairNs), PAIRNS)
ODDNS → c2
INCR(cons(z0, z1)) → c3(CONS(s(z0), n__incr(activate(z1))), ACTIVATE(z1))
INCR(z0) → c4
TAKE(0, z0) → c5
TAKE(s(z0), cons(z1, z2)) → c6(CONS(z1, n__take(z0, activate(z2))), ACTIVATE(z2))
TAKE(z0, z1) → c7
ZIP(nil, z0) → c8
ZIP(z0, nil) → c9
ZIP(cons(z0, z1), cons(z2, z3)) → c10(CONS(pair(z0, z2), n__zip(activate(z1), activate(z3))), ACTIVATE(z1), ACTIVATE(z3))
ZIP(z0, z1) → c11
TAIL(cons(z0, z1)) → c12(ACTIVATE(z1))
REPITEMS(nil) → c13
REPITEMS(cons(z0, z1)) → c14(CONS(z0, n__cons(z0, n__repItems(activate(z1)))), ACTIVATE(z1))
REPITEMS(z0) → c15
CONS(z0, z1) → c16
ACTIVATE(n__incr(z0)) → c17(INCR(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__oddNs) → c18(ODDNS)
ACTIVATE(n__take(z0, z1)) → c19(TAKE(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__zip(z0, z1)) → c20(ZIP(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c21(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__repItems(z0)) → c22(REPITEMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c23
K tuples:none
Defined Rule Symbols:

pairNs, oddNs, incr, take, zip, tail, repItems, cons, activate

Defined Pair Symbols:

PAIRNS, ODDNS, INCR, TAKE, ZIP, TAIL, REPITEMS, CONS, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 5 leading nodes:

INCR(cons(z0, z1)) → c3(CONS(s(z0), n__incr(activate(z1))), ACTIVATE(z1))
TAKE(s(z0), cons(z1, z2)) → c6(CONS(z1, n__take(z0, activate(z2))), ACTIVATE(z2))
ZIP(cons(z0, z1), cons(z2, z3)) → c10(CONS(pair(z0, z2), n__zip(activate(z1), activate(z3))), ACTIVATE(z1), ACTIVATE(z3))
TAIL(cons(z0, z1)) → c12(ACTIVATE(z1))
REPITEMS(cons(z0, z1)) → c14(CONS(z0, n__cons(z0, n__repItems(activate(z1)))), ACTIVATE(z1))
Removed 14 trailing nodes:

ZIP(z0, z1) → c11
REPITEMS(z0) → c15
TAKE(0, z0) → c5
ZIP(z0, nil) → c9
REPITEMS(nil) → c13
TAKE(z0, z1) → c7
PAIRNS → c(CONS(0, n__incr(n__oddNs)))
ODDNS → c1(INCR(pairNs), PAIRNS)
ACTIVATE(n__oddNs) → c18(ODDNS)
ACTIVATE(z0) → c23
INCR(z0) → c4
ODDNS → c2
ZIP(nil, z0) → c8
CONS(z0, z1) → c16

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

pairNs → cons(0, n__incr(n__oddNs))
oddNs → incr(pairNs)
oddNs → n__oddNs
incr(cons(z0, z1)) → cons(s(z0), n__incr(activate(z1)))
incr(z0) → n__incr(z0)
take(0, z0) → nil
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2)))
take(z0, z1) → n__take(z0, z1)
zip(nil, z0) → nil
zip(z0, nil) → nil
zip(cons(z0, z1), cons(z2, z3)) → cons(pair(z0, z2), n__zip(activate(z1), activate(z3)))
zip(z0, z1) → n__zip(z0, z1)
tail(cons(z0, z1)) → activate(z1)
repItems(nil) → nil
repItems(cons(z0, z1)) → cons(z0, n__cons(z0, n__repItems(activate(z1))))
repItems(z0) → n__repItems(z0)
cons(z0, z1) → n__cons(z0, z1)
activate(n__incr(z0)) → incr(activate(z0))
activate(n__oddNs) → oddNs
activate(n__take(z0, z1)) → take(activate(z0), activate(z1))
activate(n__zip(z0, z1)) → zip(activate(z0), activate(z1))
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
activate(n__repItems(z0)) → repItems(activate(z0))
activate(z0) → z0
Tuples:

ACTIVATE(n__incr(z0)) → c17(INCR(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c19(TAKE(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__zip(z0, z1)) → c20(ZIP(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c21(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__repItems(z0)) → c22(REPITEMS(activate(z0)), ACTIVATE(z0))
S tuples:

ACTIVATE(n__incr(z0)) → c17(INCR(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c19(TAKE(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__zip(z0, z1)) → c20(ZIP(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c21(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__repItems(z0)) → c22(REPITEMS(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

pairNs, oddNs, incr, take, zip, tail, repItems, cons, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c17, c19, c20, c21, c22

