(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(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(head(X)) → active(head(mark(X)))
mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
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)
head(mark(X)) → head(X)
head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)
2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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(head(cons(z0, z1))) → mark(z0)
active(2nd(cons(z0, z1))) → mark(head(z1))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(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(head(z0)) → active(head(mark(z0)))
mark(2nd(z0)) → active(2nd(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(sel(z0, z1)) → active(sel(mark(z0), mark(z1)))
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)
head(mark(z0)) → head(z0)
head(active(z0)) → head(z0)
2nd(mark(z0)) → 2nd(z0)
2nd(active(z0)) → 2nd(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)
sel(mark(z0), z1) → sel(z0, z1)
sel(z0, mark(z1)) → sel(z0, z1)
sel(active(z0), z1) → sel(z0, z1)
sel(z0, active(z1)) → sel(z0, z1)
Tuples:
ACTIVE(from(z0)) → c(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(head(cons(z0, z1))) → c1(MARK(z0))
ACTIVE(2nd(cons(z0, z1))) → c2(MARK(head(z1)), HEAD(z1))
ACTIVE(take(0, z0)) → c3(MARK(nil))
ACTIVE(take(s(z0), cons(z1, z2))) → c4(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
ACTIVE(sel(0, cons(z0, z1))) → c5(MARK(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c6(MARK(sel(z0, z2)), SEL(z0, z2))
MARK(from(z0)) → c7(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c8(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c9(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(head(z0)) → c10(ACTIVE(head(mark(z0))), HEAD(mark(z0)), MARK(z0))
MARK(2nd(z0)) → c11(ACTIVE(2nd(mark(z0))), 2ND(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c12(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(0) → c13(ACTIVE(0))
MARK(nil) → c14(ACTIVE(nil))
MARK(sel(z0, z1)) → c15(ACTIVE(sel(mark(z0), mark(z1))), SEL(mark(z0), mark(z1)), MARK(z0), MARK(z1))
FROM(mark(z0)) → c16(FROM(z0))
FROM(active(z0)) → c17(FROM(z0))
CONS(mark(z0), z1) → c18(CONS(z0, z1))
CONS(z0, mark(z1)) → c19(CONS(z0, z1))
CONS(active(z0), z1) → c20(CONS(z0, z1))
CONS(z0, active(z1)) → c21(CONS(z0, z1))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
HEAD(mark(z0)) → c24(HEAD(z0))
HEAD(active(z0)) → c25(HEAD(z0))
2ND(mark(z0)) → c26(2ND(z0))
2ND(active(z0)) → c27(2ND(z0))
TAKE(mark(z0), z1) → c28(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c29(TAKE(z0, z1))
TAKE(active(z0), z1) → c30(TAKE(z0, z1))
TAKE(z0, active(z1)) → c31(TAKE(z0, z1))
SEL(mark(z0), z1) → c32(SEL(z0, z1))
SEL(z0, mark(z1)) → c33(SEL(z0, z1))
SEL(active(z0), z1) → c34(SEL(z0, z1))
SEL(z0, active(z1)) → c35(SEL(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(head(cons(z0, z1))) → c1(MARK(z0))
ACTIVE(2nd(cons(z0, z1))) → c2(MARK(head(z1)), HEAD(z1))
ACTIVE(take(0, z0)) → c3(MARK(nil))
ACTIVE(take(s(z0), cons(z1, z2))) → c4(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
ACTIVE(sel(0, cons(z0, z1))) → c5(MARK(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c6(MARK(sel(z0, z2)), SEL(z0, z2))
MARK(from(z0)) → c7(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c8(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c9(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(head(z0)) → c10(ACTIVE(head(mark(z0))), HEAD(mark(z0)), MARK(z0))
MARK(2nd(z0)) → c11(ACTIVE(2nd(mark(z0))), 2ND(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c12(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(0) → c13(ACTIVE(0))
MARK(nil) → c14(ACTIVE(nil))
MARK(sel(z0, z1)) → c15(ACTIVE(sel(mark(z0), mark(z1))), SEL(mark(z0), mark(z1)), MARK(z0), MARK(z1))
FROM(mark(z0)) → c16(FROM(z0))
FROM(active(z0)) → c17(FROM(z0))
CONS(mark(z0), z1) → c18(CONS(z0, z1))
CONS(z0, mark(z1)) → c19(CONS(z0, z1))
CONS(active(z0), z1) → c20(CONS(z0, z1))
CONS(z0, active(z1)) → c21(CONS(z0, z1))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
HEAD(mark(z0)) → c24(HEAD(z0))
HEAD(active(z0)) → c25(HEAD(z0))
2ND(mark(z0)) → c26(2ND(z0))
2ND(active(z0)) → c27(2ND(z0))
TAKE(mark(z0), z1) → c28(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c29(TAKE(z0, z1))
