The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

0 CpxTRS
↳1 DependencyGraphProof (BOTH BOUNDS(ID, ID), 14 ms)
↳2 CpxTRS
↳3 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 16 ms)
↳4 CdtProblem
↳5 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳6 CdtProblem
↳7 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳8 CdtProblem
↳9 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳10 CdtProblem
↳11 CdtUsableRulesProof (⇔, 0 ms)
↳12 CdtProblem
↳13 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 72 ms)
↳14 CdtProblem
↳15 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 BOUNDS(1, 1)

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

from(X) → cons(X, n__from(n__s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(activate(XS))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
from(X) → n__from(X)
s(X) → n__s(X)
take(X1, X2) → n__take(X1, X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__take(X1, X2)) → take(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(N), cons(X, XS)) → sel(N, activate(XS))

(2) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

sel(0, cons(X, XS)) → X
head(cons(X, XS)) → X
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
activate(X) → X
activate(n__s(X)) → s(activate(X))
take(X1, X2) → n__take(X1, X2)
activate(n__from(X)) → from(activate(X))
from(X) → cons(X, n__from(n__s(X)))
take(0, XS) → nil
from(X) → n__from(X)
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
s(X) → n__s(X)
2nd(cons(X, XS)) → head(activate(XS))

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
head(cons(z0, z1)) → z0
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2)))
take(z0, z1) → n__take(z0, z1)
take(0, z0) → nil
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__from(z0)) → from(activate(z0))
activate(n__take(z0, z1)) → take(activate(z0), activate(z1))
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
s(z0) → n__s(z0)
2nd(cons(z0, z1)) → head(activate(z1))
Tuples:

SEL(0, cons(z0, z1)) → c
HEAD(cons(z0, z1)) → c1
TAKE(s(z0), cons(z1, z2)) → c2(ACTIVATE(z2))
TAKE(z0, z1) → c3
TAKE(0, z0) → c4
ACTIVATE(z0) → c5
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c7(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c8(TAKE(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
FROM(z0) → c9
FROM(z0) → c10
S(z0) → c11
2ND(cons(z0, z1)) → c12(HEAD(activate(z1)), ACTIVATE(z1))
S tuples:

SEL(0, cons(z0, z1)) → c
HEAD(cons(z0, z1)) → c1
TAKE(s(z0), cons(z1, z2)) → c2(ACTIVATE(z2))
TAKE(z0, z1) → c3
TAKE(0, z0) → c4
ACTIVATE(z0) → c5
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c7(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c8(TAKE(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
FROM(z0) → c9
FROM(z0) → c10
S(z0) → c11
2ND(cons(z0, z1)) → c12(HEAD(activate(z1)), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

sel, head, take, activate, from, s, 2nd

Defined Pair Symbols:

SEL, HEAD, TAKE, ACTIVATE, FROM, S, 2ND

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

TAKE(s(z0), cons(z1, z2)) → c2(ACTIVATE(z2))
Removed 8 trailing nodes:

ACTIVATE(z0) → c5
HEAD(cons(z0, z1)) → c1
SEL(0, cons(z0, z1)) → c
TAKE(z0, z1) → c3
FROM(z0) → c9
S(z0) → c11
FROM(z0) → c10
TAKE(0, z0) → c4

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

sel(0, cons(z0, z1)) → z0
head(cons(z0, z1)) → z0
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2)))
take(z0, z1) → n__take(z0, z1)
take(0, z0) → nil
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__from(z0)) → from(activate(z0))
activate(n__take(z0, z1)) → take(activate(z0), activate(z1))
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
s(z0) → n__s(z0)
2nd(cons(z0, z1)) → head(activate(z1))
Tuples:

ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c7(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c8(TAKE(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
2ND(cons(z0, z1)) → c12(HEAD(activate(z1)), ACTIVATE(z1))
S tuples:

ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c7(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c8(TAKE(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
2ND(cons(z0, z1)) → c12(HEAD(activate(z1)), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

sel, head, take, activate, from, s, 2nd

Defined Pair Symbols:

ACTIVATE, 2ND

Compound Symbols:

c6, c7, c8, c12

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

Removed 4 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

sel(0, cons(z0, z1)) → z0
head(cons(z0, z1)) → z0
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2)))
take(z0, z1) → n__take(z0, z1)
take(0, z0) → nil
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__from(z0)) → from(activate(z0))
activate(n__take(z0, z1)) → take(activate(z0), activate(z1))
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
s(z0) → n__s(z0)
2nd(cons(z0, z1)) → head(activate(z1))
Tuples:

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c8(ACTIVATE(z0), ACTIVATE(z1))
2ND(cons(z0, z1)) → c12(ACTIVATE(z1))
S tuples:

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c8(ACTIVATE(z0), ACTIVATE(z1))
2ND(cons(z0, z1)) → c12(ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

sel, head, take, activate, from, s, 2nd

Defined Pair Symbols:

ACTIVATE, 2ND

Compound Symbols:

c6, c7, c8, c12

(9) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

2ND(cons(z0, z1)) → c12(ACTIVATE(z1))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

sel(0, cons(z0, z1)) → z0
head(cons(z0, z1)) → z0
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2)))
take(z0, z1) → n__take(z0, z1)
take(0, z0) → nil
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__from(z0)) → from(activate(z0))
activate(n__take(z0, z1)) → take(activate(z0), activate(z1))
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
s(z0) → n__s(z0)
2nd(cons(z0, z1)) → head(activate(z1))
Tuples:

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c8(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c8(ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

sel, head, take, activate, from, s, 2nd

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c6, c7, c8

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

sel(0, cons(z0, z1)) → z0
head(cons(z0, z1)) → z0
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2)))
take(z0, z1) → n__take(z0, z1)
take(0, z0) → nil
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__from(z0)) → from(activate(z0))
activate(n__take(z0, z1)) → take(activate(z0), activate(z1))
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
s(z0) → n__s(z0)
2nd(cons(z0, z1)) → head(activate(z1))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c8(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c8(ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c6, c7, c8

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c8(ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c8(ACTIVATE(z0), ACTIVATE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = [1] + x1 + x12   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1, x2)) = x1 + x2   
POL(n__from(x1)) = [1] + x1   
POL(n__s(x1)) = [1] + x1   
POL(n__take(x1, x2)) = [1] + x1 + x2   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c8(ACTIVATE(z0), ACTIVATE(z1))
S tuples:none
K tuples:

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__take(z0, z1)) → c8(ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c6, c7, c8

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

The set S is empty

(16) BOUNDS(1, 1)