(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
from(X) → cons(X, n__from(n__s(X)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, activate(XS))
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 Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
after(0, z0) → z0
after(s(z0), cons(z1, z2)) → after(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:
FROM(z0) → c
FROM(z0) → c1
AFTER(0, z0) → c2
AFTER(s(z0), cons(z1, z2)) → c3(AFTER(z0, activate(z2)), ACTIVATE(z2))
S(z0) → c4
ACTIVATE(n__from(z0)) → c5(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c7
S tuples:
FROM(z0) → c
FROM(z0) → c1
AFTER(0, z0) → c2
AFTER(s(z0), cons(z1, z2)) → c3(AFTER(z0, activate(z2)), ACTIVATE(z2))
S(z0) → c4
ACTIVATE(n__from(z0)) → c5(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c7
K tuples:none
Defined Rule Symbols:
from, after, s, activate
Defined Pair Symbols:
FROM, AFTER, S, ACTIVATE
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7
(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
AFTER(s(z0), cons(z1, z2)) → c3(AFTER(z0, activate(z2)), ACTIVATE(z2))
Removed 5 trailing nodes:
FROM(z0) → c
FROM(z0) → c1
AFTER(0, z0) → c2
S(z0) → c4
ACTIVATE(z0) → c7
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
after(0, z0) → z0
after(s(z0), cons(z1, z2)) → after(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, after, s, activate
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c5, c6
(5) CdtRhsSimplificationProcessorProof (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)
after(0, z0) → z0
after(s(z0), cons(z1, z2)) → after(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, after, s, activate
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c5, c6
(7) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
after(0, z0) → z0
after(s(z0), cons(z1, z2)) → after(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
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:none
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:none
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c5, c6
(9) 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.
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
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:none
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:none
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c5, c6
(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(12) BOUNDS(O(1), O(1))