(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0)
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(X))
mark(nil) → active(nil)
mark(0) → active(0)
mark(length1(X)) → active(length1(X))
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)
length(mark(X)) → length(X)
length(active(X)) → length(X)
length1(mark(X)) → length1(X)
length1(active(X)) → length1(X)
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(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
mark(from(z0)) → active(from(mark(z0)))
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(z0))
mark(nil) → active(nil)
mark(0) → active(0)
mark(length1(z0)) → active(length1(z0))
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)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
length1(mark(z0)) → length1(z0)
length1(active(z0)) → length1(z0)
Tuples:
ACTIVE(from(z0)) → c(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(length(nil)) → c1(MARK(0))
ACTIVE(length(cons(z0, z1))) → c2(MARK(s(length1(z1))), S(length1(z1)), LENGTH1(z1))
ACTIVE(length1(z0)) → c3(MARK(length(z0)), LENGTH(z0))
MARK(from(z0)) → c4(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c6(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c7(ACTIVE(length(z0)), LENGTH(z0))
MARK(nil) → c8(ACTIVE(nil))
MARK(0) → c9(ACTIVE(0))
MARK(length1(z0)) → c10(ACTIVE(length1(z0)), LENGTH1(z0))
FROM(mark(z0)) → c11(FROM(z0))
FROM(active(z0)) → c12(FROM(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(z0, mark(z1)) → c14(CONS(z0, z1))
CONS(active(z0), z1) → c15(CONS(z0, z1))
CONS(z0, active(z1)) → c16(CONS(z0, z1))
S(mark(z0)) → c17(S(z0))
S(active(z0)) → c18(S(z0))
LENGTH(mark(z0)) → c19(LENGTH(z0))
LENGTH(active(z0)) → c20(LENGTH(z0))
LENGTH1(mark(z0)) → c21(LENGTH1(z0))
LENGTH1(active(z0)) → c22(LENGTH1(z0))
S tuples:
ACTIVE(from(z0)) → c(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(length(nil)) → c1(MARK(0))
ACTIVE(length(cons(z0, z1))) → c2(MARK(s(length1(z1))), S(length1(z1)), LENGTH1(z1))
ACTIVE(length1(z0)) → c3(MARK(length(z0)), LENGTH(z0))
MARK(from(z0)) → c4(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c6(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c7(ACTIVE(length(z0)), LENGTH(z0))
MARK(nil) → c8(ACTIVE(nil))
MARK(0) → c9(ACTIVE(0))
MARK(length1(z0)) → c10(ACTIVE(length1(z0)), LENGTH1(z0))
FROM(mark(z0)) → c11(FROM(z0))
FROM(active(z0)) → c12(FROM(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(z0, mark(z1)) → c14(CONS(z0, z1))
CONS(active(z0), z1) → c15(CONS(z0, z1))
CONS(z0, active(z1)) → c16(CONS(z0, z1))
S(mark(z0)) → c17(S(z0))
S(active(z0)) → c18(S(z0))
LENGTH(mark(z0)) → c19(LENGTH(z0))
LENGTH(active(z0)) → c20(LENGTH(z0))
LENGTH1(mark(z0)) → c21(LENGTH1(z0))
LENGTH1(active(z0)) → c22(LENGTH1(z0))
K tuples:none
Defined Rule Symbols:
active, mark, from, cons, s, length, length1
Defined Pair Symbols:
ACTIVE, MARK, FROM, CONS, S, LENGTH, LENGTH1
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22
(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(length(nil)) → c1(MARK(0))
ACTIVE(length(cons(z0, z1))) → c2(MARK(s(length1(z1))), S(length1(z1)), LENGTH1(z1))
ACTIVE(length1(z0)) → c3(MARK(length(z0)), LENGTH(z0))
MARK(from(z0)) → c4(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c6(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(length(z0)) → c7(ACTIVE(length(z0)), LENGTH(z0))
MARK(length1(z0)) → c10(ACTIVE(length1(z0)), LENGTH1(z0))
FROM(mark(z0)) → c11(FROM(z0))
FROM(active(z0)) → c12(FROM(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(z0, mark(z1)) → c14(CONS(z0, z1))
CONS(active(z0), z1) → c15(CONS(z0, z1))
CONS(z0, active(z1)) → c16(CONS(z0, z1))
S(mark(z0)) → c17(S(z0))
S(active(z0)) → c18(S(z0))
LENGTH(mark(z0)) → c19(LENGTH(z0))
LENGTH(active(z0)) → c20(LENGTH(z0))
LENGTH1(mark(z0)) → c21(LENGTH1(z0))
LENGTH1(active(z0)) → c22(LENGTH1(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
mark(from(z0)) → active(from(mark(z0)))
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(z0))
mark(nil) → active(nil)
mark(0) → active(0)
mark(length1(z0)) → active(length1(z0))
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)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
length1(mark(z0)) → length1(z0)
length1(active(z0)) → length1(z0)
Tuples:
MARK(nil) → c8(ACTIVE(nil))
MARK(0) → c9(ACTIVE(0))
S tuples:
MARK(nil) → c8(ACTIVE(nil))
MARK(0) → c9(ACTIVE(0))
K tuples:none
Defined Rule Symbols:
active, mark, from, cons, s, length, length1
Defined Pair Symbols:
MARK
Compound Symbols:
c8, c9
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
MARK(0) → c9(ACTIVE(0))
MARK(nil) → c8(ACTIVE(nil))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
mark(from(z0)) → active(from(mark(z0)))
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(length(z0)) → active(length(z0))
mark(nil) → active(nil)
mark(0) → active(0)
mark(length1(z0)) → active(length1(z0))
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)
length(mark(z0)) → length(z0)
length(active(z0)) → length(z0)
length1(mark(z0)) → length1(z0)
length1(active(z0)) → length1(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, from, cons, s, length, length1
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))