(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
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)
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)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(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(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(minus(z0, 0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(quot(0, s(z0))) → mark(0)
active(quot(s(z0), s(z1))) → mark(s(quot(minus(z0, z1), s(z1))))
active(zWquot(z0, nil)) → mark(nil)
active(zWquot(nil, z0)) → mark(nil)
active(zWquot(cons(z0, z1), cons(z2, z3))) → mark(cons(quot(z0, z2), zWquot(z1, z3)))
mark(from(z0)) → active(from(mark(z0)))
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(sel(z0, z1)) → active(sel(mark(z0), mark(z1)))
mark(0) → active(0)
mark(minus(z0, z1)) → active(minus(mark(z0), mark(z1)))
mark(quot(z0, z1)) → active(quot(mark(z0), mark(z1)))
mark(zWquot(z0, z1)) → active(zWquot(mark(z0), mark(z1)))
mark(nil) → active(nil)
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)
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)
minus(mark(z0), z1) → minus(z0, z1)
minus(z0, mark(z1)) → minus(z0, z1)
minus(active(z0), z1) → minus(z0, z1)
minus(z0, active(z1)) → minus(z0, z1)
quot(mark(z0), z1) → quot(z0, z1)
quot(z0, mark(z1)) → quot(z0, z1)
quot(active(z0), z1) → quot(z0, z1)
quot(z0, active(z1)) → quot(z0, z1)
zWquot(mark(z0), z1) → zWquot(z0, z1)
zWquot(z0, mark(z1)) → zWquot(z0, z1)
zWquot(active(z0), z1) → zWquot(z0, z1)
zWquot(z0, active(z1)) → zWquot(z0, z1)
Tuples:
ACTIVE(from(z0)) → c(MARK(cons(z0, from(s(z0)))), CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(sel(0, cons(z0, z1))) → c1(MARK(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c2(MARK(sel(z0, z2)), SEL(z0, z2))
ACTIVE(minus(z0, 0)) → c3(MARK(0))
ACTIVE(minus(s(z0), s(z1))) → c4(MARK(minus(z0, z1)), MINUS(z0, z1))
ACTIVE(quot(0, s(z0))) → c5(MARK(0))
ACTIVE(quot(s(z0), s(z1))) → c6(MARK(s(quot(minus(z0, z1), s(z1)))), S(quot(minus(z0, z1), s(z1))), QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1), S(z1))
ACTIVE(zWquot(z0, nil)) → c7(MARK(nil))
ACTIVE(zWquot(nil, z0)) → c8(MARK(nil))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c9(MARK(cons(quot(z0, z2), zWquot(z1, z3))), CONS(quot(z0, z2), zWquot(z1, z3)), QUOT(z0, z2), ZWQUOT(z1, z3))
MARK(from(z0)) → c10(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c11(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c12(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(sel(z0, z1)) → c13(ACTIVE(sel(mark(z0), mark(z1))), SEL(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(0) → c14(ACTIVE(0))
MARK(minus(z0, z1)) → c15(ACTIVE(minus(mark(z0), mark(z1))), MINUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(quot(z0, z1)) → c16(ACTIVE(quot(mark(z0), mark(z1))), QUOT(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(zWquot(z0, z1)) → c17(ACTIVE(zWquot(mark(z0), mark(z1))), ZWQUOT(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(nil) → c18(ACTIVE(nil))
FROM(mark(z0)) → c19(FROM(z0))
FROM(active(z0)) → c20(FROM(z0))
CONS(mark(z0), z1) → c21(CONS(z0, z1))
CONS(z0, mark(z1)) → c22(CONS(z0, z1))
CONS(active(z0), z1) → c23(CONS(z0, z1))
CONS(z0, active(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(active(z0), z1) → c29(SEL(z0, z1))
SEL(z0, active(z1)) → c30(SEL(z0, z1))
MINUS(mark(z0), z1) → c31(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c32(MINUS(z0, z1))
MINUS(active(z0), z1) → c33(MINUS(z0, z1))
MINUS(z0, active(z1)) → c34(MINUS(z0, z1))
QUOT(mark(z0), z1) → c35(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c36(QUOT(z0, z1))
QUOT(active(z0), z1) → c37(QUOT(z0, z1))
QUOT(z0, active(z1)) → c38(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c39(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c40(ZWQUOT(z0, z1))
ZWQUOT(active(z0), z1) → c41(ZWQUOT(z0, z1))
