YES(O(1), O(n^3)) 193.28/66.10 YES(O(1), O(n^3)) 193.28/66.15 193.28/66.15 193.28/66.15 193.28/66.15 193.28/66.15 193.28/66.15 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 193.28/66.15 193.28/66.15 193.28/66.15
193.28/66.15 193.28/66.15 193.28/66.15
193.28/66.15
193.28/66.15

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X)))) 193.28/66.15
active(first(0, Z)) → mark(nil) 193.28/66.15
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z))) 193.28/66.15
active(sel(0, cons(X, Z))) → mark(X) 193.28/66.15
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z)) 193.28/66.15
active(from(X)) → from(active(X)) 193.28/66.15
active(cons(X1, X2)) → cons(active(X1), X2) 193.28/66.15
active(s(X)) → s(active(X)) 193.28/66.15
active(first(X1, X2)) → first(active(X1), X2) 193.28/66.15
active(first(X1, X2)) → first(X1, active(X2)) 193.28/66.15
active(sel(X1, X2)) → sel(active(X1), X2) 193.28/66.15
active(sel(X1, X2)) → sel(X1, active(X2)) 193.28/66.15
from(mark(X)) → mark(from(X)) 193.28/66.15
cons(mark(X1), X2) → mark(cons(X1, X2)) 193.28/66.15
s(mark(X)) → mark(s(X)) 193.28/66.15
first(mark(X1), X2) → mark(first(X1, X2)) 193.28/66.15
first(X1, mark(X2)) → mark(first(X1, X2)) 193.28/66.15
sel(mark(X1), X2) → mark(sel(X1, X2)) 193.28/66.15
sel(X1, mark(X2)) → mark(sel(X1, X2)) 193.28/66.15
proper(from(X)) → from(proper(X)) 193.28/66.15
proper(cons(X1, X2)) → cons(proper(X1), proper(X2)) 193.28/66.15
proper(s(X)) → s(proper(X)) 193.28/66.15
proper(first(X1, X2)) → first(proper(X1), proper(X2)) 193.28/66.15
proper(0) → ok(0) 193.28/66.15
proper(nil) → ok(nil) 193.28/66.15
proper(sel(X1, X2)) → sel(proper(X1), proper(X2)) 193.28/66.15
from(ok(X)) → ok(from(X)) 193.28/66.15
cons(ok(X1), ok(X2)) → ok(cons(X1, X2)) 193.28/66.15
s(ok(X)) → ok(s(X)) 193.28/66.15
first(ok(X1), ok(X2)) → ok(first(X1, X2)) 193.28/66.15
sel(ok(X1), ok(X2)) → ok(sel(X1, X2)) 193.28/66.15
top(mark(X)) → top(proper(X)) 193.28/66.15
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST
193.28/66.15
193.28/66.15

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
193.28/66.15
193.28/66.15

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.28/66.15
active(first(0, z0)) → mark(nil) 193.28/66.15
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.28/66.15
active(sel(0, cons(z0, z1))) → mark(z0) 193.28/66.15
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.28/66.15
active(from(z0)) → from(active(z0)) 193.28/66.15
active(cons(z0, z1)) → cons(active(z0), z1) 193.28/66.15
active(s(z0)) → s(active(z0)) 193.28/66.15
active(first(z0, z1)) → first(active(z0), z1) 193.28/66.15
active(first(z0, z1)) → first(z0, active(z1)) 193.28/66.15
active(sel(z0, z1)) → sel(active(z0), z1) 193.28/66.15
active(sel(z0, z1)) → sel(z0, active(z1)) 193.28/66.15
from(mark(z0)) → mark(from(z0)) 193.28/66.15
from(ok(z0)) → ok(from(z0)) 193.28/66.15
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.28/66.15
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.28/66.15
s(mark(z0)) → mark(s(z0)) 193.28/66.15
s(ok(z0)) → ok(s(z0)) 193.28/66.15
first(mark(z0), z1) → mark(first(z0, z1)) 193.28/66.15
first(z0, mark(z1)) → mark(first(z0, z1)) 193.28/66.15
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.28/66.15
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.28/66.15
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.28/66.15
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.28/66.15
proper(from(z0)) → from(proper(z0)) 193.28/66.15
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.28/66.15
proper(s(z0)) → s(proper(z0)) 193.28/66.15
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.28/66.15
proper(0) → ok(0) 193.28/66.15
proper(nil) → ok(nil) 193.28/66.15
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.28/66.15
top(mark(z0)) → top(proper(z0)) 193.28/66.15
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0)) 193.28/66.15
ACTIVE(first(s(z0), cons(z1, z2))) → c2(CONS(z1, first(z0, z2)), FIRST(z0, z2)) 193.28/66.15
ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.28/66.15
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.28/66.15
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 193.28/66.15
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0)) 193.28/66.15
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.28/66.15
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.28/66.15
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.28/66.15
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.28/66.15
FROM(mark(z0)) → c12(FROM(z0)) 193.28/66.15
FROM(ok(z0)) → c13(FROM(z0)) 193.28/66.15
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.28/66.15
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.28/66.15
S(mark(z0)) → c16(S(z0)) 193.28/66.15
S(ok(z0)) → c17(S(z0)) 193.28/66.15
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.28/66.15
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.28/66.15
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.28/66.15
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.28/66.15
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.28/66.15
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.28/66.15
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0)) 193.28/66.15
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.15
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.28/66.15
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.15
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.15
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.28/66.15
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0)) 193.28/66.15
ACTIVE(first(s(z0), cons(z1, z2))) → c2(CONS(z1, first(z0, z2)), FIRST(z0, z2)) 193.28/66.15
ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.28/66.15
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.28/66.15
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 193.28/66.15
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0)) 193.28/66.15
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.28/66.15
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.28/66.15
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.28/66.15
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.28/66.15
FROM(mark(z0)) → c12(FROM(z0)) 193.28/66.15
FROM(ok(z0)) → c13(FROM(z0)) 193.28/66.15
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.28/66.15
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.28/66.15
S(mark(z0)) → c16(S(z0)) 193.28/66.15
S(ok(z0)) → c17(S(z0)) 193.28/66.15
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.28/66.15
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.28/66.15
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.28/66.15
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.28/66.15
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.28/66.15
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.28/66.15
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0)) 193.28/66.15
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.15
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.28/66.15
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.15
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.15
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.28/66.15
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, PROPER, TOP

Compound Symbols:

c, c2, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c30, c31, c32

193.28/66.15
193.28/66.15

(3) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts
193.28/66.15
193.28/66.15

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.28/66.15
active(first(0, z0)) → mark(nil) 193.28/66.15
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.28/66.15
active(sel(0, cons(z0, z1))) → mark(z0) 193.28/66.15
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.28/66.15
active(from(z0)) → from(active(z0)) 193.28/66.15
active(cons(z0, z1)) → cons(active(z0), z1) 193.28/66.15
active(s(z0)) → s(active(z0)) 193.28/66.15
active(first(z0, z1)) → first(active(z0), z1) 193.28/66.15
active(first(z0, z1)) → first(z0, active(z1)) 193.28/66.15
active(sel(z0, z1)) → sel(active(z0), z1) 193.28/66.15
active(sel(z0, z1)) → sel(z0, active(z1)) 193.28/66.15
from(mark(z0)) → mark(from(z0)) 193.28/66.15
from(ok(z0)) → ok(from(z0)) 193.28/66.15
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.28/66.15
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.28/66.15
s(mark(z0)) → mark(s(z0)) 193.28/66.15
s(ok(z0)) → ok(s(z0)) 193.28/66.15
first(mark(z0), z1) → mark(first(z0, z1)) 193.28/66.15
first(z0, mark(z1)) → mark(first(z0, z1)) 193.28/66.15
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.28/66.15
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.28/66.15
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.28/66.15
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.28/66.15
proper(from(z0)) → from(proper(z0)) 193.28/66.15
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.28/66.15
proper(s(z0)) → s(proper(z0)) 193.28/66.15
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.28/66.15
proper(0) → ok(0) 193.28/66.15
proper(nil) → ok(nil) 193.28/66.15
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.28/66.15
top(mark(z0)) → top(proper(z0)) 193.28/66.15
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(s(z0), cons(z1, z2))) → c2(CONS(z1, first(z0, z2)), FIRST(z0, z2)) 193.28/66.15
ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.28/66.15
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.28/66.15
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 193.28/66.15
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0)) 193.28/66.15
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.28/66.15
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.28/66.15
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.28/66.15
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.28/66.15
FROM(mark(z0)) → c12(FROM(z0)) 193.28/66.15
FROM(ok(z0)) → c13(FROM(z0)) 193.28/66.15
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.28/66.15
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.28/66.15
S(mark(z0)) → c16(S(z0)) 193.28/66.15
S(ok(z0)) → c17(S(z0)) 193.28/66.15
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.28/66.15
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.28/66.15
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.28/66.15
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.28/66.15
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.28/66.15
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.28/66.15
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0)) 193.28/66.15
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.15
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.28/66.15
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.15
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.15
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.28/66.15
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.28/66.18
ACTIVE(from(z0)) → c(S(z0))
S tuples:

ACTIVE(first(s(z0), cons(z1, z2))) → c2(CONS(z1, first(z0, z2)), FIRST(z0, z2)) 193.28/66.18
ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.28/66.18
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.28/66.18
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 193.28/66.18
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0)) 193.28/66.18
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.28/66.18
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.28/66.18
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.28/66.18
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.28/66.18
FROM(mark(z0)) → c12(FROM(z0)) 193.28/66.18
FROM(ok(z0)) → c13(FROM(z0)) 193.28/66.18
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.28/66.18
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.28/66.18
S(mark(z0)) → c16(S(z0)) 193.28/66.18
S(ok(z0)) → c17(S(z0)) 193.28/66.18
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.28/66.18
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.28/66.18
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.28/66.18
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.28/66.18
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.28/66.18
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.28/66.18
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0)) 193.28/66.18
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.18
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.28/66.18
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.18
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.18
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.28/66.18
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.28/66.18
ACTIVE(from(z0)) → c(S(z0))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, PROPER, TOP

Compound Symbols:

c2, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c30, c31, c32, c

193.28/66.18
193.28/66.18

(5) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC
193.28/66.18
193.28/66.18

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.28/66.18
active(first(0, z0)) → mark(nil) 193.28/66.18
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.28/66.18
active(sel(0, cons(z0, z1))) → mark(z0) 193.28/66.18
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.28/66.18
active(from(z0)) → from(active(z0)) 193.28/66.18
active(cons(z0, z1)) → cons(active(z0), z1) 193.28/66.18
active(s(z0)) → s(active(z0)) 193.28/66.18
active(first(z0, z1)) → first(active(z0), z1) 193.28/66.18
active(first(z0, z1)) → first(z0, active(z1)) 193.28/66.18
active(sel(z0, z1)) → sel(active(z0), z1) 193.28/66.18
active(sel(z0, z1)) → sel(z0, active(z1)) 193.28/66.18
from(mark(z0)) → mark(from(z0)) 193.28/66.18
from(ok(z0)) → ok(from(z0)) 193.28/66.18
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.28/66.18
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.28/66.18
s(mark(z0)) → mark(s(z0)) 193.28/66.18
s(ok(z0)) → ok(s(z0)) 193.28/66.18
first(mark(z0), z1) → mark(first(z0, z1)) 193.28/66.18
first(z0, mark(z1)) → mark(first(z0, z1)) 193.28/66.18
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.28/66.18
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.28/66.18
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.28/66.18
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.28/66.18
proper(from(z0)) → from(proper(z0)) 193.28/66.18
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.28/66.18
proper(s(z0)) → s(proper(z0)) 193.28/66.18
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.28/66.18
proper(0) → ok(0) 193.28/66.18
proper(nil) → ok(nil) 193.28/66.18
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.28/66.18
top(mark(z0)) → top(proper(z0)) 193.28/66.18
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.28/66.18
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.28/66.18
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 193.28/66.18
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0)) 193.28/66.18
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.28/66.18
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.28/66.18
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.28/66.18
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.28/66.18
FROM(mark(z0)) → c12(FROM(z0)) 193.28/66.18
FROM(ok(z0)) → c13(FROM(z0)) 193.28/66.18
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.28/66.18
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.28/66.18
S(mark(z0)) → c16(S(z0)) 193.28/66.18
S(ok(z0)) → c17(S(z0)) 193.28/66.18
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.28/66.18
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.28/66.18
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.28/66.18
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.28/66.18
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.28/66.18
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.28/66.18
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0)) 193.28/66.18
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.18
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.28/66.18
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.18
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.18
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.28/66.18
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.28/66.18
ACTIVE(from(z0)) → c(S(z0)) 193.28/66.18
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.28/66.18
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.28/66.18
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.28/66.18
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 193.28/66.18
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0)) 193.28/66.18
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.28/66.18
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.28/66.18
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.28/66.18
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.28/66.18
FROM(mark(z0)) → c12(FROM(z0)) 193.28/66.18
FROM(ok(z0)) → c13(FROM(z0)) 193.28/66.18
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.28/66.18
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.28/66.18
S(mark(z0)) → c16(S(z0)) 193.28/66.18
S(ok(z0)) → c17(S(z0)) 193.28/66.18
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.28/66.18
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.28/66.18
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.28/66.18
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.28/66.18
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.28/66.18
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.28/66.18
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0)) 193.28/66.18
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.18
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.28/66.18
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.18
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.18
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.28/66.18
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.28/66.18
ACTIVE(from(z0)) → c(S(z0)) 193.28/66.18
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.28/66.18
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, PROPER, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c30, c31, c32, c, c1

193.28/66.19
193.28/66.19

(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) by

ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.28/66.19
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.28/66.19
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.28/66.19
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.28/66.19
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.28/66.19
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.28/66.19
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.28/66.19
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.28/66.19
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.28/66.19
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.28/66.19
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.28/66.19
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
193.28/66.19
193.28/66.19

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.28/66.19
active(first(0, z0)) → mark(nil) 193.28/66.19
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.28/66.19
active(sel(0, cons(z0, z1))) → mark(z0) 193.28/66.19
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.28/66.19
active(from(z0)) → from(active(z0)) 193.28/66.19
active(cons(z0, z1)) → cons(active(z0), z1) 193.28/66.19
active(s(z0)) → s(active(z0)) 193.28/66.19
active(first(z0, z1)) → first(active(z0), z1) 193.28/66.19
active(first(z0, z1)) → first(z0, active(z1)) 193.28/66.19
active(sel(z0, z1)) → sel(active(z0), z1) 193.28/66.19
active(sel(z0, z1)) → sel(z0, active(z1)) 193.28/66.19
from(mark(z0)) → mark(from(z0)) 193.28/66.19
from(ok(z0)) → ok(from(z0)) 193.28/66.19
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.28/66.19
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.28/66.19
s(mark(z0)) → mark(s(z0)) 193.28/66.19
s(ok(z0)) → ok(s(z0)) 193.28/66.19
first(mark(z0), z1) → mark(first(z0, z1)) 193.28/66.19
first(z0, mark(z1)) → mark(first(z0, z1)) 193.28/66.19
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.28/66.19
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.28/66.19
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.28/66.19
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.28/66.19
proper(from(z0)) → from(proper(z0)) 193.28/66.19
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.28/66.19
proper(s(z0)) → s(proper(z0)) 193.28/66.19
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.28/66.19
proper(0) → ok(0) 193.28/66.19
proper(nil) → ok(nil) 193.28/66.19
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.28/66.19
top(mark(z0)) → top(proper(z0)) 193.28/66.19
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.28/66.19
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 193.28/66.19
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0)) 193.28/66.19
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.28/66.19
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.28/66.19
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.28/66.19
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.28/66.19
FROM(mark(z0)) → c12(FROM(z0)) 193.28/66.19
FROM(ok(z0)) → c13(FROM(z0)) 193.28/66.19
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.28/66.19
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.28/66.19
S(mark(z0)) → c16(S(z0)) 193.28/66.19
S(ok(z0)) → c17(S(z0)) 193.28/66.19
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.28/66.19
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.28/66.19
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.28/66.19
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.28/66.19
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.28/66.19
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.28/66.19
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0)) 193.28/66.19
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.19
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.28/66.19
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.19
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.19
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.28/66.19
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.28/66.19
ACTIVE(from(z0)) → c(S(z0)) 193.28/66.19
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.28/66.19
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.28/66.19
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.28/66.19
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.28/66.19
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.28/66.19
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.28/66.19
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.28/66.19
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.28/66.19
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.28/66.19
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.28/66.19
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.28/66.19
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.28/66.19
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.28/66.19
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.28/66.19
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 193.28/66.19
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0)) 193.28/66.19
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.28/66.19
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.28/66.19
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.28/66.19
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.28/66.19
FROM(mark(z0)) → c12(FROM(z0)) 193.28/66.19
FROM(ok(z0)) → c13(FROM(z0)) 193.28/66.19
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.28/66.19
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.28/66.19
S(mark(z0)) → c16(S(z0)) 193.28/66.19
S(ok(z0)) → c17(S(z0)) 193.28/66.19
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.28/66.19
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.28/66.19
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.28/66.19
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.28/66.19
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.28/66.19
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.28/66.19
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0)) 193.28/66.19
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.19
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.28/66.19
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.19
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.28/66.19
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.28/66.19
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.28/66.19
ACTIVE(from(z0)) → c(S(z0)) 193.28/66.19
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.28/66.19
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.28/66.19
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.28/66.19
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.28/66.19
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.28/66.19
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.28/66.19
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.28/66.19
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.28/66.19
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.28/66.19
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.28/66.19
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.28/66.19
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.22
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.22
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, PROPER, TOP

