(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(after(X1, X2)) → active(after(mark(X1), mark(X2)))
mark(0) → active(0)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
after(mark(X1), X2) → after(X1, X2)
after(X1, mark(X2)) → after(X1, X2)
after(active(X1), X2) → after(X1, X2)
after(X1, active(X2)) → after(X1, X2)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
mark(from(z0)) → active(from(mark(z0)))
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(after(z0, z1)) → active(after(mark(z0), mark(z1)))
mark(0) → active(0)
from(mark(z0)) → from(z0)
from(active(z0)) → from(z0)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
after(mark(z0), z1) → after(z0, z1)
after(z0, mark(z1)) → after(z0, z1)
after(active(z0), z1) → after(z0, z1)
after(z0, active(z1)) → after(z0, z1)
Tuples:
ACTIVE(from(z0)) → c(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(after(0, z0)) → c1(MARK(z0))
ACTIVE(after(s(z0), cons(z1, z2))) → c2(MARK(after(z0, z2)), AFTER(z0, z2))
MARK(from(z0)) → c3(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c4(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c5(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(after(z0, z1)) → c6(ACTIVE(after(mark(z0), mark(z1))), AFTER(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(0) → c7(ACTIVE(0))
FROM(mark(z0)) → c8(FROM(z0))
FROM(active(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(z0, mark(z1)) → c11(CONS(z0, z1))
CONS(active(z0), z1) → c12(CONS(z0, z1))
CONS(z0, active(z1)) → c13(CONS(z0, z1))
S(mark(z0)) → c14(S(z0))
S(active(z0)) → c15(S(z0))
AFTER(mark(z0), z1) → c16(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c17(AFTER(z0, z1))
AFTER(active(z0), z1) → c18(AFTER(z0, z1))
AFTER(z0, active(z1)) → c19(AFTER(z0, z1))
S tuples:
ACTIVE(from(z0)) → c(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(after(0, z0)) → c1(MARK(z0))
ACTIVE(after(s(z0), cons(z1, z2))) → c2(MARK(after(z0, z2)), AFTER(z0, z2))
MARK(from(z0)) → c3(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c4(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c5(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(after(z0, z1)) → c6(ACTIVE(after(mark(z0), mark(z1))), AFTER(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(0) → c7(ACTIVE(0))
FROM(mark(z0)) → c8(FROM(z0))
FROM(active(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(z0, mark(z1)) → c11(CONS(z0, z1))
CONS(active(z0), z1) → c12(CONS(z0, z1))
CONS(z0, active(z1)) → c13(CONS(z0, z1))
S(mark(z0)) → c14(S(z0))
S(active(z0)) → c15(S(z0))
AFTER(mark(z0), z1) → c16(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c17(AFTER(z0, z1))
AFTER(active(z0), z1) → c18(AFTER(z0, z1))
AFTER(z0, active(z1)) → c19(AFTER(z0, z1))
K tuples:none
Defined Rule Symbols:
active, mark, from, cons, s, after
Defined Pair Symbols:
ACTIVE, MARK, FROM, CONS, S, AFTER
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(from(z0)) → c(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(after(0, z0)) → c1(MARK(z0))
ACTIVE(after(s(z0), cons(z1, z2))) → c2(MARK(after(z0, z2)), AFTER(z0, z2))
MARK(from(z0)) → c3(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c4(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c5(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(after(z0, z1)) → c6(ACTIVE(after(mark(z0), mark(z1))), AFTER(mark(z0), mark(z1)), MARK(z0), MARK(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(active(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(z0, mark(z1)) → c11(CONS(z0, z1))
CONS(active(z0), z1) → c12(CONS(z0, z1))
CONS(z0, active(z1)) → c13(CONS(z0, z1))
S(mark(z0)) → c14(S(z0))
S(active(z0)) → c15(S(z0))
AFTER(mark(z0), z1) → c16(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c17(AFTER(z0, z1))
AFTER(active(z0), z1) → c18(AFTER(z0, z1))
AFTER(z0, active(z1)) → c19(AFTER(z0, z1))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
mark(from(z0)) → active(from(mark(z0)))
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(after(z0, z1)) → active(after(mark(z0), mark(z1)))
mark(0) → active(0)
from(mark(z0)) → from(z0)
from(active(z0)) → from(z0)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
after(mark(z0), z1) → after(z0, z1)
after(z0, mark(z1)) → after(z0, z1)
after(active(z0), z1) → after(z0, z1)
after(z0, active(z1)) → after(z0, z1)
Tuples:
MARK(0) → c7(ACTIVE(0))
S tuples:
MARK(0) → c7(ACTIVE(0))
K tuples:none
Defined Rule Symbols:
active, mark, from, cons, s, after
Defined Pair Symbols:
MARK
Compound Symbols:
c7
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
MARK(0) → c7(ACTIVE(0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
mark(from(z0)) → active(from(mark(z0)))
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(after(z0, z1)) → active(after(mark(z0), mark(z1)))
mark(0) → active(0)
from(mark(z0)) → from(z0)
from(active(z0)) → from(z0)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
after(mark(z0), z1) → after(z0, z1)
after(z0, mark(z1)) → after(z0, z1)
after(active(z0), z1) → after(z0, z1)
after(z0, active(z1)) → after(z0, z1)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, from, cons, s, after
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))