TAKE(active(z0), z1) → c30(TAKE(z0, z1))
TAKE(z0, active(z1)) → c31(TAKE(z0, z1))
SEL(mark(z0), z1) → c32(SEL(z0, z1))
SEL(z0, mark(z1)) → c33(SEL(z0, z1))
SEL(active(z0), z1) → c34(SEL(z0, z1))
SEL(z0, active(z1)) → c35(SEL(z0, z1))
K tuples:none
Defined Rule Symbols:
active, mark, from, cons, s, head, 2nd, take, sel
Defined Pair Symbols:
ACTIVE, MARK, FROM, CONS, S, HEAD, 2ND, TAKE, SEL
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, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35
(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(head(cons(z0, z1))) → c1(MARK(z0))
ACTIVE(2nd(cons(z0, z1))) → c2(MARK(head(z1)), HEAD(z1))
ACTIVE(take(0, z0)) → c3(MARK(nil))
ACTIVE(take(s(z0), cons(z1, z2))) → c4(MARK(cons(z1, take(z0, z2))), CONS(z1, take(z0, z2)), TAKE(z0, z2))
ACTIVE(sel(0, cons(z0, z1))) → c5(MARK(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c6(MARK(sel(z0, z2)), SEL(z0, z2))
MARK(from(z0)) → c7(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c8(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c9(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(head(z0)) → c10(ACTIVE(head(mark(z0))), HEAD(mark(z0)), MARK(z0))
MARK(2nd(z0)) → c11(ACTIVE(2nd(mark(z0))), 2ND(mark(z0)), MARK(z0))
MARK(take(z0, z1)) → c12(ACTIVE(take(mark(z0), mark(z1))), TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(sel(z0, z1)) → c15(ACTIVE(sel(mark(z0), mark(z1))), SEL(mark(z0), mark(z1)), MARK(z0), MARK(z1))
FROM(mark(z0)) → c16(FROM(z0))
FROM(active(z0)) → c17(FROM(z0))
CONS(mark(z0), z1) → c18(CONS(z0, z1))
CONS(z0, mark(z1)) → c19(CONS(z0, z1))
CONS(active(z0), z1) → c20(CONS(z0, z1))
CONS(z0, active(z1)) → c21(CONS(z0, z1))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
HEAD(mark(z0)) → c24(HEAD(z0))
HEAD(active(z0)) → c25(HEAD(z0))
2ND(mark(z0)) → c26(2ND(z0))
2ND(active(z0)) → c27(2ND(z0))
TAKE(mark(z0), z1) → c28(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c29(TAKE(z0, z1))
TAKE(active(z0), z1) → c30(TAKE(z0, z1))
TAKE(z0, active(z1)) → c31(TAKE(z0, z1))
SEL(mark(z0), z1) → c32(SEL(z0, z1))
SEL(z0, mark(z1)) → c33(SEL(z0, z1))
SEL(active(z0), z1) → c34(SEL(z0, z1))
SEL(z0, active(z1)) → c35(SEL(z0, z1))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(head(cons(z0, z1))) → mark(z0)
active(2nd(cons(z0, z1))) → mark(head(z1))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(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(head(z0)) → active(head(mark(z0)))
mark(2nd(z0)) → active(2nd(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(sel(z0, z1)) → active(sel(mark(z0), mark(z1)))
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)
head(mark(z0)) → head(z0)
head(active(z0)) → head(z0)
2nd(mark(z0)) → 2nd(z0)
2nd(active(z0)) → 2nd(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)
sel(mark(z0), z1) → sel(z0, z1)
sel(z0, mark(z1)) → sel(z0, z1)
sel(active(z0), z1) → sel(z0, z1)
sel(z0, active(z1)) → sel(z0, z1)
Tuples:
MARK(0) → c13(ACTIVE(0))
MARK(nil) → c14(ACTIVE(nil))
S tuples:
MARK(0) → c13(ACTIVE(0))
MARK(nil) → c14(ACTIVE(nil))
K tuples:none
Defined Rule Symbols:
active, mark, from, cons, s, head, 2nd, take, sel
Defined Pair Symbols:
MARK
Compound Symbols:
c13, c14
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
MARK(0) → c13(ACTIVE(0))
MARK(nil) → c14(ACTIVE(nil))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(head(cons(z0, z1))) → mark(z0)
active(2nd(cons(z0, z1))) → mark(head(z1))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(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(head(z0)) → active(head(mark(z0)))
mark(2nd(z0)) → active(2nd(mark(z0)))
mark(take(z0, z1)) → active(take(mark(z0), mark(z1)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(sel(z0, z1)) → active(sel(mark(z0), mark(z1)))
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)
head(mark(z0)) → head(z0)
head(active(z0)) → head(z0)
2nd(mark(z0)) → 2nd(z0)
2nd(active(z0)) → 2nd(z0)
take(mark(z0), z1) → take(z0, z1)
take(z0, mark(z1)) → take(z0, z1)
take(active(z0), z1) → take(z0, z1)
take(z0, active(z1)) → take(z0, z1)
sel(mark(z0), z1) → sel(z0, z1)
sel(z0, mark(z1)) → sel(z0, z1)
sel(active(z0), z1) → sel(z0, z1)
sel(z0, active(z1)) → sel(z0, z1)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, from, cons, s, head, 2nd, take, sel
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))