ZWQUOT(z0, active(z1)) → c42(ZWQUOT(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(sel(0, cons(z0, z1))) → c1(MARK(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c2(MARK(sel(z0, z2)), SEL(z0, z2))
ACTIVE(minus(z0, 0)) → c3(MARK(0))
ACTIVE(minus(s(z0), s(z1))) → c4(MARK(minus(z0, z1)), MINUS(z0, z1))
ACTIVE(quot(0, s(z0))) → c5(MARK(0))
ACTIVE(quot(s(z0), s(z1))) → c6(MARK(s(quot(minus(z0, z1), s(z1)))), S(quot(minus(z0, z1), s(z1))), QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1), S(z1))
ACTIVE(zWquot(z0, nil)) → c7(MARK(nil))
ACTIVE(zWquot(nil, z0)) → c8(MARK(nil))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c9(MARK(cons(quot(z0, z2), zWquot(z1, z3))), CONS(quot(z0, z2), zWquot(z1, z3)), QUOT(z0, z2), ZWQUOT(z1, z3))
MARK(from(z0)) → c10(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c11(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c12(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(sel(z0, z1)) → c13(ACTIVE(sel(mark(z0), mark(z1))), SEL(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(0) → c14(ACTIVE(0))
MARK(minus(z0, z1)) → c15(ACTIVE(minus(mark(z0), mark(z1))), MINUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(quot(z0, z1)) → c16(ACTIVE(quot(mark(z0), mark(z1))), QUOT(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(zWquot(z0, z1)) → c17(ACTIVE(zWquot(mark(z0), mark(z1))), ZWQUOT(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(nil) → c18(ACTIVE(nil))
FROM(mark(z0)) → c19(FROM(z0))
FROM(active(z0)) → c20(FROM(z0))
CONS(mark(z0), z1) → c21(CONS(z0, z1))
CONS(z0, mark(z1)) → c22(CONS(z0, z1))
CONS(active(z0), z1) → c23(CONS(z0, z1))
CONS(z0, active(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(active(z0), z1) → c29(SEL(z0, z1))
SEL(z0, active(z1)) → c30(SEL(z0, z1))
MINUS(mark(z0), z1) → c31(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c32(MINUS(z0, z1))
MINUS(active(z0), z1) → c33(MINUS(z0, z1))
MINUS(z0, active(z1)) → c34(MINUS(z0, z1))
QUOT(mark(z0), z1) → c35(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c36(QUOT(z0, z1))
QUOT(active(z0), z1) → c37(QUOT(z0, z1))
QUOT(z0, active(z1)) → c38(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c39(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c40(ZWQUOT(z0, z1))
ZWQUOT(active(z0), z1) → c41(ZWQUOT(z0, z1))
ZWQUOT(z0, active(z1)) → c42(ZWQUOT(z0, z1))
K tuples:none
Defined Rule Symbols:
active, mark, from, cons, s, sel, minus, quot, zWquot
Defined Pair Symbols:
ACTIVE, MARK, FROM, CONS, S, SEL, MINUS, QUOT, ZWQUOT
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
(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(sel(0, cons(z0, z1))) → c1(MARK(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c2(MARK(sel(z0, z2)), SEL(z0, z2))
ACTIVE(minus(z0, 0)) → c3(MARK(0))
ACTIVE(minus(s(z0), s(z1))) → c4(MARK(minus(z0, z1)), MINUS(z0, z1))
ACTIVE(quot(0, s(z0))) → c5(MARK(0))
ACTIVE(quot(s(z0), s(z1))) → c6(MARK(s(quot(minus(z0, z1), s(z1)))), S(quot(minus(z0, z1), s(z1))), QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1), S(z1))
ACTIVE(zWquot(z0, nil)) → c7(MARK(nil))
ACTIVE(zWquot(nil, z0)) → c8(MARK(nil))
ACTIVE(zWquot(cons(z0, z1), cons(z2, z3))) → c9(MARK(cons(quot(z0, z2), zWquot(z1, z3))), CONS(quot(z0, z2), zWquot(z1, z3)), QUOT(z0, z2), ZWQUOT(z1, z3))
MARK(from(z0)) → c10(ACTIVE(from(mark(z0))), FROM(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c11(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c12(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(sel(z0, z1)) → c13(ACTIVE(sel(mark(z0), mark(z1))), SEL(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(minus(z0, z1)) → c15(ACTIVE(minus(mark(z0), mark(z1))), MINUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(quot(z0, z1)) → c16(ACTIVE(quot(mark(z0), mark(z1))), QUOT(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(zWquot(z0, z1)) → c17(ACTIVE(zWquot(mark(z0), mark(z1))), ZWQUOT(mark(z0), mark(z1)), MARK(z0), MARK(z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(active(z0)) → c20(FROM(z0))
