(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(from(X)) → active(from(mark(X)))
mark(sel1(X1, X2)) → active(sel1(mark(X1), mark(X2)))
mark(quote(X)) → active(quote(X))
mark(first1(X1, X2)) → active(first1(mark(X1), mark(X2)))
mark(nil1) → active(nil1)
mark(cons1(X1, X2)) → active(cons1(mark(X1), mark(X2)))
mark(01) → active(01)
mark(quote1(X)) → active(quote1(X))
mark(s1(X)) → active(s1(mark(X)))
mark(unquote(X)) → active(unquote(mark(X)))
mark(unquote1(X)) → active(unquote1(mark(X)))
mark(fcons(X1, X2)) → active(fcons(mark(X1), mark(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)
s(mark(X)) → s(X)
s(active(X)) → s(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)
first(mark(X1), X2) → first(X1, X2)
first(X1, mark(X2)) → first(X1, X2)
first(active(X1), X2) → first(X1, X2)
first(X1, active(X2)) → first(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
sel1(mark(X1), X2) → sel1(X1, X2)
sel1(X1, mark(X2)) → sel1(X1, X2)
sel1(active(X1), X2) → sel1(X1, X2)
sel1(X1, active(X2)) → sel1(X1, X2)
quote(mark(X)) → quote(X)
quote(active(X)) → quote(X)
first1(mark(X1), X2) → first1(X1, X2)
first1(X1, mark(X2)) → first1(X1, X2)
first1(active(X1), X2) → first1(X1, X2)
first1(X1, active(X2)) → first1(X1, X2)
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)
quote1(mark(X)) → quote1(X)
quote1(active(X)) → quote1(X)
s1(mark(X)) → s1(X)
s1(active(X)) → s1(X)
unquote(mark(X)) → unquote(X)
unquote(active(X)) → unquote(X)
unquote1(mark(X)) → unquote1(X)
unquote1(active(X)) → unquote1(X)
fcons(mark(X1), X2) → fcons(X1, X2)
fcons(X1, mark(X2)) → fcons(X1, X2)
fcons(active(X1), X2) → fcons(X1, X2)
fcons(X1, active(X2)) → fcons(X1, X2)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
mark(sel(z0, z1)) → active(sel(mark(z0), mark(z1)))
mark(s(z0)) → active(s(mark(z0)))
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(first(z0, z1)) → active(first(mark(z0), mark(z1)))
mark(nil) → active(nil)
mark(from(z0)) → active(from(mark(z0)))
mark(sel1(z0, z1)) → active(sel1(mark(z0), mark(z1)))
mark(quote(z0)) → active(quote(z0))
mark(first1(z0, z1)) → active(first1(mark(z0), mark(z1)))
mark(nil1) → active(nil1)
mark(cons1(z0, z1)) → active(cons1(mark(z0), mark(z1)))
mark(01) → active(01)
mark(quote1(z0)) → active(quote1(z0))
mark(s1(z0)) → active(s1(mark(z0)))
mark(unquote(z0)) → active(unquote(mark(z0)))
mark(unquote1(z0)) → active(unquote1(mark(z0)))
mark(fcons(z0, z1)) → active(fcons(mark(z0), mark(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)
s(mark(z0)) → s(z0)
s(active(z0)) → s(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)
first(mark(z0), z1) → first(z0, z1)
first(z0, mark(z1)) → first(z0, z1)
first(active(z0), z1) → first(z0, z1)
first(z0, active(z1)) → first(z0, z1)
from(mark(z0)) → from(z0)
from(active(z0)) → from(z0)
sel1(mark(z0), z1) → sel1(z0, z1)
sel1(z0, mark(z1)) → sel1(z0, z1)
sel1(active(z0), z1) → sel1(z0, z1)
sel1(z0, active(z1)) → sel1(z0, z1)
quote(mark(z0)) → quote(z0)
quote(active(z0)) → quote(z0)
first1(mark(z0), z1) → first1(z0, z1)
first1(z0, mark(z1)) → first1(z0, z1)
first1(active(z0), z1) → first1(z0, z1)
first1(z0, active(z1)) → first1(z0, z1)
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)
quote1(mark(z0)) → quote1(z0)