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

Removed 5 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

pairNs → cons(0, n__incr(n__oddNs))
oddNs → incr(pairNs)
oddNs → n__oddNs
incr(cons(z0, z1)) → cons(s(z0), n__incr(activate(z1)))
incr(z0) → n__incr(z0)
take(0, z0) → nil
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2)))
take(z0, z1) → n__take(z0, z1)
zip(nil, z0) → nil
zip(z0, nil) → nil
zip(cons(z0, z1), cons(z2, z3)) → cons(pair(z0, z2), n__zip(activate(z1), activate(z3)))
zip(z0, z1) → n__zip(z0, z1)
tail(cons(z0, z1)) → activate(z1)
repItems(nil) → nil
repItems(cons(z0, z1)) → cons(z0, n__cons(z0, n__repItems(activate(z1))))
repItems(z0) → n__repItems(z0)
cons(z0, z1) → n__cons(z0, z1)
activate(n__incr(z0)) → incr(activate(z0))
activate(n__oddNs) → oddNs
activate(n__take(z0, z1)) → take(activate(z0), activate(z1))
activate(n__zip(z0, z1)) → zip(activate(z0), activate(z1))
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
activate(n__repItems(z0)) → repItems(activate(z0))
activate(z0) → z0
Tuples:

ACTIVATE(n__incr(z0)) → c17(ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c19(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__zip(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c21(ACTIVATE(z0))
ACTIVATE(n__repItems(z0)) → c22(ACTIVATE(z0))
S tuples:

ACTIVATE(n__incr(z0)) → c17(ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c19(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__zip(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c21(ACTIVATE(z0))
ACTIVATE(n__repItems(z0)) → c22(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

pairNs, oddNs, incr, take, zip, tail, repItems, cons, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c17, c19, c20, c21, c22

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

pairNs → cons(0, n__incr(n__oddNs))
oddNs → incr(pairNs)
oddNs → n__oddNs
incr(cons(z0, z1)) → cons(s(z0), n__incr(activate(z1)))
incr(z0) → n__incr(z0)
take(0, z0) → nil
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2)))
take(z0, z1) → n__take(z0, z1)
zip(nil, z0) → nil
zip(z0, nil) → nil
zip(cons(z0, z1), cons(z2, z3)) → cons(pair(z0, z2), n__zip(activate(z1), activate(z3)))
zip(z0, z1) → n__zip(z0, z1)
tail(cons(z0, z1)) → activate(z1)
repItems(nil) → nil
repItems(cons(z0, z1)) → cons(z0, n__cons(z0, n__repItems(activate(z1))))
repItems(z0) → n__repItems(z0)
cons(z0, z1) → n__cons(z0, z1)
activate(n__incr(z0)) → incr(activate(z0))
activate(n__oddNs) → oddNs
activate(n__take(z0, z1)) → take(activate(z0), activate(z1))
activate(n__zip(z0, z1)) → zip(activate(z0), activate(z1))
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
activate(n__repItems(z0)) → repItems(activate(z0))
activate(z0) → z0

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__incr(z0)) → c17(ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c19(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__zip(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c21(ACTIVATE(z0))
ACTIVATE(n__repItems(z0)) → c22(ACTIVATE(z0))
S tuples:

ACTIVATE(n__incr(z0)) → c17(ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c19(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__zip(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c21(ACTIVATE(z0))
ACTIVATE(n__repItems(z0)) → c22(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c17, c19, c20, c21, c22

(9) CdtRuleRemovalProof (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__incr(z0)) → c17(ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c19(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__zip(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c21(ACTIVATE(z0))
ACTIVATE(n__repItems(z0)) → c22(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__incr(z0)) → c17(ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c19(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__zip(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c21(ACTIVATE(z0))
ACTIVATE(n__repItems(z0)) → c22(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = x1   
POL(c17(x1)) = x1   
POL(c19(x1, x2)) = x1 + x2   
POL(c20(x1, x2)) = x1 + x2   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(n__cons(x1, x2)) = [1] + x1   
POL(n__incr(x1)) = [1] + x1   
POL(n__repItems(x1)) = [1] + x1   
POL(n__take(x1, x2)) = [1] + x1 + x2   
POL(n__zip(x1, x2)) = [1] + x1 + x2   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__incr(z0)) → c17(ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c19(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__zip(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c21(ACTIVATE(z0))
ACTIVATE(n__repItems(z0)) → c22(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__incr(z0)) → c17(ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c19(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__zip(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c21(ACTIVATE(z0))
ACTIVATE(n__repItems(z0)) → c22(ACTIVATE(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c17, c19, c20, c21, c22

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

The set S is empty

(12) BOUNDS(O(1), O(1))