The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(and(tt, X)) → mark(X)
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(s(length(L)))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(cons(N, take(M, IL)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(length(X)) → length(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
length(mark(X)) → mark(length(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
length(ok(X)) → ok(length(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

The following defined symbols can occur below the 0th argument of cons: active, proper, cons
The following defined symbols can occur below the 1th argument of cons: active, proper, cons
The following defined symbols can occur below the 0th argument of top: active, proper, cons
The following defined symbols can occur below the 0th argument of proper: active, proper, cons
The following defined symbols can occur below the 0th argument of active: active, proper, cons

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(and(tt, X)) → mark(X)
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(s(length(L)))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(cons(N, take(M, IL)))
active(and(X1, X2)) → and(active(X1), X2)
active(length(X)) → length(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))

(2) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

top(ok(X)) → top(active(X))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
take(X1, mark(X2)) → mark(take(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
active(zeros) → mark(cons(0, zeros))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(0) → ok(0)
length(ok(X)) → ok(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
top(mark(X)) → top(proper(X))
active(cons(X1, X2)) → cons(active(X1), X2)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))

Rewrite Strategy: INNERMOST

(3) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 5.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4, 5, 6, 7, 8]
transitions:
ok0(0) → 0
tt0() → 0
nil0() → 0
mark0(0) → 0
zeros0() → 0
00() → 0
top0(0) → 1
proper0(0) → 2
take0(0, 0) → 3
cons0(0, 0) → 4
and0(0, 0) → 5
active0(0) → 6
s0(0) → 7
length0(0) → 8
active1(0) → 9
top1(9) → 1
tt1() → 10
ok1(10) → 2
nil1() → 11
ok1(11) → 2
take1(0, 0) → 12
mark1(12) → 3
zeros1() → 13
ok1(13) → 2
cons1(0, 0) → 14
ok1(14) → 4
and1(0, 0) → 15
mark1(15) → 5
01() → 17
zeros1() → 18
cons1(17, 18) → 16
mark1(16) → 6
and1(0, 0) → 19
ok1(19) → 5
take1(0, 0) → 20
ok1(20) → 3
s1(0) → 21
ok1(21) → 7
s1(0) → 22
mark1(22) → 7
length1(0) → 23
mark1(23) → 8
01() → 24
ok1(24) → 2
length1(0) → 25
ok1(25) → 8
cons1(0, 0) → 26
mark1(26) → 4
proper1(0) → 27
top1(27) → 1
ok1(10) → 27
ok1(11) → 27
mark1(12) → 12
mark1(12) → 20
ok1(13) → 27
ok1(14) → 14
ok1(14) → 26
mark1(15) → 15
mark1(15) → 19
mark1(16) → 9
ok1(19) → 15
ok1(19) → 19
ok1(20) → 12
ok1(20) → 20
ok1(21) → 21
ok1(21) → 22
mark1(22) → 21
mark1(22) → 22
mark1(23) → 23
mark1(23) → 25
ok1(24) → 27
ok1(25) → 23
ok1(25) → 25
mark1(26) → 14
mark1(26) → 26
active2(10) → 28
top2(28) → 1
active2(11) → 28
active2(13) → 28
active2(24) → 28
proper2(16) → 29
top2(29) → 1
02() → 31
zeros2() → 32
cons2(31, 32) → 30
mark2(30) → 28
proper2(17) → 33
proper2(18) → 34
cons2(33, 34) → 29
zeros2() → 35
ok2(35) → 34
02() → 36
ok2(36) → 33
proper3(30) → 37
top3(37) → 1
cons3(36, 35) → 38
ok3(38) → 29
proper3(31) → 39
proper3(32) → 40
cons3(39, 40) → 37
zeros3() → 41
ok3(41) → 40
03() → 42
ok3(42) → 39
active3(38) → 43
top3(43) → 1
cons4(42, 41) → 44
ok4(44) → 37
active4(36) → 45
cons4(45, 35) → 43
active4(44) → 46
top4(46) → 1
active5(42) → 47
cons5(47, 41) → 46

(4) BOUNDS(1, n^1)

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

Converted Cpx (relative) TRS to CDT

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(tt) → c2
PROPER(nil) → c3
PROPER(zeros) → c4
PROPER(0) → c5
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(zeros) → c14(CONS(0, zeros))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

S tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(tt) → c2
PROPER(nil) → c3
PROPER(zeros) → c4
PROPER(0) → c5
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(zeros) → c14(CONS(0, zeros))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

K tuples:none
Defined Rule Symbols:

top, proper, take, cons, and, active, s, length

Defined Pair Symbols:

TOP, PROPER, TAKE, CONS, AND, ACTIVE, S, LENGTH

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

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

Removed 5 trailing nodes:

PROPER(zeros) → c4
ACTIVE(zeros) → c14(CONS(0, zeros))
PROPER(tt) → c2
PROPER(nil) → c3
PROPER(0) → c5

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

S tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

K tuples:none
Defined Rule Symbols:

top, proper, take, cons, and, active, s, length

Defined Pair Symbols:

TOP, PROPER, TAKE, CONS, AND, ACTIVE, S, LENGTH

Compound Symbols:

c, c1, c6, c7, c8, c9, c10, c11, c12, c13, c15, c16, c17, c18, c19

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
take(z0, mark(z1)) → mark(take(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

S tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

K tuples:none
Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, TAKE, CONS, AND, ACTIVE, S, LENGTH

Compound Symbols:

c, c1, c6, c7, c8, c9, c10, c11, c12, c13, c15, c16, c17, c18, c19

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))

We considered the (Usable) Rules:none
And the Tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ACTIVE(x₁)) = 0   
POL(AND(x₁, x₂)) = x₁   
POL(CONS(x₁, x₂)) = 0   
POL(LENGTH(x₁)) = 0   
POL(PROPER(x₁)) = 0   
POL(S(x₁)) = x₁   
POL(TAKE(x₁, x₂)) = x₂   
POL(TOP(x₁)) = 0   
POL(active(x₁)) = 0   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c15(x₁, x₂)) = x₁ + x₂   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c6(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(cons(x₁, x₂)) = 0   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   
POL(zeros) = 0   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

S tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, TAKE, CONS, AND, ACTIVE, S, LENGTH

Compound Symbols:

c, c1, c6, c7, c8, c9, c10, c11, c12, c13, c15, c16, c17, c18, c19

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

We considered the (Usable) Rules:none
And the Tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ACTIVE(x₁)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(LENGTH(x₁)) = x₁   
POL(PROPER(x₁)) = 0   
POL(S(x₁)) = 0   
POL(TAKE(x₁, x₂)) = x₁ + x₂   
POL(TOP(x₁)) = 0   
POL(active(x₁)) = 0   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c15(x₁, x₂)) = x₁ + x₂   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c6(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(cons(x₁, x₂)) = 0   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   
POL(zeros) = 0   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

S tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, TAKE, CONS, AND, ACTIVE, S, LENGTH

Compound Symbols:

c, c1, c6, c7, c8, c9, c10, c11, c12, c13, c15, c16, c17, c18, c19

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(0) → ok(0)
active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
cons(mark(z0), z1) → mark(cons(z0, z1))

And the Tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ACTIVE(x₁)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(LENGTH(x₁)) = 0   
POL(PROPER(x₁)) = 0   
POL(S(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = x₁   
POL(active(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c15(x₁, x₂)) = x₁ + x₂   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c6(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(cons(x₁, x₂)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = x₁   
POL(tt) = 0   
POL(zeros) = [1]   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

S tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, TAKE, CONS, AND, ACTIVE, S, LENGTH

Compound Symbols:

c, c1, c6, c7, c8, c9, c10, c11, c12, c13, c15, c16, c17, c18, c19

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))

We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(0) → ok(0)
active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
cons(mark(z0), z1) → mark(cons(z0, z1))

And the Tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ACTIVE(x₁)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(LENGTH(x₁)) = 0   
POL(PROPER(x₁)) = 0   
POL(S(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = x₁   
POL(active(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c15(x₁, x₂)) = x₁ + x₂   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c6(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(cons(x₁, x₂)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = [1] + x₁   
POL(tt) = 0   
POL(zeros) = [1]   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

S tuples:

PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, TAKE, CONS, AND, ACTIVE, S, LENGTH

Compound Symbols:

c, c1, c6, c7, c8, c9, c10, c11, c12, c13, c15, c16, c17, c18, c19

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(0) → ok(0)
active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
cons(mark(z0), z1) → mark(cons(z0, z1))

And the Tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ACTIVE(x₁)) = [1] + [2]x₁   
POL(AND(x₁, x₂)) = x₁·x₂   
POL(CONS(x₁, x₂)) = 0   
POL(LENGTH(x₁)) = 0   
POL(PROPER(x₁)) = 0   
POL(S(x₁)) = x₁²   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = x₁²   
POL(active(x₁)) = x₁   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c15(x₁, x₂)) = x₁ + x₂   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c6(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(cons(x₁, x₂)) = [1] + x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = [1]   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = [1] + x₁   
POL(tt) = 0   
POL(zeros) = [2]   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))

S tuples:

PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, TAKE, CONS, AND, ACTIVE, S, LENGTH

Compound Symbols:

c, c1, c6, c7, c8, c9, c10, c11, c12, c13, c15, c16, c17, c18, c19

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

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

TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))), ACTIVE(zeros))
TOP(ok(cons(z0, z1))) → c(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))), ACTIVE(zeros))
TOP(ok(cons(z0, z1))) → c(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))

S tuples:

PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, TAKE, CONS, AND, ACTIVE, S, LENGTH

Compound Symbols:

c1, c6, c7, c8, c9, c10, c11, c12, c13, c15, c16, c17, c18, c19, c

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

Removed 1 trailing tuple parts

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(cons(z0, z1))) → c(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))

S tuples:

PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, TAKE, CONS, AND, ACTIVE, S, LENGTH

Compound Symbols:

c1, c6, c7, c8, c9, c10, c11, c12, c13, c15, c16, c17, c18, c19, c, c

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

Use narrowing to replace TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(tt)) → c1(TOP(ok(tt)), PROPER(tt))
TOP(mark(nil)) → c1(TOP(ok(nil)), PROPER(nil))
TOP(mark(zeros)) → c1(TOP(ok(zeros)), PROPER(zeros))
TOP(mark(0)) → c1(TOP(ok(0)), PROPER(0))
TOP(mark(cons(z0, z1))) → c1(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(cons(z0, z1))) → c(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(tt)) → c1(TOP(ok(tt)), PROPER(tt))
TOP(mark(nil)) → c1(TOP(ok(nil)), PROPER(nil))
TOP(mark(zeros)) → c1(TOP(ok(zeros)), PROPER(zeros))
TOP(mark(0)) → c1(TOP(ok(0)), PROPER(0))
TOP(mark(cons(z0, z1))) → c1(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))

S tuples:

PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c(TOP(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

PROPER, TAKE, CONS, AND, ACTIVE, S, LENGTH, TOP

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13, c15, c16, c17, c18, c19, c, c, c1

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

Removed 3 trailing nodes:

TOP(mark(tt)) → c1(TOP(ok(tt)), PROPER(tt))
TOP(mark(0)) → c1(TOP(ok(0)), PROPER(0))
TOP(mark(nil)) → c1(TOP(ok(nil)), PROPER(nil))

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(cons(z0, z1))) → c(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)), PROPER(zeros))
TOP(mark(cons(z0, z1))) → c1(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))

S tuples:

PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

PROPER, TAKE, CONS, AND, ACTIVE, S, LENGTH, TOP

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13, c15, c16, c17, c18, c19, c, c, c1

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

Removed 1 trailing tuple parts

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(cons(z0, z1))) → c(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(cons(z0, z1))) → c1(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))

S tuples:

PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

PROPER, TAKE, CONS, AND, ACTIVE, S, LENGTH, TOP

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13, c15, c16, c17, c18, c19, c, c, c1, c1

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

Use narrowing to replace PROPER(cons(z0, z1)) → c6(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0), PROPER(zeros))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, cons(z0, z1))) → c6(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(zeros), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c6(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(cons(z0, z1))) → c(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(cons(z0, z1))) → c1(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0), PROPER(zeros))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, cons(z0, z1))) → c6(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(zeros), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c6(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0), PROPER(zeros))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, cons(z0, z1))) → c6(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(zeros), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c6(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, ACTIVE, S, LENGTH, TOP, PROPER

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c15, c16, c17, c18, c19, c, c, c1, c1, c6

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

Removed 8 trailing tuple parts

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(cons(z0, z1))) → c(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(cons(z0, z1))) → c1(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(x0, cons(z0, z1))) → c6(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(cons(z0, z1), x1)) → c6(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, cons(z0, z1))) → c6(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(cons(z0, z1), x1)) → c6(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, ACTIVE, S, LENGTH, TOP, PROPER

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c15, c16, c17, c18, c19, c, c, c1, c1, c6, c6

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

Use narrowing to replace ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0)) by

ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1), ACTIVE(zeros))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(cons(z0, z1))) → c(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(cons(z0, z1))) → c1(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(x0, cons(z0, z1))) → c6(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(cons(z0, z1), x1)) → c6(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1), ACTIVE(zeros))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, cons(z0, z1))) → c6(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(cons(z0, z1), x1)) → c6(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c, c1, c1, c6, c6, c15

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

Removed 1 trailing tuple parts

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(cons(z0, z1))) → c(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(cons(z0, z1))) → c1(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(x0, cons(z0, z1))) → c6(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(cons(z0, z1), x1)) → c6(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, cons(z0, z1))) → c6(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(cons(z0, z1), x1)) → c6(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c, c1, c1, c6, c6, c15, c15

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

Use narrowing to replace TOP(ok(cons(z0, z1))) → c(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) by

TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(cons(z0, z1))) → c1(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(x0, cons(z0, z1))) → c6(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(cons(z0, z1), x1)) → c6(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, cons(z0, z1))) → c6(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(cons(z0, z1), x1)) → c6(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c1, c6, c6, c15, c15, c

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

Use narrowing to replace TOP(mark(cons(z0, z1))) → c1(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) by

TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(x0, cons(z0, z1))) → c6(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(cons(z0, z1), x1)) → c6(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, cons(z0, z1))) → c6(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(cons(z0, z1), x1)) → c6(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c6, c15, c15, c, c1

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

Use narrowing to replace PROPER(cons(x0, cons(z0, z1))) → c6(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) by

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(tt), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(nil), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(zeros), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(0), PROPER(cons(x1, x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(cons(z0, z1), x1)) → c6(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(tt), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(nil), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(zeros), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(0), PROPER(cons(x1, x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(cons(z0, z1), x1)) → c6(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(tt), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(nil), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(zeros), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(0), PROPER(cons(x1, x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c6, c15, c15, c, c1

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

Removed 4 trailing tuple parts

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(cons(z0, z1), x1)) → c6(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(cons(z0, z1), x1)) → c6(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c6, c15, c15, c, c1

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

Use narrowing to replace PROPER(cons(cons(z0, z1), x1)) → c6(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) by

PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)), PROPER(tt))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)), PROPER(nil))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)), PROPER(zeros))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)), PROPER(0))
PROPER(cons(cons(x0, x1), cons(z0, z1))) → c6(CONS(cons(proper(x0), proper(x1)), cons(proper(z0), proper(z1))), PROPER(cons(x0, x1)), PROPER(cons(z0, z1)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)), PROPER(tt))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)), PROPER(nil))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)), PROPER(zeros))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)), PROPER(0))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)), PROPER(tt))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)), PROPER(nil))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)), PROPER(zeros))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)), PROPER(0))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c15, c15, c, c1, c6

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

Removed 4 trailing tuple parts

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c15, c15, c, c1, c6

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

Use narrowing to replace PROPER(cons(x0, tt)) → c6(CONS(proper(x0), ok(tt)), PROPER(x0)) by

PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)), PROPER(tt))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)), PROPER(nil))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)), PROPER(zeros))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)), PROPER(0))
PROPER(cons(cons(z0, z1), tt)) → c6(CONS(cons(proper(z0), proper(z1)), ok(tt)), PROPER(cons(z0, z1)))

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)), PROPER(tt))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)), PROPER(nil))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)), PROPER(zeros))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)), PROPER(0))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)), PROPER(tt))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)), PROPER(nil))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)), PROPER(zeros))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)), PROPER(0))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c15, c15, c, c1, c6

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

Removed 4 trailing tuple parts

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c15, c15, c, c1, c6, c6

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

Use narrowing to replace PROPER(cons(x0, nil)) → c6(CONS(proper(x0), ok(nil)), PROPER(x0)) by

PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)), PROPER(tt))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)), PROPER(nil))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)), PROPER(zeros))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)), PROPER(0))
PROPER(cons(cons(z0, z1), nil)) → c6(CONS(cons(proper(z0), proper(z1)), ok(nil)), PROPER(cons(z0, z1)))

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)), PROPER(tt))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)), PROPER(nil))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)), PROPER(zeros))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)), PROPER(0))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)), PROPER(tt))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)), PROPER(nil))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)), PROPER(zeros))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)), PROPER(0))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c15, c15, c, c1, c6, c6

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

Removed 4 trailing tuple parts

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c15, c15, c, c1, c6, c6

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

Use narrowing to replace PROPER(cons(x0, zeros)) → c6(CONS(proper(x0), ok(zeros)), PROPER(x0)) by

PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)), PROPER(tt))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)), PROPER(nil))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)), PROPER(zeros))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)), PROPER(0))
PROPER(cons(cons(z0, z1), zeros)) → c6(CONS(cons(proper(z0), proper(z1)), ok(zeros)), PROPER(cons(z0, z1)))

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)), PROPER(tt))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)), PROPER(nil))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)), PROPER(zeros))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)), PROPER(0))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)), PROPER(tt))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)), PROPER(nil))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)), PROPER(zeros))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)), PROPER(0))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c15, c15, c, c1, c6, c6

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

Removed 4 trailing tuple parts

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c15, c15, c, c1, c6, c6

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

Use narrowing to replace PROPER(cons(x0, 0)) → c6(CONS(proper(x0), ok(0)), PROPER(x0)) by

PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)), PROPER(tt))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)), PROPER(nil))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)), PROPER(zeros))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)), PROPER(0))
PROPER(cons(cons(z0, z1), 0)) → c6(CONS(cons(proper(z0), proper(z1)), ok(0)), PROPER(cons(z0, z1)))

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)), PROPER(tt))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)), PROPER(nil))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)), PROPER(zeros))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)), PROPER(0))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)), PROPER(tt))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)), PROPER(nil))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)), PROPER(zeros))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)), PROPER(0))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c15, c15, c, c1, c6, c6

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

Removed 4 trailing tuple parts

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c15, c15, c, c1, c6, c6

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

Use narrowing to replace PROPER(cons(tt, x1)) → c6(CONS(ok(tt), proper(x1)), PROPER(x1)) by

PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)), PROPER(tt))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)), PROPER(nil))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)), PROPER(zeros))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)), PROPER(0))
PROPER(cons(tt, cons(z0, z1))) → c6(CONS(ok(tt), cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)), PROPER(tt))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)), PROPER(nil))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)), PROPER(zeros))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)), PROPER(0))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)), PROPER(tt))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)), PROPER(nil))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)), PROPER(zeros))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)), PROPER(0))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c15, c15, c, c1, c6, c6

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

Removed 4 trailing tuple parts

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c15, c15, c, c1, c6, c6

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

Use narrowing to replace PROPER(cons(nil, x1)) → c6(CONS(ok(nil), proper(x1)), PROPER(x1)) by

PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)), PROPER(tt))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)), PROPER(nil))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)), PROPER(zeros))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)), PROPER(0))
PROPER(cons(nil, cons(z0, z1))) → c6(CONS(ok(nil), cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)), PROPER(tt))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)), PROPER(nil))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)), PROPER(zeros))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)), PROPER(0))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)), PROPER(tt))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)), PROPER(nil))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)), PROPER(zeros))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)), PROPER(0))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c15, c15, c, c1, c6, c6

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

Removed 4 trailing tuple parts

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c15, c15, c, c1, c6, c6

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

Use narrowing to replace PROPER(cons(zeros, x1)) → c6(CONS(ok(zeros), proper(x1)), PROPER(x1)) by

PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)), PROPER(tt))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)), PROPER(nil))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)), PROPER(zeros))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)), PROPER(0))
PROPER(cons(zeros, cons(z0, z1))) → c6(CONS(ok(zeros), cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)), PROPER(tt))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)), PROPER(nil))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)), PROPER(zeros))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)), PROPER(0))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)), PROPER(tt))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)), PROPER(nil))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)), PROPER(zeros))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)), PROPER(0))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c15, c15, c, c1, c6, c6

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

Removed 4 trailing tuple parts

(78) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, PROPER, ACTIVE

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c6, c15, c15, c, c1, c6, c6

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

Use narrowing to replace PROPER(cons(0, x1)) → c6(CONS(ok(0), proper(x1)), PROPER(x1)) by

PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)), PROPER(tt))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)), PROPER(nil))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)), PROPER(zeros))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)), PROPER(0))
PROPER(cons(0, cons(z0, z1))) → c6(CONS(ok(0), cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))

(80) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)), PROPER(tt))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)), PROPER(nil))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)), PROPER(zeros))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)), PROPER(0))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)), PROPER(tt))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)), PROPER(nil))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)), PROPER(zeros))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)), PROPER(0))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, ACTIVE, PROPER

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c15, c15, c, c1, c6, c6, c6

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

Removed 4 trailing tuple parts

(82) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, ACTIVE, PROPER

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c15, c15, c, c1, c6, c6, c6

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

Use narrowing to replace ACTIVE(cons(cons(z0, z1), x1)) → c15(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) by

ACTIVE(cons(cons(zeros, x1), x2)) → c15(CONS(cons(mark(cons(0, zeros)), x1), x2), ACTIVE(cons(zeros, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))

(84) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))
ACTIVE(cons(cons(zeros, x1), x2)) → c15(CONS(cons(mark(cons(0, zeros)), x1), x2), ACTIVE(cons(zeros, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, ACTIVE, PROPER

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c15, c, c1, c6, c6, c6, c15

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

Split RHS of tuples not part of any SCC

(86) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(CONS(cons(mark(cons(0, zeros)), x1), x2))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(ACTIVE(cons(zeros, x1)))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, ACTIVE, PROPER

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c15, c, c1, c6, c6, c6, c15, c2

(87) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace TOP(ok(cons(zeros, x1))) → c(TOP(cons(mark(cons(0, zeros)), x1)), ACTIVE(cons(zeros, x1))) by TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))), ACTIVE(cons(zeros, z0)))

(88) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(CONS(cons(mark(cons(0, zeros)), x1), x2))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(ACTIVE(cons(zeros, x1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))), ACTIVE(cons(zeros, z0)))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))

K tuples:

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, ACTIVE, PROPER

Compound Symbols:

c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c15, c, c1, c6, c6, c6, c15, c2

(89) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace TAKE(z0, mark(z1)) → c7(TAKE(z0, z1)) by

TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))

(90) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(CONS(cons(mark(cons(0, zeros)), x1), x2))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(ACTIVE(cons(zeros, x1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))), ACTIVE(cons(zeros, z0)))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))

K tuples:

TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
TAKE(mark(z0), z1) → c8(TAKE(z0, z1))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, ACTIVE, PROPER

Compound Symbols:

c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c15, c, c1, c6, c6, c6, c15, c2, c7

