(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
from(X) → cons(X, n__from(n__s(X)))
first(0, Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
sel(0, cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
s(X) → n__s(X)
first(X1, X2) → n__first(X1, X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
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)
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
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(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0
Tuples:
FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__from(z0)) → c8(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
S tuples:
FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__from(z0)) → c8(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:
from, first, sel, s, activate
Defined Pair Symbols:
FIRST, SEL, ACTIVATE
Compound Symbols:
c3, c6, c8, c9, c10
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(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)
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
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(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0
Tuples:
ACTIVATE(n__from(z0)) → c8(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
S tuples:
ACTIVATE(n__from(z0)) → c8(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:
from, first, sel, s, activate
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c8, c9, c10
(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
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(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0
Tuples:
ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
S tuples:
ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:
from, first, sel, s, activate
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c8, c9, c10
(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)) → c8(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:none
And the Tuples:
ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ACTIVATE(x1)) = [5] + [4]x1
POL(c10(x1, x2)) = x1 + x2
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(n__first(x1, x2)) = [4] + x1 + x2
POL(n__from(x1)) = [5] + x1
POL(n__s(x1)) = [2] + x1
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
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(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0
Tuples:
ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
S tuples:none
K tuples:
ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:
from, first, sel, s, activate
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c8, c9, c10
(9) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(10) BOUNDS(O(1), O(1))