Compound Symbols:

c4, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c30, c31, c32, c, c1, c5

193.64/66.22
193.64/66.22

(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) by

ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.22
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.22
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.22
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.22
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.22
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.22
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.22
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.22
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.22
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.22
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.22
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
193.64/66.22
193.64/66.22

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.64/66.22
active(first(0, z0)) → mark(nil) 193.64/66.22
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.64/66.22
active(sel(0, cons(z0, z1))) → mark(z0) 193.64/66.22
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.64/66.22
active(from(z0)) → from(active(z0)) 193.64/66.22
active(cons(z0, z1)) → cons(active(z0), z1) 193.64/66.22
active(s(z0)) → s(active(z0)) 193.64/66.22
active(first(z0, z1)) → first(active(z0), z1) 193.64/66.22
active(first(z0, z1)) → first(z0, active(z1)) 193.64/66.22
active(sel(z0, z1)) → sel(active(z0), z1) 193.64/66.22
active(sel(z0, z1)) → sel(z0, active(z1)) 193.64/66.22
from(mark(z0)) → mark(from(z0)) 193.64/66.22
from(ok(z0)) → ok(from(z0)) 193.64/66.22
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.64/66.22
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.64/66.22
s(mark(z0)) → mark(s(z0)) 193.64/66.22
s(ok(z0)) → ok(s(z0)) 193.64/66.22
first(mark(z0), z1) → mark(first(z0, z1)) 193.64/66.22
first(z0, mark(z1)) → mark(first(z0, z1)) 193.64/66.22
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.64/66.22
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.64/66.22
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.64/66.22
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.64/66.22
proper(from(z0)) → from(proper(z0)) 193.64/66.22
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.64/66.22
proper(s(z0)) → s(proper(z0)) 193.64/66.22
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.64/66.22
proper(0) → ok(0) 193.64/66.22
proper(nil) → ok(nil) 193.64/66.22
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.64/66.22
top(mark(z0)) → top(proper(z0)) 193.64/66.22
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.22
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0)) 193.64/66.22
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.22
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.22
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.22
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.22
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.22
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.22
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.22
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.22
S(mark(z0)) → c16(S(z0)) 193.64/66.22
S(ok(z0)) → c17(S(z0)) 193.64/66.22
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.22
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.22
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.22
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.22
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.22
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.22
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0)) 193.64/66.22
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.22
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.64/66.22
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.22
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.22
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.22
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.22
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.22
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.22
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.22
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.22
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.22
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.22
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.22
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.22
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.22
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.22
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.22
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.22
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.22
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.22
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.22
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.22
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.22
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.22
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.22
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.22
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.22
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.22
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.22
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.22
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.22
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.22
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.22
ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0)) 193.64/66.22
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.22
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.22
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.22
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.22
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.22
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.22
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.22
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.22
S(mark(z0)) → c16(S(z0)) 193.64/66.22
S(ok(z0)) → c17(S(z0)) 193.64/66.22
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.22
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.22
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.22
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.22
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.22
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.22
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0)) 193.64/66.22
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.22
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.64/66.22
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.22
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.22
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.22
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.22
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.22
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.22
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.22
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.22
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.22
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.22
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.22
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.22
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.22
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.22
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.22
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.22
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.22
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.22
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.22
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.22
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.22
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.22
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.22
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.22
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.22
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.22
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.22
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.22
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.22
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.22
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, PROPER, TOP

Compound Symbols:

c4, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c30, c31, c32, c, c1, c5, c6

193.64/66.22
193.64/66.22

(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(s(z0)) → c7(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.22
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.22
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.22
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.22
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.22
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.22
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.22
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.22
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.22
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
193.64/66.23
193.64/66.23

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.64/66.23
active(first(0, z0)) → mark(nil) 193.64/66.23
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.64/66.23
active(sel(0, cons(z0, z1))) → mark(z0) 193.64/66.23
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.64/66.23
active(from(z0)) → from(active(z0)) 193.64/66.23
active(cons(z0, z1)) → cons(active(z0), z1) 193.64/66.23
active(s(z0)) → s(active(z0)) 193.64/66.23
active(first(z0, z1)) → first(active(z0), z1) 193.64/66.23
active(first(z0, z1)) → first(z0, active(z1)) 193.64/66.23
active(sel(z0, z1)) → sel(active(z0), z1) 193.64/66.23
active(sel(z0, z1)) → sel(z0, active(z1)) 193.64/66.23
from(mark(z0)) → mark(from(z0)) 193.64/66.23
from(ok(z0)) → ok(from(z0)) 193.64/66.23
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.64/66.23
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.64/66.23
s(mark(z0)) → mark(s(z0)) 193.64/66.23
s(ok(z0)) → ok(s(z0)) 193.64/66.23
first(mark(z0), z1) → mark(first(z0, z1)) 193.64/66.23
first(z0, mark(z1)) → mark(first(z0, z1)) 193.64/66.23
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.64/66.23
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.64/66.23
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.64/66.23
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.64/66.23
proper(from(z0)) → from(proper(z0)) 193.64/66.23
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.64/66.23
proper(s(z0)) → s(proper(z0)) 193.64/66.23
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.64/66.23
proper(0) → ok(0) 193.64/66.23
proper(nil) → ok(nil) 193.64/66.23
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.64/66.23
top(mark(z0)) → top(proper(z0)) 193.64/66.23
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.23
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.23
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.23
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.23
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.23
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.23
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.23
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.23
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.23
S(mark(z0)) → c16(S(z0)) 193.64/66.23
S(ok(z0)) → c17(S(z0)) 193.64/66.23
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.23
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.23
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.23
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.23
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.23
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.23
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0)) 193.64/66.23
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.23
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.64/66.23
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.23
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.23
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.23
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.23
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.23
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.23
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.23
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.23
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.23
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.23
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.23
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.23
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.23
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.23
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.23
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.23
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.23
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.23
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.23
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.23
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.23
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.23
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.23
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.23
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.23
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.23
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.23
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.23
S(mark(z0)) → c16(S(z0)) 193.64/66.23
S(ok(z0)) → c17(S(z0)) 193.64/66.23
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.23
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.23
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.23
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.23
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.23
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.23
PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0)) 193.64/66.23
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.23
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.64/66.23
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.23
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.23
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.23
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.23
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.23
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.23
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.23
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.23
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.23
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.23
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.23
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.23
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.23
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.23
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.23
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.23
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.23
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.23
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.23
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, PROPER, TOP

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c30, c31, c32, c, c1, c5, c6, c7

193.64/66.23
193.64/66.23

(13) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(from(z0)) → c24(FROM(proper(z0)), PROPER(z0)) by

PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.23
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.23
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.23
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.23
PROPER(from(0)) → c24(FROM(ok(0)), PROPER(0)) 193.64/66.23
PROPER(from(nil)) → c24(FROM(ok(nil)), PROPER(nil)) 193.64/66.23
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
193.64/66.23
193.64/66.23

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.64/66.23
active(first(0, z0)) → mark(nil) 193.64/66.23
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.64/66.23
active(sel(0, cons(z0, z1))) → mark(z0) 193.64/66.23
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.64/66.23
active(from(z0)) → from(active(z0)) 193.64/66.23
active(cons(z0, z1)) → cons(active(z0), z1) 193.64/66.23
active(s(z0)) → s(active(z0)) 193.64/66.23
active(first(z0, z1)) → first(active(z0), z1) 193.64/66.23
active(first(z0, z1)) → first(z0, active(z1)) 193.64/66.23
active(sel(z0, z1)) → sel(active(z0), z1) 193.64/66.23
active(sel(z0, z1)) → sel(z0, active(z1)) 193.64/66.23
from(mark(z0)) → mark(from(z0)) 193.64/66.23
from(ok(z0)) → ok(from(z0)) 193.64/66.23
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.64/66.23
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.64/66.23
s(mark(z0)) → mark(s(z0)) 193.64/66.23
s(ok(z0)) → ok(s(z0)) 193.64/66.23
first(mark(z0), z1) → mark(first(z0, z1)) 193.64/66.23
first(z0, mark(z1)) → mark(first(z0, z1)) 193.64/66.23
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.64/66.23
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.64/66.23
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.64/66.23
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.64/66.23
proper(from(z0)) → from(proper(z0)) 193.64/66.23
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.64/66.23
proper(s(z0)) → s(proper(z0)) 193.64/66.23
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.64/66.23
proper(0) → ok(0) 193.64/66.23
proper(nil) → ok(nil) 193.64/66.23
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.64/66.23
top(mark(z0)) → top(proper(z0)) 193.64/66.23
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.23
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.23
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.23
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.23
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.23
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.23
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.23
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.23
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.23
S(mark(z0)) → c16(S(z0)) 193.64/66.23
S(ok(z0)) → c17(S(z0)) 193.64/66.23
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.23
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.23
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.23
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.23
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.23
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.23
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.23
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.64/66.23
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.23
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.23
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.23
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.23
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.23
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.23
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.23
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.23
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.23
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.23
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.23
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.23
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.23
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.23
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.23
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.23
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.23
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.23
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.23
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.23
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.23
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.23
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.23
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.23
PROPER(from(0)) → c24(FROM(ok(0)), PROPER(0)) 193.64/66.23
PROPER(from(nil)) → c24(FROM(ok(nil)), PROPER(nil)) 193.64/66.23
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.23
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.23
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.23
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.23
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.23
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.23
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.23
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.23
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.23
S(mark(z0)) → c16(S(z0)) 193.64/66.23
S(ok(z0)) → c17(S(z0)) 193.64/66.23
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.23
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.23
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.23
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.23
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.23
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.23
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.23
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.64/66.23
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.23
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.23
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.23
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.23
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.23
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.23
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.23
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.23
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.23
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.23
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.23
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.23
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.23
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.23
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.23
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.23
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.23
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.23
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.23
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.23
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.23
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.23
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.23
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.23
PROPER(from(0)) → c24(FROM(ok(0)), PROPER(0)) 193.64/66.23
PROPER(from(nil)) → c24(FROM(ok(nil)), PROPER(nil)) 193.64/66.23
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, PROPER, TOP

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c25, c26, c27, c30, c31, c32, c, c1, c5, c6, c7, c24

193.64/66.23
193.64/66.23

(15) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts
193.64/66.23
193.64/66.23

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.64/66.23
active(first(0, z0)) → mark(nil) 193.64/66.23
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.64/66.23
active(sel(0, cons(z0, z1))) → mark(z0) 193.64/66.23
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.64/66.23
active(from(z0)) → from(active(z0)) 193.64/66.23
active(cons(z0, z1)) → cons(active(z0), z1) 193.64/66.23
active(s(z0)) → s(active(z0)) 193.64/66.23
active(first(z0, z1)) → first(active(z0), z1) 193.64/66.23
active(first(z0, z1)) → first(z0, active(z1)) 193.64/66.23
active(sel(z0, z1)) → sel(active(z0), z1) 193.64/66.23
active(sel(z0, z1)) → sel(z0, active(z1)) 193.64/66.23
from(mark(z0)) → mark(from(z0)) 193.64/66.23
from(ok(z0)) → ok(from(z0)) 193.64/66.23
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.64/66.23
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.64/66.23
s(mark(z0)) → mark(s(z0)) 193.64/66.23
s(ok(z0)) → ok(s(z0)) 193.64/66.23
first(mark(z0), z1) → mark(first(z0, z1)) 193.64/66.23
first(z0, mark(z1)) → mark(first(z0, z1)) 193.64/66.23
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.64/66.23
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.64/66.23
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.64/66.23
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.64/66.23
proper(from(z0)) → from(proper(z0)) 193.64/66.23
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.64/66.23
proper(s(z0)) → s(proper(z0)) 193.64/66.23
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.64/66.23
proper(0) → ok(0) 193.64/66.23
proper(nil) → ok(nil) 193.64/66.23
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.64/66.23
top(mark(z0)) → top(proper(z0)) 193.64/66.23
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.23
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.23
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.23
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.23
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.23
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.23
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.23
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.23
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.23
S(mark(z0)) → c16(S(z0)) 193.64/66.23
S(ok(z0)) → c17(S(z0)) 193.64/66.23
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.23
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.23
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.23
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.23
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.23
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.23
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.23
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.64/66.23
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.23
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.23
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.23
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.23
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.23
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.23
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.23
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.23
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.23
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.23
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.23
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.23
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.23
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.23
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.23
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.23
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.23
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.23
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.23
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.23
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.23
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.23
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.23
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.23
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.23
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.23
PROPER(from(nil)) → c24(FROM(ok(nil)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.23
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.23
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.23
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.23
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.23
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.23
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.23
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.23
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.23
S(mark(z0)) → c16(S(z0)) 193.64/66.23
S(ok(z0)) → c17(S(z0)) 193.64/66.23
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.23
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.23
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.23
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.23
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.23
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.23
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.23
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.64/66.23
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.23
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.23
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.23
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.23
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.23
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.23
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.23
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.23
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.23
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.23
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.23
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.23
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.23
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.23
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.23
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.23
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.23
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.23
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.23
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.26
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.26
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.26
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.26
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.26
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.26
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.26
PROPER(from(nil)) → c24(FROM(ok(nil)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, PROPER, TOP

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c25, c26, c27, c30, c31, c32, c, c1, c5, c6, c7, c24, c24

193.64/66.26
193.64/66.26

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.26
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.26
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.26
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.26
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 193.64/66.26
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 193.64/66.26
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.26
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.26
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.26
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.26
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.26
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 193.64/66.26
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 193.64/66.26
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
193.64/66.26
193.64/66.26

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.64/66.26
active(first(0, z0)) → mark(nil) 193.64/66.26
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.64/66.26
active(sel(0, cons(z0, z1))) → mark(z0) 193.64/66.26
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.64/66.26
active(from(z0)) → from(active(z0)) 193.64/66.26
active(cons(z0, z1)) → cons(active(z0), z1) 193.64/66.26
active(s(z0)) → s(active(z0)) 193.64/66.26
active(first(z0, z1)) → first(active(z0), z1) 193.64/66.26
active(first(z0, z1)) → first(z0, active(z1)) 193.64/66.26
active(sel(z0, z1)) → sel(active(z0), z1) 193.64/66.26
active(sel(z0, z1)) → sel(z0, active(z1)) 193.64/66.26
from(mark(z0)) → mark(from(z0)) 193.64/66.26
from(ok(z0)) → ok(from(z0)) 193.64/66.26
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.64/66.26
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.64/66.26
s(mark(z0)) → mark(s(z0)) 193.64/66.26
s(ok(z0)) → ok(s(z0)) 193.64/66.26
first(mark(z0), z1) → mark(first(z0, z1)) 193.64/66.26
first(z0, mark(z1)) → mark(first(z0, z1)) 193.64/66.26
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.64/66.26
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.64/66.26
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.64/66.26
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.64/66.26
proper(from(z0)) → from(proper(z0)) 193.64/66.26
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.64/66.26
proper(s(z0)) → s(proper(z0)) 193.64/66.26
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.64/66.26
proper(0) → ok(0) 193.64/66.26
proper(nil) → ok(nil) 193.64/66.26
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.64/66.26
top(mark(z0)) → top(proper(z0)) 193.64/66.26
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.26
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.26
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.26
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.26
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.26
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.26
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.26
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.26
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.26
S(mark(z0)) → c16(S(z0)) 193.64/66.26
S(ok(z0)) → c17(S(z0)) 193.64/66.26
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.26
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.26
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.26
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.26
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.26
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.26
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.64/66.26
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.26
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.26
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.26
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.26
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.26
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.26
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.26
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.26
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.26
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.26
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.26
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.26
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.26
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.26
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.26
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.26
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.26
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.26
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.26
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 193.64/66.26
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 193.64/66.26
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.26
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.26
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.26
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.26
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.26
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 193.64/66.26
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 193.64/66.26
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.26
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.26
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.26
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.26
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.26
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.26
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.26
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.26
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.26
S(mark(z0)) → c16(S(z0)) 193.64/66.26
S(ok(z0)) → c17(S(z0)) 193.64/66.26
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.26
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.26
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.26
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.26
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.26
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.26
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.64/66.26
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.26
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.26
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.26
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.26
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.26
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.26
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.26
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.26
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.26
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.26
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.26
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.26
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.26
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.26
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.26
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.26
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.26
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.26
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.26
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 193.64/66.26
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 193.64/66.26
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.26
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.26
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.26
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.26
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.26
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 193.64/66.26
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 193.64/66.26
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, PROPER, TOP

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c30, c31, c32, c, c1, c5, c6, c7, c24, c24, c25

193.64/66.26
193.64/66.26

(19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts
193.64/66.26
193.64/66.26

