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(n__add(n__sqr(activate(X)), n__dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
s(X) → n__s(X)
add(X1, X2) → n__add(X1, X2)
sqr(X) → n__sqr(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(X)
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(n__sqr(X)) → sqr(activate(X))
activate(n__dbl(X)) → dbl(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 Cpx (relative) TRS 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(n__add(n__sqr(activate(z0)), n__dbl(activate(z0))))
sqr(z0) → n__sqr(z0)
dbl(0) → 0
dbl(s(z0)) → s(n__s(n__dbl(activate(z0))))
dbl(z0) → n__dbl(z0)
add(0, z0) → z0
add(s(z0), z1) → s(n__add(activate(z0), z1))
add(z0, z1) → n__add(z0, z1)
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(activate(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(z0)
activate(n__add(z0, z1)) → add(activate(z0), activate(z1))
activate(n__sqr(z0)) → sqr(activate(z0))
activate(n__dbl(z0)) → dbl(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0

Tuples:

TERMS(z0) → c(SQR(z0))
TERMS(z0) → c1
SQR(0) → c2
SQR(s(z0)) → c3(S(n__add(n__sqr(activate(z0)), n__dbl(activate(z0)))), ACTIVATE(z0), ACTIVATE(z0))
SQR(z0) → c4
DBL(0) → c5
DBL(s(z0)) → c6(S(n__s(n__dbl(activate(z0)))), ACTIVATE(z0))
DBL(z0) → c7
ADD(0, z0) → c8
ADD(s(z0), z1) → c9(S(n__add(activate(z0), z1)), ACTIVATE(z0))
ADD(z0, z1) → c10
FIRST(0, z0) → c11
FIRST(s(z0), cons(z1, z2)) → c12(ACTIVATE(z0), ACTIVATE(z2))
FIRST(z0, z1) → c13
S(z0) → c14
ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(S(z0))
ACTIVATE(n__add(z0, z1)) → c17(ADD(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__sqr(z0)) → c18(SQR(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__dbl(z0)) → c19(DBL(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c20(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(z0) → c21

S tuples:

TERMS(z0) → c(SQR(z0))
TERMS(z0) → c1
SQR(0) → c2
SQR(s(z0)) → c3(S(n__add(n__sqr(activate(z0)), n__dbl(activate(z0)))), ACTIVATE(z0), ACTIVATE(z0))
SQR(z0) → c4
DBL(0) → c5
DBL(s(z0)) → c6(S(n__s(n__dbl(activate(z0)))), ACTIVATE(z0))
DBL(z0) → c7
ADD(0, z0) → c8
ADD(s(z0), z1) → c9(S(n__add(activate(z0), z1)), ACTIVATE(z0))
ADD(z0, z1) → c10
FIRST(0, z0) → c11
FIRST(s(z0), cons(z1, z2)) → c12(ACTIVATE(z0), ACTIVATE(z2))
FIRST(z0, z1) → c13
S(z0) → c14
ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(S(z0))
ACTIVATE(n__add(z0, z1)) → c17(ADD(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__sqr(z0)) → c18(SQR(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__dbl(z0)) → c19(DBL(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c20(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(z0) → c21

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

TERMS, SQR, DBL, ADD, FIRST, S, 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

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 4 leading nodes:

SQR(s(z0)) → c3(S(n__add(n__sqr(activate(z0)), n__dbl(activate(z0)))), ACTIVATE(z0), ACTIVATE(z0))
DBL(s(z0)) → c6(S(n__s(n__dbl(activate(z0)))), ACTIVATE(z0))
ADD(s(z0), z1) → c9(S(n__add(activate(z0), z1)), ACTIVATE(z0))
FIRST(s(z0), cons(z1, z2)) → c12(ACTIVATE(z0), ACTIVATE(z2))

Removed 13 trailing nodes:

FIRST(0, z0) → c11
DBL(0) → c5
FIRST(z0, z1) → c13
DBL(z0) → c7
ADD(z0, z1) → c10
S(z0) → c14
TERMS(z0) → c(SQR(z0))
TERMS(z0) → c1
ACTIVATE(z0) → c21
SQR(z0) → c4
SQR(0) → c2
ADD(0, z0) → c8
ACTIVATE(n__s(z0)) → c16(S(z0))

(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(n__add(n__sqr(activate(z0)), n__dbl(activate(z0))))
sqr(z0) → n__sqr(z0)
dbl(0) → 0
dbl(s(z0)) → s(n__s(n__dbl(activate(z0))))
dbl(z0) → n__dbl(z0)
add(0, z0) → z0
add(s(z0), z1) → s(n__add(activate(z0), z1))
add(z0, z1) → n__add(z0, z1)
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(activate(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(z0)
activate(n__add(z0, z1)) → add(activate(z0), activate(z1))
activate(n__sqr(z0)) → sqr(activate(z0))
activate(n__dbl(z0)) → dbl(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0

Tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__add(z0, z1)) → c17(ADD(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__sqr(z0)) → c18(SQR(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__dbl(z0)) → c19(DBL(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c20(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

S tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__add(z0, z1)) → c17(ADD(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__sqr(z0)) → c18(SQR(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__dbl(z0)) → c19(DBL(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c20(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:

c15, c17, c18, c19, c20

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

Removed 5 trailing tuple parts

(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(n__add(n__sqr(activate(z0)), n__dbl(activate(z0))))
sqr(z0) → n__sqr(z0)
dbl(0) → 0
dbl(s(z0)) → s(n__s(n__dbl(activate(z0))))
dbl(z0) → n__dbl(z0)
add(0, z0) → z0
add(s(z0), z1) → s(n__add(activate(z0), z1))
add(z0, z1) → n__add(z0, z1)
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(activate(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(z0)
activate(n__add(z0, z1)) → add(activate(z0), activate(z1))
activate(n__sqr(z0)) → sqr(activate(z0))
activate(n__dbl(z0)) → dbl(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0

Tuples:

ACTIVATE(n__terms(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0))
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))

S tuples:

ACTIVATE(n__terms(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0))
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c15, c17, c18, c19, c20

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__terms(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0))
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))

S tuples:

ACTIVATE(n__terms(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0))
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c15, c17, c18, c19, c20

(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__dbl(z0)) → c19(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))

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

ACTIVATE(n__terms(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0))
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))

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

POL(ACTIVATE(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c17(x₁, x₂)) = x₁ + x₂   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁, x₂)) = x₁ + x₂   
POL(n__add(x₁, x₂)) = x₁ + x₂   
POL(n__dbl(x₁)) = [2] + x₁   
POL(n__first(x₁, x₂)) = [1] + x₁ + x₂   
POL(n__sqr(x₁)) = x₁   
POL(n__terms(x₁)) = x₁   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__terms(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0))
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))

S tuples:

ACTIVATE(n__terms(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0))

K tuples:

ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))

Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c15, c17, c18, c19, c20

(11) 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__terms(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0))

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

ACTIVATE(n__terms(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0))
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))

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

POL(ACTIVATE(x₁)) = [5] + [4]x₁   
POL(c15(x₁)) = x₁   
POL(c17(x₁, x₂)) = x₁ + x₂   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁, x₂)) = x₁ + x₂   
POL(n__add(x₁, x₂)) = [4] + x₁ + x₂   
POL(n__dbl(x₁)) = x₁   
POL(n__first(x₁, x₂)) = [4] + x₁ + x₂   
POL(n__sqr(x₁)) = [5] + x₁   
POL(n__terms(x₁)) = [5] + x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__terms(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0))
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))

S tuples:none
K tuples:

ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__terms(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c15, c17, c18, c19, c20

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

The set S is empty

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