quote1(active(z0)) → quote1(z0)
s1(mark(z0)) → s1(z0)
s1(active(z0)) → s1(z0)
unquote(mark(z0)) → unquote(z0)
unquote(active(z0)) → unquote(z0)
unquote1(mark(z0)) → unquote1(z0)
unquote1(active(z0)) → unquote1(z0)
fcons(mark(z0), z1) → fcons(z0, z1)
fcons(z0, mark(z1)) → fcons(z0, z1)
fcons(active(z0), z1) → fcons(z0, z1)
fcons(z0, active(z1)) → fcons(z0, z1)
Tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c(MARK(sel(z0, z2)), SEL(z0, z2))
ACTIVE(sel(0, cons(z0, z1))) → c1(MARK(z0))
ACTIVE(first(0, z0)) → c2(MARK(nil))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(MARK(cons(z1, first(z0, z2))), CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(from(z0)) → c4(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(MARK(sel1(z0, z2)), SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(MARK(quote(z0)), QUOTE(z0))
ACTIVE(first1(0, z0)) → c7(MARK(nil1))
ACTIVE(first1(s(z0), cons(z1, z2))) → c8(MARK(cons1(quote(z1), first1(z0, z2))), CONS1(quote(z1), first1(z0, z2)), QUOTE(z1), FIRST1(z0, z2))
ACTIVE(quote(0)) → c9(MARK(01))
ACTIVE(quote1(cons(z0, z1))) → c10(MARK(cons1(quote(z0), quote1(z1))), CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote1(nil)) → c11(MARK(nil1))
ACTIVE(quote(s(z0))) → c12(MARK(s1(quote(z0))), S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(sel(z0, z1))) → c13(MARK(sel1(z0, z1)), SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(MARK(first1(z0, z1)), FIRST1(z0, z1))
ACTIVE(unquote(01)) → c15(MARK(0))
ACTIVE(unquote(s1(z0))) → c16(MARK(s(unquote(z0))), S(unquote(z0)), UNQUOTE(z0))
ACTIVE(unquote1(nil1)) → c17(MARK(nil))
ACTIVE(unquote1(cons1(z0, z1))) → c18(MARK(fcons(unquote(z0), unquote1(z1))), FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(MARK(cons(z0, z1)), CONS(z0, z1))
MARK(sel(z0, z1)) → c20(ACTIVE(sel(mark(z0), mark(z1))), SEL(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(s(z0)) → c21(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c22(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(0) → c23(ACTIVE(0))
MARK(first(z0, z1)) → c24(ACTIVE(first(mark(z0), mark(z1))), FIRST(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(nil) → c25(ACTIVE(nil))
MARK(from(z0)) → c26(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(sel1(z0, z1)) → c27(ACTIVE(sel1(mark(z0), mark(z1))), SEL1(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(quote(z0)) → c28(ACTIVE(quote(z0)), QUOTE(z0))
MARK(first1(z0, z1)) → c29(ACTIVE(first1(mark(z0), mark(z1))), FIRST1(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(nil1) → c30(ACTIVE(nil1))
MARK(cons1(z0, z1)) → c31(ACTIVE(cons1(mark(z0), mark(z1))), CONS1(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(01) → c32(ACTIVE(01))
MARK(quote1(z0)) → c33(ACTIVE(quote1(z0)), QUOTE1(z0))
MARK(s1(z0)) → c34(ACTIVE(s1(mark(z0))), S1(mark(z0)), MARK(z0))
MARK(unquote(z0)) → c35(ACTIVE(unquote(mark(z0))), UNQUOTE(mark(z0)), MARK(z0))
MARK(unquote1(z0)) → c36(ACTIVE(unquote1(mark(z0))), UNQUOTE1(mark(z0)), MARK(z0))
MARK(fcons(z0, z1)) → c37(ACTIVE(fcons(mark(z0), mark(z1))), FCONS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(active(z0), z1) → c40(SEL(z0, z1))
SEL(z0, active(z1)) → c41(SEL(z0, z1))
S(mark(z0)) → c42(S(z0))
S(active(z0)) → c43(S(z0))