(91) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace TAKE(mark(z0), z1) → c8(TAKE(z0, z1)) by

TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))

(92) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(CONS(cons(mark(cons(0, zeros)), x1), x2))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(ACTIVE(cons(zeros, x1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))), ACTIVE(cons(zeros, z0)))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))

K tuples:

TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TAKE, CONS, AND, S, LENGTH, TOP, ACTIVE, PROPER

Compound Symbols:

c9, c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c15, c, c1, c6, c6, c6, c15, c2, c7, c8

(93) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace TAKE(ok(z0), ok(z1)) → c9(TAKE(z0, z1)) by

TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))

(94) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(CONS(cons(mark(cons(0, zeros)), x1), x2))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(ACTIVE(cons(zeros, x1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))), ACTIVE(cons(zeros, z0)))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))

S tuples:

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))

K tuples:

AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

CONS, AND, S, LENGTH, TOP, ACTIVE, PROPER, TAKE

Compound Symbols:

c10, c11, c12, c13, c16, c17, c18, c19, c, c1, c15, c, c1, c6, c6, c6, c15, c2, c7, c8, c9

(95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1)) by

CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))

(96) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(CONS(cons(mark(cons(0, zeros)), x1), x2))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(ACTIVE(cons(zeros, x1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))), ACTIVE(cons(zeros, z0)))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))

S tuples:

CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(tt, cons(x1, x2))) → c6(CONS(ok(tt), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(CONS(ok(nil), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(CONS(ok(zeros), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(CONS(ok(0), cons(proper(x1), proper(x2))), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
PROPER(cons(tt, tt)) → c6(CONS(ok(tt), ok(tt)))
PROPER(cons(nil, tt)) → c6(CONS(ok(nil), ok(tt)))
PROPER(cons(zeros, tt)) → c6(CONS(ok(zeros), ok(tt)))
PROPER(cons(0, tt)) → c6(CONS(ok(0), ok(tt)))
PROPER(cons(tt, nil)) → c6(CONS(ok(tt), ok(nil)))
PROPER(cons(nil, nil)) → c6(CONS(ok(nil), ok(nil)))
PROPER(cons(zeros, nil)) → c6(CONS(ok(zeros), ok(nil)))
PROPER(cons(0, nil)) → c6(CONS(ok(0), ok(nil)))
PROPER(cons(tt, zeros)) → c6(CONS(ok(tt), ok(zeros)))
PROPER(cons(nil, zeros)) → c6(CONS(ok(nil), ok(zeros)))
PROPER(cons(zeros, zeros)) → c6(CONS(ok(zeros), ok(zeros)))
PROPER(cons(0, zeros)) → c6(CONS(ok(0), ok(zeros)))
PROPER(cons(tt, 0)) → c6(CONS(ok(tt), ok(0)))
PROPER(cons(nil, 0)) → c6(CONS(ok(nil), ok(0)))
PROPER(cons(zeros, 0)) → c6(CONS(ok(zeros), ok(0)))
PROPER(cons(0, 0)) → c6(CONS(ok(0), ok(0)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))

K tuples:

AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

CONS, AND, S, LENGTH, TOP, ACTIVE, PROPER, TAKE

Compound Symbols:

c11, c12, c13, c16, c17, c18, c19, c, c1, c15, c, c1, c6, c6, c6, c15, c2, c7, c8, c9, c10

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

Removed 40 trailing tuple parts

(98) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

CONS(mark(z0), z1) → c11(CONS(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(CONS(cons(mark(cons(0, zeros)), x1), x2))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(ACTIVE(cons(zeros, x1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))), ACTIVE(cons(zeros, z0)))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6

S tuples:

CONS(mark(z0), z1) → c11(CONS(z0, z1))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6

K tuples:

AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

CONS, AND, S, LENGTH, TOP, ACTIVE, PROPER, TAKE

Compound Symbols:

c11, c12, c13, c16, c17, c18, c19, c, c1, c15, c, c1, c6, c6, c15, c2, c7, c8, c9, c10, c6, c6

(99) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace CONS(mark(z0), z1) → c11(CONS(z0, z1)) by

CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

(100) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(CONS(cons(mark(cons(0, zeros)), x1), x2))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(ACTIVE(cons(zeros, x1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))), ACTIVE(cons(zeros, z0)))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c15(CONS(active(z0), z1), ACTIVE(z0))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

AND, S, LENGTH, TOP, ACTIVE, PROPER, TAKE, CONS

Compound Symbols:

c12, c13, c16, c17, c18, c19, c, c1, c15, c, c1, c6, c6, c15, c2, c7, c8, c9, c10, c6, c6, c11

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

Removed 2 trailing nodes:

ACTIVE(cons(cons(zeros, x1), x2)) → c2(ACTIVE(cons(zeros, x1)))
ACTIVE(cons(zeros, x1)) → c15(CONS(mark(cons(0, zeros)), x1))

(102) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(CONS(cons(mark(cons(0, zeros)), x1), x2))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))), ACTIVE(cons(zeros, z0)))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

AND, S, LENGTH, TOP, PROPER, ACTIVE, TAKE, CONS

Compound Symbols:

c12, c13, c16, c17, c18, c19, c, c1, c, c1, c6, c6, c15, c2, c7, c8, c9, c10, c6, c6, c11

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

Removed 1 trailing tuple parts

(104) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(CONS(cons(mark(cons(0, zeros)), x1), x2))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

AND, S, LENGTH, TOP, PROPER, ACTIVE, TAKE, CONS

Compound Symbols:

c12, c13, c16, c17, c18, c19, c, c1, c, c1, c6, c6, c15, c2, c7, c8, c9, c10, c6, c6, c11

(105) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace AND(mark(z0), z1) → c12(AND(z0, z1)) by

AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))

(106) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(CONS(cons(mark(cons(0, zeros)), x1), x2))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

AND, S, LENGTH, TOP, PROPER, ACTIVE, TAKE, CONS

Compound Symbols:

c13, c16, c17, c18, c19, c, c1, c, c1, c6, c6, c15, c2, c7, c8, c9, c10, c6, c6, c11, c12

(107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace AND(ok(z0), ok(z1)) → c13(AND(z0, z1)) by

AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))

(108) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(CONS(cons(mark(cons(0, zeros)), x1), x2))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

S(ok(z0)) → c16(S(z0))
S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

S, LENGTH, TOP, PROPER, ACTIVE, TAKE, CONS, AND

Compound Symbols:

c16, c17, c18, c19, c, c1, c, c1, c6, c6, c15, c2, c7, c8, c9, c10, c6, c6, c11, c12, c13

(109) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace S(ok(z0)) → c16(S(z0)) by

S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))

(110) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(CONS(cons(mark(cons(0, zeros)), x1), x2))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

S(mark(z0)) → c17(S(z0))
LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

S, LENGTH, TOP, PROPER, ACTIVE, TAKE, CONS, AND

Compound Symbols:

c17, c18, c19, c, c1, c, c1, c6, c6, c15, c2, c7, c8, c9, c10, c6, c6, c11, c12, c13, c16

(111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace S(mark(z0)) → c17(S(z0)) by

S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))

(112) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(CONS(cons(mark(cons(0, zeros)), x1), x2))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

LENGTH(mark(z0)) → c18(LENGTH(z0))
LENGTH(ok(z0)) → c19(LENGTH(z0))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

LENGTH, TOP, PROPER, ACTIVE, TAKE, CONS, AND, S

Compound Symbols:

c18, c19, c, c1, c, c1, c6, c6, c15, c2, c7, c8, c9, c10, c6, c6, c11, c12, c13, c16, c17

(113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace LENGTH(mark(z0)) → c18(LENGTH(z0)) by

LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))

(114) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

LENGTH(ok(z0)) → c19(LENGTH(z0))
TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(CONS(cons(mark(cons(0, zeros)), x1), x2))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

LENGTH(ok(z0)) → c19(LENGTH(z0))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

LENGTH, TOP, PROPER, ACTIVE, TAKE, CONS, AND, S

Compound Symbols:

c19, c, c1, c, c1, c6, c6, c15, c2, c7, c8, c9, c10, c6, c6, c11, c12, c13, c16, c17, c18

(115) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace LENGTH(ok(z0)) → c19(LENGTH(z0)) by

LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))

(116) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
ACTIVE(cons(cons(zeros, x1), x2)) → c2(CONS(cons(mark(cons(0, zeros)), x1), x2))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, ACTIVE, TAKE, CONS, AND, S, LENGTH

Compound Symbols:

c, c1, c, c1, c6, c6, c15, c2, c7, c8, c9, c10, c6, c6, c11, c12, c13, c16, c17, c18, c19

(117) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace ACTIVE(cons(cons(zeros, x1), x2)) → c2(CONS(cons(mark(cons(0, zeros)), x1), x2)) by ACTIVE(cons(cons(zeros, z0), z1)) → c2(CONS(mark(cons(cons(0, zeros), z0)), z1))

(118) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
ACTIVE(cons(cons(zeros, z0), z1)) → c2(CONS(mark(cons(cons(0, zeros), z0)), z1))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, ACTIVE, TAKE, CONS, AND, S, LENGTH

Compound Symbols:

c, c1, c, c1, c6, c6, c15, c7, c8, c9, c10, c6, c6, c11, c12, c13, c16, c17, c18, c19, c2

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

Removed 1 trailing nodes:

ACTIVE(cons(cons(zeros, z0), z1)) → c2(CONS(mark(cons(cons(0, zeros), z0)), z1))

(120) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, ACTIVE, TAKE, CONS, AND, S, LENGTH

Compound Symbols:

c, c1, c, c1, c6, c6, c15, c7, c8, c9, c10, c6, c6, c11, c12, c13, c16, c17, c18, c19

(121) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace TAKE(z0, mark(mark(y1))) → c7(TAKE(z0, mark(y1))) by

TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))

(122) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, ACTIVE, TAKE, CONS, AND, S, LENGTH

Compound Symbols:

c, c1, c, c1, c6, c6, c15, c7, c8, c9, c10, c6, c6, c11, c12, c13, c16, c17, c18, c19

(123) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace TAKE(mark(y0), mark(z1)) → c7(TAKE(mark(y0), z1)) by

TAKE(mark(z0), mark(mark(y1))) → c7(TAKE(mark(z0), mark(y1)))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(y1)))) → c7(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(y0)), mark(mark(y1))) → c7(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))

(124) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, ACTIVE, TAKE, CONS, AND, S, LENGTH

Compound Symbols:

c, c1, c, c1, c6, c6, c15, c7, c8, c9, c10, c6, c6, c11, c12, c13, c16, c17, c18, c19

(125) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace TAKE(ok(y0), mark(ok(y1))) → c7(TAKE(ok(y0), ok(y1))) by

TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))

(126) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, ACTIVE, TAKE, CONS, AND, S, LENGTH

Compound Symbols:

c, c1, c, c1, c6, c6, c15, c8, c9, c10, c6, c6, c11, c12, c13, c16, c17, c18, c19, c7

(127) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace TAKE(mark(mark(y0)), z1) → c8(TAKE(mark(y0), z1)) by

TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))

(128) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, ACTIVE, TAKE, CONS, AND, S, LENGTH

Compound Symbols:

c, c1, c, c1, c6, c6, c15, c8, c9, c10, c6, c6, c11, c12, c13, c16, c17, c18, c19, c7

(129) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace TAKE(mark(ok(y0)), ok(y1)) → c8(TAKE(ok(y0), ok(y1))) by

TAKE(mark(ok(ok(y0))), ok(ok(y1))) → c8(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(mark(ok(z0)), ok(mark(mark(y1)))) → c8(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(y0))), ok(mark(y1))) → c8(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(mark(ok(ok(y0))), ok(mark(ok(y1)))) → c8(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(z1)) → c8(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(mark(ok(mark(ok(y0)))), ok(ok(y1))) → c8(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(mark(ok(mark(y0))), ok(mark(mark(y1)))) → c8(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(mark(y1))) → c8(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(mark(ok(mark(ok(y0)))), ok(mark(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))