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.64/66.26
active(first(0, z0)) → mark(nil) 193.64/66.26
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.64/66.26
active(sel(0, cons(z0, z1))) → mark(z0) 193.64/66.26
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.64/66.26
active(from(z0)) → from(active(z0)) 193.64/66.26
active(cons(z0, z1)) → cons(active(z0), z1) 193.64/66.26
active(s(z0)) → s(active(z0)) 193.64/66.26
active(first(z0, z1)) → first(active(z0), z1) 193.64/66.26
active(first(z0, z1)) → first(z0, active(z1)) 193.64/66.26
active(sel(z0, z1)) → sel(active(z0), z1) 193.64/66.26
active(sel(z0, z1)) → sel(z0, active(z1)) 193.64/66.26
from(mark(z0)) → mark(from(z0)) 193.64/66.26
from(ok(z0)) → ok(from(z0)) 193.64/66.26
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.64/66.26
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.64/66.26
s(mark(z0)) → mark(s(z0)) 193.64/66.26
s(ok(z0)) → ok(s(z0)) 193.64/66.26
first(mark(z0), z1) → mark(first(z0, z1)) 193.64/66.26
first(z0, mark(z1)) → mark(first(z0, z1)) 193.64/66.26
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.64/66.26
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.64/66.26
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.64/66.26
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.64/66.26
proper(from(z0)) → from(proper(z0)) 193.64/66.26
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.64/66.26
proper(s(z0)) → s(proper(z0)) 193.64/66.26
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.64/66.26
proper(0) → ok(0) 193.64/66.26
proper(nil) → ok(nil) 193.64/66.26
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.64/66.26
top(mark(z0)) → top(proper(z0)) 193.64/66.26
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.26
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.26
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.26
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.26
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.26
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.26
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.26
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.26
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.26
S(mark(z0)) → c16(S(z0)) 193.64/66.26
S(ok(z0)) → c17(S(z0)) 193.64/66.26
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.26
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.26
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.26
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.26
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.26
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.26
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.64/66.26
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.26
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.26
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.26
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.26
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.26
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.26
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.26
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.26
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.26
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.26
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.26
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.26
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.26
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.26
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.26
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.26
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.26
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.26
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.26
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.26
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.26
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.26
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.26
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.26
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.26
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.26
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.26
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.26
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.26
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.26
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.26
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.26
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.26
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.26
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.26
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.26
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.26
S(mark(z0)) → c16(S(z0)) 193.64/66.26
S(ok(z0)) → c17(S(z0)) 193.64/66.26
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.26
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.26
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.26
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.26
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.26
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.26
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 193.64/66.26
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.26
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.26
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.26
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.26
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.26
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.26
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.26
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.26
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.26
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.26
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.26
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.26
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.26
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.26
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.26
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.26
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.26
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.26
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.26
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.26
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.26
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.26
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.26
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.26
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.26
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.26
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.26
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.26
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, PROPER, TOP

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c30, c31, c32, c, c1, c5, c6, c7, c24, c24, c25, c25

193.64/66.26
193.64/66.26

(21) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) by

PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.64/66.26
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.26
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.64/66.26
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.26
PROPER(s(0)) → c26(S(ok(0)), PROPER(0)) 193.64/66.26
PROPER(s(nil)) → c26(S(ok(nil)), PROPER(nil)) 193.64/66.26
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
193.64/66.26
193.64/66.26

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.64/66.26
active(first(0, z0)) → mark(nil) 193.64/66.26
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.64/66.26
active(sel(0, cons(z0, z1))) → mark(z0) 193.64/66.26
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.64/66.26
active(from(z0)) → from(active(z0)) 193.64/66.26
active(cons(z0, z1)) → cons(active(z0), z1) 193.64/66.26
active(s(z0)) → s(active(z0)) 193.64/66.26
active(first(z0, z1)) → first(active(z0), z1) 193.64/66.26
active(first(z0, z1)) → first(z0, active(z1)) 193.64/66.26
active(sel(z0, z1)) → sel(active(z0), z1) 193.64/66.26
active(sel(z0, z1)) → sel(z0, active(z1)) 193.64/66.26
from(mark(z0)) → mark(from(z0)) 193.64/66.26
from(ok(z0)) → ok(from(z0)) 193.64/66.26
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.64/66.26
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.64/66.26
s(mark(z0)) → mark(s(z0)) 193.64/66.26
s(ok(z0)) → ok(s(z0)) 193.64/66.26
first(mark(z0), z1) → mark(first(z0, z1)) 193.64/66.26
first(z0, mark(z1)) → mark(first(z0, z1)) 193.64/66.26
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.64/66.26
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.64/66.26
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.64/66.26
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.64/66.26
proper(from(z0)) → from(proper(z0)) 193.64/66.26
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.64/66.26
proper(s(z0)) → s(proper(z0)) 193.64/66.26
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.64/66.26
proper(0) → ok(0) 193.64/66.26
proper(nil) → ok(nil) 193.64/66.26
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.64/66.26
top(mark(z0)) → top(proper(z0)) 193.64/66.26
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.26
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.26
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.26
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.26
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.26
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.26
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.26
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.26
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.26
S(mark(z0)) → c16(S(z0)) 193.64/66.26
S(ok(z0)) → c17(S(z0)) 193.64/66.26
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.26
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.26
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.26
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.26
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.26
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.26
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.26
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.26
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.26
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.26
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.26
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.26
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.26
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.26
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.26
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.26
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.26
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.26
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.26
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.26
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.26
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.26
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.26
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.26
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.26
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.26
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.26
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.26
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.26
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.26
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.26
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.26
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.26
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.26
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.26
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.64/66.26
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.26
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.64/66.26
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.26
PROPER(s(0)) → c26(S(ok(0)), PROPER(0)) 193.64/66.26
PROPER(s(nil)) → c26(S(ok(nil)), PROPER(nil)) 193.64/66.26
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.26
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.26
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.26
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.26
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.26
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.26
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.26
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.26
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.26
S(mark(z0)) → c16(S(z0)) 193.64/66.26
S(ok(z0)) → c17(S(z0)) 193.64/66.26
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.26
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.26
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.26
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.26
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.26
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.26
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.26
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.26
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.26
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.26
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.26
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.26
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.26
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.26
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.26
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.26
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.26
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.26
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.26
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.26
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.26
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.26
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.26
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.26
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.26
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.26
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.26
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.26
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.26
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.26
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.26
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.26
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.26
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.26
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.27
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.27
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.27
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.27
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.64/66.27
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.64/66.27
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.27
PROPER(s(0)) → c26(S(ok(0)), PROPER(0)) 193.64/66.27
PROPER(s(nil)) → c26(S(ok(nil)), PROPER(nil)) 193.64/66.27
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, PROPER, TOP

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c27, c30, c31, c32, c, c1, c5, c6, c7, c24, c24, c25, c25, c26

193.64/66.27
193.64/66.27

(23) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts
193.64/66.27
193.64/66.27

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.64/66.27
active(first(0, z0)) → mark(nil) 193.64/66.27
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.64/66.27
active(sel(0, cons(z0, z1))) → mark(z0) 193.64/66.27
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.64/66.27
active(from(z0)) → from(active(z0)) 193.64/66.27
active(cons(z0, z1)) → cons(active(z0), z1) 193.64/66.27
active(s(z0)) → s(active(z0)) 193.64/66.27
active(first(z0, z1)) → first(active(z0), z1) 193.64/66.27
active(first(z0, z1)) → first(z0, active(z1)) 193.64/66.27
active(sel(z0, z1)) → sel(active(z0), z1) 193.64/66.27
active(sel(z0, z1)) → sel(z0, active(z1)) 193.64/66.27
from(mark(z0)) → mark(from(z0)) 193.64/66.27
from(ok(z0)) → ok(from(z0)) 193.64/66.27
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.64/66.27
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.64/66.27
s(mark(z0)) → mark(s(z0)) 193.64/66.27
s(ok(z0)) → ok(s(z0)) 193.64/66.27
first(mark(z0), z1) → mark(first(z0, z1)) 193.64/66.27
first(z0, mark(z1)) → mark(first(z0, z1)) 193.64/66.27
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.64/66.27
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.64/66.27
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.64/66.27
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.64/66.27
proper(from(z0)) → from(proper(z0)) 193.64/66.27
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.64/66.27
proper(s(z0)) → s(proper(z0)) 193.64/66.27
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.64/66.27
proper(0) → ok(0) 193.64/66.27
proper(nil) → ok(nil) 193.64/66.27
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.64/66.27
top(mark(z0)) → top(proper(z0)) 193.64/66.27
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.27
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.27
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.27
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.27
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.27
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.27
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.27
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.27
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.27
S(mark(z0)) → c16(S(z0)) 193.64/66.27
S(ok(z0)) → c17(S(z0)) 193.64/66.27
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.27
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.27
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.27
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.27
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.27
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.27
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.27
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.27
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.27
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.27
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.27
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.27
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.27
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.27
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.27
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.27
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.27
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.27
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.27
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.27
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.27
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.27
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.27
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.27
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.27
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.27
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.64/66.27
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.64/66.27
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.27
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(s(0)) → c26(S(ok(0))) 193.64/66.27
PROPER(s(nil)) → c26(S(ok(nil)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.27
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.27
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.27
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.27
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.27
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.27
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.27
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.27
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.27
S(mark(z0)) → c16(S(z0)) 193.64/66.27
S(ok(z0)) → c17(S(z0)) 193.64/66.27
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.27
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.27
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.27
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.27
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.27
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.27
PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.27
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.27
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.27
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.27
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.27
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.27
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.27
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.27
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.27
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.27
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.27
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.27
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.27
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.27
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.27
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.27
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.27
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.27
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.27
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.27
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.64/66.27
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.64/66.27
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.27
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(s(0)) → c26(S(ok(0))) 193.64/66.27
PROPER(s(nil)) → c26(S(ok(nil)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, PROPER, TOP

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c27, c30, c31, c32, c, c1, c5, c6, c7, c24, c24, c25, c25, c26, c26

193.64/66.27
193.64/66.27

(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(first(z0, z1)) → c27(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.27
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.27
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.27
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 193.64/66.27
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 193.64/66.27
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.27
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.27
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 193.64/66.27
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 193.64/66.27
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
193.64/66.27
193.64/66.27

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.64/66.27
active(first(0, z0)) → mark(nil) 193.64/66.27
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.64/66.27
active(sel(0, cons(z0, z1))) → mark(z0) 193.64/66.27
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.64/66.27
active(from(z0)) → from(active(z0)) 193.64/66.27
active(cons(z0, z1)) → cons(active(z0), z1) 193.64/66.27
active(s(z0)) → s(active(z0)) 193.64/66.27
active(first(z0, z1)) → first(active(z0), z1) 193.64/66.27
active(first(z0, z1)) → first(z0, active(z1)) 193.64/66.27
active(sel(z0, z1)) → sel(active(z0), z1) 193.64/66.27
active(sel(z0, z1)) → sel(z0, active(z1)) 193.64/66.27
from(mark(z0)) → mark(from(z0)) 193.64/66.27
from(ok(z0)) → ok(from(z0)) 193.64/66.27
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.64/66.27
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.64/66.27
s(mark(z0)) → mark(s(z0)) 193.64/66.27
s(ok(z0)) → ok(s(z0)) 193.64/66.27
first(mark(z0), z1) → mark(first(z0, z1)) 193.64/66.27
first(z0, mark(z1)) → mark(first(z0, z1)) 193.64/66.27
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.64/66.27
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.64/66.27
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.64/66.27
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.64/66.27
proper(from(z0)) → from(proper(z0)) 193.64/66.27
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.64/66.27
proper(s(z0)) → s(proper(z0)) 193.64/66.27
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.64/66.27
proper(0) → ok(0) 193.64/66.27
proper(nil) → ok(nil) 193.64/66.27
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.64/66.27
top(mark(z0)) → top(proper(z0)) 193.64/66.27
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.27
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.27
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.27
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.27
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.27
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.27
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.27
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.27
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.27
S(mark(z0)) → c16(S(z0)) 193.64/66.27
S(ok(z0)) → c17(S(z0)) 193.64/66.27
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.27
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.27
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.27
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.27
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.27
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.27
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.27
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.27
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.27
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.27
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.27
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.27
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.27
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.27
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.27
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.27
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.27
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.27
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.27
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.27
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.27
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.27
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.27
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.27
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.27
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.64/66.27
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.64/66.27
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.27
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(s(0)) → c26(S(ok(0))) 193.64/66.27
PROPER(s(nil)) → c26(S(ok(nil))) 193.64/66.27
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.27
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.27
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.27
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 193.64/66.27
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 193.64/66.27
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.27
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.27
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 193.64/66.27
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 193.64/66.27
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.27
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.27
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.27
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.27
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.27
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.27
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.27
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.27
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.27
S(mark(z0)) → c16(S(z0)) 193.64/66.27
S(ok(z0)) → c17(S(z0)) 193.64/66.27
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.27
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.27
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.27
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.27
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.27
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.27
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.27
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.27
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.27
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.27
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.27
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.27
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.27
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.27
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.27
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.27
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.27
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.27
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.27
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.27
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.27
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.27
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.27
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.27
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.27
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.64/66.27
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.64/66.27
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.27
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(s(0)) → c26(S(ok(0))) 193.64/66.27
PROPER(s(nil)) → c26(S(ok(nil))) 193.64/66.27
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.27
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.27
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.27
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 193.64/66.27
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 193.64/66.27
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.27
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.27
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 193.64/66.27
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 193.64/66.27
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, PROPER, TOP

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c30, c31, c32, c, c1, c5, c6, c7, c24, c24, c25, c25, c26, c26, c27

193.64/66.27
193.64/66.27

(27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts
193.64/66.27
193.64/66.27

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.64/66.27
active(first(0, z0)) → mark(nil) 193.64/66.27
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.64/66.27
active(sel(0, cons(z0, z1))) → mark(z0) 193.64/66.27
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.64/66.27
active(from(z0)) → from(active(z0)) 193.64/66.27
active(cons(z0, z1)) → cons(active(z0), z1) 193.64/66.27
active(s(z0)) → s(active(z0)) 193.64/66.27
active(first(z0, z1)) → first(active(z0), z1) 193.64/66.27
active(first(z0, z1)) → first(z0, active(z1)) 193.64/66.27
active(sel(z0, z1)) → sel(active(z0), z1) 193.64/66.27
active(sel(z0, z1)) → sel(z0, active(z1)) 193.64/66.27
from(mark(z0)) → mark(from(z0)) 193.64/66.27
from(ok(z0)) → ok(from(z0)) 193.64/66.27
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.64/66.27
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.64/66.27
s(mark(z0)) → mark(s(z0)) 193.64/66.27
s(ok(z0)) → ok(s(z0)) 193.64/66.27
first(mark(z0), z1) → mark(first(z0, z1)) 193.64/66.27
first(z0, mark(z1)) → mark(first(z0, z1)) 193.64/66.27
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.64/66.27
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.64/66.27
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.64/66.27
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.64/66.27
proper(from(z0)) → from(proper(z0)) 193.64/66.27
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.64/66.27
proper(s(z0)) → s(proper(z0)) 193.64/66.27
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.64/66.27
proper(0) → ok(0) 193.64/66.27
proper(nil) → ok(nil) 193.64/66.27
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.64/66.27
top(mark(z0)) → top(proper(z0)) 193.64/66.27
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.27
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.27
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.27
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.27
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.27
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.27
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.27
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.27
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.27
S(mark(z0)) → c16(S(z0)) 193.64/66.27
S(ok(z0)) → c17(S(z0)) 193.64/66.27
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.27
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.27
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.27
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.27
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.27
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.27
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.27
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.27
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.27
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.27
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.27
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.27
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.27
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.27
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.27
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.27
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.27
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.27
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.27
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.27
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.27
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.27
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.27
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.27
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.27
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.64/66.27
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.64/66.27
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.27
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(s(0)) → c26(S(ok(0))) 193.64/66.27
PROPER(s(nil)) → c26(S(ok(nil))) 193.64/66.27
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.27
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.27
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.27
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.27
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.27
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0)) 193.64/66.27
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.27
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.27
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.27
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.27
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.27
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.27
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.27
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.27
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.27
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.27
S(mark(z0)) → c16(S(z0)) 193.64/66.27
S(ok(z0)) → c17(S(z0)) 193.64/66.27
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.27
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.27
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.27
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.27
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.27
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.27
PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.64/66.27
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.27
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.27
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.27
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.27
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.27
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.27
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.27
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.27
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.27
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.27
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.27
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.27
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.27
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.27
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.27
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.27
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.27
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.27
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.64/66.27
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.64/66.27
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.27
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(s(0)) → c26(S(ok(0))) 193.64/66.27
PROPER(s(nil)) → c26(S(ok(nil))) 193.64/66.27
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.27
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.27
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.27
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.27
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.27
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0)) 193.64/66.27
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.27
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, PROPER, TOP

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c30, c31, c32, c, c1, c5, c6, c7, c24, c24, c25, c25, c26, c26, c27, c27

193.64/66.27
193.64/66.27

(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(sel(z0, z1)) → c30(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(sel(x0, from(z0))) → c30(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.27
PROPER(sel(x0, cons(z0, z1))) → c30(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(sel(x0, s(z0))) → c30(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.27
PROPER(sel(x0, first(z0, z1))) → c30(SEL(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.27
PROPER(sel(x0, 0)) → c30(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 193.64/66.27
PROPER(sel(x0, nil)) → c30(SEL(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 193.64/66.27
PROPER(sel(x0, sel(z0, z1))) → c30(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(sel(from(z0), x1)) → c30(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.27
PROPER(sel(cons(z0, z1), x1)) → c30(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(sel(s(z0), x1)) → c30(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.27
PROPER(sel(first(z0, z1), x1)) → c30(SEL(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(sel(0, x1)) → c30(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 193.64/66.27
PROPER(sel(nil, x1)) → c30(SEL(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 193.64/66.27
PROPER(sel(sel(z0, z1), x1)) → c30(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
193.64/66.27
193.64/66.27

