(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:
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) DependencyGraphProof (BOTH BOUNDS(ID, ID) transformation)
The following rules are not reachable from basic terms in the dependency graph and can be removed:
2nd(cons(X, n__cons(Y, Z))) → activate(Y)
(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:
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__cons(X1, X2)) → cons(activate(X1), X2)
cons(X1, X2) → n__cons(X1, X2)
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:
activate(n__from(z0)) → from(activate(z0))
activate(z0) → z0
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
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)
cons(z0, z1) → n__cons(z0, z1)
Tuples:
ACTIVATE(n__from(z0)) → c(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c1
ACTIVATE(n__cons(z0, z1)) → c2(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(S(activate(z0)), ACTIVATE(z0))
FROM(z0) → c4(CONS(z0, n__from(n__s(z0))))
FROM(z0) → c5
S(z0) → c6
CONS(z0, z1) → c7
S tuples:
ACTIVATE(n__from(z0)) → c(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c1
ACTIVATE(n__cons(z0, z1)) → c2(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(S(activate(z0)), ACTIVATE(z0))
FROM(z0) → c4(CONS(z0, n__from(n__s(z0))))
FROM(z0) → c5
S(z0) → c6
CONS(z0, z1) → c7
K tuples:none
Defined Rule Symbols:
activate, from, s, cons
Defined Pair Symbols:
ACTIVATE, FROM, S, CONS
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 5 trailing nodes:
FROM(z0) → c5
S(z0) → c6
CONS(z0, z1) → c7
ACTIVATE(z0) → c1
FROM(z0) → c4(CONS(z0, n__from(n__s(z0))))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
activate(n__from(z0)) → from(activate(z0))
activate(z0) → z0
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
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)
cons(z0, z1) → n__cons(z0, z1)
Tuples:
ACTIVATE(n__from(z0)) → c(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c2(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(S(activate(z0)), ACTIVATE(z0))
S tuples:
ACTIVATE(n__from(z0)) → c(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c2(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:
activate, from, s, cons
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c, c2, c3
(7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing tuple parts
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
activate(n__from(z0)) → from(activate(z0))
activate(z0) → z0
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
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)
cons(z0, z1) → n__cons(z0, z1)
Tuples:
ACTIVATE(n__from(z0)) → c(ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c2(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
S tuples:
ACTIVATE(n__from(z0)) → c(ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c2(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:
activate, from, s, cons
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c, c2, c3
(9) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
activate(n__from(z0)) → from(activate(z0))
activate(z0) → z0
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
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)
cons(z0, z1) → n__cons(z0, z1)
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
ACTIVATE(n__from(z0)) → c(ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c2(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
S tuples:
ACTIVATE(n__from(z0)) → c(ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c2(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c, c2, 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)) → c(ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c2(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:
ACTIVATE(n__from(z0)) → c(ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c2(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(c(x1)) = x1
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(n__cons(x1, x2)) = [2] + x1
POL(n__from(x1)) = [2] + x1
POL(n__s(x1)) = [2] + x1
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
ACTIVATE(n__from(z0)) → c(ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c2(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
S tuples:none
K tuples:
ACTIVATE(n__from(z0)) → c(ACTIVATE(z0))
ACTIVATE(n__cons(z0, z1)) → c2(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c3(ACTIVATE(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c, c2, c3
(13) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(14) BOUNDS(1, 1)