(130) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), ok(ok(y1))) → c8(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(mark(ok(z0)), ok(mark(mark(y1)))) → c8(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(y0))), ok(mark(y1))) → c8(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(mark(ok(ok(y0))), ok(mark(ok(y1)))) → c8(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(z1)) → c8(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(mark(ok(mark(ok(y0)))), ok(ok(y1))) → c8(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(mark(ok(mark(y0))), ok(mark(mark(y1)))) → c8(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(mark(y1))) → c8(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(mark(ok(mark(ok(y0)))), ok(mark(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1))))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), ok(ok(y1))) → c8(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(mark(ok(z0)), ok(mark(mark(y1)))) → c8(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(y0))), ok(mark(y1))) → c8(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(mark(ok(ok(y0))), ok(mark(ok(y1)))) → c8(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(z1)) → c8(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(mark(ok(mark(ok(y0)))), ok(ok(y1))) → c8(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(mark(ok(mark(y0))), ok(mark(mark(y1)))) → c8(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(mark(y1))) → c8(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(mark(ok(mark(ok(y0)))), ok(mark(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, ACTIVE, TAKE, CONS, AND, S, LENGTH

Compound Symbols:

c, c1, c, c1, c6, c6, c15, c8, c9, c10, c6, c6, c11, c12, c13, c16, c17, c18, c19, c7

(131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace TAKE(mark(z0), mark(mark(y1))) → c8(TAKE(z0, mark(mark(y1)))) by

TAKE(mark(mark(y0)), mark(mark(z1))) → c8(TAKE(mark(y0), mark(mark(z1))))
TAKE(mark(mark(mark(y0))), mark(mark(z1))) → c8(TAKE(mark(mark(y0)), mark(mark(z1))))
TAKE(mark(z0), mark(mark(mark(y1)))) → c8(TAKE(z0, mark(mark(mark(y1)))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c8(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c8(TAKE(mark(y0), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(y0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))

(132) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), ok(ok(y1))) → c8(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(mark(ok(z0)), ok(mark(mark(y1)))) → c8(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(y0))), ok(mark(y1))) → c8(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(mark(ok(ok(y0))), ok(mark(ok(y1)))) → c8(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(z1)) → c8(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(mark(ok(mark(ok(y0)))), ok(ok(y1))) → c8(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(mark(ok(mark(y0))), ok(mark(mark(y1)))) → c8(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(mark(y1))) → c8(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(mark(ok(mark(ok(y0)))), ok(mark(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(z0), mark(mark(mark(y1)))) → c8(TAKE(z0, mark(mark(mark(y1)))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c8(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), ok(ok(y1))) → c8(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(mark(ok(z0)), ok(mark(mark(y1)))) → c8(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(y0))), ok(mark(y1))) → c8(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(mark(ok(ok(y0))), ok(mark(ok(y1)))) → c8(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(z1)) → c8(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(mark(ok(mark(ok(y0)))), ok(ok(y1))) → c8(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(mark(ok(mark(y0))), ok(mark(mark(y1)))) → c8(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(mark(y1))) → c8(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(mark(ok(mark(ok(y0)))), ok(mark(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(z0), mark(mark(mark(y1)))) → c8(TAKE(z0, mark(mark(mark(y1)))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c8(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, ACTIVE, TAKE, CONS, AND, S, LENGTH

Compound Symbols:

c, c1, c, c1, c6, c6, c15, c8, c9, c10, c6, c6, c11, c12, c13, c16, c17, c18, c19, c7

(133) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace TAKE(mark(mark(y0)), mark(y1)) → c8(TAKE(mark(y0), mark(y1))) by

TAKE(mark(mark(mark(y0))), mark(z1)) → c8(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(z1)) → c8(TAKE(mark(mark(mark(y0))), mark(z1)))
TAKE(mark(mark(mark(ok(y0)))), mark(ok(y1))) → c8(TAKE(mark(mark(ok(y0))), mark(ok(y1))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(y1))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(y1))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(y1))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(y1))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))

(134) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), ok(ok(y1))) → c8(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(mark(ok(z0)), ok(mark(mark(y1)))) → c8(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(y0))), ok(mark(y1))) → c8(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(mark(ok(ok(y0))), ok(mark(ok(y1)))) → c8(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(z1)) → c8(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(mark(ok(mark(ok(y0)))), ok(ok(y1))) → c8(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(mark(ok(mark(y0))), ok(mark(mark(y1)))) → c8(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(mark(y1))) → c8(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(mark(ok(mark(ok(y0)))), ok(mark(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(z0), mark(mark(mark(y1)))) → c8(TAKE(z0, mark(mark(mark(y1)))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c8(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(z1)) → c8(TAKE(mark(mark(mark(y0))), mark(z1)))
TAKE(mark(mark(mark(ok(y0)))), mark(ok(y1))) → c8(TAKE(mark(mark(ok(y0))), mark(ok(y1))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(y1))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(y1))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), ok(ok(y1))) → c8(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(mark(ok(z0)), ok(mark(mark(y1)))) → c8(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(y0))), ok(mark(y1))) → c8(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(mark(ok(ok(y0))), ok(mark(ok(y1)))) → c8(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(z1)) → c8(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(mark(ok(mark(ok(y0)))), ok(ok(y1))) → c8(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(mark(ok(mark(y0))), ok(mark(mark(y1)))) → c8(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(mark(y1))) → c8(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(mark(ok(mark(ok(y0)))), ok(mark(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(z0), mark(mark(mark(y1)))) → c8(TAKE(z0, mark(mark(mark(y1)))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c8(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(z1)) → c8(TAKE(mark(mark(mark(y0))), mark(z1)))
TAKE(mark(mark(mark(ok(y0)))), mark(ok(y1))) → c8(TAKE(mark(mark(ok(y0))), mark(ok(y1))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(y1))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(y1))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, ACTIVE, TAKE, CONS, AND, S, LENGTH

Compound Symbols:

c, c1, c, c1, c6, c6, c15, c8, c9, c10, c6, c6, c11, c12, c13, c16, c17, c18, c19, c7

(135) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace TAKE(mark(ok(y0)), mark(ok(y1))) → c8(TAKE(ok(y0), mark(ok(y1)))) by

TAKE(mark(ok(ok(y0))), mark(ok(ok(y1)))) → c8(TAKE(ok(ok(y0)), mark(ok(ok(y1)))))
TAKE(mark(ok(z0)), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(z0), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(y1)))))
TAKE(mark(ok(ok(y0))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(z1))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(z1))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))))

(136) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), ok(ok(y1))) → c8(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(mark(ok(z0)), ok(mark(mark(y1)))) → c8(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(y0))), ok(mark(y1))) → c8(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(mark(ok(ok(y0))), ok(mark(ok(y1)))) → c8(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(z1)) → c8(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(mark(ok(mark(ok(y0)))), ok(ok(y1))) → c8(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(mark(ok(mark(y0))), ok(mark(mark(y1)))) → c8(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(mark(y1))) → c8(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(mark(ok(mark(ok(y0)))), ok(mark(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(z0), mark(mark(mark(y1)))) → c8(TAKE(z0, mark(mark(mark(y1)))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c8(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(z1)) → c8(TAKE(mark(mark(mark(y0))), mark(z1)))
TAKE(mark(mark(mark(ok(y0)))), mark(ok(y1))) → c8(TAKE(mark(mark(ok(y0))), mark(ok(y1))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(y1))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(y1))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), mark(ok(ok(y1)))) → c8(TAKE(ok(ok(y0)), mark(ok(ok(y1)))))
TAKE(mark(ok(z0)), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(z0), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(y1)))))
TAKE(mark(ok(ok(y0))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(z1))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(z1))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), ok(ok(y1))) → c8(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(mark(ok(z0)), ok(mark(mark(y1)))) → c8(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(y0))), ok(mark(y1))) → c8(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(mark(ok(ok(y0))), ok(mark(ok(y1)))) → c8(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(z1)) → c8(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(mark(ok(mark(ok(y0)))), ok(ok(y1))) → c8(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(mark(ok(mark(y0))), ok(mark(mark(y1)))) → c8(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(mark(y1))) → c8(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(mark(ok(mark(ok(y0)))), ok(mark(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(z0), mark(mark(mark(y1)))) → c8(TAKE(z0, mark(mark(mark(y1)))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c8(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(z1)) → c8(TAKE(mark(mark(mark(y0))), mark(z1)))
TAKE(mark(mark(mark(ok(y0)))), mark(ok(y1))) → c8(TAKE(mark(mark(ok(y0))), mark(ok(y1))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(y1))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(y1))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), mark(ok(ok(y1)))) → c8(TAKE(ok(ok(y0)), mark(ok(ok(y1)))))
TAKE(mark(ok(z0)), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(z0), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(y1)))))
TAKE(mark(ok(ok(y0))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(z1))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(z1))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, ACTIVE, TAKE, CONS, AND, S, LENGTH

Compound Symbols:

c, c1, c, c1, c6, c6, c15, c9, c10, c6, c6, c11, c12, c13, c16, c17, c18, c19, c7, c8

(137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace TAKE(ok(ok(y0)), ok(ok(y1))) → c9(TAKE(ok(y0), ok(y1))) by

TAKE(ok(ok(ok(y0))), ok(ok(ok(y1)))) → c9(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(ok(z0)), ok(ok(mark(mark(y1))))) → c9(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(ok(mark(y0))), ok(ok(mark(y1)))) → c9(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(ok(y0))), ok(ok(mark(ok(y1))))) → c9(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(ok(mark(mark(y0)))), ok(ok(z1))) → c9(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(ok(mark(ok(y0)))), ok(ok(ok(y1)))) → c9(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(ok(mark(y0))), ok(ok(mark(mark(y1))))) → c9(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(ok(mark(mark(y0)))), ok(ok(mark(y1)))) → c9(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(ok(mark(ok(y0)))), ok(ok(mark(ok(y1))))) → c9(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))