CONS(mark(z0), z1) → c44(CONS(z0, z1))
CONS(z0, mark(z1)) → c45(CONS(z0, z1))
CONS(active(z0), z1) → c46(CONS(z0, z1))
CONS(z0, active(z1)) → c47(CONS(z0, z1))
FIRST(mark(z0), z1) → c48(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c49(FIRST(z0, z1))
FIRST(active(z0), z1) → c50(FIRST(z0, z1))
FIRST(z0, active(z1)) → c51(FIRST(z0, z1))
FROM(mark(z0)) → c52(FROM(z0))
FROM(active(z0)) → c53(FROM(z0))
SEL1(mark(z0), z1) → c54(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c55(SEL1(z0, z1))
SEL1(active(z0), z1) → c56(SEL1(z0, z1))
SEL1(z0, active(z1)) → c57(SEL1(z0, z1))
QUOTE(mark(z0)) → c58(QUOTE(z0))
QUOTE(active(z0)) → c59(QUOTE(z0))
FIRST1(mark(z0), z1) → c60(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c61(FIRST1(z0, z1))
FIRST1(active(z0), z1) → c62(FIRST1(z0, z1))
FIRST1(z0, active(z1)) → c63(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c64(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c65(CONS1(z0, z1))
CONS1(active(z0), z1) → c66(CONS1(z0, z1))
CONS1(z0, active(z1)) → c67(CONS1(z0, z1))
QUOTE1(mark(z0)) → c68(QUOTE1(z0))
QUOTE1(active(z0)) → c69(QUOTE1(z0))
S1(mark(z0)) → c70(S1(z0))
S1(active(z0)) → c71(S1(z0))
UNQUOTE(mark(z0)) → c72(UNQUOTE(z0))
UNQUOTE(active(z0)) → c73(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c74(UNQUOTE1(z0))
UNQUOTE1(active(z0)) → c75(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c76(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c77(FCONS(z0, z1))
FCONS(active(z0), z1) → c78(FCONS(z0, z1))
FCONS(z0, active(z1)) → c79(FCONS(z0, z1))
S tuples:
ACTIVE(sel(s(z0), cons(z1, z2))) → c(MARK(sel(z0, z2)), SEL(z0, z2))
ACTIVE(sel(0, cons(z0, z1))) → c1(MARK(z0))
ACTIVE(first(0, z0)) → c2(MARK(nil))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(MARK(cons(z1, first(z0, z2))), CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(from(z0)) → c4(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(MARK(sel1(z0, z2)), SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(MARK(quote(z0)), QUOTE(z0))
ACTIVE(first1(0, z0)) → c7(MARK(nil1))
ACTIVE(first1(s(z0), cons(z1, z2))) → c8(MARK(cons1(quote(z1), first1(z0, z2))), CONS1(quote(z1), first1(z0, z2)), QUOTE(z1), FIRST1(z0, z2))
ACTIVE(quote(0)) → c9(MARK(01))
ACTIVE(quote1(cons(z0, z1))) → c10(MARK(cons1(quote(z0), quote1(z1))), CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote1(nil)) → c11(MARK(nil1))
ACTIVE(quote(s(z0))) → c12(MARK(s1(quote(z0))), S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(sel(z0, z1))) → c13(MARK(sel1(z0, z1)), SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(MARK(first1(z0, z1)), FIRST1(z0, z1))
ACTIVE(unquote(01)) → c15(MARK(0))
ACTIVE(unquote(s1(z0))) → c16(MARK(s(unquote(z0))), S(unquote(z0)), UNQUOTE(z0))
ACTIVE(unquote1(nil1)) → c17(MARK(nil))
ACTIVE(unquote1(cons1(z0, z1))) → c18(MARK(fcons(unquote(z0), unquote1(z1))), FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(MARK(cons(z0, z1)), CONS(z0, z1))
MARK(sel(z0, z1)) → c20(ACTIVE(sel(mark(z0), mark(z1))), SEL(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(s(z0)) → c21(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c22(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(0) → c23(ACTIVE(0))
