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), 15 ms)
↳2 CpxTRS
↳3 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 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)), 52 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)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
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, Y)) → X
activate(n__from(X)) → from(activate(X))
from(X) → cons(X, n__from(n__s(X)))
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
activate(n__from(z0)) → from(activate(z0))
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
ACTIVATE(n__from(z0)) → c1(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c2
ACTIVATE(n__s(z0)) → c3(S(activate(z0)), ACTIVATE(z0))
FROM(z0) → c4
FROM(z0) → c5
S(z0) → c6
S tuples:

SEL(0, cons(z0, z1)) → c
ACTIVATE(n__from(z0)) → c1(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c2
ACTIVATE(n__s(z0)) → c3(S(activate(z0)), ACTIVATE(z0))
FROM(z0) → c4
FROM(z0) → c5
S(z0) → c6
K tuples:none
Defined Rule Symbols:

sel, activate, from, s

Defined Pair Symbols:

SEL, ACTIVATE, FROM, S

Compound Symbols:

c, c1, c2, c3, c4, c5, c6

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

Removed 5 trailing nodes:

FROM(z0) → c5
S(z0) → c6
SEL(0, cons(z0, z1)) → c
FROM(z0) → c4
ACTIVATE(z0) → c2

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

sel(0, cons(z0, z1)) → z0
activate(n__from(z0)) → from(activate(z0))
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)) → c1(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(S(activate(z0)), ACTIVATE(z0))
S tuples:

ACTIVATE(n__from(z0)) → c1(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

sel, activate, from, s

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c1, c3

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

Removed 2 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

sel(0, cons(z0, z1)) → z0
activate(n__from(z0)) → from(activate(z0))
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)) → c1(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
S tuples:

ACTIVATE(n__from(z0)) → c1(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

sel, activate, from, s

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c1, c3

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

sel(0, cons(z0, z1)) → z0
activate(n__from(z0)) → from(activate(z0))
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)) → c1(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
S tuples:

ACTIVATE(n__from(z0)) → c1(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c1, c3

(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)) → c1(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__from(z0)) → c1(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = [2]x1   
POL(c1(x1)) = x1   
POL(c3(x1)) = x1   
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)) → c1(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__from(z0)) → c1(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c1, c3

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

The set S is empty

(14) BOUNDS(1, 1)