(138) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), ok(ok(y1))) → c8(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(mark(ok(z0)), ok(mark(mark(y1)))) → c8(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(y0))), ok(mark(y1))) → c8(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(mark(ok(ok(y0))), ok(mark(ok(y1)))) → c8(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(z1)) → c8(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(mark(ok(mark(ok(y0)))), ok(ok(y1))) → c8(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(mark(ok(mark(y0))), ok(mark(mark(y1)))) → c8(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(mark(y1))) → c8(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(mark(ok(mark(ok(y0)))), ok(mark(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(z0), mark(mark(mark(y1)))) → c8(TAKE(z0, mark(mark(mark(y1)))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c8(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(z1)) → c8(TAKE(mark(mark(mark(y0))), mark(z1)))
TAKE(mark(mark(mark(ok(y0)))), mark(ok(y1))) → c8(TAKE(mark(mark(ok(y0))), mark(ok(y1))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(y1))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(y1))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), mark(ok(ok(y1)))) → c8(TAKE(ok(ok(y0)), mark(ok(ok(y1)))))
TAKE(mark(ok(z0)), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(z0), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(y1)))))
TAKE(mark(ok(ok(y0))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(z1))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(z1))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))))
TAKE(ok(ok(ok(y0))), ok(ok(ok(y1)))) → c9(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(ok(z0)), ok(ok(mark(mark(y1))))) → c9(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(ok(mark(y0))), ok(ok(mark(y1)))) → c9(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(ok(y0))), ok(ok(mark(ok(y1))))) → c9(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(ok(mark(mark(y0)))), ok(ok(z1))) → c9(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(ok(mark(ok(y0)))), ok(ok(ok(y1)))) → c9(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(ok(mark(y0))), ok(ok(mark(mark(y1))))) → c9(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(ok(mark(mark(y0)))), ok(ok(mark(y1)))) → c9(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(ok(mark(ok(y0)))), ok(ok(mark(ok(y1))))) → c9(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1))))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), ok(ok(y1))) → c8(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(mark(ok(z0)), ok(mark(mark(y1)))) → c8(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(y0))), ok(mark(y1))) → c8(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(mark(ok(ok(y0))), ok(mark(ok(y1)))) → c8(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(z1)) → c8(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(mark(ok(mark(ok(y0)))), ok(ok(y1))) → c8(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(mark(ok(mark(y0))), ok(mark(mark(y1)))) → c8(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(mark(y1))) → c8(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(mark(ok(mark(ok(y0)))), ok(mark(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(z0), mark(mark(mark(y1)))) → c8(TAKE(z0, mark(mark(mark(y1)))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c8(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(z1)) → c8(TAKE(mark(mark(mark(y0))), mark(z1)))
TAKE(mark(mark(mark(ok(y0)))), mark(ok(y1))) → c8(TAKE(mark(mark(ok(y0))), mark(ok(y1))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(y1))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(y1))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), mark(ok(ok(y1)))) → c8(TAKE(ok(ok(y0)), mark(ok(ok(y1)))))
TAKE(mark(ok(z0)), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(z0), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(y1)))))
TAKE(mark(ok(ok(y0))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(z1))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(z1))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))))
TAKE(ok(ok(ok(y0))), ok(ok(ok(y1)))) → c9(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(ok(z0)), ok(ok(mark(mark(y1))))) → c9(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(ok(mark(y0))), ok(ok(mark(y1)))) → c9(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(ok(y0))), ok(ok(mark(ok(y1))))) → c9(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(ok(mark(mark(y0)))), ok(ok(z1))) → c9(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(ok(mark(ok(y0)))), ok(ok(ok(y1)))) → c9(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(ok(mark(y0))), ok(ok(mark(mark(y1))))) → c9(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(ok(mark(mark(y0)))), ok(ok(mark(y1)))) → c9(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(ok(mark(ok(y0)))), ok(ok(mark(ok(y1))))) → c9(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, ACTIVE, TAKE, CONS, AND, S, LENGTH

Compound Symbols:

c, c1, c, c1, c6, c6, c15, c9, c10, c6, c6, c11, c12, c13, c16, c17, c18, c19, c7, c8

(139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace TAKE(ok(z0), ok(mark(mark(y1)))) → c9(TAKE(z0, mark(mark(y1)))) by

TAKE(ok(z0), ok(mark(mark(mark(y1))))) → c9(TAKE(z0, mark(mark(mark(y1)))))
TAKE(ok(mark(y0)), ok(mark(mark(z1)))) → c9(TAKE(mark(y0), mark(mark(z1))))
TAKE(ok(ok(y0)), ok(mark(mark(ok(y1))))) → c9(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(y0)), mark(mark(z1))))
TAKE(ok(mark(y0)), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(y0), mark(mark(mark(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(mark(y0)), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(y0), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(mark(mark(y0)))))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(z1))))
TAKE(ok(mark(mark(mark(mark(ok(y0)))))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(mark(mark(y0)))))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(mark(mark(ok(y0)))))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))))

(140) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), ok(ok(y1))) → c8(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(mark(ok(z0)), ok(mark(mark(y1)))) → c8(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(y0))), ok(mark(y1))) → c8(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(mark(ok(ok(y0))), ok(mark(ok(y1)))) → c8(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(z1)) → c8(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(mark(ok(mark(ok(y0)))), ok(ok(y1))) → c8(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(mark(ok(mark(y0))), ok(mark(mark(y1)))) → c8(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(mark(y1))) → c8(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(mark(ok(mark(ok(y0)))), ok(mark(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(z0), mark(mark(mark(y1)))) → c8(TAKE(z0, mark(mark(mark(y1)))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c8(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(z1)) → c8(TAKE(mark(mark(mark(y0))), mark(z1)))
TAKE(mark(mark(mark(ok(y0)))), mark(ok(y1))) → c8(TAKE(mark(mark(ok(y0))), mark(ok(y1))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(y1))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(y1))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), mark(ok(ok(y1)))) → c8(TAKE(ok(ok(y0)), mark(ok(ok(y1)))))
TAKE(mark(ok(z0)), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(z0), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(y1)))))
TAKE(mark(ok(ok(y0))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(z1))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(z1))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))))
TAKE(ok(ok(ok(y0))), ok(ok(ok(y1)))) → c9(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(ok(z0)), ok(ok(mark(mark(y1))))) → c9(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(ok(mark(y0))), ok(ok(mark(y1)))) → c9(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(ok(y0))), ok(ok(mark(ok(y1))))) → c9(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(ok(mark(mark(y0)))), ok(ok(z1))) → c9(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(ok(mark(ok(y0)))), ok(ok(ok(y1)))) → c9(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(ok(mark(y0))), ok(ok(mark(mark(y1))))) → c9(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(ok(mark(mark(y0)))), ok(ok(mark(y1)))) → c9(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(ok(mark(ok(y0)))), ok(ok(mark(ok(y1))))) → c9(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(ok(z0), ok(mark(mark(mark(y1))))) → c9(TAKE(z0, mark(mark(mark(y1)))))
TAKE(ok(ok(y0)), ok(mark(mark(ok(y1))))) → c9(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(y0)), mark(mark(z1))))
TAKE(ok(mark(y0)), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(y0), mark(mark(mark(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(mark(y0)), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(y0), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(mark(mark(y0)))))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(z1))))
TAKE(ok(mark(mark(mark(mark(ok(y0)))))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(mark(mark(y0)))))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(mark(mark(ok(y0)))))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), ok(ok(y1))) → c8(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(mark(ok(z0)), ok(mark(mark(y1)))) → c8(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(y0))), ok(mark(y1))) → c8(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(mark(ok(ok(y0))), ok(mark(ok(y1)))) → c8(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(z1)) → c8(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(mark(ok(mark(ok(y0)))), ok(ok(y1))) → c8(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(mark(ok(mark(y0))), ok(mark(mark(y1)))) → c8(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(mark(y1))) → c8(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(mark(ok(mark(ok(y0)))), ok(mark(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(z0), mark(mark(mark(y1)))) → c8(TAKE(z0, mark(mark(mark(y1)))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c8(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(z1)) → c8(TAKE(mark(mark(mark(y0))), mark(z1)))
TAKE(mark(mark(mark(ok(y0)))), mark(ok(y1))) → c8(TAKE(mark(mark(ok(y0))), mark(ok(y1))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(y1))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(y1))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), mark(ok(ok(y1)))) → c8(TAKE(ok(ok(y0)), mark(ok(ok(y1)))))
TAKE(mark(ok(z0)), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(z0), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(y1)))))
TAKE(mark(ok(ok(y0))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(z1))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(z1))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))))
TAKE(ok(ok(ok(y0))), ok(ok(ok(y1)))) → c9(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(ok(z0)), ok(ok(mark(mark(y1))))) → c9(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(ok(mark(y0))), ok(ok(mark(y1)))) → c9(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(ok(y0))), ok(ok(mark(ok(y1))))) → c9(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(ok(mark(mark(y0)))), ok(ok(z1))) → c9(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(ok(mark(ok(y0)))), ok(ok(ok(y1)))) → c9(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(ok(mark(y0))), ok(ok(mark(mark(y1))))) → c9(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(ok(mark(mark(y0)))), ok(ok(mark(y1)))) → c9(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(ok(mark(ok(y0)))), ok(ok(mark(ok(y1))))) → c9(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(ok(z0), ok(mark(mark(mark(y1))))) → c9(TAKE(z0, mark(mark(mark(y1)))))
TAKE(ok(ok(y0)), ok(mark(mark(ok(y1))))) → c9(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(y0)), mark(mark(z1))))
TAKE(ok(mark(y0)), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(y0), mark(mark(mark(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(mark(y0)), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(y0), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(mark(mark(y0)))))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(z1))))
TAKE(ok(mark(mark(mark(mark(ok(y0)))))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(mark(mark(y0)))))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(mark(mark(ok(y0)))))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, ACTIVE, TAKE, CONS, AND, S, LENGTH

Compound Symbols:

c, c1, c, c1, c6, c6, c15, c9, c10, c6, c6, c11, c12, c13, c16, c17, c18, c19, c7, c8

(141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace TAKE(ok(mark(y0)), ok(mark(y1))) → c9(TAKE(mark(y0), mark(y1))) by

TAKE(ok(mark(z0)), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(ok(mark(z0)), ok(mark(mark(y1)))) → c9(TAKE(mark(z0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(y1)))) → c9(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(ok(mark(ok(y0))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), ok(mark(z1))) → c9(TAKE(mark(mark(y0)), mark(z1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(ok(mark(z0)), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(z1))) → c9(TAKE(mark(mark(mark(y0))), mark(z1)))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(ok(y1)))) → c9(TAKE(mark(mark(ok(y0))), mark(ok(y1))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(y1)))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(y1))))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(y1)))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(y1))))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(z1))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(z1)))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(ok(y1)))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(ok(y1))))
TAKE(ok(mark(mark(mark(mark(mark(y0)))))), ok(mark(mark(y1)))) → c9(TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(y1))))
TAKE(ok(mark(mark(mark(mark(ok(y0)))))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(mark(mark(y0)))))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(mark(mark(ok(y0)))))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(ok(ok(y0)))), ok(mark(ok(ok(y1))))) → c9(TAKE(mark(ok(ok(y0))), mark(ok(ok(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(ok(mark(mark(y1)))))) → c9(TAKE(mark(ok(y0)), mark(ok(mark(mark(y1))))))
TAKE(ok(mark(ok(mark(y0)))), ok(mark(ok(mark(y1))))) → c9(TAKE(mark(ok(mark(y0))), mark(ok(mark(y1)))))
TAKE(ok(mark(ok(ok(y0)))), ok(mark(ok(mark(ok(y1)))))) → c9(TAKE(mark(ok(ok(y0))), mark(ok(mark(ok(y1))))))
TAKE(ok(mark(ok(mark(mark(y0))))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(mark(mark(y0)))), mark(ok(y1))))
TAKE(ok(mark(ok(mark(ok(y0))))), ok(mark(ok(ok(y1))))) → c9(TAKE(mark(ok(mark(ok(y0)))), mark(ok(ok(y1)))))
TAKE(ok(mark(ok(mark(y0)))), ok(mark(ok(mark(mark(y1)))))) → c9(TAKE(mark(ok(mark(y0))), mark(ok(mark(mark(y1))))))
TAKE(ok(mark(ok(mark(mark(y0))))), ok(mark(ok(mark(y1))))) → c9(TAKE(mark(ok(mark(mark(y0)))), mark(ok(mark(y1)))))
TAKE(ok(mark(ok(mark(ok(y0))))), ok(mark(ok(mark(ok(y1)))))) → c9(TAKE(mark(ok(mark(ok(y0)))), mark(ok(mark(ok(y1))))))