MARK(first(z0, z1)) → c24(ACTIVE(first(mark(z0), mark(z1))), FIRST(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(nil) → c25(ACTIVE(nil))
MARK(from(z0)) → c26(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(sel1(z0, z1)) → c27(ACTIVE(sel1(mark(z0), mark(z1))), SEL1(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(quote(z0)) → c28(ACTIVE(quote(z0)), QUOTE(z0))
MARK(first1(z0, z1)) → c29(ACTIVE(first1(mark(z0), mark(z1))), FIRST1(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(nil1) → c30(ACTIVE(nil1))
MARK(cons1(z0, z1)) → c31(ACTIVE(cons1(mark(z0), mark(z1))), CONS1(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(01) → c32(ACTIVE(01))
MARK(quote1(z0)) → c33(ACTIVE(quote1(z0)), QUOTE1(z0))
MARK(s1(z0)) → c34(ACTIVE(s1(mark(z0))), S1(mark(z0)), MARK(z0))
MARK(unquote(z0)) → c35(ACTIVE(unquote(mark(z0))), UNQUOTE(mark(z0)), MARK(z0))
MARK(unquote1(z0)) → c36(ACTIVE(unquote1(mark(z0))), UNQUOTE1(mark(z0)), MARK(z0))
MARK(fcons(z0, z1)) → c37(ACTIVE(fcons(mark(z0), mark(z1))), FCONS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(active(z0), z1) → c40(SEL(z0, z1))
SEL(z0, active(z1)) → c41(SEL(z0, z1))
S(mark(z0)) → c42(S(z0))
S(active(z0)) → c43(S(z0))
CONS(mark(z0), z1) → c44(CONS(z0, z1))
CONS(z0, mark(z1)) → c45(CONS(z0, z1))
CONS(active(z0), z1) → c46(CONS(z0, z1))
CONS(z0, active(z1)) → c47(CONS(z0, z1))
FIRST(mark(z0), z1) → c48(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c49(FIRST(z0, z1))
FIRST(active(z0), z1) → c50(FIRST(z0, z1))
FIRST(z0, active(z1)) → c51(FIRST(z0, z1))
FROM(mark(z0)) → c52(FROM(z0))
FROM(active(z0)) → c53(FROM(z0))
SEL1(mark(z0), z1) → c54(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c55(SEL1(z0, z1))
SEL1(active(z0), z1) → c56(SEL1(z0, z1))
SEL1(z0, active(z1)) → c57(SEL1(z0, z1))
QUOTE(mark(z0)) → c58(QUOTE(z0))
QUOTE(active(z0)) → c59(QUOTE(z0))
FIRST1(mark(z0), z1) → c60(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c61(FIRST1(z0, z1))
FIRST1(active(z0), z1) → c62(FIRST1(z0, z1))
FIRST1(z0, active(z1)) → c63(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c64(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c65(CONS1(z0, z1))
CONS1(active(z0), z1) → c66(CONS1(z0, z1))
CONS1(z0, active(z1)) → c67(CONS1(z0, z1))
QUOTE1(mark(z0)) → c68(QUOTE1(z0))
QUOTE1(active(z0)) → c69(QUOTE1(z0))
S1(mark(z0)) → c70(S1(z0))
S1(active(z0)) → c71(S1(z0))
UNQUOTE(mark(z0)) → c72(UNQUOTE(z0))
UNQUOTE(active(z0)) → c73(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c74(UNQUOTE1(z0))
UNQUOTE1(active(z0)) → c75(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c76(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c77(FCONS(z0, z1))
FCONS(active(z0), z1) → c78(FCONS(z0, z1))
FCONS(z0, active(z1)) → c79(FCONS(z0, z1))
K tuples:none
Defined Rule Symbols:
active, mark, sel, s, cons, first, from, sel1, quote, first1, cons1, quote1, s1, unquote, unquote1, fcons
Defined Pair Symbols:
ACTIVE, MARK, SEL, S, CONS, FIRST, FROM, SEL1, QUOTE, FIRST1, CONS1, QUOTE1, S1, UNQUOTE, UNQUOTE1, FCONS
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, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c78, c79
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(sel(s(z0), cons(z1, z2))) → c(MARK(sel(z0, z2)), SEL(z0, z2))
