(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(from(z0)) → c3(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
S tuples:
ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(from(z0)) → c3(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, PROPER, TOP
Compound Symbols:
c, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c22, c23
(3) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(from(z0)) → c3(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(from(z0)) → c3(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, PROPER, TOP
Compound Symbols:
c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c22, c23, c
(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
from(
z0)) →
c3(
FROM(
active(
z0)),
ACTIVE(
z0)) by
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(from(x0)) → c3
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(from(x0)) → c3
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(from(x0)) → c3
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, PROPER, TOP
Compound Symbols:
c2, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c22, c23, c, c3, c3
(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
ACTIVE(from(x0)) → c3
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, PROPER, TOP
Compound Symbols:
c2, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c22, c23, c, c3
(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
cons(
z0,
z1)) →
c4(
CONS(
active(
z0),
z1),
ACTIVE(
z0)) by
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(x0, x1)) → c4
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(x0, x1)) → c4
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(x0, x1)) → c4
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, PROPER, TOP
Compound Symbols:
c2, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c22, c23, c, c3, c4, c4
(11) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
ACTIVE(cons(x0, x1)) → c4
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, PROPER, TOP
Compound Symbols:
c2, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c22, c23, c, c3, c4
(13) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
s(
z0)) →
c5(
S(
active(
z0)),
ACTIVE(
z0)) by
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(s(x0)) → c5
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(s(x0)) → c5
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(s(x0)) → c5
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, PROPER, TOP
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c22, c23, c, c3, c4, c5, c5
(15) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
ACTIVE(s(x0)) → c5
(16) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, PROPER, TOP
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c22, c23, c, c3, c4, c5
(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
from(
z0)) →
c17(
FROM(
proper(
z0)),
PROPER(
z0)) by
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)), PROPER(0))
PROPER(from(x0)) → c17
(18) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)), PROPER(0))
PROPER(from(x0)) → c17
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)), PROPER(0))
PROPER(from(x0)) → c17
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, PROPER, TOP
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c18, c19, c20, c22, c23, c, c3, c4, c5, c17, c17
(19) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
PROPER(from(x0)) → c17
(20) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)), PROPER(0))
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)), PROPER(0))
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, PROPER, TOP
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c18, c19, c20, c22, c23, c, c3, c4, c5, c17
(21) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(22) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, PROPER, TOP
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c18, c19, c20, c22, c23, c, c3, c4, c5, c17, c17
(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
cons(
z0,
z1)) →
c18(
CONS(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(x0, x1)) → c18
(24) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(x0, x1)) → c18
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(x0, x1)) → c18
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, PROPER, TOP
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c19, c20, c22, c23, c, c3, c4, c5, c17, c17, c18, c18
(25) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
PROPER(cons(x0, x1)) → c18
(26) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, PROPER, TOP
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c19, c20, c22, c23, c, c3, c4, c5, c17, c17, c18
(27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(28) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, PROPER, TOP
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c19, c20, c22, c23, c, c3, c4, c5, c17, c17, c18, c18
(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
s(
z0)) →
c19(
S(
proper(
z0)),
PROPER(
z0)) by
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)), PROPER(0))
PROPER(s(x0)) → c19
(30) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)), PROPER(0))
PROPER(s(x0)) → c19
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)), PROPER(0))
PROPER(s(x0)) → c19
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, PROPER, TOP
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c20, c22, c23, c, c3, c4, c5, c17, c17, c18, c18, c19, c19
