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

(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0)))
terms(z0) → n__terms(z0)
sqr(0) → 0
sqr(s(z0)) → s(add(sqr(z0), dbl(z0)))
dbl(0) → 0
dbl(s(z0)) → s(s(dbl(z0)))
add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
s(z0) → n__s(z0)
activate(n__terms(z0)) → terms(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0

Tuples:

TERMS(z0) → c(SQR(z0))
SQR(s(z0)) → c3(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0))
DBL(s(z0)) → c5(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ADD(s(z0), z1) → c7(S(add(z0, z1)), ADD(z0, z1))
FIRST(s(z0), cons(z1, z2)) → c9(ACTIVATE(z2))
ACTIVATE(n__terms(z0)) → c12(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

S tuples:

TERMS(z0) → c(SQR(z0))
SQR(s(z0)) → c3(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0))
DBL(s(z0)) → c5(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ADD(s(z0), z1) → c7(S(add(z0, z1)), ADD(z0, z1))
FIRST(s(z0), cons(z1, z2)) → c9(ACTIVATE(z2))
ACTIVATE(n__terms(z0)) → c12(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(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, c7, c9, c12, c13, c14

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

SQR(s(z0)) → c3(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0))
DBL(s(z0)) → c5(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ADD(s(z0), z1) → c7(S(add(z0, z1)), ADD(z0, z1))
FIRST(s(z0), cons(z1, z2)) → c9(ACTIVATE(z2))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0)))
terms(z0) → n__terms(z0)
sqr(0) → 0
sqr(s(z0)) → s(add(sqr(z0), dbl(z0)))
dbl(0) → 0
dbl(s(z0)) → s(s(dbl(z0)))
add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
s(z0) → n__s(z0)
activate(n__terms(z0)) → terms(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0

Tuples:

TERMS(z0) → c(SQR(z0))
ACTIVATE(n__terms(z0)) → c12(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

S tuples:

TERMS(z0) → c(SQR(z0))
ACTIVATE(n__terms(z0)) → c12(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

TERMS, ACTIVATE

Compound Symbols:

c, c12, c13, c14

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 4 dangling nodes:

TERMS(z0) → c(SQR(z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0)))
terms(z0) → n__terms(z0)
sqr(0) → 0
sqr(s(z0)) → s(add(sqr(z0), dbl(z0)))
dbl(0) → 0
dbl(s(z0)) → s(s(dbl(z0)))
add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
s(z0) → n__s(z0)
activate(n__terms(z0)) → terms(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0

Tuples:

ACTIVATE(n__terms(z0)) → c12(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

S tuples:

ACTIVATE(n__terms(z0)) → c12(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c12, c13, c14

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

Removed 3 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0)))
terms(z0) → n__terms(z0)
sqr(0) → 0
sqr(s(z0)) → s(add(sqr(z0), dbl(z0)))
dbl(0) → 0
dbl(s(z0)) → s(s(dbl(z0)))
add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
s(z0) → n__s(z0)
activate(n__terms(z0)) → terms(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0

Tuples:

ACTIVATE(n__terms(z0)) → c12(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(ACTIVATE(z0), ACTIVATE(z1))

S tuples:

ACTIVATE(n__terms(z0)) → c12(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(ACTIVATE(z0), ACTIVATE(z1))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c12, c13, c14

(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__terms(z0)) → c12(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))

We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__terms(z0)) → c12(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(ACTIVATE(z0), ACTIVATE(z1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x₁)) = [5] + [4]x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁, x₂)) = x₁ + x₂   
POL(n__first(x₁, x₂)) = [4] + x₁ + x₂   
POL(n__s(x₁)) = [2] + x₁   
POL(n__terms(x₁)) = [5] + x₁   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0)))
terms(z0) → n__terms(z0)
sqr(0) → 0
sqr(s(z0)) → s(add(sqr(z0), dbl(z0)))
dbl(0) → 0
dbl(s(z0)) → s(s(dbl(z0)))
add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
s(z0) → n__s(z0)
activate(n__terms(z0)) → terms(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0

Tuples:

ACTIVATE(n__terms(z0)) → c12(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(ACTIVATE(z0), ACTIVATE(z1))

S tuples:

ACTIVATE(n__first(z0, z1)) → c14(ACTIVATE(z0), ACTIVATE(z1))

K tuples:

ACTIVATE(n__terms(z0)) → c12(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c12, c13, c14

(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__first(z0, z1)) → c14(ACTIVATE(z0), ACTIVATE(z1))

We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__terms(z0)) → c12(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(ACTIVATE(z0), ACTIVATE(z1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x₁)) = [3] + [4]x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁, x₂)) = x₁ + x₂   
POL(n__first(x₁, x₂)) = [4] + x₁ + x₂   
POL(n__s(x₁)) = x₁   
POL(n__terms(x₁)) = x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0)))
terms(z0) → n__terms(z0)
sqr(0) → 0
sqr(s(z0)) → s(add(sqr(z0), dbl(z0)))
dbl(0) → 0
dbl(s(z0)) → s(s(dbl(z0)))
add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
s(z0) → n__s(z0)
activate(n__terms(z0)) → terms(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0

Tuples:

ACTIVATE(n__terms(z0)) → c12(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(ACTIVATE(z0), ACTIVATE(z1))

S tuples:none
K tuples:

ACTIVATE(n__terms(z0)) → c12(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(ACTIVATE(z0), ACTIVATE(z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c12, c13, c14

(13) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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