(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(2nd(cons1(X, cons(Y, Z)))) → mark(Y)
active(2nd(cons(X, X1))) → mark(2nd(cons1(X, X1)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(2nd(X)) → active(2nd(mark(X)))
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
2nd(mark(X)) → 2nd(X)
2nd(active(X)) → 2nd(X)
cons1(mark(X1), X2) → cons1(X1, X2)
cons1(X1, mark(X2)) → cons1(X1, X2)
cons1(active(X1), X2) → cons1(X1, X2)
cons1(X1, active(X2)) → cons1(X1, X2)
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)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
mark(2nd(z0)) → active(2nd(mark(z0)))
mark(cons1(z0, z1)) → active(cons1(mark(z0), mark(z1)))
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(from(z0)) → active(from(mark(z0)))
mark(s(z0)) → active(s(mark(z0)))
2nd(mark(z0)) → 2nd(z0)
2nd(active(z0)) → 2nd(z0)
cons1(mark(z0), z1) → cons1(z0, z1)
cons1(z0, mark(z1)) → cons1(z0, z1)
cons1(active(z0), z1) → cons1(z0, z1)
cons1(z0, active(z1)) → cons1(z0, z1)
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)
from(mark(z0)) → from(z0)
from(active(z0)) → from(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
Tuples:
ACTIVE(2nd(cons1(z0, cons(z1, z2)))) → c(MARK(z1))
ACTIVE(2nd(cons(z0, z1))) → c1(MARK(2nd(cons1(z0, z1))), 2ND(cons1(z0, z1)), CONS1(z0, z1))
ACTIVE(from(z0)) → c2(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
MARK(2nd(z0)) → c3(ACTIVE(2nd(mark(z0))), 2ND(mark(z0)), MARK(z0))
MARK(cons1(z0, z1)) → c4(ACTIVE(cons1(mark(z0), mark(z1))), CONS1(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(from(z0)) → c6(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(s(z0)) → c7(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
2ND(mark(z0)) → c8(2ND(z0))
2ND(active(z0)) → c9(2ND(z0))
CONS1(mark(z0), z1) → c10(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c11(CONS1(z0, z1))
CONS1(active(z0), z1) → c12(CONS1(z0, z1))
CONS1(z0, active(z1)) → c13(CONS1(z0, z1))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
FROM(mark(z0)) → c18(FROM(z0))
FROM(active(z0)) → c19(FROM(z0))
S(mark(z0)) → c20(S(z0))
S(active(z0)) → c21(S(z0))
S tuples:
ACTIVE(2nd(cons1(z0, cons(z1, z2)))) → c(MARK(z1))
ACTIVE(2nd(cons(z0, z1))) → c1(MARK(2nd(cons1(z0, z1))), 2ND(cons1(z0, z1)), CONS1(z0, z1))
ACTIVE(from(z0)) → c2(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
MARK(2nd(z0)) → c3(ACTIVE(2nd(mark(z0))), 2ND(mark(z0)), MARK(z0))
MARK(cons1(z0, z1)) → c4(ACTIVE(cons1(mark(z0), mark(z1))), CONS1(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(from(z0)) → c6(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(s(z0)) → c7(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
2ND(mark(z0)) → c8(2ND(z0))
2ND(active(z0)) → c9(2ND(z0))
CONS1(mark(z0), z1) → c10(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c11(CONS1(z0, z1))
CONS1(active(z0), z1) → c12(CONS1(z0, z1))
CONS1(z0, active(z1)) → c13(CONS1(z0, z1))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
FROM(mark(z0)) → c18(FROM(z0))
FROM(active(z0)) → c19(FROM(z0))
S(mark(z0)) → c20(S(z0))
S(active(z0)) → c21(S(z0))
K tuples:none
Defined Rule Symbols:
active, mark, 2nd, cons1, cons, from, s
Defined Pair Symbols:
ACTIVE, MARK, 2ND, CONS1, CONS, FROM, S
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(2nd(cons1(z0, cons(z1, z2)))) → c(MARK(z1))
ACTIVE(2nd(cons(z0, z1))) → c1(MARK(2nd(cons1(z0, z1))), 2ND(cons1(z0, z1)), CONS1(z0, z1))
ACTIVE(from(z0)) → c2(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
MARK(2nd(z0)) → c3(ACTIVE(2nd(mark(z0))), 2ND(mark(z0)), MARK(z0))
MARK(cons1(z0, z1)) → c4(ACTIVE(cons1(mark(z0), mark(z1))), CONS1(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(from(z0)) → c6(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(s(z0)) → c7(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
2ND(mark(z0)) → c8(2ND(z0))
2ND(active(z0)) → c9(2ND(z0))
CONS1(mark(z0), z1) → c10(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c11(CONS1(z0, z1))
CONS1(active(z0), z1) → c12(CONS1(z0, z1))
CONS1(z0, active(z1)) → c13(CONS1(z0, z1))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(z0, mark(z1)) → c15(CONS(z0, z1))
CONS(active(z0), z1) → c16(CONS(z0, z1))
CONS(z0, active(z1)) → c17(CONS(z0, z1))
FROM(mark(z0)) → c18(FROM(z0))
FROM(active(z0)) → c19(FROM(z0))
S(mark(z0)) → c20(S(z0))
S(active(z0)) → c21(S(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1)
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
mark(2nd(z0)) → active(2nd(mark(z0)))
mark(cons1(z0, z1)) → active(cons1(mark(z0), mark(z1)))
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(from(z0)) → active(from(mark(z0)))
mark(s(z0)) → active(s(mark(z0)))
2nd(mark(z0)) → 2nd(z0)
2nd(active(z0)) → 2nd(z0)
cons1(mark(z0), z1) → cons1(z0, z1)
cons1(z0, mark(z1)) → cons1(z0, z1)
cons1(active(z0), z1) → cons1(z0, z1)
cons1(z0, active(z1)) → cons1(z0, z1)
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)
from(mark(z0)) → from(z0)
from(active(z0)) → from(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, 2nd, cons1, cons, from, s
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(6) BOUNDS(O(1), O(1))