(31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
PROPER(s(x0)) → c19
(32) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)), PROPER(0))
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)), PROPER(0))
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, PROPER, TOP
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c20, c22, c23, c, c3, c4, c5, c17, c17, c18, c18, c19
(33) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(34) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, PROPER, TOP
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c20, c22, c23, c, c3, c4, c5, c17, c17, c18, c18, c19, c19
(35) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
after(
z0,
z1)) →
c20(
AFTER(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(after(x0, x1)) → c20
(36) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(after(x0, x1)) → c20
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(after(x0, x1)) → c20
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, TOP, PROPER
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c22, c23, c, c3, c4, c5, c17, c17, c18, c18, c19, c19, c20, c20
(37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
PROPER(after(x0, x1)) → c20
(38) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(0), PROPER(x1))
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(0), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, TOP, PROPER
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c22, c23, c, c3, c4, c5, c17, c17, c18, c18, c19, c19, c20
(39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(40) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, TOP, PROPER
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c22, c23, c, c3, c4, c5, c17, c17, c18, c18, c19, c19, c20, c20
(41) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
TOP(
mark(
z0)) →
c22(
TOP(
proper(
z0)),
PROPER(
z0)) by
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))
TOP(mark(x0)) → c22
(42) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))
TOP(mark(x0)) → c22
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))
TOP(mark(x0)) → c22
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, TOP, PROPER
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c23, c, c3, c4, c5, c17, c17, c18, c18, c19, c19, c20, c20, c22, c22
(43) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
TOP(mark(x0)) → c22
(44) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, TOP, PROPER
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c23, c, c3, c4, c5, c17, c17, c18, c18, c19, c19, c20, c20, c22
(45) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(46) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)))
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)))
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, TOP, PROPER
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c23, c, c3, c4, c5, c17, c17, c18, c18, c19, c19, c20, c20, c22, c22
(47) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
TOP(mark(0)) → c22(TOP(ok(0)))
We considered the (Usable) Rules:
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
And the Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(ACTIVE(x1)) = 0
POL(AFTER(x1, x2)) = 0
POL(CONS(x1, x2)) = 0
POL(FROM(x1)) = 0
POL(PROPER(x1)) = 0
POL(S(x1)) = 0
POL(TOP(x1)) = x1
POL(active(x1)) = x1
POL(after(x1, x2)) = [1]
POL(c(x1)) = x1
POL(c10(x1)) = x1
POL(c11(x1)) = x1
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c17(x1)) = x1
POL(c17(x1, x2)) = x1 + x2
POL(c18(x1, x2)) = x1 + x2
POL(c18(x1, x2, x3)) = x1 + x2 + x3
POL(c19(x1)) = x1
POL(c19(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c20(x1, x2)) = x1 + x2
POL(c20(x1, x2, x3)) = x1 + x2 + x3
POL(c22(x1)) = x1
POL(c22(x1, x2)) = x1 + x2
POL(c23(x1, x2)) = x1 + x2
POL(c3(x1, x2)) = x1 + x2
POL(c4(x1, x2)) = x1 + x2
POL(c5(x1, x2)) = x1 + x2
POL(c6(x1, x2)) = x1 + x2
POL(c7(x1, x2)) = x1 + x2
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(cons(x1, x2)) = [1]
POL(from(x1)) = [1]
POL(mark(x1)) = [1]
POL(ok(x1)) = x1
POL(proper(x1)) = 0
POL(s(x1)) = [1]
(48) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)))
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
K tuples:
TOP(mark(0)) → c22(TOP(ok(0)))
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, TOP, PROPER
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c23, c, c3, c4, c5, c17, c17, c18, c18, c19, c19, c20, c20, c22, c22
(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
TOP(
ok(
z0)) →
c23(
TOP(
active(
z0)),
ACTIVE(
z0)) by
TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(after(0, z0))) → c23(TOP(mark(z0)), ACTIVE(after(0, z0)))
TOP(ok(after(s(z0), cons(z1, z2)))) → c23(TOP(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(after(z0, z1))) → c23(TOP(after(active(z0), z1)), ACTIVE(after(z0, z1)))
TOP(ok(after(z0, z1))) → c23(TOP(after(z0, active(z1))), ACTIVE(after(z0, z1)))
TOP(ok(x0)) → c23
(50) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
TOP(mark(0)) → c22(TOP(ok(0)))
TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(after(0, z0))) → c23(TOP(mark(z0)), ACTIVE(after(0, z0)))
TOP(ok(after(s(z0), cons(z1, z2)))) → c23(TOP(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(after(z0, z1))) → c23(TOP(after(active(z0), z1)), ACTIVE(after(z0, z1)))
TOP(ok(after(z0, z1))) → c23(TOP(after(z0, active(z1))), ACTIVE(after(z0, z1)))
TOP(ok(x0)) → c23
S tuples:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(after(0, z0))) → c23(TOP(mark(z0)), ACTIVE(after(0, z0)))