ACTIVE(sel(0, cons(z0, z1))) → c1(MARK(z0))
ACTIVE(first(0, z0)) → c2(MARK(nil))
ACTIVE(first(s(z0), cons(z1, z2))) → c3(MARK(cons(z1, first(z0, z2))), CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(from(z0)) → c4(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(sel1(s(z0), cons(z1, z2))) → c5(MARK(sel1(z0, z2)), SEL1(z0, z2))
ACTIVE(sel1(0, cons(z0, z1))) → c6(MARK(quote(z0)), QUOTE(z0))
ACTIVE(first1(0, z0)) → c7(MARK(nil1))
ACTIVE(first1(s(z0), cons(z1, z2))) → c8(MARK(cons1(quote(z1), first1(z0, z2))), CONS1(quote(z1), first1(z0, z2)), QUOTE(z1), FIRST1(z0, z2))
ACTIVE(quote(0)) → c9(MARK(01))
ACTIVE(quote1(cons(z0, z1))) → c10(MARK(cons1(quote(z0), quote1(z1))), CONS1(quote(z0), quote1(z1)), QUOTE(z0), QUOTE1(z1))
ACTIVE(quote1(nil)) → c11(MARK(nil1))
ACTIVE(quote(s(z0))) → c12(MARK(s1(quote(z0))), S1(quote(z0)), QUOTE(z0))
ACTIVE(quote(sel(z0, z1))) → c13(MARK(sel1(z0, z1)), SEL1(z0, z1))
ACTIVE(quote1(first(z0, z1))) → c14(MARK(first1(z0, z1)), FIRST1(z0, z1))
ACTIVE(unquote(01)) → c15(MARK(0))
ACTIVE(unquote(s1(z0))) → c16(MARK(s(unquote(z0))), S(unquote(z0)), UNQUOTE(z0))
ACTIVE(unquote1(nil1)) → c17(MARK(nil))
ACTIVE(unquote1(cons1(z0, z1))) → c18(MARK(fcons(unquote(z0), unquote1(z1))), FCONS(unquote(z0), unquote1(z1)), UNQUOTE(z0), UNQUOTE1(z1))
ACTIVE(fcons(z0, z1)) → c19(MARK(cons(z0, z1)), CONS(z0, z1))
MARK(sel(z0, z1)) → c20(ACTIVE(sel(mark(z0), mark(z1))), SEL(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(s(z0)) → c21(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c22(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(first(z0, z1)) → c24(ACTIVE(first(mark(z0), mark(z1))), FIRST(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(from(z0)) → c26(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(sel1(z0, z1)) → c27(ACTIVE(sel1(mark(z0), mark(z1))), SEL1(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(quote(z0)) → c28(ACTIVE(quote(z0)), QUOTE(z0))
MARK(first1(z0, z1)) → c29(ACTIVE(first1(mark(z0), mark(z1))), FIRST1(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(cons1(z0, z1)) → c31(ACTIVE(cons1(mark(z0), mark(z1))), CONS1(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(quote1(z0)) → c33(ACTIVE(quote1(z0)), QUOTE1(z0))
MARK(s1(z0)) → c34(ACTIVE(s1(mark(z0))), S1(mark(z0)), MARK(z0))
MARK(unquote(z0)) → c35(ACTIVE(unquote(mark(z0))), UNQUOTE(mark(z0)), MARK(z0))
MARK(unquote1(z0)) → c36(ACTIVE(unquote1(mark(z0))), UNQUOTE1(mark(z0)), MARK(z0))
MARK(fcons(z0, z1)) → c37(ACTIVE(fcons(mark(z0), mark(z1))), FCONS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
SEL(mark(z0), z1) → c38(SEL(z0, z1))
SEL(z0, mark(z1)) → c39(SEL(z0, z1))
SEL(active(z0), z1) → c40(SEL(z0, z1))
SEL(z0, active(z1)) → c41(SEL(z0, z1))
S(mark(z0)) → c42(S(z0))
S(active(z0)) → c43(S(z0))
CONS(mark(z0), z1) → c44(CONS(z0, z1))
CONS(z0, mark(z1)) → c45(CONS(z0, z1))
CONS(active(z0), z1) → c46(CONS(z0, z1))
CONS(z0, active(z1)) → c47(CONS(z0, z1))
FIRST(mark(z0), z1) → c48(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c49(FIRST(z0, z1))
FIRST(active(z0), z1) → c50(FIRST(z0, z1))
FIRST(z0, active(z1)) → c51(FIRST(z0, z1))
FROM(mark(z0)) → c52(FROM(z0))
FROM(active(z0)) → c53(FROM(z0))
SEL1(mark(z0), z1) → c54(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c55(SEL1(z0, z1))