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.64/66.27
active(first(0, z0)) → mark(nil) 193.64/66.27
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.64/66.27
active(sel(0, cons(z0, z1))) → mark(z0) 193.64/66.27
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.64/66.27
active(from(z0)) → from(active(z0)) 193.64/66.27
active(cons(z0, z1)) → cons(active(z0), z1) 193.64/66.27
active(s(z0)) → s(active(z0)) 193.64/66.27
active(first(z0, z1)) → first(active(z0), z1) 193.64/66.27
active(first(z0, z1)) → first(z0, active(z1)) 193.64/66.27
active(sel(z0, z1)) → sel(active(z0), z1) 193.64/66.27
active(sel(z0, z1)) → sel(z0, active(z1)) 193.64/66.27
from(mark(z0)) → mark(from(z0)) 193.64/66.27
from(ok(z0)) → ok(from(z0)) 193.64/66.27
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.64/66.27
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.64/66.27
s(mark(z0)) → mark(s(z0)) 193.64/66.27
s(ok(z0)) → ok(s(z0)) 193.64/66.27
first(mark(z0), z1) → mark(first(z0, z1)) 193.64/66.27
first(z0, mark(z1)) → mark(first(z0, z1)) 193.64/66.27
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.64/66.27
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.64/66.27
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.64/66.27
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.64/66.27
proper(from(z0)) → from(proper(z0)) 193.64/66.27
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.64/66.27
proper(s(z0)) → s(proper(z0)) 193.64/66.27
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.64/66.27
proper(0) → ok(0) 193.64/66.27
proper(nil) → ok(nil) 193.64/66.27
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.64/66.27
top(mark(z0)) → top(proper(z0)) 193.64/66.27
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.27
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.27
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.27
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.27
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.27
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.27
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.27
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.27
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.27
S(mark(z0)) → c16(S(z0)) 193.64/66.27
S(ok(z0)) → c17(S(z0)) 193.64/66.27
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.27
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.27
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.27
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.27
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.27
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.27
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.27
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.27
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.27
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.27
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.27
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.27
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.27
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.27
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.27
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.27
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.27
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.27
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.27
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.27
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.27
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.27
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.27
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.27
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.64/66.27
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.64/66.27
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.27
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(s(0)) → c26(S(ok(0))) 193.64/66.27
PROPER(s(nil)) → c26(S(ok(nil))) 193.64/66.27
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.27
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.27
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.27
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.27
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.27
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0)) 193.64/66.27
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.27
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(sel(x0, from(z0))) → c30(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.27
PROPER(sel(x0, cons(z0, z1))) → c30(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(sel(x0, s(z0))) → c30(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.27
PROPER(sel(x0, first(z0, z1))) → c30(SEL(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.27
PROPER(sel(x0, 0)) → c30(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 193.64/66.27
PROPER(sel(x0, nil)) → c30(SEL(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 193.64/66.27
PROPER(sel(x0, sel(z0, z1))) → c30(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(sel(from(z0), x1)) → c30(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.27
PROPER(sel(cons(z0, z1), x1)) → c30(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(sel(s(z0), x1)) → c30(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.27
PROPER(sel(first(z0, z1), x1)) → c30(SEL(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(sel(0, x1)) → c30(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 193.64/66.27
PROPER(sel(nil, x1)) → c30(SEL(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 193.64/66.27
PROPER(sel(sel(z0, z1), x1)) → c30(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.27
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.27
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.27
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.27
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.27
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.27
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.27
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.27
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.27
S(mark(z0)) → c16(S(z0)) 193.64/66.27
S(ok(z0)) → c17(S(z0)) 193.64/66.27
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.27
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.27
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.27
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.27
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.27
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.27
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.27
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.27
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.27
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.27
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.27
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.27
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.27
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.27
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.27
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.27
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.27
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.27
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.27
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.27
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.27
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.27
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.27
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.27
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.27
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.27
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.27
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.27
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.27
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.27
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.27
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.27
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.27
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.64/66.27
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.64/66.27
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.27
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(s(0)) → c26(S(ok(0))) 193.64/66.27
PROPER(s(nil)) → c26(S(ok(nil))) 193.64/66.27
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.27
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.27
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.27
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.27
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.27
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.27
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.27
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.27
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.28
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0)) 193.64/66.28
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.28
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(x1)) 193.64/66.28
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.28
PROPER(sel(x0, from(z0))) → c30(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.28
PROPER(sel(x0, cons(z0, z1))) → c30(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.28
PROPER(sel(x0, s(z0))) → c30(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.28
PROPER(sel(x0, first(z0, z1))) → c30(SEL(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.28
PROPER(sel(x0, 0)) → c30(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 193.64/66.28
PROPER(sel(x0, nil)) → c30(SEL(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 193.64/66.28
PROPER(sel(x0, sel(z0, z1))) → c30(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.28
PROPER(sel(from(z0), x1)) → c30(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.28
PROPER(sel(cons(z0, z1), x1)) → c30(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.28
PROPER(sel(s(z0), x1)) → c30(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.28
PROPER(sel(first(z0, z1), x1)) → c30(SEL(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.28
PROPER(sel(0, x1)) → c30(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 193.64/66.28
PROPER(sel(nil, x1)) → c30(SEL(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 193.64/66.28
PROPER(sel(sel(z0, z1), x1)) → c30(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, TOP, PROPER

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c31, c32, c, c1, c5, c6, c7, c24, c24, c25, c25, c26, c26, c27, c27, c30

193.64/66.28
193.64/66.28

(31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts
193.64/66.28
193.64/66.28

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.64/66.28
active(first(0, z0)) → mark(nil) 193.64/66.28
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.64/66.28
active(sel(0, cons(z0, z1))) → mark(z0) 193.64/66.28
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.64/66.28
active(from(z0)) → from(active(z0)) 193.64/66.28
active(cons(z0, z1)) → cons(active(z0), z1) 193.64/66.28
active(s(z0)) → s(active(z0)) 193.64/66.28
active(first(z0, z1)) → first(active(z0), z1) 193.64/66.28
active(first(z0, z1)) → first(z0, active(z1)) 193.64/66.28
active(sel(z0, z1)) → sel(active(z0), z1) 193.64/66.28
active(sel(z0, z1)) → sel(z0, active(z1)) 193.64/66.28
from(mark(z0)) → mark(from(z0)) 193.64/66.28
from(ok(z0)) → ok(from(z0)) 193.64/66.28
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.64/66.28
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.64/66.28
s(mark(z0)) → mark(s(z0)) 193.64/66.28
s(ok(z0)) → ok(s(z0)) 193.64/66.28
first(mark(z0), z1) → mark(first(z0, z1)) 193.64/66.28
first(z0, mark(z1)) → mark(first(z0, z1)) 193.64/66.28
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.64/66.28
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.64/66.28
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.64/66.28
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.64/66.28
proper(from(z0)) → from(proper(z0)) 193.64/66.28
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.64/66.28
proper(s(z0)) → s(proper(z0)) 193.64/66.28
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.64/66.28
proper(0) → ok(0) 193.64/66.28
proper(nil) → ok(nil) 193.64/66.28
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.64/66.28
top(mark(z0)) → top(proper(z0)) 193.64/66.28
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.28
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.28
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.28
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.28
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.28
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.28
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.28
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.28
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.28
S(mark(z0)) → c16(S(z0)) 193.64/66.28
S(ok(z0)) → c17(S(z0)) 193.64/66.28
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.28
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.28
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.28
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.28
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.28
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.28
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.28
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.28
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.28
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.28
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.28
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.28
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.28
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.28
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.28
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.28
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.28
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.28
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.28
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.28
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.28
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.28
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.28
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.28
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.28
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.28
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.28
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.28
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.28
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.28
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.28
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.28
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.28
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.28
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.28
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.28
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.28
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.28
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.28
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.28
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.28
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.28
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.28
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.28
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.28
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.28
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.28
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.28
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.28
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.28
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.28
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.28
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.28
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.28
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.28
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.28
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.28
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.28
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.28
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.28
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.28
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.28
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.28
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.28
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.28
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.28
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.28
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.28
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.64/66.28
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.28
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.64/66.28
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.28
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.28
PROPER(s(0)) → c26(S(ok(0))) 193.64/66.28
PROPER(s(nil)) → c26(S(ok(nil))) 193.64/66.28
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.28
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.28
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.28
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.28
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.28
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.28
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.28
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.28
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.28
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.28
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0)) 193.64/66.28
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.28
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(x1)) 193.64/66.28
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.28
PROPER(sel(x0, from(z0))) → c30(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.28
PROPER(sel(x0, cons(z0, z1))) → c30(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.28
PROPER(sel(x0, s(z0))) → c30(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.28
PROPER(sel(x0, first(z0, z1))) → c30(SEL(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.28
PROPER(sel(x0, sel(z0, z1))) → c30(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.28
PROPER(sel(from(z0), x1)) → c30(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.28
PROPER(sel(cons(z0, z1), x1)) → c30(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.28
PROPER(sel(s(z0), x1)) → c30(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.28
PROPER(sel(first(z0, z1), x1)) → c30(SEL(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.28
PROPER(sel(sel(z0, z1), x1)) → c30(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.28
PROPER(sel(x0, 0)) → c30(SEL(proper(x0), ok(0)), PROPER(x0)) 193.64/66.28
PROPER(sel(x0, nil)) → c30(SEL(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.28
PROPER(sel(0, x1)) → c30(SEL(ok(0), proper(x1)), PROPER(x1)) 193.64/66.28
PROPER(sel(nil, x1)) → c30(SEL(ok(nil), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.28
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.28
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.28
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.28
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.28
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.28
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.28
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.28
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.28
S(mark(z0)) → c16(S(z0)) 193.64/66.28
S(ok(z0)) → c17(S(z0)) 193.64/66.28
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.28
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.28
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.28
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.28
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.28
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.28
TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) 193.64/66.28
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.28
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.28
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.28
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.28
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.28
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.28
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.28
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.28
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.28
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.28
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.28
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.28
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.28
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.28
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.28
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.28
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.28
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.28
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.28
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.28
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.28
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.28
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.28
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.28
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.28
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.28
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.28
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.28
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.28
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.28
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.28
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.28
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.28
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.28
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.28
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.28
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.28
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.28
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.28
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.28
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.28
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.28
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.28
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.28
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.28
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.28
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.28
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.28
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.28
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.28
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.28
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.28
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.28
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.28
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.28
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.28
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.28
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.28
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.28
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.28
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.28
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.64/66.28
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.28
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.64/66.28
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.28
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.28
PROPER(s(0)) → c26(S(ok(0))) 193.64/66.28
PROPER(s(nil)) → c26(S(ok(nil))) 193.64/66.28
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.28
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.28
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.28
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.28
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.28
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.28
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.28
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.28
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.28
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.28
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0)) 193.64/66.28
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.28
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(x1)) 193.64/66.28
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.29
PROPER(sel(x0, from(z0))) → c30(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.29
PROPER(sel(x0, cons(z0, z1))) → c30(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.29
PROPER(sel(x0, s(z0))) → c30(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.29
PROPER(sel(x0, first(z0, z1))) → c30(SEL(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.29
PROPER(sel(x0, sel(z0, z1))) → c30(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.29
PROPER(sel(from(z0), x1)) → c30(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.29
PROPER(sel(cons(z0, z1), x1)) → c30(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(sel(s(z0), x1)) → c30(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.29
PROPER(sel(first(z0, z1), x1)) → c30(SEL(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(sel(sel(z0, z1), x1)) → c30(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(sel(x0, 0)) → c30(SEL(proper(x0), ok(0)), PROPER(x0)) 193.64/66.29
PROPER(sel(x0, nil)) → c30(SEL(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.29
PROPER(sel(0, x1)) → c30(SEL(ok(0), proper(x1)), PROPER(x1)) 193.64/66.29
PROPER(sel(nil, x1)) → c30(SEL(ok(nil), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, TOP, PROPER

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c31, c32, c, c1, c5, c6, c7, c24, c24, c25, c25, c26, c26, c27, c27, c30, c30

193.64/66.29
193.64/66.29

(33) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace TOP(mark(z0)) → c31(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(from(z0))) → c31(TOP(from(proper(z0))), PROPER(from(z0))) 193.64/66.29
TOP(mark(cons(z0, z1))) → c31(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.29
TOP(mark(s(z0))) → c31(TOP(s(proper(z0))), PROPER(s(z0))) 193.64/66.29
TOP(mark(first(z0, z1))) → c31(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.29
TOP(mark(0)) → c31(TOP(ok(0)), PROPER(0)) 193.64/66.29
TOP(mark(nil)) → c31(TOP(ok(nil)), PROPER(nil)) 193.64/66.29
TOP(mark(sel(z0, z1))) → c31(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
193.64/66.29
193.64/66.29

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.64/66.29
active(first(0, z0)) → mark(nil) 193.64/66.29
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.64/66.29
active(sel(0, cons(z0, z1))) → mark(z0) 193.64/66.29
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.64/66.29
active(from(z0)) → from(active(z0)) 193.64/66.29
active(cons(z0, z1)) → cons(active(z0), z1) 193.64/66.29
active(s(z0)) → s(active(z0)) 193.64/66.29
active(first(z0, z1)) → first(active(z0), z1) 193.64/66.29
active(first(z0, z1)) → first(z0, active(z1)) 193.64/66.29
active(sel(z0, z1)) → sel(active(z0), z1) 193.64/66.29
active(sel(z0, z1)) → sel(z0, active(z1)) 193.64/66.29
from(mark(z0)) → mark(from(z0)) 193.64/66.29
from(ok(z0)) → ok(from(z0)) 193.64/66.29
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.64/66.29
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.64/66.29
s(mark(z0)) → mark(s(z0)) 193.64/66.29
s(ok(z0)) → ok(s(z0)) 193.64/66.29
first(mark(z0), z1) → mark(first(z0, z1)) 193.64/66.29
first(z0, mark(z1)) → mark(first(z0, z1)) 193.64/66.29
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.64/66.29
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.64/66.29
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.64/66.29
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.64/66.29
proper(from(z0)) → from(proper(z0)) 193.64/66.29
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.64/66.29
proper(s(z0)) → s(proper(z0)) 193.64/66.29
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.64/66.29
proper(0) → ok(0) 193.64/66.29
proper(nil) → ok(nil) 193.64/66.29
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.64/66.29
top(mark(z0)) → top(proper(z0)) 193.64/66.29
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.29
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.29
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.29
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.29
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.29
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.29
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.29
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.29
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.29
S(mark(z0)) → c16(S(z0)) 193.64/66.29
S(ok(z0)) → c17(S(z0)) 193.64/66.29
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.29
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.29
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.29
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.29
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.29
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.29
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.29
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.29
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.29
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.29
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.29
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.29
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.29
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.29
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.29
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.29
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.29
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.29
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.29
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.29
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.29
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.29
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.29
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.29
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.29
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.29
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.29
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.29
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.29
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.29
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.29
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.29
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.29
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.29
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.29
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.29
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.29
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.29
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.29
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.29
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.29
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.29
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.29
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.29
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.29
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.29
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.64/66.29
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.29
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.64/66.29
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.29
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.29
PROPER(s(0)) → c26(S(ok(0))) 193.64/66.29
PROPER(s(nil)) → c26(S(ok(nil))) 193.64/66.29
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.29
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.29
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.29
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.29
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.29
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.29
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.29
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0)) 193.64/66.29
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.29
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(x1)) 193.64/66.29
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.29
PROPER(sel(x0, from(z0))) → c30(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.29
PROPER(sel(x0, cons(z0, z1))) → c30(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.29
PROPER(sel(x0, s(z0))) → c30(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.29
PROPER(sel(x0, first(z0, z1))) → c30(SEL(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.29
PROPER(sel(x0, sel(z0, z1))) → c30(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.29
PROPER(sel(from(z0), x1)) → c30(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.29
PROPER(sel(cons(z0, z1), x1)) → c30(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(sel(s(z0), x1)) → c30(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.29
PROPER(sel(first(z0, z1), x1)) → c30(SEL(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(sel(sel(z0, z1), x1)) → c30(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(sel(x0, 0)) → c30(SEL(proper(x0), ok(0)), PROPER(x0)) 193.64/66.29
PROPER(sel(x0, nil)) → c30(SEL(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.29
PROPER(sel(0, x1)) → c30(SEL(ok(0), proper(x1)), PROPER(x1)) 193.64/66.29
PROPER(sel(nil, x1)) → c30(SEL(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.29
TOP(mark(from(z0))) → c31(TOP(from(proper(z0))), PROPER(from(z0))) 193.64/66.29
TOP(mark(cons(z0, z1))) → c31(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.29
TOP(mark(s(z0))) → c31(TOP(s(proper(z0))), PROPER(s(z0))) 193.64/66.29
TOP(mark(first(z0, z1))) → c31(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.29
TOP(mark(0)) → c31(TOP(ok(0)), PROPER(0)) 193.64/66.29
TOP(mark(nil)) → c31(TOP(ok(nil)), PROPER(nil)) 193.64/66.29
TOP(mark(sel(z0, z1))) → c31(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.29
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.29
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.29
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.29
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.29
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.29
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.29
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.29
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.29
S(mark(z0)) → c16(S(z0)) 193.64/66.29
S(ok(z0)) → c17(S(z0)) 193.64/66.29
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.29
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.29
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.29
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.29
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.29
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.29
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.29
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.29
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.29
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.29
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.29
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.29
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.29
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.29
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.29
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.29
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.29
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.29
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.29
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.29
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.29
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.29
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.29
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.29
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.29
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.29
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.29
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.29
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.29
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.29
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.29
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.29
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.29
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.29
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.29
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.29
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.29
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.29
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.29
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.29
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.29
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.29
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.29
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.29
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.29
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.29
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.64/66.29
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.29
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.64/66.29
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.29
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.29
PROPER(s(0)) → c26(S(ok(0))) 193.64/66.29
PROPER(s(nil)) → c26(S(ok(nil))) 193.64/66.29
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.29
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.29
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.29
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.29
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.29
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.29
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.29
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0)) 193.64/66.29
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.29
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(x1)) 193.64/66.29
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.29
PROPER(sel(x0, from(z0))) → c30(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.29
PROPER(sel(x0, cons(z0, z1))) → c30(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.29
PROPER(sel(x0, s(z0))) → c30(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.29
PROPER(sel(x0, first(z0, z1))) → c30(SEL(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.29
PROPER(sel(x0, sel(z0, z1))) → c30(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.29
PROPER(sel(from(z0), x1)) → c30(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.29
PROPER(sel(cons(z0, z1), x1)) → c30(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(sel(s(z0), x1)) → c30(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.29
PROPER(sel(first(z0, z1), x1)) → c30(SEL(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(sel(sel(z0, z1), x1)) → c30(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(sel(x0, 0)) → c30(SEL(proper(x0), ok(0)), PROPER(x0)) 193.64/66.29
PROPER(sel(x0, nil)) → c30(SEL(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.29
PROPER(sel(0, x1)) → c30(SEL(ok(0), proper(x1)), PROPER(x1)) 193.64/66.29
PROPER(sel(nil, x1)) → c30(SEL(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.29
TOP(mark(from(z0))) → c31(TOP(from(proper(z0))), PROPER(from(z0))) 193.64/66.29
TOP(mark(cons(z0, z1))) → c31(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.29
TOP(mark(s(z0))) → c31(TOP(s(proper(z0))), PROPER(s(z0))) 193.64/66.29
TOP(mark(first(z0, z1))) → c31(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.29
TOP(mark(0)) → c31(TOP(ok(0)), PROPER(0)) 193.64/66.29
TOP(mark(nil)) → c31(TOP(ok(nil)), PROPER(nil)) 193.64/66.29
TOP(mark(sel(z0, z1))) → c31(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, TOP, PROPER

