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

0 CpxTRS
↳1 DependencyGraphProof (BOTH BOUNDS(ID, ID), 25 ms)
↳2 CpxTRS
↳3 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳6 CdtProblem
↳7 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳8 CdtProblem
↳9 CdtUsableRulesProof (⇔, 0 ms)
↳10 CdtProblem
↳11 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 54 ms)
↳12 CdtProblem
↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 BOUNDS(1, 1)

(0) Obligation:

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


The TRS R consists of the following rules:

from(X) → cons(X, n__from(n__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)
s(X) → n__s(X)
first(X1, X2) → n__first(X1, X2)
activate(n__from(X)) → from(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) DependencyGraphProof (BOTH BOUNDS(ID, ID) transformation)

The following rules are not reachable from basic terms in the dependency graph and can be removed:
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))

(2) Obligation:

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


The TRS R consists of the following rules:

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

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

SEL(0, cons(z0, z1)) → c
FIRST(z0, z1) → c1
FIRST(s(z0), cons(z1, z2)) → c2(ACTIVATE(z2))
FIRST(0, z0) → c3
ACTIVATE(n__from(z0)) → c4(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c5(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(z0) → c6
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
FROM(z0) → c8
FROM(z0) → c9
S(z0) → c10
S tuples:

SEL(0, cons(z0, z1)) → c
FIRST(z0, z1) → c1
FIRST(s(z0), cons(z1, z2)) → c2(ACTIVATE(z2))
FIRST(0, z0) → c3
ACTIVATE(n__from(z0)) → c4(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c5(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(z0) → c6
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
FROM(z0) → c8
FROM(z0) → c9
S(z0) → c10
K tuples:none
Defined Rule Symbols:

sel, first, activate, from, s

Defined Pair Symbols:

SEL, FIRST, ACTIVATE, FROM, S

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

FIRST(s(z0), cons(z1, z2)) → c2(ACTIVATE(z2))
Removed 7 trailing nodes:

S(z0) → c10
FROM(z0) → c9
ACTIVATE(z0) → c6
SEL(0, cons(z0, z1)) → c
FROM(z0) → c8
FIRST(0, z0) → c3
FIRST(z0, z1) → c1

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVATE(n__from(z0)) → c4(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c5(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
S tuples:

ACTIVATE(n__from(z0)) → c4(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c5(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

sel, first, activate, from, s

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c4, c5, c7

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

Removed 3 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVATE(n__from(z0)) → c4(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c5(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
S tuples:

ACTIVATE(n__from(z0)) → c4(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c5(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

sel, first, activate, from, s

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c4, c5, c7

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__from(z0)) → c4(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c5(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
S tuples:

ACTIVATE(n__from(z0)) → c4(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c5(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c4, c5, c7

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

ACTIVATE(n__from(z0)) → c4(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c5(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__from(z0)) → c4(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c5(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1, x2)) = x1 + x2   
POL(c7(x1)) = x1   
POL(n__first(x1, x2)) = [1] + x1 + x2   
POL(n__from(x1)) = [1] + x1   
POL(n__s(x1)) = [1] + x1   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__from(z0)) → c4(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c5(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__from(z0)) → c4(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c5(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c4, c5, c7

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

The set S is empty

(14) BOUNDS(1, 1)