(142) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), ok(ok(y1))) → c8(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(mark(ok(z0)), ok(mark(mark(y1)))) → c8(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(y0))), ok(mark(y1))) → c8(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(mark(ok(ok(y0))), ok(mark(ok(y1)))) → c8(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(z1)) → c8(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(mark(ok(mark(ok(y0)))), ok(ok(y1))) → c8(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(mark(ok(mark(y0))), ok(mark(mark(y1)))) → c8(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(mark(y1))) → c8(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(mark(ok(mark(ok(y0)))), ok(mark(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(z0), mark(mark(mark(y1)))) → c8(TAKE(z0, mark(mark(mark(y1)))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c8(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(z1)) → c8(TAKE(mark(mark(mark(y0))), mark(z1)))
TAKE(mark(mark(mark(ok(y0)))), mark(ok(y1))) → c8(TAKE(mark(mark(ok(y0))), mark(ok(y1))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(y1))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(y1))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), mark(ok(ok(y1)))) → c8(TAKE(ok(ok(y0)), mark(ok(ok(y1)))))
TAKE(mark(ok(z0)), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(z0), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(y1)))))
TAKE(mark(ok(ok(y0))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(z1))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(z1))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))))
TAKE(ok(ok(ok(y0))), ok(ok(ok(y1)))) → c9(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(ok(z0)), ok(ok(mark(mark(y1))))) → c9(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(ok(mark(y0))), ok(ok(mark(y1)))) → c9(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(ok(y0))), ok(ok(mark(ok(y1))))) → c9(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(ok(mark(mark(y0)))), ok(ok(z1))) → c9(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(ok(mark(ok(y0)))), ok(ok(ok(y1)))) → c9(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(ok(mark(y0))), ok(ok(mark(mark(y1))))) → c9(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(ok(mark(mark(y0)))), ok(ok(mark(y1)))) → c9(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(ok(mark(ok(y0)))), ok(ok(mark(ok(y1))))) → c9(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(ok(z0), ok(mark(mark(mark(y1))))) → c9(TAKE(z0, mark(mark(mark(y1)))))
TAKE(ok(ok(y0)), ok(mark(mark(ok(y1))))) → c9(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(y0)), mark(mark(z1))))
TAKE(ok(mark(y0)), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(y0), mark(mark(mark(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(mark(y0)), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(y0), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(mark(mark(y0)))))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(z1))))
TAKE(ok(mark(mark(mark(mark(ok(y0)))))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(mark(mark(y0)))))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(mark(mark(ok(y0)))))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(z1))) → c9(TAKE(mark(mark(mark(y0))), mark(z1)))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(ok(y1)))) → c9(TAKE(mark(mark(ok(y0))), mark(ok(y1))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(z1))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(z1)))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(ok(y1)))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(ok(y1))))
TAKE(ok(mark(ok(ok(y0)))), ok(mark(ok(ok(y1))))) → c9(TAKE(mark(ok(ok(y0))), mark(ok(ok(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(ok(mark(mark(y1)))))) → c9(TAKE(mark(ok(y0)), mark(ok(mark(mark(y1))))))
TAKE(ok(mark(ok(mark(y0)))), ok(mark(ok(mark(y1))))) → c9(TAKE(mark(ok(mark(y0))), mark(ok(mark(y1)))))
TAKE(ok(mark(ok(ok(y0)))), ok(mark(ok(mark(ok(y1)))))) → c9(TAKE(mark(ok(ok(y0))), mark(ok(mark(ok(y1))))))
TAKE(ok(mark(ok(mark(mark(y0))))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(mark(mark(y0)))), mark(ok(y1))))
TAKE(ok(mark(ok(mark(ok(y0))))), ok(mark(ok(ok(y1))))) → c9(TAKE(mark(ok(mark(ok(y0)))), mark(ok(ok(y1)))))
TAKE(ok(mark(ok(mark(y0)))), ok(mark(ok(mark(mark(y1)))))) → c9(TAKE(mark(ok(mark(y0))), mark(ok(mark(mark(y1))))))
TAKE(ok(mark(ok(mark(mark(y0))))), ok(mark(ok(mark(y1))))) → c9(TAKE(mark(ok(mark(mark(y0)))), mark(ok(mark(y1)))))
TAKE(ok(mark(ok(mark(ok(y0))))), ok(mark(ok(mark(ok(y1)))))) → c9(TAKE(mark(ok(mark(ok(y0)))), mark(ok(mark(ok(y1))))))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1))))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), ok(ok(y1))) → c8(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(mark(ok(z0)), ok(mark(mark(y1)))) → c8(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(y0))), ok(mark(y1))) → c8(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(mark(ok(ok(y0))), ok(mark(ok(y1)))) → c8(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(z1)) → c8(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(mark(ok(mark(ok(y0)))), ok(ok(y1))) → c8(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(mark(ok(mark(y0))), ok(mark(mark(y1)))) → c8(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(mark(y1))) → c8(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(mark(ok(mark(ok(y0)))), ok(mark(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(z0), mark(mark(mark(y1)))) → c8(TAKE(z0, mark(mark(mark(y1)))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c8(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(z1)) → c8(TAKE(mark(mark(mark(y0))), mark(z1)))
TAKE(mark(mark(mark(ok(y0)))), mark(ok(y1))) → c8(TAKE(mark(mark(ok(y0))), mark(ok(y1))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(y1))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(y1))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), mark(ok(ok(y1)))) → c8(TAKE(ok(ok(y0)), mark(ok(ok(y1)))))
TAKE(mark(ok(z0)), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(z0), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(y1)))))
TAKE(mark(ok(ok(y0))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(z1))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(z1))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))))
TAKE(ok(ok(ok(y0))), ok(ok(ok(y1)))) → c9(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(ok(z0)), ok(ok(mark(mark(y1))))) → c9(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(ok(mark(y0))), ok(ok(mark(y1)))) → c9(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(ok(y0))), ok(ok(mark(ok(y1))))) → c9(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(ok(mark(mark(y0)))), ok(ok(z1))) → c9(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(ok(mark(ok(y0)))), ok(ok(ok(y1)))) → c9(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(ok(mark(y0))), ok(ok(mark(mark(y1))))) → c9(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(ok(mark(mark(y0)))), ok(ok(mark(y1)))) → c9(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(ok(mark(ok(y0)))), ok(ok(mark(ok(y1))))) → c9(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(ok(z0), ok(mark(mark(mark(y1))))) → c9(TAKE(z0, mark(mark(mark(y1)))))
TAKE(ok(ok(y0)), ok(mark(mark(ok(y1))))) → c9(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(y0)), mark(mark(z1))))
TAKE(ok(mark(y0)), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(y0), mark(mark(mark(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(mark(y0)), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(y0), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(mark(mark(y0)))))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(z1))))
TAKE(ok(mark(mark(mark(mark(ok(y0)))))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(mark(mark(y0)))))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(mark(mark(ok(y0)))))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(z1))) → c9(TAKE(mark(mark(mark(y0))), mark(z1)))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(ok(y1)))) → c9(TAKE(mark(mark(ok(y0))), mark(ok(y1))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(z1))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(z1)))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(ok(y1)))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(ok(y1))))
TAKE(ok(mark(ok(ok(y0)))), ok(mark(ok(ok(y1))))) → c9(TAKE(mark(ok(ok(y0))), mark(ok(ok(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(ok(mark(mark(y1)))))) → c9(TAKE(mark(ok(y0)), mark(ok(mark(mark(y1))))))
TAKE(ok(mark(ok(mark(y0)))), ok(mark(ok(mark(y1))))) → c9(TAKE(mark(ok(mark(y0))), mark(ok(mark(y1)))))
TAKE(ok(mark(ok(ok(y0)))), ok(mark(ok(mark(ok(y1)))))) → c9(TAKE(mark(ok(ok(y0))), mark(ok(mark(ok(y1))))))
TAKE(ok(mark(ok(mark(mark(y0))))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(mark(mark(y0)))), mark(ok(y1))))
TAKE(ok(mark(ok(mark(ok(y0))))), ok(mark(ok(ok(y1))))) → c9(TAKE(mark(ok(mark(ok(y0)))), mark(ok(ok(y1)))))
TAKE(ok(mark(ok(mark(y0)))), ok(mark(ok(mark(mark(y1)))))) → c9(TAKE(mark(ok(mark(y0))), mark(ok(mark(mark(y1))))))
TAKE(ok(mark(ok(mark(mark(y0))))), ok(mark(ok(mark(y1))))) → c9(TAKE(mark(ok(mark(mark(y0)))), mark(ok(mark(y1)))))
TAKE(ok(mark(ok(mark(ok(y0))))), ok(mark(ok(mark(ok(y1)))))) → c9(TAKE(mark(ok(mark(ok(y0)))), mark(ok(mark(ok(y1))))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, ACTIVE, TAKE, CONS, AND, S, LENGTH

Compound Symbols:

c, c1, c, c1, c6, c6, c15, c9, c10, c6, c6, c11, c12, c13, c16, c17, c18, c19, c7, c8

(143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace TAKE(ok(ok(y0)), ok(mark(ok(y1)))) → c9(TAKE(ok(y0), mark(ok(y1)))) by

TAKE(ok(ok(ok(y0))), ok(mark(ok(ok(y1))))) → c9(TAKE(ok(ok(y0)), mark(ok(ok(y1)))))
TAKE(ok(ok(z0)), ok(mark(ok(mark(mark(y1)))))) → c9(TAKE(ok(z0), mark(ok(mark(mark(y1))))))
TAKE(ok(ok(mark(y0))), ok(mark(ok(mark(y1))))) → c9(TAKE(ok(mark(y0)), mark(ok(mark(y1)))))
TAKE(ok(ok(ok(y0))), ok(mark(ok(mark(ok(y1)))))) → c9(TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))))
TAKE(ok(ok(mark(mark(y0)))), ok(mark(ok(z1)))) → c9(TAKE(ok(mark(mark(y0))), mark(ok(z1))))
TAKE(ok(ok(mark(ok(y0)))), ok(mark(ok(ok(y1))))) → c9(TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))))
TAKE(ok(ok(mark(y0))), ok(mark(ok(mark(mark(y1)))))) → c9(TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))))
TAKE(ok(ok(mark(mark(y0)))), ok(mark(ok(mark(y1))))) → c9(TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))))
TAKE(ok(ok(mark(ok(y0)))), ok(mark(ok(mark(ok(y1)))))) → c9(TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))))

(144) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(zeros) → mark(cons(0, zeros))
active(cons(z0, z1)) → cons(active(z0), z1)
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))

Tuples:

TOP(ok(zeros)) → c(TOP(mark(cons(0, zeros))))
TOP(mark(zeros)) → c1(TOP(ok(zeros)))
TOP(ok(cons(cons(z0, z1), x1))) → c(TOP(cons(cons(active(z0), z1), x1)), ACTIVE(cons(cons(z0, z1), x1)))
TOP(mark(cons(x0, tt))) → c1(TOP(cons(proper(x0), ok(tt))), PROPER(cons(x0, tt)))
TOP(mark(cons(x0, nil))) → c1(TOP(cons(proper(x0), ok(nil))), PROPER(cons(x0, nil)))
TOP(mark(cons(x0, zeros))) → c1(TOP(cons(proper(x0), ok(zeros))), PROPER(cons(x0, zeros)))
TOP(mark(cons(x0, 0))) → c1(TOP(cons(proper(x0), ok(0))), PROPER(cons(x0, 0)))
TOP(mark(cons(x0, cons(z0, z1)))) → c1(TOP(cons(proper(x0), cons(proper(z0), proper(z1)))), PROPER(cons(x0, cons(z0, z1))))
TOP(mark(cons(tt, x1))) → c1(TOP(cons(ok(tt), proper(x1))), PROPER(cons(tt, x1)))
TOP(mark(cons(nil, x1))) → c1(TOP(cons(ok(nil), proper(x1))), PROPER(cons(nil, x1)))
TOP(mark(cons(zeros, x1))) → c1(TOP(cons(ok(zeros), proper(x1))), PROPER(cons(zeros, x1)))
TOP(mark(cons(0, x1))) → c1(TOP(cons(ok(0), proper(x1))), PROPER(cons(0, x1)))
TOP(mark(cons(cons(z0, z1), x1))) → c1(TOP(cons(cons(proper(z0), proper(z1)), proper(x1))), PROPER(cons(cons(z0, z1), x1)))
PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
ACTIVE(cons(cons(cons(z0, z1), x1), x2)) → c15(CONS(cons(cons(active(z0), z1), x1), x2), ACTIVE(cons(cons(z0, z1), x1)))
TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))
TOP(ok(cons(zeros, z0))) → c(TOP(mark(cons(cons(0, zeros), z0))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), ok(ok(y1))) → c8(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(mark(ok(z0)), ok(mark(mark(y1)))) → c8(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(y0))), ok(mark(y1))) → c8(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(mark(ok(ok(y0))), ok(mark(ok(y1)))) → c8(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(z1)) → c8(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(mark(ok(mark(ok(y0)))), ok(ok(y1))) → c8(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(mark(ok(mark(y0))), ok(mark(mark(y1)))) → c8(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(mark(y1))) → c8(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(mark(ok(mark(ok(y0)))), ok(mark(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(z0), mark(mark(mark(y1)))) → c8(TAKE(z0, mark(mark(mark(y1)))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c8(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(z1)) → c8(TAKE(mark(mark(mark(y0))), mark(z1)))
TAKE(mark(mark(mark(ok(y0)))), mark(ok(y1))) → c8(TAKE(mark(mark(ok(y0))), mark(ok(y1))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(y1))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(y1))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), mark(ok(ok(y1)))) → c8(TAKE(ok(ok(y0)), mark(ok(ok(y1)))))
TAKE(mark(ok(z0)), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(z0), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(y1)))))
TAKE(mark(ok(ok(y0))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(z1))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(z1))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))))
TAKE(ok(ok(ok(y0))), ok(ok(ok(y1)))) → c9(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(ok(z0)), ok(ok(mark(mark(y1))))) → c9(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(ok(mark(y0))), ok(ok(mark(y1)))) → c9(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(ok(y0))), ok(ok(mark(ok(y1))))) → c9(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(ok(mark(mark(y0)))), ok(ok(z1))) → c9(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(ok(mark(ok(y0)))), ok(ok(ok(y1)))) → c9(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(ok(mark(y0))), ok(ok(mark(mark(y1))))) → c9(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(ok(mark(mark(y0)))), ok(ok(mark(y1)))) → c9(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(ok(mark(ok(y0)))), ok(ok(mark(ok(y1))))) → c9(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(ok(z0), ok(mark(mark(mark(y1))))) → c9(TAKE(z0, mark(mark(mark(y1)))))
TAKE(ok(ok(y0)), ok(mark(mark(ok(y1))))) → c9(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(y0)), mark(mark(z1))))
TAKE(ok(mark(y0)), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(y0), mark(mark(mark(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(mark(y0)), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(y0), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(mark(mark(y0)))))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(z1))))
TAKE(ok(mark(mark(mark(mark(ok(y0)))))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(mark(mark(y0)))))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(mark(mark(ok(y0)))))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(z1))) → c9(TAKE(mark(mark(mark(y0))), mark(z1)))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(ok(y1)))) → c9(TAKE(mark(mark(ok(y0))), mark(ok(y1))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(z1))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(z1)))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(ok(y1)))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(ok(y1))))
TAKE(ok(mark(ok(ok(y0)))), ok(mark(ok(ok(y1))))) → c9(TAKE(mark(ok(ok(y0))), mark(ok(ok(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(ok(mark(mark(y1)))))) → c9(TAKE(mark(ok(y0)), mark(ok(mark(mark(y1))))))
TAKE(ok(mark(ok(mark(y0)))), ok(mark(ok(mark(y1))))) → c9(TAKE(mark(ok(mark(y0))), mark(ok(mark(y1)))))
TAKE(ok(mark(ok(ok(y0)))), ok(mark(ok(mark(ok(y1)))))) → c9(TAKE(mark(ok(ok(y0))), mark(ok(mark(ok(y1))))))
TAKE(ok(mark(ok(mark(mark(y0))))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(mark(mark(y0)))), mark(ok(y1))))
TAKE(ok(mark(ok(mark(ok(y0))))), ok(mark(ok(ok(y1))))) → c9(TAKE(mark(ok(mark(ok(y0)))), mark(ok(ok(y1)))))
TAKE(ok(mark(ok(mark(y0)))), ok(mark(ok(mark(mark(y1)))))) → c9(TAKE(mark(ok(mark(y0))), mark(ok(mark(mark(y1))))))
TAKE(ok(mark(ok(mark(mark(y0))))), ok(mark(ok(mark(y1))))) → c9(TAKE(mark(ok(mark(mark(y0)))), mark(ok(mark(y1)))))
TAKE(ok(mark(ok(mark(ok(y0))))), ok(mark(ok(mark(ok(y1)))))) → c9(TAKE(mark(ok(mark(ok(y0)))), mark(ok(mark(ok(y1))))))
TAKE(ok(ok(ok(y0))), ok(mark(ok(ok(y1))))) → c9(TAKE(ok(ok(y0)), mark(ok(ok(y1)))))
TAKE(ok(ok(z0)), ok(mark(ok(mark(mark(y1)))))) → c9(TAKE(ok(z0), mark(ok(mark(mark(y1))))))
TAKE(ok(ok(mark(y0))), ok(mark(ok(mark(y1))))) → c9(TAKE(ok(mark(y0)), mark(ok(mark(y1)))))
TAKE(ok(ok(ok(y0))), ok(mark(ok(mark(ok(y1)))))) → c9(TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))))
TAKE(ok(ok(mark(mark(y0)))), ok(mark(ok(z1)))) → c9(TAKE(ok(mark(mark(y0))), mark(ok(z1))))
TAKE(ok(ok(mark(ok(y0)))), ok(mark(ok(ok(y1))))) → c9(TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))))
TAKE(ok(ok(mark(y0))), ok(mark(ok(mark(mark(y1)))))) → c9(TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))))
TAKE(ok(ok(mark(mark(y0)))), ok(mark(ok(mark(y1))))) → c9(TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))))
TAKE(ok(ok(mark(ok(y0)))), ok(mark(ok(mark(ok(y1)))))) → c9(TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))))

S tuples:

PROPER(cons(x0, cons(x1, tt))) → c6(CONS(proper(x0), cons(proper(x1), ok(tt))), PROPER(x0), PROPER(cons(x1, tt)))
PROPER(cons(x0, cons(x1, nil))) → c6(CONS(proper(x0), cons(proper(x1), ok(nil))), PROPER(x0), PROPER(cons(x1, nil)))
PROPER(cons(x0, cons(x1, zeros))) → c6(CONS(proper(x0), cons(proper(x1), ok(zeros))), PROPER(x0), PROPER(cons(x1, zeros)))
PROPER(cons(x0, cons(x1, 0))) → c6(CONS(proper(x0), cons(proper(x1), ok(0))), PROPER(x0), PROPER(cons(x1, 0)))
PROPER(cons(x0, cons(x1, cons(z0, z1)))) → c6(CONS(proper(x0), cons(proper(x1), cons(proper(z0), proper(z1)))), PROPER(x0), PROPER(cons(x1, cons(z0, z1))))
PROPER(cons(x0, cons(tt, x2))) → c6(CONS(proper(x0), cons(ok(tt), proper(x2))), PROPER(x0), PROPER(cons(tt, x2)))
PROPER(cons(x0, cons(nil, x2))) → c6(CONS(proper(x0), cons(ok(nil), proper(x2))), PROPER(x0), PROPER(cons(nil, x2)))
PROPER(cons(x0, cons(zeros, x2))) → c6(CONS(proper(x0), cons(ok(zeros), proper(x2))), PROPER(x0), PROPER(cons(zeros, x2)))
PROPER(cons(x0, cons(0, x2))) → c6(CONS(proper(x0), cons(ok(0), proper(x2))), PROPER(x0), PROPER(cons(0, x2)))
PROPER(cons(x0, cons(cons(z0, z1), x2))) → c6(CONS(proper(x0), cons(cons(proper(z0), proper(z1)), proper(x2))), PROPER(x0), PROPER(cons(cons(z0, z1), x2)))
PROPER(cons(cons(z0, z1), cons(x1, x2))) → c6(CONS(cons(proper(z0), proper(z1)), cons(proper(x1), proper(x2))), PROPER(cons(z0, z1)), PROPER(cons(x1, x2)))
PROPER(cons(cons(x0, tt), x2)) → c6(CONS(cons(proper(x0), ok(tt)), proper(x2)), PROPER(cons(x0, tt)), PROPER(x2))
PROPER(cons(cons(x0, nil), x2)) → c6(CONS(cons(proper(x0), ok(nil)), proper(x2)), PROPER(cons(x0, nil)), PROPER(x2))
PROPER(cons(cons(x0, zeros), x2)) → c6(CONS(cons(proper(x0), ok(zeros)), proper(x2)), PROPER(cons(x0, zeros)), PROPER(x2))
PROPER(cons(cons(x0, 0), x2)) → c6(CONS(cons(proper(x0), ok(0)), proper(x2)), PROPER(cons(x0, 0)), PROPER(x2))
PROPER(cons(cons(x0, cons(z0, z1)), x2)) → c6(CONS(cons(proper(x0), cons(proper(z0), proper(z1))), proper(x2)), PROPER(cons(x0, cons(z0, z1))), PROPER(x2))
PROPER(cons(cons(tt, x1), x2)) → c6(CONS(cons(ok(tt), proper(x1)), proper(x2)), PROPER(cons(tt, x1)), PROPER(x2))
PROPER(cons(cons(nil, x1), x2)) → c6(CONS(cons(ok(nil), proper(x1)), proper(x2)), PROPER(cons(nil, x1)), PROPER(x2))
PROPER(cons(cons(zeros, x1), x2)) → c6(CONS(cons(ok(zeros), proper(x1)), proper(x2)), PROPER(cons(zeros, x1)), PROPER(x2))
PROPER(cons(cons(0, x1), x2)) → c6(CONS(cons(ok(0), proper(x1)), proper(x2)), PROPER(cons(0, x1)), PROPER(x2))
PROPER(cons(cons(cons(z0, z1), x1), x2)) → c6(CONS(cons(cons(proper(z0), proper(z1)), proper(x1)), proper(x2)), PROPER(cons(cons(z0, z1), x1)), PROPER(x2))
PROPER(cons(cons(x0, x1), tt)) → c6(CONS(cons(proper(x0), proper(x1)), ok(tt)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), nil)) → c6(CONS(cons(proper(x0), proper(x1)), ok(nil)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), zeros)) → c6(CONS(cons(proper(x0), proper(x1)), ok(zeros)), PROPER(cons(x0, x1)))
PROPER(cons(cons(x0, x1), 0)) → c6(CONS(cons(proper(x0), proper(x1)), ok(0)), PROPER(cons(x0, x1)))
CONS(ok(ok(y0)), ok(ok(y1))) → c10(CONS(ok(y0), ok(y1)))
CONS(ok(mark(y0)), ok(z1)) → c10(CONS(mark(y0), z1))
PROPER(cons(tt, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(nil, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(zeros, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(0, cons(x1, x2))) → c6(PROPER(cons(x1, x2)))
PROPER(cons(tt, tt)) → c6
PROPER(cons(nil, tt)) → c6
PROPER(cons(zeros, tt)) → c6
PROPER(cons(0, tt)) → c6
PROPER(cons(tt, nil)) → c6
PROPER(cons(nil, nil)) → c6
PROPER(cons(zeros, nil)) → c6
PROPER(cons(0, nil)) → c6
PROPER(cons(tt, zeros)) → c6
PROPER(cons(nil, zeros)) → c6
PROPER(cons(zeros, zeros)) → c6
PROPER(cons(0, zeros)) → c6
PROPER(cons(tt, 0)) → c6
PROPER(cons(nil, 0)) → c6
PROPER(cons(zeros, 0)) → c6
PROPER(cons(0, 0)) → c6
CONS(mark(mark(y0)), z1) → c11(CONS(mark(y0), z1))
CONS(mark(ok(ok(y0))), ok(ok(y1))) → c11(CONS(ok(ok(y0)), ok(ok(y1))))
CONS(mark(ok(mark(y0))), ok(y1)) → c11(CONS(ok(mark(y0)), ok(y1)))