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c32, c, c1, c5, c6, c7, c24, c24, c25, c25, c26, c26, c27, c27, c30, c30, c31

193.64/66.29
193.64/66.29

(35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts
193.64/66.29
193.64/66.29

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.64/66.29
active(first(0, z0)) → mark(nil) 193.64/66.29
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.64/66.29
active(sel(0, cons(z0, z1))) → mark(z0) 193.64/66.29
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.64/66.29
active(from(z0)) → from(active(z0)) 193.64/66.29
active(cons(z0, z1)) → cons(active(z0), z1) 193.64/66.29
active(s(z0)) → s(active(z0)) 193.64/66.29
active(first(z0, z1)) → first(active(z0), z1) 193.64/66.29
active(first(z0, z1)) → first(z0, active(z1)) 193.64/66.29
active(sel(z0, z1)) → sel(active(z0), z1) 193.64/66.29
active(sel(z0, z1)) → sel(z0, active(z1)) 193.64/66.29
from(mark(z0)) → mark(from(z0)) 193.64/66.29
from(ok(z0)) → ok(from(z0)) 193.64/66.29
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.64/66.29
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.64/66.29
s(mark(z0)) → mark(s(z0)) 193.64/66.29
s(ok(z0)) → ok(s(z0)) 193.64/66.29
first(mark(z0), z1) → mark(first(z0, z1)) 193.64/66.29
first(z0, mark(z1)) → mark(first(z0, z1)) 193.64/66.29
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.64/66.29
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.64/66.29
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.64/66.29
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.64/66.29
proper(from(z0)) → from(proper(z0)) 193.64/66.29
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.64/66.29
proper(s(z0)) → s(proper(z0)) 193.64/66.29
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.64/66.29
proper(0) → ok(0) 193.64/66.29
proper(nil) → ok(nil) 193.64/66.29
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.64/66.29
top(mark(z0)) → top(proper(z0)) 193.64/66.29
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.29
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.29
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.29
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.29
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.29
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.29
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.29
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.29
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.29
S(mark(z0)) → c16(S(z0)) 193.64/66.29
S(ok(z0)) → c17(S(z0)) 193.64/66.29
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.29
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.29
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.29
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.29
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.29
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.29
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.29
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.29
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.29
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.29
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.29
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.29
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.29
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.29
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.29
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.29
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.29
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.29
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.29
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.29
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.29
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.29
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.29
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.29
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.29
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.29
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.29
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.29
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.29
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.29
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.29
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.29
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.29
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.29
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.29
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.29
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.29
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.29
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.29
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.29
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.29
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.29
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.29
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.29
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.29
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.29
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.64/66.29
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.29
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.64/66.29
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.29
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.29
PROPER(s(0)) → c26(S(ok(0))) 193.64/66.29
PROPER(s(nil)) → c26(S(ok(nil))) 193.64/66.29
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.29
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.29
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.29
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.29
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.29
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.29
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.29
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0)) 193.64/66.29
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.29
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(x1)) 193.64/66.29
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.29
PROPER(sel(x0, from(z0))) → c30(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.29
PROPER(sel(x0, cons(z0, z1))) → c30(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.29
PROPER(sel(x0, s(z0))) → c30(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.29
PROPER(sel(x0, first(z0, z1))) → c30(SEL(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.29
PROPER(sel(x0, sel(z0, z1))) → c30(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.29
PROPER(sel(from(z0), x1)) → c30(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.29
PROPER(sel(cons(z0, z1), x1)) → c30(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(sel(s(z0), x1)) → c30(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.29
PROPER(sel(first(z0, z1), x1)) → c30(SEL(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(sel(sel(z0, z1), x1)) → c30(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.29
PROPER(sel(x0, 0)) → c30(SEL(proper(x0), ok(0)), PROPER(x0)) 193.64/66.29
PROPER(sel(x0, nil)) → c30(SEL(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.29
PROPER(sel(0, x1)) → c30(SEL(ok(0), proper(x1)), PROPER(x1)) 193.64/66.29
PROPER(sel(nil, x1)) → c30(SEL(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.29
TOP(mark(from(z0))) → c31(TOP(from(proper(z0))), PROPER(from(z0))) 193.64/66.29
TOP(mark(cons(z0, z1))) → c31(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.29
TOP(mark(s(z0))) → c31(TOP(s(proper(z0))), PROPER(s(z0))) 193.64/66.29
TOP(mark(first(z0, z1))) → c31(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.29
TOP(mark(sel(z0, z1))) → c31(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.29
TOP(mark(0)) → c31(TOP(ok(0))) 193.64/66.29
TOP(mark(nil)) → c31(TOP(ok(nil)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.29
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.29
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.29
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.29
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.29
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.29
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.29
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.29
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.29
S(mark(z0)) → c16(S(z0)) 193.64/66.29
S(ok(z0)) → c17(S(z0)) 193.64/66.29
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.29
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.29
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.29
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.29
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.29
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.29
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.29
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.29
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.29
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.29
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.29
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.29
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.29
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.29
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.29
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.29
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.29
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.29
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.29
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.30
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.30
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.30
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.30
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.30
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.30
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.30
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.30
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.30
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.30
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.30
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.30
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.30
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.30
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.30
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.30
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.30
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.30
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.30
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.30
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.30
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.30
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.30
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.30
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.30
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.30
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.30
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.30
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.30
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.30
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.30
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.30
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.30
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.30
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.30
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.30
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.30
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.30
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.30
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.30
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.30
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.30
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.30
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.30
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.64/66.30
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.30
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.64/66.30
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.30
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.30
PROPER(s(0)) → c26(S(ok(0))) 193.64/66.30
PROPER(s(nil)) → c26(S(ok(nil))) 193.64/66.30
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.30
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.30
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.30
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.30
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.30
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.30
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.30
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.30
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.30
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.30
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0)) 193.64/66.30
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.30
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(x1)) 193.64/66.30
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.30
PROPER(sel(x0, from(z0))) → c30(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.30
PROPER(sel(x0, cons(z0, z1))) → c30(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.30
PROPER(sel(x0, s(z0))) → c30(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.30
PROPER(sel(x0, first(z0, z1))) → c30(SEL(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.30
PROPER(sel(x0, sel(z0, z1))) → c30(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.30
PROPER(sel(from(z0), x1)) → c30(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.30
PROPER(sel(cons(z0, z1), x1)) → c30(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.30
PROPER(sel(s(z0), x1)) → c30(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.30
PROPER(sel(first(z0, z1), x1)) → c30(SEL(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.30
PROPER(sel(sel(z0, z1), x1)) → c30(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.30
PROPER(sel(x0, 0)) → c30(SEL(proper(x0), ok(0)), PROPER(x0)) 193.64/66.30
PROPER(sel(x0, nil)) → c30(SEL(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.30
PROPER(sel(0, x1)) → c30(SEL(ok(0), proper(x1)), PROPER(x1)) 193.64/66.30
PROPER(sel(nil, x1)) → c30(SEL(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.30
TOP(mark(from(z0))) → c31(TOP(from(proper(z0))), PROPER(from(z0))) 193.64/66.30
TOP(mark(cons(z0, z1))) → c31(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.30
TOP(mark(s(z0))) → c31(TOP(s(proper(z0))), PROPER(s(z0))) 193.64/66.30
TOP(mark(first(z0, z1))) → c31(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.30
TOP(mark(sel(z0, z1))) → c31(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.30
TOP(mark(0)) → c31(TOP(ok(0))) 193.64/66.30
TOP(mark(nil)) → c31(TOP(ok(nil)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, TOP, PROPER

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c32, c, c1, c5, c6, c7, c24, c24, c25, c25, c26, c26, c27, c27, c30, c30, c31, c31

193.64/66.30
193.64/66.30

(37) 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)) → c31(TOP(ok(0))) 193.64/66.30
TOP(mark(nil)) → c31(TOP(ok(nil)))
We considered the (Usable) Rules:

proper(from(z0)) → from(proper(z0)) 193.64/66.30
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.64/66.30
proper(s(z0)) → s(proper(z0)) 193.64/66.30
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.64/66.30
proper(0) → ok(0) 193.64/66.30
proper(nil) → ok(nil) 193.64/66.30
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.64/66.30
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.64/66.30
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.64/66.30
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.64/66.30
first(mark(z0), z1) → mark(first(z0, z1)) 193.64/66.30
first(z0, mark(z1)) → mark(first(z0, z1)) 193.64/66.30
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.64/66.30
s(mark(z0)) → mark(s(z0)) 193.64/66.30
s(ok(z0)) → ok(s(z0)) 193.64/66.30
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.64/66.30
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.64/66.30
from(mark(z0)) → mark(from(z0)) 193.64/66.30
from(ok(z0)) → ok(from(z0)) 193.64/66.30
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.64/66.30
active(first(0, z0)) → mark(nil) 193.64/66.30
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.64/66.30
active(sel(0, cons(z0, z1))) → mark(z0) 193.64/66.30
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.64/66.30
active(from(z0)) → from(active(z0)) 193.64/66.30
active(cons(z0, z1)) → cons(active(z0), z1) 193.64/66.30
active(s(z0)) → s(active(z0)) 193.64/66.30
active(first(z0, z1)) → first(active(z0), z1) 193.64/66.30
active(first(z0, z1)) → first(z0, active(z1)) 193.64/66.30
active(sel(z0, z1)) → sel(active(z0), z1) 193.64/66.30
active(sel(z0, z1)) → sel(z0, active(z1))
And the Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.30
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.30
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.30
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.30
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.30
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.30
S(mark(z0)) → c16(S(z0)) 193.64/66.30
S(ok(z0)) → c17(S(z0)) 193.64/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.30
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.30
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.30
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.30
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.30
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.30
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.30
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.30
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.30
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.30
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.30
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.30
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.30
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.30
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.30
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.30
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.30
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.30
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.30
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.30
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.30
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.30
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.30
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.30
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.30
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.30
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.30
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.30
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.30
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.30
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.30
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.30
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.30
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.30
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.30
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.30
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.30
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.30
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.30
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.30
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.30
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.64/66.30
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.30
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.64/66.30
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.30
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.30
PROPER(from(0)) → c24(FROM(ok(0))) 193.64/66.30
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.64/66.30
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.30
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.30
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.30
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.30
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.30
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.30
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.30
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.30
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.30
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.30
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.64/66.30
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.30
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.64/66.30
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.30
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.64/66.30
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.30
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.64/66.30
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.30
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.30
PROPER(s(0)) → c26(S(ok(0))) 193.64/66.30
PROPER(s(nil)) → c26(S(ok(nil))) 193.64/66.30
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.30
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.30
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.30
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.30
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.30
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.30
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.30
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.30
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.30
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.30
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0)) 193.64/66.30
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.30
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(x1)) 193.64/66.30
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.30
PROPER(sel(x0, from(z0))) → c30(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.64/66.30
PROPER(sel(x0, cons(z0, z1))) → c30(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.64/66.30
PROPER(sel(x0, s(z0))) → c30(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.64/66.30
PROPER(sel(x0, first(z0, z1))) → c30(SEL(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.64/66.30
PROPER(sel(x0, sel(z0, z1))) → c30(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.64/66.30
PROPER(sel(from(z0), x1)) → c30(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.64/66.30
PROPER(sel(cons(z0, z1), x1)) → c30(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.64/66.30
PROPER(sel(s(z0), x1)) → c30(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.64/66.30
PROPER(sel(first(z0, z1), x1)) → c30(SEL(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.64/66.30
PROPER(sel(sel(z0, z1), x1)) → c30(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.64/66.30
PROPER(sel(x0, 0)) → c30(SEL(proper(x0), ok(0)), PROPER(x0)) 193.64/66.30
PROPER(sel(x0, nil)) → c30(SEL(proper(x0), ok(nil)), PROPER(x0)) 193.64/66.30
PROPER(sel(0, x1)) → c30(SEL(ok(0), proper(x1)), PROPER(x1)) 193.64/66.30
PROPER(sel(nil, x1)) → c30(SEL(ok(nil), proper(x1)), PROPER(x1)) 193.64/66.30
TOP(mark(from(z0))) → c31(TOP(from(proper(z0))), PROPER(from(z0))) 193.64/66.30
TOP(mark(cons(z0, z1))) → c31(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.64/66.30
TOP(mark(s(z0))) → c31(TOP(s(proper(z0))), PROPER(s(z0))) 193.64/66.30
TOP(mark(first(z0, z1))) → c31(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.64/66.30
TOP(mark(sel(z0, z1))) → c31(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.64/66.30
TOP(mark(0)) → c31(TOP(ok(0))) 193.64/66.30
TOP(mark(nil)) → c31(TOP(ok(nil)))
The order we found is given by the following interpretation:
Polynomial interpretation : 193.64/66.30

POL(0) = 0    193.64/66.30
POL(ACTIVE(x1)) = 0    193.64/66.30
POL(CONS(x1, x2)) = 0    193.64/66.30
POL(FIRST(x1, x2)) = 0    193.64/66.30
POL(FROM(x1)) = 0    193.64/66.30
POL(PROPER(x1)) = 0    193.64/66.30
POL(S(x1)) = 0    193.64/66.30
POL(SEL(x1, x2)) = 0    193.64/66.30
POL(TOP(x1)) = [4]x1    193.64/66.30
POL(active(x1)) = x1    193.64/66.30
POL(c(x1)) = x1    193.64/66.30
POL(c1(x1)) = x1    193.64/66.30
POL(c10(x1, x2)) = x1 + x2    193.64/66.30
POL(c11(x1, x2)) = x1 + x2    193.64/66.30
POL(c12(x1)) = x1    193.64/66.30
POL(c13(x1)) = x1    193.64/66.30
POL(c14(x1)) = x1    193.64/66.30
POL(c15(x1)) = x1    193.64/66.30
POL(c16(x1)) = x1    193.64/66.30
POL(c17(x1)) = x1    193.64/66.30
POL(c18(x1)) = x1    193.64/66.30
POL(c19(x1)) = x1    193.64/66.30
POL(c20(x1)) = x1    193.64/66.30
POL(c21(x1)) = x1    193.64/66.30
POL(c22(x1)) = x1    193.64/66.30
POL(c23(x1)) = x1    193.64/66.30
POL(c24(x1)) = x1    193.64/66.30
POL(c24(x1, x2)) = x1 + x2    193.64/66.30
POL(c25(x1, x2)) = x1 + x2    193.64/66.30
POL(c25(x1, x2, x3)) = x1 + x2 + x3    193.64/66.30
POL(c26(x1)) = x1    193.64/66.30
POL(c26(x1, x2)) = x1 + x2    193.64/66.30
POL(c27(x1, x2)) = x1 + x2    193.64/66.30
POL(c27(x1, x2, x3)) = x1 + x2 + x3    193.64/66.30
POL(c30(x1, x2)) = x1 + x2    193.64/66.30
POL(c30(x1, x2, x3)) = x1 + x2 + x3    193.64/66.30
POL(c31(x1)) = x1    193.64/66.30
POL(c31(x1, x2)) = x1 + x2    193.64/66.30
POL(c32(x1, x2)) = x1 + x2    193.64/66.30
POL(c4(x1)) = x1    193.64/66.30
POL(c5(x1, x2)) = x1 + x2    193.64/66.30
POL(c6(x1, x2)) = x1 + x2    193.64/66.30
POL(c7(x1, x2)) = x1 + x2    193.64/66.30
POL(c8(x1, x2)) = x1 + x2    193.64/66.30
POL(c9(x1, x2)) = x1 + x2    193.64/66.30
POL(cons(x1, x2)) = [1]    193.64/66.30
POL(first(x1, x2)) = [1]    193.64/66.30
POL(from(x1)) = [1]    193.64/66.30
POL(mark(x1)) = [1]    193.64/66.30
POL(nil) = 0    193.64/66.30
POL(ok(x1)) = x1    193.64/66.30
POL(proper(x1)) = 0    193.64/66.30
POL(s(x1)) = [1]    193.64/66.30
POL(sel(x1, x2)) = [1]   
193.64/66.30
193.64/66.30