CONS(mark(z0), z1) → c21(CONS(z0, z1))
CONS(z0, mark(z1)) → c22(CONS(z0, z1))
CONS(active(z0), z1) → c23(CONS(z0, z1))
CONS(z0, active(z1)) → c24(CONS(z0, z1))
S(mark(z0)) → c25(S(z0))
S(active(z0)) → c26(S(z0))
SEL(mark(z0), z1) → c27(SEL(z0, z1))
SEL(z0, mark(z1)) → c28(SEL(z0, z1))
SEL(active(z0), z1) → c29(SEL(z0, z1))
SEL(z0, active(z1)) → c30(SEL(z0, z1))
MINUS(mark(z0), z1) → c31(MINUS(z0, z1))
MINUS(z0, mark(z1)) → c32(MINUS(z0, z1))
MINUS(active(z0), z1) → c33(MINUS(z0, z1))
MINUS(z0, active(z1)) → c34(MINUS(z0, z1))
QUOT(mark(z0), z1) → c35(QUOT(z0, z1))
QUOT(z0, mark(z1)) → c36(QUOT(z0, z1))
QUOT(active(z0), z1) → c37(QUOT(z0, z1))
QUOT(z0, active(z1)) → c38(QUOT(z0, z1))
ZWQUOT(mark(z0), z1) → c39(ZWQUOT(z0, z1))
ZWQUOT(z0, mark(z1)) → c40(ZWQUOT(z0, z1))
ZWQUOT(active(z0), z1) → c41(ZWQUOT(z0, z1))
ZWQUOT(z0, active(z1)) → c42(ZWQUOT(z0, z1))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(minus(z0, 0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(quot(0, s(z0))) → mark(0)
active(quot(s(z0), s(z1))) → mark(s(quot(minus(z0, z1), s(z1))))
active(zWquot(z0, nil)) → mark(nil)
active(zWquot(nil, z0)) → mark(nil)
active(zWquot(cons(z0, z1), cons(z2, z3))) → mark(cons(quot(z0, z2), zWquot(z1, z3)))
mark(from(z0)) → active(from(mark(z0)))
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(sel(z0, z1)) → active(sel(mark(z0), mark(z1)))
mark(0) → active(0)
mark(minus(z0, z1)) → active(minus(mark(z0), mark(z1)))
mark(quot(z0, z1)) → active(quot(mark(z0), mark(z1)))
mark(zWquot(z0, z1)) → active(zWquot(mark(z0), mark(z1)))
mark(nil) → active(nil)
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)
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)
minus(mark(z0), z1) → minus(z0, z1)
minus(z0, mark(z1)) → minus(z0, z1)
minus(active(z0), z1) → minus(z0, z1)
minus(z0, active(z1)) → minus(z0, z1)
quot(mark(z0), z1) → quot(z0, z1)
quot(z0, mark(z1)) → quot(z0, z1)
quot(active(z0), z1) → quot(z0, z1)
quot(z0, active(z1)) → quot(z0, z1)
zWquot(mark(z0), z1) → zWquot(z0, z1)
zWquot(z0, mark(z1)) → zWquot(z0, z1)
zWquot(active(z0), z1) → zWquot(z0, z1)
zWquot(z0, active(z1)) → zWquot(z0, z1)
Tuples:
MARK(0) → c14(ACTIVE(0))
MARK(nil) → c18(ACTIVE(nil))
S tuples:
MARK(0) → c14(ACTIVE(0))
MARK(nil) → c18(ACTIVE(nil))
K tuples:none
Defined Rule Symbols:
active, mark, from, cons, s, sel, minus, quot, zWquot
Defined Pair Symbols:
MARK
Compound Symbols:
c14, c18
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
MARK(0) → c14(ACTIVE(0))
MARK(nil) → c18(ACTIVE(nil))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(minus(z0, 0)) → mark(0)
active(minus(s(z0), s(z1))) → mark(minus(z0, z1))
active(quot(0, s(z0))) → mark(0)
active(quot(s(z0), s(z1))) → mark(s(quot(minus(z0, z1), s(z1))))
active(zWquot(z0, nil)) → mark(nil)
active(zWquot(nil, z0)) → mark(nil)
active(zWquot(cons(z0, z1), cons(z2, z3))) → mark(cons(quot(z0, z2), zWquot(z1, z3)))
mark(from(z0)) → active(from(mark(z0)))
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(sel(z0, z1)) → active(sel(mark(z0), mark(z1)))
mark(0) → active(0)
mark(minus(z0, z1)) → active(minus(mark(z0), mark(z1)))
mark(quot(z0, z1)) → active(quot(mark(z0), mark(z1)))
mark(zWquot(z0, z1)) → active(zWquot(mark(z0), mark(z1)))
mark(nil) → active(nil)
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)
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)
minus(mark(z0), z1) → minus(z0, z1)
minus(z0, mark(z1)) → minus(z0, z1)
minus(active(z0), z1) → minus(z0, z1)
minus(z0, active(z1)) → minus(z0, z1)
quot(mark(z0), z1) → quot(z0, z1)
quot(z0, mark(z1)) → quot(z0, z1)
quot(active(z0), z1) → quot(z0, z1)
quot(z0, active(z1)) → quot(z0, z1)
zWquot(mark(z0), z1) → zWquot(z0, z1)
zWquot(z0, mark(z1)) → zWquot(z0, z1)
zWquot(active(z0), z1) → zWquot(z0, z1)
zWquot(z0, active(z1)) → zWquot(z0, z1)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, from, cons, s, sel, minus, quot, zWquot
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))