SEL1(active(z0), z1) → c56(SEL1(z0, z1))
SEL1(z0, active(z1)) → c57(SEL1(z0, z1))
QUOTE(mark(z0)) → c58(QUOTE(z0))
QUOTE(active(z0)) → c59(QUOTE(z0))
FIRST1(mark(z0), z1) → c60(FIRST1(z0, z1))
FIRST1(z0, mark(z1)) → c61(FIRST1(z0, z1))
FIRST1(active(z0), z1) → c62(FIRST1(z0, z1))
FIRST1(z0, active(z1)) → c63(FIRST1(z0, z1))
CONS1(mark(z0), z1) → c64(CONS1(z0, z1))
CONS1(z0, mark(z1)) → c65(CONS1(z0, z1))
CONS1(active(z0), z1) → c66(CONS1(z0, z1))
CONS1(z0, active(z1)) → c67(CONS1(z0, z1))
QUOTE1(mark(z0)) → c68(QUOTE1(z0))
QUOTE1(active(z0)) → c69(QUOTE1(z0))
S1(mark(z0)) → c70(S1(z0))
S1(active(z0)) → c71(S1(z0))
UNQUOTE(mark(z0)) → c72(UNQUOTE(z0))
UNQUOTE(active(z0)) → c73(UNQUOTE(z0))
UNQUOTE1(mark(z0)) → c74(UNQUOTE1(z0))
UNQUOTE1(active(z0)) → c75(UNQUOTE1(z0))
FCONS(mark(z0), z1) → c76(FCONS(z0, z1))
FCONS(z0, mark(z1)) → c77(FCONS(z0, z1))
FCONS(active(z0), z1) → c78(FCONS(z0, z1))
FCONS(z0, active(z1)) → c79(FCONS(z0, z1))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
mark(sel(z0, z1)) → active(sel(mark(z0), mark(z1)))
mark(s(z0)) → active(s(mark(z0)))
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(first(z0, z1)) → active(first(mark(z0), mark(z1)))
mark(nil) → active(nil)
mark(from(z0)) → active(from(mark(z0)))
mark(sel1(z0, z1)) → active(sel1(mark(z0), mark(z1)))
mark(quote(z0)) → active(quote(z0))
mark(first1(z0, z1)) → active(first1(mark(z0), mark(z1)))
mark(nil1) → active(nil1)
mark(cons1(z0, z1)) → active(cons1(mark(z0), mark(z1)))
mark(01) → active(01)
mark(quote1(z0)) → active(quote1(z0))
mark(s1(z0)) → active(s1(mark(z0)))
mark(unquote(z0)) → active(unquote(mark(z0)))
mark(unquote1(z0)) → active(unquote1(mark(z0)))
mark(fcons(z0, z1)) → active(fcons(mark(z0), mark(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)
s(mark(z0)) → s(z0)
s(active(z0)) → s(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)
first(mark(z0), z1) → first(z0, z1)
first(z0, mark(z1)) → first(z0, z1)
first(active(z0), z1) → first(z0, z1)
first(z0, active(z1)) → first(z0, z1)
from(mark(z0)) → from(z0)
from(active(z0)) → from(z0)
sel1(mark(z0), z1) → sel1(z0, z1)
sel1(z0, mark(z1)) → sel1(z0, z1)
sel1(active(z0), z1) → sel1(z0, z1)
sel1(z0, active(z1)) → sel1(z0, z1)
quote(mark(z0)) → quote(z0)
quote(active(z0)) → quote(z0)
first1(mark(z0), z1) → first1(z0, z1)
first1(z0, mark(z1)) → first1(z0, z1)
first1(active(z0), z1) → first1(z0, z1)
first1(z0, active(z1)) → first1(z0, z1)
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)
quote1(mark(z0)) → quote1(z0)
quote1(active(z0)) → quote1(z0)
s1(mark(z0)) → s1(z0)
s1(active(z0)) → s1(z0)
unquote(mark(z0)) → unquote(z0)
unquote(active(z0)) → unquote(z0)
unquote1(mark(z0)) → unquote1(z0)
unquote1(active(z0)) → unquote1(z0)
fcons(mark(z0), z1) → fcons(z0, z1)
fcons(z0, mark(z1)) → fcons(z0, z1)
fcons(active(z0), z1) → fcons(z0, z1)
fcons(z0, active(z1)) → fcons(z0, z1)
Tuples:
MARK(0) → c23(ACTIVE(0))
MARK(nil) → c25(ACTIVE(nil))
MARK(nil1) → c30(ACTIVE(nil1))
MARK(01) → c32(ACTIVE(01))
S tuples:
MARK(0) → c23(ACTIVE(0))
MARK(nil) → c25(ACTIVE(nil))
MARK(nil1) → c30(ACTIVE(nil1))
MARK(01) → c32(ACTIVE(01))
K tuples:none
Defined Rule Symbols:
active, mark, sel, s, cons, first, from, sel1, quote, first1, cons1, quote1, s1, unquote, unquote1, fcons
Defined Pair Symbols:
MARK
Compound Symbols:
c23, c25, c30, c32
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing nodes:
MARK(nil) → c25(ACTIVE(nil))
MARK(01) → c32(ACTIVE(01))
MARK(nil1) → c30(ACTIVE(nil1))
MARK(0) → c23(ACTIVE(0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(sel(0, cons(z0, z1))) → mark(z0)
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel1(s(z0), cons(z1, z2))) → mark(sel1(z0, z2))
active(sel1(0, cons(z0, z1))) → mark(quote(z0))
active(first1(0, z0)) → mark(nil1)
active(first1(s(z0), cons(z1, z2))) → mark(cons1(quote(z1), first1(z0, z2)))
active(quote(0)) → mark(01)
active(quote1(cons(z0, z1))) → mark(cons1(quote(z0), quote1(z1)))
active(quote1(nil)) → mark(nil1)
active(quote(s(z0))) → mark(s1(quote(z0)))
active(quote(sel(z0, z1))) → mark(sel1(z0, z1))
active(quote1(first(z0, z1))) → mark(first1(z0, z1))
active(unquote(01)) → mark(0)
active(unquote(s1(z0))) → mark(s(unquote(z0)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(z0, z1))) → mark(fcons(unquote(z0), unquote1(z1)))
active(fcons(z0, z1)) → mark(cons(z0, z1))
mark(sel(z0, z1)) → active(sel(mark(z0), mark(z1)))
mark(s(z0)) → active(s(mark(z0)))
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(0) → active(0)
mark(first(z0, z1)) → active(first(mark(z0), mark(z1)))
mark(nil) → active(nil)
mark(from(z0)) → active(from(mark(z0)))
mark(sel1(z0, z1)) → active(sel1(mark(z0), mark(z1)))
mark(quote(z0)) → active(quote(z0))
mark(first1(z0, z1)) → active(first1(mark(z0), mark(z1)))
mark(nil1) → active(nil1)
mark(cons1(z0, z1)) → active(cons1(mark(z0), mark(z1)))
mark(01) → active(01)
mark(quote1(z0)) → active(quote1(z0))
mark(s1(z0)) → active(s1(mark(z0)))
mark(unquote(z0)) → active(unquote(mark(z0)))
mark(unquote1(z0)) → active(unquote1(mark(z0)))
mark(fcons(z0, z1)) → active(fcons(mark(z0), mark(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)
s(mark(z0)) → s(z0)
s(active(z0)) → s(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)
first(mark(z0), z1) → first(z0, z1)
first(z0, mark(z1)) → first(z0, z1)
first(active(z0), z1) → first(z0, z1)
first(z0, active(z1)) → first(z0, z1)
from(mark(z0)) → from(z0)
from(active(z0)) → from(z0)
sel1(mark(z0), z1) → sel1(z0, z1)
sel1(z0, mark(z1)) → sel1(z0, z1)
sel1(active(z0), z1) → sel1(z0, z1)
sel1(z0, active(z1)) → sel1(z0, z1)
quote(mark(z0)) → quote(z0)
quote(active(z0)) → quote(z0)
first1(mark(z0), z1) → first1(z0, z1)
first1(z0, mark(z1)) → first1(z0, z1)
first1(active(z0), z1) → first1(z0, z1)
first1(z0, active(z1)) → first1(z0, z1)
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)
quote1(mark(z0)) → quote1(z0)
quote1(active(z0)) → quote1(z0)
s1(mark(z0)) → s1(z0)
s1(active(z0)) → s1(z0)
unquote(mark(z0)) → unquote(z0)
unquote(active(z0)) → unquote(z0)
unquote1(mark(z0)) → unquote1(z0)
unquote1(active(z0)) → unquote1(z0)
fcons(mark(z0), z1) → fcons(z0, z1)
fcons(z0, mark(z1)) → fcons(z0, z1)
fcons(active(z0), z1) → fcons(z0, z1)
fcons(z0, active(z1)) → fcons(z0, z1)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, sel, s, cons, first, from, sel1, quote, first1, cons1, quote1, s1, unquote, unquote1, fcons
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))