TOP(ok(after(s(z0), cons(z1, z2)))) → c23(TOP(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(after(z0, z1))) → c23(TOP(after(active(z0), z1)), ACTIVE(after(z0, z1)))
TOP(ok(after(z0, z1))) → c23(TOP(after(z0, active(z1))), ACTIVE(after(z0, z1)))
TOP(ok(x0)) → c23
K tuples:
TOP(mark(0)) → c22(TOP(ok(0)))
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
ACTIVE, FROM, CONS, S, AFTER, PROPER, TOP
Compound Symbols:
c2, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c, c3, c4, c5, c17, c17, c18, c18, c19, c19, c20, c20, c22, c22, c23, c23
(51) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2))
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0))
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0)))
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1)))
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0)))
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1)))
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(from(0)) → c17(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
PROPER(s(0)) → c19(S(ok(0)))
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1)))
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1))
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0))
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(after(0, z0))) → c23(TOP(mark(z0)), ACTIVE(after(0, z0)))
TOP(ok(after(s(z0), cons(z1, z2)))) → c23(TOP(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2))))
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(after(z0, z1))) → c23(TOP(after(active(z0), z1)), ACTIVE(after(z0, z1)))
TOP(ok(after(z0, z1))) → c23(TOP(after(z0, active(z1))), ACTIVE(after(z0, z1)))
(52) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
TOP(mark(0)) → c22(TOP(ok(0)))
TOP(ok(x0)) → c23
S tuples:
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
TOP(ok(x0)) → c23
K tuples:
TOP(mark(0)) → c22(TOP(ok(0)))
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
FROM, CONS, S, AFTER, TOP
Compound Symbols:
c8, c9, c10, c11, c12, c13, c14, c15, c16, c22, c23
(53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
TOP(ok(x0)) → c23
TOP(mark(0)) → c22(TOP(ok(0)))
(54) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
S tuples:
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
K tuples:none
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
FROM, CONS, S, AFTER
Compound Symbols:
c8, c9, c10, c11, c12, c13, c14, c15, c16
(55) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
CONS(mark(z0), z1) → c10(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(AFTER(x1, x2)) = [2]x1 + [4]x2
POL(CONS(x1, x2)) = [2]x1 + [3]x2
POL(FROM(x1)) = 0
POL(S(x1)) = [3]x1
POL(c10(x1)) = x1
POL(c11(x1)) = x1
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(mark(x1)) = [1] + x1
POL(ok(x1)) = x1
(56) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
S tuples:
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(ok(z0)) → c13(S(z0))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
K tuples:
CONS(mark(z0), z1) → c10(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
FROM, CONS, S, AFTER
Compound Symbols:
c8, c9, c10, c11, c12, c13, c14, c15, c16
(57) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(ok(z0)) → c13(S(z0))
We considered the (Usable) Rules:none
And the Tuples:
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(AFTER(x1, x2)) = 0
POL(CONS(x1, x2)) = [5]x1 + [3]x2
POL(FROM(x1)) = [2]x1
POL(S(x1)) = [5]x1
POL(c10(x1)) = x1
POL(c11(x1)) = x1
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(mark(x1)) = [4] + x1
POL(ok(x1)) = [2] + x1
(58) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
S tuples:
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
K tuples:
CONS(mark(z0), z1) → c10(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(ok(z0)) → c13(S(z0))
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
FROM, CONS, S, AFTER
Compound Symbols:
c8, c9, c10, c11, c12, c13, c14, c15, c16
(59) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(AFTER(x1, x2)) = x2
POL(CONS(x1, x2)) = x1·x2
POL(FROM(x1)) = [2]x1 + x12
POL(S(x1)) = 0
POL(c10(x1)) = x1
POL(c11(x1)) = x1
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c14(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(mark(x1)) = [3] + x1
POL(ok(x1)) = [1] + x1
(60) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(after(0, z0)) → mark(z0)
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
active(after(z0, z1)) → after(active(z0), z1)
active(after(z0, z1)) → after(z0, active(z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
after(mark(z0), z1) → mark(after(z0, z1))
after(z0, mark(z1)) → mark(after(z0, z1))
after(ok(z0), ok(z1)) → ok(after(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(after(z0, z1)) → after(proper(z0), proper(z1))
proper(0) → ok(0)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
S tuples:none
K tuples:
CONS(mark(z0), z1) → c10(CONS(z0, z1))
S(mark(z0)) → c12(S(z0))
AFTER(mark(z0), z1) → c14(AFTER(z0, z1))
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1))
FROM(mark(z0)) → c8(FROM(z0))
FROM(ok(z0)) → c9(FROM(z0))
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
S(ok(z0)) → c13(S(z0))
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
Defined Rule Symbols:
active, from, cons, s, after, proper, top
Defined Pair Symbols:
FROM, CONS, S, AFTER
Compound Symbols:
c8, c9, c10, c11, c12, c13, c14, c15, c16
(61) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(62) BOUNDS(O(1), O(1))