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.64/66.30
active(first(0, z0)) → mark(nil) 193.64/66.30
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.64/66.30
active(sel(0, cons(z0, z1))) → mark(z0) 193.64/66.30
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.64/66.30
active(from(z0)) → from(active(z0)) 193.64/66.30
active(cons(z0, z1)) → cons(active(z0), z1) 193.64/66.30
active(s(z0)) → s(active(z0)) 193.64/66.30
active(first(z0, z1)) → first(active(z0), z1) 193.64/66.30
active(first(z0, z1)) → first(z0, active(z1)) 193.64/66.30
active(sel(z0, z1)) → sel(active(z0), z1) 193.64/66.30
active(sel(z0, z1)) → sel(z0, active(z1)) 193.64/66.30
from(mark(z0)) → mark(from(z0)) 193.64/66.30
from(ok(z0)) → ok(from(z0)) 193.64/66.30
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.64/66.30
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.64/66.30
s(mark(z0)) → mark(s(z0)) 193.64/66.30
s(ok(z0)) → ok(s(z0)) 193.64/66.30
first(mark(z0), z1) → mark(first(z0, z1)) 193.64/66.30
first(z0, mark(z1)) → mark(first(z0, z1)) 193.64/66.30
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.64/66.30
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.64/66.30
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.64/66.30
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.64/66.30
proper(from(z0)) → from(proper(z0)) 193.64/66.30
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.64/66.30
proper(s(z0)) → s(proper(z0)) 193.64/66.30
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.64/66.30
proper(0) → ok(0) 193.64/66.30
proper(nil) → ok(nil) 193.64/66.30
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.64/66.30
top(mark(z0)) → top(proper(z0)) 193.64/66.30
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.64/66.30
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.64/66.30
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.64/66.30
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.64/66.30
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.64/66.30
FROM(mark(z0)) → c12(FROM(z0)) 193.64/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.64/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.64/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.64/66.30
S(mark(z0)) → c16(S(z0)) 193.64/66.30
S(ok(z0)) → c17(S(z0)) 193.64/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.64/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.64/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.64/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.64/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.64/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.64/66.30
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.64/66.30
ACTIVE(from(z0)) → c(S(z0)) 193.64/66.30
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.64/66.30
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.64/66.30
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.30
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.30
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.30
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.30
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.30
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.64/66.30
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.30
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.64/66.30
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.30
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.30
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.30
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.64/66.30
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.64/66.30
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.64/66.30
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.30
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.30
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.30
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.64/66.30
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.64/66.30
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.64/66.30
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.64/66.30
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.64/66.30
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.64/66.30
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.64/66.30
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.64/66.30
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.64/66.30
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.64/66.30
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.64/66.30
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.64/66.30
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.64/66.30
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.64/66.30
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.64/66.30
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.64/66.30
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.64/66.30
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.64/66.30
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.96/66.30
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.96/66.30
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.96/66.30
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.96/66.30
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(from(0)) → c24(FROM(ok(0))) 193.96/66.30
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.96/66.30
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.96/66.30
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.96/66.30
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.96/66.30
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.96/66.30
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.96/66.30
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.96/66.30
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.96/66.30
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.96/66.30
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.96/66.30
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.96/66.30
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(s(0)) → c26(S(ok(0))) 193.96/66.30
PROPER(s(nil)) → c26(S(ok(nil))) 193.96/66.30
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.96/66.30
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.96/66.30
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.96/66.30
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.96/66.30
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.96/66.30
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0)) 193.96/66.30
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0)) 193.96/66.30
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(sel(x0, from(z0))) → c30(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.96/66.30
PROPER(sel(x0, cons(z0, z1))) → c30(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(sel(x0, s(z0))) → c30(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.96/66.30
PROPER(sel(x0, first(z0, z1))) → c30(SEL(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.96/66.30
PROPER(sel(x0, sel(z0, z1))) → c30(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(sel(from(z0), x1)) → c30(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.96/66.30
PROPER(sel(cons(z0, z1), x1)) → c30(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(sel(s(z0), x1)) → c30(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.96/66.30
PROPER(sel(first(z0, z1), x1)) → c30(SEL(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(sel(sel(z0, z1), x1)) → c30(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(sel(x0, 0)) → c30(SEL(proper(x0), ok(0)), PROPER(x0)) 193.96/66.30
PROPER(sel(x0, nil)) → c30(SEL(proper(x0), ok(nil)), PROPER(x0)) 193.96/66.30
PROPER(sel(0, x1)) → c30(SEL(ok(0), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(sel(nil, x1)) → c30(SEL(ok(nil), proper(x1)), PROPER(x1)) 193.96/66.30
TOP(mark(from(z0))) → c31(TOP(from(proper(z0))), PROPER(from(z0))) 193.96/66.30
TOP(mark(cons(z0, z1))) → c31(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.96/66.30
TOP(mark(s(z0))) → c31(TOP(s(proper(z0))), PROPER(s(z0))) 193.96/66.30
TOP(mark(first(z0, z1))) → c31(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.96/66.30
TOP(mark(sel(z0, z1))) → c31(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.96/66.30
TOP(mark(0)) → c31(TOP(ok(0))) 193.96/66.30
TOP(mark(nil)) → c31(TOP(ok(nil)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.96/66.30
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.96/66.30
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.96/66.30
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.96/66.30
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.96/66.30
FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.96/66.30
TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) 193.96/66.30
ACTIVE(from(z0)) → c(S(z0)) 193.96/66.30
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.96/66.30
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.96/66.30
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.96/66.30
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.96/66.30
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.96/66.30
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.96/66.30
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.96/66.30
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.96/66.30
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.96/66.30
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.96/66.30
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.96/66.30
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.96/66.30
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.96/66.30
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.96/66.30
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.96/66.30
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.96/66.30
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.96/66.30
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.96/66.30
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(from(0)) → c24(FROM(ok(0))) 193.96/66.30
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.96/66.30
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.96/66.30
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.96/66.30
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.96/66.30
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.96/66.30
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.96/66.30
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.96/66.30
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.96/66.30
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.96/66.30
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.96/66.30
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.96/66.30
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(s(0)) → c26(S(ok(0))) 193.96/66.30
PROPER(s(nil)) → c26(S(ok(nil))) 193.96/66.30
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.96/66.30
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.96/66.30
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.96/66.30
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.96/66.30
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.96/66.30
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0)) 193.96/66.30
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0)) 193.96/66.30
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(sel(x0, from(z0))) → c30(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.96/66.30
PROPER(sel(x0, cons(z0, z1))) → c30(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(sel(x0, s(z0))) → c30(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.96/66.30
PROPER(sel(x0, first(z0, z1))) → c30(SEL(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.96/66.30
PROPER(sel(x0, sel(z0, z1))) → c30(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(sel(from(z0), x1)) → c30(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.96/66.30
PROPER(sel(cons(z0, z1), x1)) → c30(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(sel(s(z0), x1)) → c30(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.96/66.30
PROPER(sel(first(z0, z1), x1)) → c30(SEL(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(sel(sel(z0, z1), x1)) → c30(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(sel(x0, 0)) → c30(SEL(proper(x0), ok(0)), PROPER(x0)) 193.96/66.30
PROPER(sel(x0, nil)) → c30(SEL(proper(x0), ok(nil)), PROPER(x0)) 193.96/66.30
PROPER(sel(0, x1)) → c30(SEL(ok(0), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(sel(nil, x1)) → c30(SEL(ok(nil), proper(x1)), PROPER(x1)) 193.96/66.30
TOP(mark(from(z0))) → c31(TOP(from(proper(z0))), PROPER(from(z0))) 193.96/66.30
TOP(mark(cons(z0, z1))) → c31(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.96/66.30
TOP(mark(s(z0))) → c31(TOP(s(proper(z0))), PROPER(s(z0))) 193.96/66.30
TOP(mark(first(z0, z1))) → c31(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.96/66.30
TOP(mark(sel(z0, z1))) → c31(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
K tuples:

TOP(mark(0)) → c31(TOP(ok(0))) 193.96/66.30
TOP(mark(nil)) → c31(TOP(ok(nil)))
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, TOP, PROPER

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c32, c, c1, c5, c6, c7, c24, c24, c25, c25, c26, c26, c27, c27, c30, c30, c31, c31

193.96/66.30
193.96/66.30

(39) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace TOP(ok(z0)) → c32(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(from(z0))) → c32(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.96/66.30
TOP(ok(first(0, z0))) → c32(TOP(mark(nil)), ACTIVE(first(0, z0))) 193.96/66.30
TOP(ok(first(s(z0), cons(z1, z2)))) → c32(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.96/66.30
TOP(ok(sel(0, cons(z0, z1)))) → c32(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.96/66.30
TOP(ok(sel(s(z0), cons(z1, z2)))) → c32(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.96/66.30
TOP(ok(from(z0))) → c32(TOP(from(active(z0))), ACTIVE(from(z0))) 193.96/66.30
TOP(ok(cons(z0, z1))) → c32(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.96/66.30
TOP(ok(s(z0))) → c32(TOP(s(active(z0))), ACTIVE(s(z0))) 193.96/66.30
TOP(ok(first(z0, z1))) → c32(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.96/66.30
TOP(ok(first(z0, z1))) → c32(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.96/66.30
TOP(ok(sel(z0, z1))) → c32(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.96/66.30
TOP(ok(sel(z0, z1))) → c32(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
193.96/66.30
193.96/66.30

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.96/66.30
active(first(0, z0)) → mark(nil) 193.96/66.30
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.96/66.30
active(sel(0, cons(z0, z1))) → mark(z0) 193.96/66.30
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.96/66.30
active(from(z0)) → from(active(z0)) 193.96/66.30
active(cons(z0, z1)) → cons(active(z0), z1) 193.96/66.30
active(s(z0)) → s(active(z0)) 193.96/66.30
active(first(z0, z1)) → first(active(z0), z1) 193.96/66.30
active(first(z0, z1)) → first(z0, active(z1)) 193.96/66.30
active(sel(z0, z1)) → sel(active(z0), z1) 193.96/66.30
active(sel(z0, z1)) → sel(z0, active(z1)) 193.96/66.30
from(mark(z0)) → mark(from(z0)) 193.96/66.30
from(ok(z0)) → ok(from(z0)) 193.96/66.30
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.96/66.30
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.96/66.30
s(mark(z0)) → mark(s(z0)) 193.96/66.30
s(ok(z0)) → ok(s(z0)) 193.96/66.30
first(mark(z0), z1) → mark(first(z0, z1)) 193.96/66.30
first(z0, mark(z1)) → mark(first(z0, z1)) 193.96/66.30
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.96/66.30
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.96/66.30
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.96/66.30
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.96/66.30
proper(from(z0)) → from(proper(z0)) 193.96/66.30
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.96/66.30
proper(s(z0)) → s(proper(z0)) 193.96/66.30
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.96/66.30
proper(0) → ok(0) 193.96/66.30
proper(nil) → ok(nil) 193.96/66.30
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.96/66.30
top(mark(z0)) → top(proper(z0)) 193.96/66.30
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.96/66.30
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.96/66.30
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.96/66.30
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.96/66.30
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.96/66.30
FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.96/66.30
ACTIVE(from(z0)) → c(S(z0)) 193.96/66.30
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.96/66.30
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.96/66.30
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.96/66.30
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.96/66.30
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.96/66.30
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.96/66.30
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.96/66.30
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.96/66.30
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.96/66.30
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.96/66.30
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.96/66.30
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.96/66.30
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.96/66.30
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.96/66.30
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.96/66.30
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.96/66.30
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.96/66.30
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.96/66.30
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(from(0)) → c24(FROM(ok(0))) 193.96/66.30
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.96/66.30
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.96/66.30
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.96/66.30
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.96/66.30
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.96/66.30
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.96/66.30
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.96/66.30
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.96/66.30
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.96/66.30
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.96/66.30
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.96/66.30
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(s(0)) → c26(S(ok(0))) 193.96/66.30
PROPER(s(nil)) → c26(S(ok(nil))) 193.96/66.30
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.96/66.30
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.96/66.30
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.96/66.30
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.96/66.30
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.96/66.30
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0)) 193.96/66.30
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0)) 193.96/66.30
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(sel(x0, from(z0))) → c30(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.96/66.30
PROPER(sel(x0, cons(z0, z1))) → c30(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(sel(x0, s(z0))) → c30(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.96/66.30
PROPER(sel(x0, first(z0, z1))) → c30(SEL(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.96/66.30
PROPER(sel(x0, sel(z0, z1))) → c30(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(sel(from(z0), x1)) → c30(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.96/66.30
PROPER(sel(cons(z0, z1), x1)) → c30(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(sel(s(z0), x1)) → c30(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.96/66.30
PROPER(sel(first(z0, z1), x1)) → c30(SEL(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(sel(sel(z0, z1), x1)) → c30(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(sel(x0, 0)) → c30(SEL(proper(x0), ok(0)), PROPER(x0)) 193.96/66.30
PROPER(sel(x0, nil)) → c30(SEL(proper(x0), ok(nil)), PROPER(x0)) 193.96/66.30
PROPER(sel(0, x1)) → c30(SEL(ok(0), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(sel(nil, x1)) → c30(SEL(ok(nil), proper(x1)), PROPER(x1)) 193.96/66.30
TOP(mark(from(z0))) → c31(TOP(from(proper(z0))), PROPER(from(z0))) 193.96/66.30
TOP(mark(cons(z0, z1))) → c31(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.96/66.30
TOP(mark(s(z0))) → c31(TOP(s(proper(z0))), PROPER(s(z0))) 193.96/66.30
TOP(mark(first(z0, z1))) → c31(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.96/66.30
TOP(mark(sel(z0, z1))) → c31(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.96/66.30
TOP(mark(0)) → c31(TOP(ok(0))) 193.96/66.30
TOP(mark(nil)) → c31(TOP(ok(nil))) 193.96/66.30
TOP(ok(from(z0))) → c32(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.96/66.30
TOP(ok(first(0, z0))) → c32(TOP(mark(nil)), ACTIVE(first(0, z0))) 193.96/66.30
TOP(ok(first(s(z0), cons(z1, z2)))) → c32(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.96/66.30
TOP(ok(sel(0, cons(z0, z1)))) → c32(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.96/66.30
TOP(ok(sel(s(z0), cons(z1, z2)))) → c32(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.96/66.30
TOP(ok(from(z0))) → c32(TOP(from(active(z0))), ACTIVE(from(z0))) 193.96/66.30
TOP(ok(cons(z0, z1))) → c32(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.96/66.30
TOP(ok(s(z0))) → c32(TOP(s(active(z0))), ACTIVE(s(z0))) 193.96/66.30
TOP(ok(first(z0, z1))) → c32(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.96/66.30
TOP(ok(first(z0, z1))) → c32(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.96/66.30
TOP(ok(sel(z0, z1))) → c32(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.96/66.30
TOP(ok(sel(z0, z1))) → c32(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.96/66.30
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.96/66.30
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.96/66.30
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.96/66.30
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.96/66.30
FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.96/66.30
ACTIVE(from(z0)) → c(S(z0)) 193.96/66.30
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.96/66.30
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.96/66.30
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.96/66.30
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.96/66.30
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.96/66.30
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.96/66.30
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.96/66.30
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.96/66.30
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.96/66.30
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.96/66.30
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.96/66.30
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.96/66.30
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.96/66.30
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.96/66.30
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.96/66.30
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.96/66.30
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.96/66.30
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.96/66.30
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(from(0)) → c24(FROM(ok(0))) 193.96/66.30
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.96/66.30
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.96/66.30
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.96/66.30
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.96/66.30
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.96/66.30
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.96/66.30
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.96/66.30
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.96/66.30
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.96/66.30
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.96/66.30
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.96/66.30
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(s(0)) → c26(S(ok(0))) 193.96/66.30
PROPER(s(nil)) → c26(S(ok(nil))) 193.96/66.30
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.96/66.30
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.96/66.30
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.96/66.30
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.96/66.30
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.96/66.30
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0)) 193.96/66.30
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0)) 193.96/66.30
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(sel(x0, from(z0))) → c30(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.96/66.30
PROPER(sel(x0, cons(z0, z1))) → c30(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(sel(x0, s(z0))) → c30(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.96/66.30
PROPER(sel(x0, first(z0, z1))) → c30(SEL(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.96/66.30
PROPER(sel(x0, sel(z0, z1))) → c30(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(sel(from(z0), x1)) → c30(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.96/66.30
PROPER(sel(cons(z0, z1), x1)) → c30(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(sel(s(z0), x1)) → c30(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.96/66.30
PROPER(sel(first(z0, z1), x1)) → c30(SEL(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(sel(sel(z0, z1), x1)) → c30(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(sel(x0, 0)) → c30(SEL(proper(x0), ok(0)), PROPER(x0)) 193.96/66.30
PROPER(sel(x0, nil)) → c30(SEL(proper(x0), ok(nil)), PROPER(x0)) 193.96/66.30
PROPER(sel(0, x1)) → c30(SEL(ok(0), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(sel(nil, x1)) → c30(SEL(ok(nil), proper(x1)), PROPER(x1)) 193.96/66.30
TOP(mark(from(z0))) → c31(TOP(from(proper(z0))), PROPER(from(z0))) 193.96/66.30
TOP(mark(cons(z0, z1))) → c31(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.96/66.30
TOP(mark(s(z0))) → c31(TOP(s(proper(z0))), PROPER(s(z0))) 193.96/66.30
TOP(mark(first(z0, z1))) → c31(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.96/66.30
TOP(mark(sel(z0, z1))) → c31(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.96/66.30
TOP(ok(from(z0))) → c32(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.96/66.30
TOP(ok(first(0, z0))) → c32(TOP(mark(nil)), ACTIVE(first(0, z0))) 193.96/66.30
TOP(ok(first(s(z0), cons(z1, z2)))) → c32(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.96/66.30
TOP(ok(sel(0, cons(z0, z1)))) → c32(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.96/66.30
TOP(ok(sel(s(z0), cons(z1, z2)))) → c32(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.96/66.30
TOP(ok(from(z0))) → c32(TOP(from(active(z0))), ACTIVE(from(z0))) 193.96/66.30
TOP(ok(cons(z0, z1))) → c32(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.96/66.30
TOP(ok(s(z0))) → c32(TOP(s(active(z0))), ACTIVE(s(z0))) 193.96/66.30
TOP(ok(first(z0, z1))) → c32(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.96/66.30
TOP(ok(first(z0, z1))) → c32(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.96/66.30
TOP(ok(sel(z0, z1))) → c32(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.96/66.30
TOP(ok(sel(z0, z1))) → c32(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
K tuples:

