(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(X)) → active(dbl(mark(X)))
mark(0) → active(0)
mark(s(X)) → active(s(X))
mark(dbls(X)) → active(dbls(mark(X)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(X1, X2))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(indx(X1, X2)) → active(indx(mark(X1), X2))
mark(from(X)) → active(from(X))
dbl(mark(X)) → dbl(X)
dbl(active(X)) → dbl(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
dbls(mark(X)) → dbls(X)
dbls(active(X)) → dbls(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)
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)
indx(mark(X1), X2) → indx(X1, X2)
indx(X1, mark(X2)) → indx(X1, X2)
indx(active(X1), X2) → indx(X1, X2)
indx(X1, active(X2)) → indx(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
mark(dbl(z0)) → active(dbl(mark(z0)))
mark(0) → active(0)
mark(s(z0)) → active(s(z0))
mark(dbls(z0)) → active(dbls(mark(z0)))
mark(nil) → active(nil)
mark(cons(z0, z1)) → active(cons(z0, z1))
mark(sel(z0, z1)) → active(sel(mark(z0), mark(z1)))
mark(indx(z0, z1)) → active(indx(mark(z0), z1))
mark(from(z0)) → active(from(z0))
dbl(mark(z0)) → dbl(z0)
dbl(active(z0)) → dbl(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
dbls(mark(z0)) → dbls(z0)
dbls(active(z0)) → dbls(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)
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)
indx(mark(z0), z1) → indx(z0, z1)
indx(z0, mark(z1)) → indx(z0, z1)
indx(active(z0), z1) → indx(z0, z1)
indx(z0, active(z1)) → indx(z0, z1)
from(mark(z0)) → from(z0)
from(active(z0)) → from(z0)
Tuples:
ACTIVE(dbl(0)) → c(MARK(0))
ACTIVE(dbl(s(z0))) → c1(MARK(s(s(dbl(z0)))), S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(dbls(nil)) → c2(MARK(nil))
ACTIVE(dbls(cons(z0, z1))) → c3(MARK(cons(dbl(z0), dbls(z1))), CONS(dbl(z0), dbls(z1)), DBL(z0), DBLS(z1))
ACTIVE(sel(0, cons(z0, z1))) → c4(MARK(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(MARK(sel(z0, z2)), SEL(z0, z2))
ACTIVE(indx(nil, z0)) → c6(MARK(nil))
ACTIVE(indx(cons(z0, z1), z2)) → c7(MARK(cons(sel(z0, z2), indx(z1, z2))), CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(from(z0)) → c8(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
MARK(dbl(z0)) → c9(ACTIVE(dbl(mark(z0))), DBL(mark(z0)), MARK(z0))
MARK(0) → c10(ACTIVE(0))
MARK(s(z0)) → c11(ACTIVE(s(z0)), S(z0))
MARK(dbls(z0)) → c12(ACTIVE(dbls(mark(z0))), DBLS(mark(z0)), MARK(z0))
MARK(nil) → c13(ACTIVE(nil))
MARK(cons(z0, z1)) → c14(ACTIVE(cons(z0, z1)), CONS(z0, z1))
MARK(sel(z0, z1)) → c15(ACTIVE(sel(mark(z0), mark(z1))), SEL(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(indx(z0, z1)) → c16(ACTIVE(indx(mark(z0), z1)), INDX(mark(z0), z1), MARK(z0))
MARK(from(z0)) → c17(ACTIVE(from(z0)), FROM(z0))
DBL(mark(z0)) → c18(DBL(z0))
DBL(active(z0)) → c19(DBL(z0))
S(mark(z0)) → c20(S(z0))
S(active(z0)) → c21(S(z0))
DBLS(mark(z0)) → c22(DBLS(z0))
DBLS(active(z0)) → c23(DBLS(z0))
CONS(mark(z0), z1) → c24(CONS(z0, z1))
CONS(z0, mark(z1)) → c25(CONS(z0, z1))
CONS(active(z0), z1) → c26(CONS(z0, z1))
CONS(z0, active(z1)) → c27(CONS(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SEL(z0, mark(z1)) → c29(SEL(z0, z1))
SEL(active(z0), z1) → c30(SEL(z0, z1))
SEL(z0, active(z1)) → c31(SEL(z0, z1))
INDX(mark(z0), z1) → c32(INDX(z0, z1))
INDX(z0, mark(z1)) → c33(INDX(z0, z1))
