(0) Obligation:
Runtime Complexity TRS:
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) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
sel(0, cons(z0, z1)) → z0
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2))
s(z0) → n__s(z0)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:
SEL(s(z0), cons(z1, z2)) → c3(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__from(z0)) → c5(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
S tuples:
SEL(s(z0), cons(z1, z2)) → c3(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__from(z0)) → c5(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:
from, sel, s, activate
Defined Pair Symbols:
SEL, ACTIVATE
Compound Symbols:
c3, c5, c6
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
SEL(s(z0), cons(z1, z2)) → c3(SEL(z0, activate(z2)), ACTIVATE(z2))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
sel(0, cons(z0, z1)) → z0
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2))
s(z0) → n__s(z0)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:
ACTIVATE(n__from(z0)) → c5(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
S tuples:
ACTIVATE(n__from(z0)) → c5(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:
from, sel, s, activate
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c5, c6
(5) CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
sel(0, cons(z0, z1)) → z0
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2))
s(z0) → n__s(z0)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:
ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
S tuples:
ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:
from, sel, s, activate
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c5, c6
(7) CdtPolyRedPairProof (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__from(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:
ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ACTIVATE(x1)) = [4]x1
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(n__from(x1)) = [1] + x1
POL(n__s(x1)) = [4] + x1
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
sel(0, cons(z0, z1)) → z0
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2))
s(z0) → n__s(z0)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:
ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
S tuples:none
K tuples:
ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
Defined Rule Symbols:
from, sel, s, activate
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c5, c6
(9) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(10) BOUNDS(O(1), O(1))