### (0) Obligation:

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

from(X) → cons(X, n__from(s(X)))
first(0, Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
sel(0, cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
first(X1, X2) → n__first(X1, X2)
activate(n__from(X)) → from(X)
activate(n__first(X1, X2)) → first(X1, 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:

from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
sel(0, cons(z0, z1)) → z0
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2))
activate(n__from(z0)) → from(z0)
activate(n__first(z0, z1)) → first(z0, z1)
activate(z0) → z0
Tuples:

FROM(z0) → c
FROM(z0) → c1
FIRST(0, z0) → c2
FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
FIRST(z0, z1) → c4
SEL(0, cons(z0, z1)) → c5
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__from(z0)) → c7(FROM(z0))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))
ACTIVATE(z0) → c9
S tuples:

FROM(z0) → c
FROM(z0) → c1
FIRST(0, z0) → c2
FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
FIRST(z0, z1) → c4
SEL(0, cons(z0, z1)) → c5
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__from(z0)) → c7(FROM(z0))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))
ACTIVATE(z0) → c9
K tuples:none
Defined Rule Symbols:

from, first, sel, activate

Defined Pair Symbols:

FROM, FIRST, SEL, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9

### (3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 7 trailing nodes:

FROM(z0) → c
FROM(z0) → c1
SEL(0, cons(z0, z1)) → c5
FIRST(0, z0) → c2
FIRST(z0, z1) → c4
ACTIVATE(z0) → c9
ACTIVATE(n__from(z0)) → c7(FROM(z0))

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
sel(0, cons(z0, z1)) → z0
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2))
activate(n__from(z0)) → from(z0)
activate(n__first(z0, z1)) → first(z0, z1)
activate(z0) → z0
Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))
S tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))
K tuples:none
Defined Rule Symbols:

from, first, sel, activate

Defined Pair Symbols:

FIRST, SEL, ACTIVATE

Compound Symbols:

c3, c6, c8

### (5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

sel(0, cons(z0, z1)) → z0
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2))

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

activate(n__from(z0)) → from(z0)
activate(n__first(z0, z1)) → first(z0, z1)
activate(z0) → z0
from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))
S tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))
K tuples:none
Defined Rule Symbols:

activate, from, first

Defined Pair Symbols:

FIRST, SEL, ACTIVATE

Compound Symbols:

c3, c6, c8

### (7) 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.

SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
We considered the (Usable) Rules:none
And the Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(ACTIVATE(x1)) = 0
POL(FIRST(x1, x2)) = 0
POL(SEL(x1, x2)) = x1
POL(activate(x1)) = [4]x1
POL(c3(x1)) = x1
POL(c6(x1, x2)) = x1 + x2
POL(c8(x1)) = x1
POL(cons(x1, x2)) = [3] + x1
POL(first(x1, x2)) = [3] + [2]x1 + [2]x2
POL(from(x1)) = [4] + [2]x1
POL(n__first(x1, x2)) = [1] + x1 + x2
POL(n__from(x1)) = [3] + x1
POL(nil) = [3]
POL(s(x1)) = [1] + x1

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

activate(n__from(z0)) → from(z0)
activate(n__first(z0, z1)) → first(z0, z1)
activate(z0) → z0
from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))
S tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))
K tuples:

SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
Defined Rule Symbols:

activate, from, first

Defined Pair Symbols:

FIRST, SEL, ACTIVATE

Compound Symbols:

c3, c6, c8

### (9) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))
We considered the (Usable) Rules:

activate(n__first(z0, z1)) → first(z0, z1)
first(z0, z1) → n__first(z0, z1)
activate(z0) → z0
activate(n__from(z0)) → from(z0)
from(z0) → n__from(z0)
first(0, z0) → nil
from(z0) → cons(z0, n__from(s(z0)))
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
And the Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(ACTIVATE(x1)) = [1] + [2]x1
POL(FIRST(x1, x2)) = [2]x1 + [2]x2
POL(SEL(x1, x2)) = x1·x2 + [2]x12
POL(activate(x1)) = [2] + x1
POL(c3(x1)) = x1
POL(c6(x1, x2)) = x1 + x2
POL(c8(x1)) = x1
POL(cons(x1, x2)) = x2
POL(first(x1, x2)) = x1 + x2
POL(from(x1)) = [2]
POL(n__first(x1, x2)) = x1 + x2
POL(n__from(x1)) = 0
POL(nil) = 0
POL(s(x1)) = [2] + x1

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

activate(n__from(z0)) → from(z0)
activate(n__first(z0, z1)) → first(z0, z1)
activate(z0) → z0
from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))
S tuples:none
K tuples:

SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))
Defined Rule Symbols:

activate, from, first

Defined Pair Symbols:

FIRST, SEL, ACTIVATE

Compound Symbols:

c3, c6, c8

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

The set S is empty