TOP(mark(0)) → c31(TOP(ok(0))) 193.96/66.30
TOP(mark(nil)) → c31(TOP(ok(nil)))
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, FIRST, SEL, PROPER, TOP

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c, c1, c5, c6, c7, c24, c24, c25, c25, c26, c26, c27, c27, c30, c30, c31, c31, c32

193.96/66.30
193.96/66.30

(41) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 193.96/66.30
ACTIVE(first(z0, z1)) → c8(FIRST(active(z0), z1), ACTIVE(z0)) 193.96/66.30
ACTIVE(first(z0, z1)) → c9(FIRST(z0, active(z1)), ACTIVE(z1)) 193.96/66.30
ACTIVE(sel(z0, z1)) → c10(SEL(active(z0), z1), ACTIVE(z0)) 193.96/66.30
ACTIVE(sel(z0, z1)) → c11(SEL(z0, active(z1)), ACTIVE(z1)) 193.96/66.30
ACTIVE(from(z0)) → c(S(z0)) 193.96/66.30
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2))) 193.96/66.30
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2)) 193.96/66.30
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(from(first(0, z0))) → c5(FROM(mark(nil)), ACTIVE(first(0, z0))) 193.96/66.30
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c5(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(from(sel(0, cons(z0, z1)))) → c5(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.96/66.30
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c5(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.96/66.30
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.96/66.30
ACTIVE(from(first(z0, z1))) → c5(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(from(first(z0, z1))) → c5(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(from(sel(z0, z1))) → c5(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 193.96/66.30
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 193.96/66.30
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.96/66.30
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.96/66.30
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(s(from(z0))) → c7(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(s(first(0, z0))) → c7(S(mark(nil)), ACTIVE(first(0, z0))) 193.96/66.30
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c7(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(s(sel(0, cons(z0, z1)))) → c7(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.96/66.30
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c7(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.96/66.30
ACTIVE(s(from(z0))) → c7(S(from(active(z0))), ACTIVE(from(z0))) 193.96/66.30
ACTIVE(s(cons(z0, z1))) → c7(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.96/66.30
ACTIVE(s(s(z0))) → c7(S(s(active(z0))), ACTIVE(s(z0))) 193.96/66.30
ACTIVE(s(first(z0, z1))) → c7(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(s(first(z0, z1))) → c7(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.96/66.30
ACTIVE(s(sel(z0, z1))) → c7(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.96/66.30
ACTIVE(s(sel(z0, z1))) → c7(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 193.96/66.30
PROPER(from(from(z0))) → c24(FROM(from(proper(z0))), PROPER(from(z0))) 193.96/66.30
PROPER(from(cons(z0, z1))) → c24(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(from(s(z0))) → c24(FROM(s(proper(z0))), PROPER(s(z0))) 193.96/66.30
PROPER(from(first(z0, z1))) → c24(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.96/66.30
PROPER(from(sel(z0, z1))) → c24(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(from(0)) → c24(FROM(ok(0))) 193.96/66.30
PROPER(from(nil)) → c24(FROM(ok(nil))) 193.96/66.30
PROPER(cons(x0, from(z0))) → c25(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.96/66.30
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.96/66.30
PROPER(cons(x0, first(z0, z1))) → c25(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.96/66.30
PROPER(cons(x0, sel(z0, z1))) → c25(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(cons(from(z0), x1)) → c25(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.96/66.30
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.96/66.30
PROPER(cons(first(z0, z1), x1)) → c25(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(cons(sel(z0, z1), x1)) → c25(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0)) 193.96/66.30
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0)) 193.96/66.30
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(s(from(z0))) → c26(S(from(proper(z0))), PROPER(from(z0))) 193.96/66.30
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 193.96/66.30
PROPER(s(first(z0, z1))) → c26(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.96/66.30
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(s(0)) → c26(S(ok(0))) 193.96/66.30
PROPER(s(nil)) → c26(S(ok(nil))) 193.96/66.30
PROPER(first(x0, from(z0))) → c27(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.96/66.30
PROPER(first(x0, cons(z0, z1))) → c27(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(first(x0, s(z0))) → c27(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.96/66.30
PROPER(first(x0, first(z0, z1))) → c27(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.96/66.30
PROPER(first(x0, sel(z0, z1))) → c27(FIRST(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(first(from(z0), x1)) → c27(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.96/66.30
PROPER(first(cons(z0, z1), x1)) → c27(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(first(s(z0), x1)) → c27(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.96/66.30
PROPER(first(first(z0, z1), x1)) → c27(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(first(sel(z0, z1), x1)) → c27(FIRST(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(first(x0, 0)) → c27(FIRST(proper(x0), ok(0)), PROPER(x0)) 193.96/66.30
PROPER(first(x0, nil)) → c27(FIRST(proper(x0), ok(nil)), PROPER(x0)) 193.96/66.30
PROPER(first(0, x1)) → c27(FIRST(ok(0), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(first(nil, x1)) → c27(FIRST(ok(nil), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(sel(x0, from(z0))) → c30(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.96/66.30
PROPER(sel(x0, cons(z0, z1))) → c30(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.96/66.30
PROPER(sel(x0, s(z0))) → c30(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.96/66.30
PROPER(sel(x0, first(z0, z1))) → c30(SEL(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 193.96/66.30
PROPER(sel(x0, sel(z0, z1))) → c30(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 193.96/66.30
PROPER(sel(from(z0), x1)) → c30(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.96/66.30
PROPER(sel(cons(z0, z1), x1)) → c30(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(sel(s(z0), x1)) → c30(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.96/66.30
PROPER(sel(first(z0, z1), x1)) → c30(SEL(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(sel(sel(z0, z1), x1)) → c30(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 193.96/66.30
PROPER(sel(x0, 0)) → c30(SEL(proper(x0), ok(0)), PROPER(x0)) 193.96/66.30
PROPER(sel(x0, nil)) → c30(SEL(proper(x0), ok(nil)), PROPER(x0)) 193.96/66.30
PROPER(sel(0, x1)) → c30(SEL(ok(0), proper(x1)), PROPER(x1)) 193.96/66.30
PROPER(sel(nil, x1)) → c30(SEL(ok(nil), proper(x1)), PROPER(x1)) 193.96/66.30
TOP(mark(from(z0))) → c31(TOP(from(proper(z0))), PROPER(from(z0))) 193.96/66.30
TOP(mark(cons(z0, z1))) → c31(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.96/66.30
TOP(mark(s(z0))) → c31(TOP(s(proper(z0))), PROPER(s(z0))) 193.96/66.30
TOP(mark(first(z0, z1))) → c31(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 193.96/66.30
TOP(mark(sel(z0, z1))) → c31(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 193.96/66.30
TOP(ok(from(z0))) → c32(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.96/66.30
TOP(ok(first(0, z0))) → c32(TOP(mark(nil)), ACTIVE(first(0, z0))) 193.96/66.30
TOP(ok(first(s(z0), cons(z1, z2)))) → c32(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 193.96/66.30
TOP(ok(sel(0, cons(z0, z1)))) → c32(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 193.96/66.30
TOP(ok(sel(s(z0), cons(z1, z2)))) → c32(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 193.96/66.30
TOP(ok(from(z0))) → c32(TOP(from(active(z0))), ACTIVE(from(z0))) 193.96/66.30
TOP(ok(cons(z0, z1))) → c32(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.96/66.30
TOP(ok(s(z0))) → c32(TOP(s(active(z0))), ACTIVE(s(z0))) 193.96/66.30
TOP(ok(first(z0, z1))) → c32(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1))) 193.96/66.30
TOP(ok(first(z0, z1))) → c32(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1))) 193.96/66.30
TOP(ok(sel(z0, z1))) → c32(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 193.96/66.30
TOP(ok(sel(z0, z1))) → c32(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
193.96/66.30
193.96/66.30

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.96/66.30
active(first(0, z0)) → mark(nil) 193.96/66.30
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.96/66.30
active(sel(0, cons(z0, z1))) → mark(z0) 193.96/66.30
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.96/66.30
active(from(z0)) → from(active(z0)) 193.96/66.30
active(cons(z0, z1)) → cons(active(z0), z1) 193.96/66.30
active(s(z0)) → s(active(z0)) 193.96/66.30
active(first(z0, z1)) → first(active(z0), z1) 193.96/66.30
active(first(z0, z1)) → first(z0, active(z1)) 193.96/66.30
active(sel(z0, z1)) → sel(active(z0), z1) 193.96/66.30
active(sel(z0, z1)) → sel(z0, active(z1)) 193.96/66.30
from(mark(z0)) → mark(from(z0)) 193.96/66.30
from(ok(z0)) → ok(from(z0)) 193.96/66.30
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.96/66.30
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.96/66.30
s(mark(z0)) → mark(s(z0)) 193.96/66.30
s(ok(z0)) → ok(s(z0)) 193.96/66.30
first(mark(z0), z1) → mark(first(z0, z1)) 193.96/66.30
first(z0, mark(z1)) → mark(first(z0, z1)) 193.96/66.30
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.96/66.30
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.96/66.30
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.96/66.30
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.96/66.30
proper(from(z0)) → from(proper(z0)) 193.96/66.30
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.96/66.30
proper(s(z0)) → s(proper(z0)) 193.96/66.30
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.96/66.30
proper(0) → ok(0) 193.96/66.30
proper(nil) → ok(nil) 193.96/66.30
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.96/66.30
top(mark(z0)) → top(proper(z0)) 193.96/66.30
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.96/66.30
TOP(mark(0)) → c31(TOP(ok(0))) 193.96/66.30
TOP(mark(nil)) → c31(TOP(ok(nil)))
S tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
K tuples:

TOP(mark(0)) → c31(TOP(ok(0))) 193.96/66.30
TOP(mark(nil)) → c31(TOP(ok(nil)))
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, FIRST, SEL, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c31

193.96/66.30
193.96/66.30

(43) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts
193.96/66.30
193.96/66.30

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.96/66.30
active(first(0, z0)) → mark(nil) 193.96/66.30
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.96/66.30
active(sel(0, cons(z0, z1))) → mark(z0) 193.96/66.30
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.96/66.30
active(from(z0)) → from(active(z0)) 193.96/66.30
active(cons(z0, z1)) → cons(active(z0), z1) 193.96/66.30
active(s(z0)) → s(active(z0)) 193.96/66.30
active(first(z0, z1)) → first(active(z0), z1) 193.96/66.30
active(first(z0, z1)) → first(z0, active(z1)) 193.96/66.30
active(sel(z0, z1)) → sel(active(z0), z1) 193.96/66.30
active(sel(z0, z1)) → sel(z0, active(z1)) 193.96/66.30
from(mark(z0)) → mark(from(z0)) 193.96/66.30
from(ok(z0)) → ok(from(z0)) 193.96/66.30
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.96/66.30
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.96/66.30
s(mark(z0)) → mark(s(z0)) 193.96/66.30
s(ok(z0)) → ok(s(z0)) 193.96/66.30
first(mark(z0), z1) → mark(first(z0, z1)) 193.96/66.30
first(z0, mark(z1)) → mark(first(z0, z1)) 193.96/66.30
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.96/66.30
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.96/66.30
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.96/66.30
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.96/66.30
proper(from(z0)) → from(proper(z0)) 193.96/66.30
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.96/66.30
proper(s(z0)) → s(proper(z0)) 193.96/66.30
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.96/66.30
proper(0) → ok(0) 193.96/66.30
proper(nil) → ok(nil) 193.96/66.30
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.96/66.30
top(mark(z0)) → top(proper(z0)) 193.96/66.30
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.96/66.30
TOP(mark(0)) → c31 193.96/66.30
TOP(mark(nil)) → c31
S tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
K tuples:

TOP(mark(0)) → c31 193.96/66.30
TOP(mark(nil)) → c31
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, FIRST, SEL, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c31

193.96/66.30
193.96/66.30

(45) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

TOP(mark(0)) → c31 193.96/66.30
TOP(mark(nil)) → c31
193.96/66.30
193.96/66.30

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.96/66.30
active(first(0, z0)) → mark(nil) 193.96/66.30
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.96/66.30
active(sel(0, cons(z0, z1))) → mark(z0) 193.96/66.30
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.96/66.30
active(from(z0)) → from(active(z0)) 193.96/66.30
active(cons(z0, z1)) → cons(active(z0), z1) 193.96/66.30
active(s(z0)) → s(active(z0)) 193.96/66.30
active(first(z0, z1)) → first(active(z0), z1) 193.96/66.30
active(first(z0, z1)) → first(z0, active(z1)) 193.96/66.30
active(sel(z0, z1)) → sel(active(z0), z1) 193.96/66.30
active(sel(z0, z1)) → sel(z0, active(z1)) 193.96/66.30
from(mark(z0)) → mark(from(z0)) 193.96/66.30
from(ok(z0)) → ok(from(z0)) 193.96/66.30
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.96/66.30
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.96/66.30
s(mark(z0)) → mark(s(z0)) 193.96/66.30
s(ok(z0)) → ok(s(z0)) 193.96/66.30
first(mark(z0), z1) → mark(first(z0, z1)) 193.96/66.30
first(z0, mark(z1)) → mark(first(z0, z1)) 193.96/66.30
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.96/66.30
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.96/66.30
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.96/66.30
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.96/66.30
proper(from(z0)) → from(proper(z0)) 193.96/66.30
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.96/66.30
proper(s(z0)) → s(proper(z0)) 193.96/66.30
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.96/66.30
proper(0) → ok(0) 193.96/66.30
proper(nil) → ok(nil) 193.96/66.30
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.96/66.30
top(mark(z0)) → top(proper(z0)) 193.96/66.30
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
S tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, FIRST, SEL

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23

193.96/66.30
193.96/66.30

(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.

FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 193.96/66.30

POL(CONS(x1, x2)) = 0    193.96/66.30
POL(FIRST(x1, x2)) = [4]x1    193.96/66.30
POL(FROM(x1)) = 0    193.96/66.30
POL(S(x1)) = 0    193.96/66.30
POL(SEL(x1, x2)) = x1    193.96/66.30
POL(c12(x1)) = x1    193.96/66.30
POL(c13(x1)) = x1    193.96/66.30
POL(c14(x1)) = x1    193.96/66.30
POL(c15(x1)) = x1    193.96/66.30
POL(c16(x1)) = x1    193.96/66.30
POL(c17(x1)) = x1    193.96/66.30
POL(c18(x1)) = x1    193.96/66.30
POL(c19(x1)) = x1    193.96/66.30
POL(c20(x1)) = x1    193.96/66.30
POL(c21(x1)) = x1    193.96/66.30
POL(c22(x1)) = x1    193.96/66.30
POL(c23(x1)) = x1    193.96/66.30
POL(mark(x1)) = [4] + x1    193.96/66.30
POL(ok(x1)) = [2] + x1   
193.96/66.30
193.96/66.30

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.96/66.30
active(first(0, z0)) → mark(nil) 193.96/66.30
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.96/66.30
active(sel(0, cons(z0, z1))) → mark(z0) 193.96/66.30
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.96/66.30
active(from(z0)) → from(active(z0)) 193.96/66.30
active(cons(z0, z1)) → cons(active(z0), z1) 193.96/66.30
active(s(z0)) → s(active(z0)) 193.96/66.30
active(first(z0, z1)) → first(active(z0), z1) 193.96/66.30
active(first(z0, z1)) → first(z0, active(z1)) 193.96/66.30
active(sel(z0, z1)) → sel(active(z0), z1) 193.96/66.30
active(sel(z0, z1)) → sel(z0, active(z1)) 193.96/66.30
from(mark(z0)) → mark(from(z0)) 193.96/66.30
from(ok(z0)) → ok(from(z0)) 193.96/66.30
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.96/66.30
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.96/66.30
s(mark(z0)) → mark(s(z0)) 193.96/66.30
s(ok(z0)) → ok(s(z0)) 193.96/66.30
first(mark(z0), z1) → mark(first(z0, z1)) 193.96/66.30
first(z0, mark(z1)) → mark(first(z0, z1)) 193.96/66.30
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.96/66.30
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.96/66.30
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.96/66.30
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.96/66.30
proper(from(z0)) → from(proper(z0)) 193.96/66.30
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.96/66.30
proper(s(z0)) → s(proper(z0)) 193.96/66.30
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.96/66.30
proper(0) → ok(0) 193.96/66.30
proper(nil) → ok(nil) 193.96/66.30
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.96/66.30
top(mark(z0)) → top(proper(z0)) 193.96/66.30
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
S tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
K tuples:

FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, FIRST, SEL

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23

193.96/66.30
193.96/66.30

(49) 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)) → c12(FROM(z0))
We considered the (Usable) Rules:none
And the Tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 193.96/66.30

POL(CONS(x1, x2)) = 0    193.96/66.30
POL(FIRST(x1, x2)) = [3]x1    193.96/66.30
POL(FROM(x1)) = [2]x1    193.96/66.30
POL(S(x1)) = 0    193.96/66.30
POL(SEL(x1, x2)) = [3]x1    193.96/66.30
POL(c12(x1)) = x1    193.96/66.30
POL(c13(x1)) = x1    193.96/66.30
POL(c14(x1)) = x1    193.96/66.30
POL(c15(x1)) = x1    193.96/66.30
POL(c16(x1)) = x1    193.96/66.30
POL(c17(x1)) = x1    193.96/66.30
POL(c18(x1)) = x1    193.96/66.30
POL(c19(x1)) = x1    193.96/66.30
POL(c20(x1)) = x1    193.96/66.30
POL(c21(x1)) = x1    193.96/66.30
POL(c22(x1)) = x1    193.96/66.30
POL(c23(x1)) = x1    193.96/66.30
POL(mark(x1)) = [1] + x1    193.96/66.30
POL(ok(x1)) = x1   
193.96/66.30
193.96/66.30

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.96/66.30
active(first(0, z0)) → mark(nil) 193.96/66.30
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.96/66.30
active(sel(0, cons(z0, z1))) → mark(z0) 193.96/66.30
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.96/66.30
active(from(z0)) → from(active(z0)) 193.96/66.30
active(cons(z0, z1)) → cons(active(z0), z1) 193.96/66.30
active(s(z0)) → s(active(z0)) 193.96/66.30
active(first(z0, z1)) → first(active(z0), z1) 193.96/66.30
active(first(z0, z1)) → first(z0, active(z1)) 193.96/66.30
active(sel(z0, z1)) → sel(active(z0), z1) 193.96/66.30
active(sel(z0, z1)) → sel(z0, active(z1)) 193.96/66.30
from(mark(z0)) → mark(from(z0)) 193.96/66.30
from(ok(z0)) → ok(from(z0)) 193.96/66.30
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.96/66.30
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.96/66.30
s(mark(z0)) → mark(s(z0)) 193.96/66.30
s(ok(z0)) → ok(s(z0)) 193.96/66.30
first(mark(z0), z1) → mark(first(z0, z1)) 193.96/66.30
first(z0, mark(z1)) → mark(first(z0, z1)) 193.96/66.30
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.96/66.30
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.96/66.30
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.96/66.30
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.96/66.30
proper(from(z0)) → from(proper(z0)) 193.96/66.30
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.96/66.30
proper(s(z0)) → s(proper(z0)) 193.96/66.30
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.96/66.30
proper(0) → ok(0) 193.96/66.30
proper(nil) → ok(nil) 193.96/66.30
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.96/66.30
top(mark(z0)) → top(proper(z0)) 193.96/66.30
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
S tuples:

FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
K tuples:

FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.96/66.30
FROM(mark(z0)) → c12(FROM(z0))
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, FIRST, SEL

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23

193.96/66.30
193.96/66.30

(51) 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(ok(z0)) → c13(FROM(z0))
We considered the (Usable) Rules:none
And the Tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 193.96/66.30

POL(CONS(x1, x2)) = 0    193.96/66.30
POL(FIRST(x1, x2)) = [3]x1    193.96/66.30
POL(FROM(x1)) = [2]x1    193.96/66.30
POL(S(x1)) = 0    193.96/66.30
POL(SEL(x1, x2)) = [5]x1    193.96/66.30
POL(c12(x1)) = x1    193.96/66.30
POL(c13(x1)) = x1    193.96/66.30
POL(c14(x1)) = x1    193.96/66.30
POL(c15(x1)) = x1    193.96/66.30
POL(c16(x1)) = x1    193.96/66.30
POL(c17(x1)) = x1    193.96/66.30
POL(c18(x1)) = x1    193.96/66.30
POL(c19(x1)) = x1    193.96/66.30
POL(c20(x1)) = x1    193.96/66.30
POL(c21(x1)) = x1    193.96/66.30
POL(c22(x1)) = x1    193.96/66.30
POL(c23(x1)) = x1    193.96/66.30
POL(mark(x1)) = x1    193.96/66.30
POL(ok(x1)) = [1] + x1   
193.96/66.30
193.96/66.30

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.96/66.30
active(first(0, z0)) → mark(nil) 193.96/66.30
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.96/66.30
active(sel(0, cons(z0, z1))) → mark(z0) 193.96/66.30
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.96/66.30
active(from(z0)) → from(active(z0)) 193.96/66.30
active(cons(z0, z1)) → cons(active(z0), z1) 193.96/66.30
active(s(z0)) → s(active(z0)) 193.96/66.30
active(first(z0, z1)) → first(active(z0), z1) 193.96/66.30
active(first(z0, z1)) → first(z0, active(z1)) 193.96/66.30
active(sel(z0, z1)) → sel(active(z0), z1) 193.96/66.30
active(sel(z0, z1)) → sel(z0, active(z1)) 193.96/66.30
from(mark(z0)) → mark(from(z0)) 193.96/66.30
from(ok(z0)) → ok(from(z0)) 193.96/66.30
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.96/66.30
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.96/66.30
s(mark(z0)) → mark(s(z0)) 193.96/66.30
s(ok(z0)) → ok(s(z0)) 193.96/66.30
first(mark(z0), z1) → mark(first(z0, z1)) 193.96/66.30
first(z0, mark(z1)) → mark(first(z0, z1)) 193.96/66.30
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.96/66.30
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.96/66.30
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.96/66.30
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.96/66.30
proper(from(z0)) → from(proper(z0)) 193.96/66.30
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.96/66.30
proper(s(z0)) → s(proper(z0)) 193.96/66.30
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.96/66.30
proper(0) → ok(0) 193.96/66.30
proper(nil) → ok(nil) 193.96/66.30
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.96/66.30
top(mark(z0)) → top(proper(z0)) 193.96/66.30
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
S tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
K tuples:

FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.96/66.30
FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0))
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, FIRST, SEL

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23

193.96/66.30
193.96/66.30

(53) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 193.96/66.30

POL(CONS(x1, x2)) = x1 + x22 + x1·x2 + x12 + x13 + x12·x2 + x1·x22 + x23    193.96/66.30
POL(FIRST(x1, x2)) = x1 + x22 + x1·x2 + x12 + x13 + x12·x2 + x1·x22 + x23    193.96/66.30
POL(FROM(x1)) = x1 + x12 + x13    193.96/66.30
POL(S(x1)) = x1 + x12 + x13    193.96/66.30
POL(SEL(x1, x2)) = x2 + x22 + x1·x2 + x12 + x13 + x12·x2 + x1·x22 + x23    193.96/66.30
POL(c12(x1)) = x1    193.96/66.30
POL(c13(x1)) = x1    193.96/66.30
POL(c14(x1)) = x1    193.96/66.30
POL(c15(x1)) = x1    193.96/66.30
POL(c16(x1)) = x1    193.96/66.30
POL(c17(x1)) = x1    193.96/66.30
POL(c18(x1)) = x1    193.96/66.30
POL(c19(x1)) = x1    193.96/66.30
POL(c20(x1)) = x1    193.96/66.30
POL(c21(x1)) = x1    193.96/66.30
POL(c22(x1)) = x1    193.96/66.30
POL(c23(x1)) = x1    193.96/66.30
POL(mark(x1)) = [1] + x1    193.96/66.30
POL(ok(x1)) = x1   
193.96/66.30
193.96/66.30

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.96/66.30
active(first(0, z0)) → mark(nil) 193.96/66.30
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.96/66.30
active(sel(0, cons(z0, z1))) → mark(z0) 193.96/66.30
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.96/66.30
active(from(z0)) → from(active(z0)) 193.96/66.30
active(cons(z0, z1)) → cons(active(z0), z1) 193.96/66.30
active(s(z0)) → s(active(z0)) 193.96/66.30
active(first(z0, z1)) → first(active(z0), z1) 193.96/66.30
active(first(z0, z1)) → first(z0, active(z1)) 193.96/66.30
active(sel(z0, z1)) → sel(active(z0), z1) 193.96/66.30
active(sel(z0, z1)) → sel(z0, active(z1)) 193.96/66.30
from(mark(z0)) → mark(from(z0)) 193.96/66.30
from(ok(z0)) → ok(from(z0)) 193.96/66.30
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.96/66.30
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.96/66.30
s(mark(z0)) → mark(s(z0)) 193.96/66.30
s(ok(z0)) → ok(s(z0)) 193.96/66.30
first(mark(z0), z1) → mark(first(z0, z1)) 193.96/66.30
first(z0, mark(z1)) → mark(first(z0, z1)) 193.96/66.30
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.96/66.30
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.96/66.30
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.96/66.30
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.96/66.30
proper(from(z0)) → from(proper(z0)) 193.96/66.30
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.96/66.30
proper(s(z0)) → s(proper(z0)) 193.96/66.30
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.96/66.30
proper(0) → ok(0) 193.96/66.30
proper(nil) → ok(nil) 193.96/66.30
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.96/66.30
top(mark(z0)) → top(proper(z0)) 193.96/66.30
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
S tuples:

CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(ok(z0)) → c17(S(z0))
K tuples:

FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.96/66.30
FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1))
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, FIRST, SEL

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23

193.96/66.30
193.96/66.30

(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(ok(z0), ok(z1)) → c15(CONS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 193.96/66.30

POL(CONS(x1, x2)) = x2    193.96/66.30
POL(FIRST(x1, x2)) = [3]x1 + [3]x2    193.96/66.30
POL(FROM(x1)) = [5]x1    193.96/66.30
POL(S(x1)) = 0    193.96/66.30
POL(SEL(x1, x2)) = [5]x1 + [3]x2    193.96/66.30
POL(c12(x1)) = x1    193.96/66.30
POL(c13(x1)) = x1    193.96/66.30
POL(c14(x1)) = x1    193.96/66.30
POL(c15(x1)) = x1    193.96/66.30
POL(c16(x1)) = x1    193.96/66.30
POL(c17(x1)) = x1    193.96/66.30
POL(c18(x1)) = x1    193.96/66.30
POL(c19(x1)) = x1    193.96/66.30
POL(c20(x1)) = x1    193.96/66.30
POL(c21(x1)) = x1    193.96/66.30
POL(c22(x1)) = x1    193.96/66.30
POL(c23(x1)) = x1    193.96/66.30
POL(mark(x1)) = x1    193.96/66.30
POL(ok(x1)) = [1] + x1   
193.96/66.30
193.96/66.30

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.96/66.30
active(first(0, z0)) → mark(nil) 193.96/66.30
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.96/66.30
active(sel(0, cons(z0, z1))) → mark(z0) 193.96/66.30
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.96/66.30
active(from(z0)) → from(active(z0)) 193.96/66.30
active(cons(z0, z1)) → cons(active(z0), z1) 193.96/66.30
active(s(z0)) → s(active(z0)) 193.96/66.30
active(first(z0, z1)) → first(active(z0), z1) 193.96/66.30
active(first(z0, z1)) → first(z0, active(z1)) 193.96/66.30
active(sel(z0, z1)) → sel(active(z0), z1) 193.96/66.30
active(sel(z0, z1)) → sel(z0, active(z1)) 193.96/66.30
from(mark(z0)) → mark(from(z0)) 193.96/66.30
from(ok(z0)) → ok(from(z0)) 193.96/66.30
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.96/66.30
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.96/66.30
s(mark(z0)) → mark(s(z0)) 193.96/66.30
s(ok(z0)) → ok(s(z0)) 193.96/66.30
first(mark(z0), z1) → mark(first(z0, z1)) 193.96/66.30
first(z0, mark(z1)) → mark(first(z0, z1)) 193.96/66.30
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.96/66.30
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.96/66.30
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.96/66.30
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.96/66.30
proper(from(z0)) → from(proper(z0)) 193.96/66.30
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.96/66.30
proper(s(z0)) → s(proper(z0)) 193.96/66.30
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.96/66.30
proper(0) → ok(0) 193.96/66.30
proper(nil) → ok(nil) 193.96/66.30
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.96/66.30
top(mark(z0)) → top(proper(z0)) 193.96/66.30
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
S tuples:

S(ok(z0)) → c17(S(z0))
K tuples:

FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.96/66.30
FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, FIRST, SEL

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23

193.96/66.30
193.96/66.30

(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.

S(ok(z0)) → c17(S(z0))
We considered the (Usable) Rules:none
And the Tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 193.96/66.30

POL(CONS(x1, x2)) = [5]x1 + [5]x2    193.96/66.30
POL(FIRST(x1, x2)) = [3]x1 + [3]x2    193.96/66.30
POL(FROM(x1)) = [3]x1    193.96/66.30
POL(S(x1)) = [3]x1    193.96/66.30
POL(SEL(x1, x2)) = [5]x1 + [5]x2    193.96/66.30
POL(c12(x1)) = x1    193.96/66.30
POL(c13(x1)) = x1    193.96/66.30
POL(c14(x1)) = x1    193.96/66.30
POL(c15(x1)) = x1    193.96/66.30
POL(c16(x1)) = x1    193.96/66.30
POL(c17(x1)) = x1    193.96/66.30
POL(c18(x1)) = x1    193.96/66.30
POL(c19(x1)) = x1    193.96/66.30
POL(c20(x1)) = x1    193.96/66.30
POL(c21(x1)) = x1    193.96/66.30
POL(c22(x1)) = x1    193.96/66.30
POL(c23(x1)) = x1    193.96/66.30
POL(mark(x1)) = x1    193.96/66.30
POL(ok(x1)) = [1] + x1   
193.96/66.30
193.96/66.30

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.96/66.30
active(first(0, z0)) → mark(nil) 193.96/66.30
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 193.96/66.30
active(sel(0, cons(z0, z1))) → mark(z0) 193.96/66.30
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 193.96/66.30
active(from(z0)) → from(active(z0)) 193.96/66.30
active(cons(z0, z1)) → cons(active(z0), z1) 193.96/66.30
active(s(z0)) → s(active(z0)) 193.96/66.30
active(first(z0, z1)) → first(active(z0), z1) 193.96/66.30
active(first(z0, z1)) → first(z0, active(z1)) 193.96/66.30
active(sel(z0, z1)) → sel(active(z0), z1) 193.96/66.30
active(sel(z0, z1)) → sel(z0, active(z1)) 193.96/66.30
from(mark(z0)) → mark(from(z0)) 193.96/66.30
from(ok(z0)) → ok(from(z0)) 193.96/66.30
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.96/66.30
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.96/66.30
s(mark(z0)) → mark(s(z0)) 193.96/66.30
s(ok(z0)) → ok(s(z0)) 193.96/66.30
first(mark(z0), z1) → mark(first(z0, z1)) 193.96/66.30
first(z0, mark(z1)) → mark(first(z0, z1)) 193.96/66.30
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 193.96/66.30
sel(mark(z0), z1) → mark(sel(z0, z1)) 193.96/66.30
sel(z0, mark(z1)) → mark(sel(z0, z1)) 193.96/66.30
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 193.96/66.30
proper(from(z0)) → from(proper(z0)) 193.96/66.30
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.96/66.30
proper(s(z0)) → s(proper(z0)) 193.96/66.30
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 193.96/66.30
proper(0) → ok(0) 193.96/66.30
proper(nil) → ok(nil) 193.96/66.30
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 193.96/66.30
top(mark(z0)) → top(proper(z0)) 193.96/66.30
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
S(ok(z0)) → c17(S(z0)) 193.96/66.30
FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1))
S tuples:none
K tuples:

FIRST(mark(z0), z1) → c18(FIRST(z0, z1)) 193.96/66.30
FIRST(ok(z0), ok(z1)) → c20(FIRST(z0, z1)) 193.96/66.30
SEL(mark(z0), z1) → c21(SEL(z0, z1)) 193.96/66.30
SEL(ok(z0), ok(z1)) → c23(SEL(z0, z1)) 193.96/66.30
FROM(mark(z0)) → c12(FROM(z0)) 193.96/66.30
FROM(ok(z0)) → c13(FROM(z0)) 193.96/66.30
CONS(mark(z0), z1) → c14(CONS(z0, z1)) 193.96/66.30
S(mark(z0)) → c16(S(z0)) 193.96/66.30
FIRST(z0, mark(z1)) → c19(FIRST(z0, z1)) 193.96/66.30
SEL(z0, mark(z1)) → c22(SEL(z0, z1)) 193.96/66.30
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1)) 193.96/66.30
S(ok(z0)) → c17(S(z0))
Defined Rule Symbols:

active, from, cons, s, first, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, FIRST, SEL

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23

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(59) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty
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(60) BOUNDS(O(1), O(1))

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193.96/66.37 EOF