(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
2nd(cons(X, n__cons(Y, Z))) → activate(Y)
from(X) → cons(X, n__from(n__s(X)))
cons(X1, X2) → n__cons(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
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:
2nd(cons(z0, n__cons(z1, z2))) → activate(z1)
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
cons(z0, z1) → n__cons(z0, z1)
s(z0) → n__s(z0)
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:
2ND(cons(z0, n__cons(z1, z2))) → c(ACTIVATE(z1))
FROM(z0) → c1(CONS(z0, n__from(n__s(z0))))
FROM(z0) → c2
CONS(z0, z1) → c3
S(z0) → c4
ACTIVATE(n__cons(z0, z1)) → c5(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c8
S tuples:
2ND(cons(z0, n__cons(z1, z2))) → c(ACTIVATE(z1))
FROM(z0) → c1(CONS(z0, n__from(n__s(z0))))
FROM(z0) → c2
CONS(z0, z1) → c3
S(z0) → c4
ACTIVATE(n__cons(z0, z1)) → c5(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c8
K tuples:none
Defined Rule Symbols:
2nd, from, cons, s, activate
Defined Pair Symbols:
2ND, FROM, CONS, S, ACTIVATE
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7, c8
(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
2ND(cons(z0, n__cons(z1, z2))) → c(ACTIVATE(z1))
Removed 5 trailing nodes:
FROM(z0) → c1(CONS(z0, n__from(n__s(z0))))
FROM(z0) → c2
CONS(z0, z1) → c3
S(z0) → c4
ACTIVATE(z0) → c8
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
2nd(cons(z0, n__cons(z1, z2))) → activate(z1)
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
cons(z0, z1) → n__cons(z0, z1)
s(z0) → n__s(z0)
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:
ACTIVATE(n__cons(z0, z1)) → c5(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
S tuples:
ACTIVATE(n__cons(z0, z1)) → c5(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:
2nd, from, cons, s, activate
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c5, c6, c7
(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
2nd(cons(z0, n__cons(z1, z2))) → activate(z1)
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
cons(z0, z1) → n__cons(z0, z1)
s(z0) → n__s(z0)
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:
ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
S tuples:
ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:
2nd, from, cons, s, activate
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c5, c6, c7
(7) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
2nd(cons(z0, n__cons(z1, z2))) → activate(z1)
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
cons(z0, z1) → n__cons(z0, z1)
s(z0) → n__s(z0)
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
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__cons(z0, z1)) → c5(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
S tuples:
ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c5, c6, c7
(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__cons(z0, z1)) → c5(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:
ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(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(c7(x1)) = x1
POL(n__cons(x1, x2)) = [2] + x1
POL(n__from(x1)) = [5] + x1
POL(n__s(x1)) = [4] + x1
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
S tuples:none
K tuples:
ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c5, c6, c7
(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(12) BOUNDS(O(1), O(1))