INDX(active(z0), z1) → c34(INDX(z0, z1))
INDX(z0, active(z1)) → c35(INDX(z0, z1))
FROM(mark(z0)) → c36(FROM(z0))
FROM(active(z0)) → c37(FROM(z0))
S tuples:
ACTIVE(dbl(0)) → c(MARK(0))
ACTIVE(dbl(s(z0))) → c1(MARK(s(s(dbl(z0)))), S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(dbls(nil)) → c2(MARK(nil))
ACTIVE(dbls(cons(z0, z1))) → c3(MARK(cons(dbl(z0), dbls(z1))), CONS(dbl(z0), dbls(z1)), DBL(z0), DBLS(z1))
ACTIVE(sel(0, cons(z0, z1))) → c4(MARK(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(MARK(sel(z0, z2)), SEL(z0, z2))
ACTIVE(indx(nil, z0)) → c6(MARK(nil))
ACTIVE(indx(cons(z0, z1), z2)) → c7(MARK(cons(sel(z0, z2), indx(z1, z2))), CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(from(z0)) → c8(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
MARK(dbl(z0)) → c9(ACTIVE(dbl(mark(z0))), DBL(mark(z0)), MARK(z0))
MARK(0) → c10(ACTIVE(0))
MARK(s(z0)) → c11(ACTIVE(s(z0)), S(z0))
MARK(dbls(z0)) → c12(ACTIVE(dbls(mark(z0))), DBLS(mark(z0)), MARK(z0))
MARK(nil) → c13(ACTIVE(nil))
MARK(cons(z0, z1)) → c14(ACTIVE(cons(z0, z1)), CONS(z0, z1))
MARK(sel(z0, z1)) → c15(ACTIVE(sel(mark(z0), mark(z1))), SEL(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(indx(z0, z1)) → c16(ACTIVE(indx(mark(z0), z1)), INDX(mark(z0), z1), MARK(z0))
MARK(from(z0)) → c17(ACTIVE(from(z0)), FROM(z0))
DBL(mark(z0)) → c18(DBL(z0))
DBL(active(z0)) → c19(DBL(z0))
S(mark(z0)) → c20(S(z0))
S(active(z0)) → c21(S(z0))
DBLS(mark(z0)) → c22(DBLS(z0))
DBLS(active(z0)) → c23(DBLS(z0))
CONS(mark(z0), z1) → c24(CONS(z0, z1))
CONS(z0, mark(z1)) → c25(CONS(z0, z1))
CONS(active(z0), z1) → c26(CONS(z0, z1))
CONS(z0, active(z1)) → c27(CONS(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SEL(z0, mark(z1)) → c29(SEL(z0, z1))
SEL(active(z0), z1) → c30(SEL(z0, z1))
SEL(z0, active(z1)) → c31(SEL(z0, z1))
INDX(mark(z0), z1) → c32(INDX(z0, z1))
INDX(z0, mark(z1)) → c33(INDX(z0, z1))
INDX(active(z0), z1) → c34(INDX(z0, z1))
INDX(z0, active(z1)) → c35(INDX(z0, z1))
FROM(mark(z0)) → c36(FROM(z0))
FROM(active(z0)) → c37(FROM(z0))
K tuples:none
Defined Rule Symbols:
active, mark, dbl, s, dbls, cons, sel, indx, from
Defined Pair Symbols:
ACTIVE, MARK, DBL, S, DBLS, CONS, SEL, INDX, FROM
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, c36, c37
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(dbl(0)) → c(MARK(0))
ACTIVE(dbl(s(z0))) → c1(MARK(s(s(dbl(z0)))), S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(dbls(nil)) → c2(MARK(nil))
ACTIVE(dbls(cons(z0, z1))) → c3(MARK(cons(dbl(z0), dbls(z1))), CONS(dbl(z0), dbls(z1)), DBL(z0), DBLS(z1))
ACTIVE(sel(0, cons(z0, z1))) → c4(MARK(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c5(MARK(sel(z0, z2)), SEL(z0, z2))
ACTIVE(indx(nil, z0)) → c6(MARK(nil))
ACTIVE(indx(cons(z0, z1), z2)) → c7(MARK(cons(sel(z0, z2), indx(z1, z2))), CONS(sel(z0, z2), indx(z1, z2)), SEL(z0, z2), INDX(z1, z2))
ACTIVE(from(z0)) → c8(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
MARK(dbl(z0)) → c9(ACTIVE(dbl(mark(z0))), DBL(mark(z0)), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(z0)), S(z0))
MARK(dbls(z0)) → c12(ACTIVE(dbls(mark(z0))), DBLS(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c14(ACTIVE(cons(z0, z1)), CONS(z0, z1))
MARK(sel(z0, z1)) → c15(ACTIVE(sel(mark(z0), mark(z1))), SEL(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(indx(z0, z1)) → c16(ACTIVE(indx(mark(z0), z1)), INDX(mark(z0), z1), MARK(z0))
MARK(from(z0)) → c17(ACTIVE(from(z0)), FROM(z0))
DBL(mark(z0)) → c18(DBL(z0))
DBL(active(z0)) → c19(DBL(z0))
S(mark(z0)) → c20(S(z0))
S(active(z0)) → c21(S(z0))
DBLS(mark(z0)) → c22(DBLS(z0))
DBLS(active(z0)) → c23(DBLS(z0))
CONS(mark(z0), z1) → c24(CONS(z0, z1))
CONS(z0, mark(z1)) → c25(CONS(z0, z1))
CONS(active(z0), z1) → c26(CONS(z0, z1))
CONS(z0, active(z1)) → c27(CONS(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SEL(z0, mark(z1)) → c29(SEL(z0, z1))
SEL(active(z0), z1) → c30(SEL(z0, z1))
SEL(z0, active(z1)) → c31(SEL(z0, z1))
INDX(mark(z0), z1) → c32(INDX(z0, z1))
INDX(z0, mark(z1)) → c33(INDX(z0, z1))
INDX(active(z0), z1) → c34(INDX(z0, z1))
INDX(z0, active(z1)) → c35(INDX(z0, z1))
FROM(mark(z0)) → c36(FROM(z0))
FROM(active(z0)) → c37(FROM(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
mark(dbl(z0)) → active(dbl(mark(z0)))
mark(0) → active(0)
mark(s(z0)) → active(s(z0))
mark(dbls(z0)) → active(dbls(mark(z0)))
mark(nil) → active(nil)
mark(cons(z0, z1)) → active(cons(z0, z1))
mark(sel(z0, z1)) → active(sel(mark(z0), mark(z1)))
mark(indx(z0, z1)) → active(indx(mark(z0), z1))
mark(from(z0)) → active(from(z0))
dbl(mark(z0)) → dbl(z0)
dbl(active(z0)) → dbl(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
dbls(mark(z0)) → dbls(z0)
dbls(active(z0)) → dbls(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)
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)
indx(mark(z0), z1) → indx(z0, z1)
indx(z0, mark(z1)) → indx(z0, z1)
indx(active(z0), z1) → indx(z0, z1)
indx(z0, active(z1)) → indx(z0, z1)
from(mark(z0)) → from(z0)
from(active(z0)) → from(z0)
Tuples:
MARK(0) → c10(ACTIVE(0))
MARK(nil) → c13(ACTIVE(nil))
S tuples:
MARK(0) → c10(ACTIVE(0))
MARK(nil) → c13(ACTIVE(nil))
K tuples:none
Defined Rule Symbols:
active, mark, dbl, s, dbls, cons, sel, indx, from
Defined Pair Symbols:
MARK
Compound Symbols:
c10, c13
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
MARK(nil) → c13(ACTIVE(nil))
MARK(0) → c10(ACTIVE(0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(z0, z1))) → mark(cons(dbl(z0), dbls(z1)))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(indx(nil, z0)) → mark(nil)
active(indx(cons(z0, z1), z2)) → mark(cons(sel(z0, z2), indx(z1, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
mark(dbl(z0)) → active(dbl(mark(z0)))
mark(0) → active(0)
mark(s(z0)) → active(s(z0))
mark(dbls(z0)) → active(dbls(mark(z0)))
mark(nil) → active(nil)
mark(cons(z0, z1)) → active(cons(z0, z1))
mark(sel(z0, z1)) → active(sel(mark(z0), mark(z1)))
mark(indx(z0, z1)) → active(indx(mark(z0), z1))
mark(from(z0)) → active(from(z0))
dbl(mark(z0)) → dbl(z0)
dbl(active(z0)) → dbl(z0)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
dbls(mark(z0)) → dbls(z0)
dbls(active(z0)) → dbls(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)
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)
indx(mark(z0), z1) → indx(z0, z1)
indx(z0, mark(z1)) → indx(z0, z1)
indx(active(z0), z1) → indx(z0, z1)
indx(z0, active(z1)) → indx(z0, z1)
from(mark(z0)) → from(z0)
from(active(z0)) → from(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, dbl, s, dbls, cons, sel, indx, from
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))