K tuples:

TAKE(ok(mark(mark(y0))), ok(z1)) → c9(TAKE(mark(mark(y0)), z1))
TAKE(ok(mark(ok(y0))), ok(ok(y1))) → c9(TAKE(mark(ok(y0)), ok(y1)))
TAKE(ok(mark(y0)), ok(mark(mark(y1)))) → c9(TAKE(mark(y0), mark(mark(y1))))
TAKE(ok(mark(mark(y0))), ok(mark(y1))) → c9(TAKE(mark(mark(y0)), mark(y1)))
TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(y0)), mark(ok(y1))))
AND(mark(mark(y0)), z1) → c12(AND(mark(y0), z1))
AND(mark(ok(y0)), ok(y1)) → c12(AND(ok(y0), ok(y1)))
AND(ok(ok(y0)), ok(ok(y1))) → c13(AND(ok(y0), ok(y1)))
AND(ok(mark(mark(y0))), ok(z1)) → c13(AND(mark(mark(y0)), z1))
AND(ok(mark(ok(y0))), ok(ok(y1))) → c13(AND(mark(ok(y0)), ok(y1)))
S(ok(ok(y0))) → c16(S(ok(y0)))
S(ok(mark(y0))) → c16(S(mark(y0)))
S(mark(mark(y0))) → c17(S(mark(y0)))
S(mark(ok(ok(y0)))) → c17(S(ok(ok(y0))))
S(mark(ok(mark(y0)))) → c17(S(ok(mark(y0))))
LENGTH(mark(mark(y0))) → c18(LENGTH(mark(y0)))
LENGTH(mark(ok(y0))) → c18(LENGTH(ok(y0)))
LENGTH(ok(ok(y0))) → c19(LENGTH(ok(y0)))
LENGTH(ok(mark(mark(y0)))) → c19(LENGTH(mark(mark(y0))))
LENGTH(ok(mark(ok(y0)))) → c19(LENGTH(mark(ok(y0))))
TAKE(z0, mark(mark(mark(y1)))) → c7(TAKE(z0, mark(mark(y1))))
TAKE(mark(y0), mark(mark(z1))) → c7(TAKE(mark(y0), mark(z1)))
TAKE(ok(y0), mark(mark(ok(y1)))) → c7(TAKE(ok(y0), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(mark(z1))) → c7(TAKE(mark(mark(y0)), mark(z1)))
TAKE(mark(y0), mark(mark(mark(y1)))) → c7(TAKE(mark(y0), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c7(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(y0)), mark(z1)) → c7(TAKE(mark(mark(y0)), z1))
TAKE(mark(ok(y0)), mark(ok(y1))) → c7(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(z0), mark(mark(mark(mark(y1))))) → c7(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(y0)), mark(mark(mark(y1)))) → c7(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))) → c7(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(ok(y0)), mark(ok(ok(y1)))) → c7(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(z0), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(mark(y0)), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(z1))) → c7(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))) → c7(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))) → c7(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))) → c7(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))) → c7(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), z1) → c8(TAKE(mark(mark(y0)), z1))
TAKE(mark(mark(ok(y0))), ok(y1)) → c8(TAKE(mark(ok(y0)), ok(y1)))
TAKE(mark(mark(z0)), mark(mark(y1))) → c8(TAKE(mark(z0), mark(mark(y1))))
TAKE(mark(mark(mark(y0))), mark(y1)) → c8(TAKE(mark(mark(y0)), mark(y1)))
TAKE(mark(mark(ok(y0))), mark(ok(y1))) → c8(TAKE(mark(ok(y0)), mark(ok(y1))))
TAKE(mark(mark(z0)), mark(mark(mark(y1)))) → c8(TAKE(mark(z0), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(y1))) → c8(TAKE(mark(mark(y0)), mark(mark(y1))))
TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))) → c8(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(mark(mark(z0)), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(z0), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), ok(ok(y1))) → c8(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(mark(ok(z0)), ok(mark(mark(y1)))) → c8(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(y0))), ok(mark(y1))) → c8(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(mark(ok(ok(y0))), ok(mark(ok(y1)))) → c8(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(z1)) → c8(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(mark(ok(mark(ok(y0)))), ok(ok(y1))) → c8(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(mark(ok(mark(y0))), ok(mark(mark(y1)))) → c8(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(mark(ok(mark(mark(y0)))), ok(mark(y1))) → c8(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(mark(ok(mark(ok(y0)))), ok(mark(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(mark(z0), mark(mark(mark(y1)))) → c8(TAKE(z0, mark(mark(mark(y1)))))
TAKE(mark(ok(y0)), mark(mark(ok(y1)))) → c8(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(mark(mark(mark(mark(y0)))), mark(z1)) → c8(TAKE(mark(mark(mark(y0))), mark(z1)))
TAKE(mark(mark(mark(ok(y0)))), mark(ok(y1))) → c8(TAKE(mark(mark(ok(y0))), mark(ok(y1))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(y1))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(y1))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))) → c8(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))) → c8(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))) → c8(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))
TAKE(mark(ok(ok(y0))), mark(ok(ok(y1)))) → c8(TAKE(ok(ok(y0)), mark(ok(ok(y1)))))
TAKE(mark(ok(z0)), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(z0), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(y1)))))
TAKE(mark(ok(ok(y0))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(z1))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(z1))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(ok(y1)))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))))
TAKE(mark(ok(mark(y0))), mark(ok(mark(mark(y1))))) → c8(TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))))
TAKE(mark(ok(mark(mark(y0)))), mark(ok(mark(y1)))) → c8(TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))))
TAKE(mark(ok(mark(ok(y0)))), mark(ok(mark(ok(y1))))) → c8(TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))))
TAKE(ok(ok(ok(y0))), ok(ok(ok(y1)))) → c9(TAKE(ok(ok(y0)), ok(ok(y1))))
TAKE(ok(ok(z0)), ok(ok(mark(mark(y1))))) → c9(TAKE(ok(z0), ok(mark(mark(y1)))))
TAKE(ok(ok(mark(y0))), ok(ok(mark(y1)))) → c9(TAKE(ok(mark(y0)), ok(mark(y1))))
TAKE(ok(ok(ok(y0))), ok(ok(mark(ok(y1))))) → c9(TAKE(ok(ok(y0)), ok(mark(ok(y1)))))
TAKE(ok(ok(mark(mark(y0)))), ok(ok(z1))) → c9(TAKE(ok(mark(mark(y0))), ok(z1)))
TAKE(ok(ok(mark(ok(y0)))), ok(ok(ok(y1)))) → c9(TAKE(ok(mark(ok(y0))), ok(ok(y1))))
TAKE(ok(ok(mark(y0))), ok(ok(mark(mark(y1))))) → c9(TAKE(ok(mark(y0)), ok(mark(mark(y1)))))
TAKE(ok(ok(mark(mark(y0)))), ok(ok(mark(y1)))) → c9(TAKE(ok(mark(mark(y0))), ok(mark(y1))))
TAKE(ok(ok(mark(ok(y0)))), ok(ok(mark(ok(y1))))) → c9(TAKE(ok(mark(ok(y0))), ok(mark(ok(y1)))))
TAKE(ok(z0), ok(mark(mark(mark(y1))))) → c9(TAKE(z0, mark(mark(mark(y1)))))
TAKE(ok(ok(y0)), ok(mark(mark(ok(y1))))) → c9(TAKE(ok(y0), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(y0)), mark(mark(z1))))
TAKE(ok(mark(y0)), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(y0), mark(mark(mark(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(ok(y0)), mark(mark(ok(y1)))))
TAKE(ok(mark(y0)), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(y0), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(y0)), mark(mark(mark(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(ok(y0)), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(z1))))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(ok(y0))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(y0))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(y0)), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(ok(y0))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(z1))))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(mark(y0))), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(mark(mark(y0)))))), ok(mark(mark(z1)))) → c9(TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(z1))))
TAKE(ok(mark(mark(mark(mark(ok(y0)))))), ok(mark(mark(ok(y1))))) → c9(TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(ok(y1)))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(mark(mark(mark(y1)))))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(mark(mark(mark(y1))))))
TAKE(ok(mark(mark(mark(mark(mark(y0)))))), ok(mark(mark(mark(y1))))) → c9(TAKE(mark(mark(mark(mark(mark(y0))))), mark(mark(mark(y1)))))
TAKE(ok(mark(mark(mark(mark(ok(y0)))))), ok(mark(mark(mark(ok(y1)))))) → c9(TAKE(mark(mark(mark(mark(ok(y0))))), mark(mark(mark(ok(y1))))))
TAKE(ok(mark(mark(mark(y0)))), ok(mark(z1))) → c9(TAKE(mark(mark(mark(y0))), mark(z1)))
TAKE(ok(mark(mark(ok(y0)))), ok(mark(ok(y1)))) → c9(TAKE(mark(mark(ok(y0))), mark(ok(y1))))
TAKE(ok(mark(mark(mark(mark(y0))))), ok(mark(z1))) → c9(TAKE(mark(mark(mark(mark(y0)))), mark(z1)))
TAKE(ok(mark(mark(mark(ok(y0))))), ok(mark(ok(y1)))) → c9(TAKE(mark(mark(mark(ok(y0)))), mark(ok(y1))))
TAKE(ok(mark(ok(ok(y0)))), ok(mark(ok(ok(y1))))) → c9(TAKE(mark(ok(ok(y0))), mark(ok(ok(y1)))))
TAKE(ok(mark(ok(y0))), ok(mark(ok(mark(mark(y1)))))) → c9(TAKE(mark(ok(y0)), mark(ok(mark(mark(y1))))))
TAKE(ok(mark(ok(mark(y0)))), ok(mark(ok(mark(y1))))) → c9(TAKE(mark(ok(mark(y0))), mark(ok(mark(y1)))))
TAKE(ok(mark(ok(ok(y0)))), ok(mark(ok(mark(ok(y1)))))) → c9(TAKE(mark(ok(ok(y0))), mark(ok(mark(ok(y1))))))
TAKE(ok(mark(ok(mark(mark(y0))))), ok(mark(ok(y1)))) → c9(TAKE(mark(ok(mark(mark(y0)))), mark(ok(y1))))
TAKE(ok(mark(ok(mark(ok(y0))))), ok(mark(ok(ok(y1))))) → c9(TAKE(mark(ok(mark(ok(y0)))), mark(ok(ok(y1)))))
TAKE(ok(mark(ok(mark(y0)))), ok(mark(ok(mark(mark(y1)))))) → c9(TAKE(mark(ok(mark(y0))), mark(ok(mark(mark(y1))))))
TAKE(ok(mark(ok(mark(mark(y0))))), ok(mark(ok(mark(y1))))) → c9(TAKE(mark(ok(mark(mark(y0)))), mark(ok(mark(y1)))))
TAKE(ok(mark(ok(mark(ok(y0))))), ok(mark(ok(mark(ok(y1)))))) → c9(TAKE(mark(ok(mark(ok(y0)))), mark(ok(mark(ok(y1))))))
TAKE(ok(ok(ok(y0))), ok(mark(ok(ok(y1))))) → c9(TAKE(ok(ok(y0)), mark(ok(ok(y1)))))
TAKE(ok(ok(z0)), ok(mark(ok(mark(mark(y1)))))) → c9(TAKE(ok(z0), mark(ok(mark(mark(y1))))))
TAKE(ok(ok(mark(y0))), ok(mark(ok(mark(y1))))) → c9(TAKE(ok(mark(y0)), mark(ok(mark(y1)))))
TAKE(ok(ok(ok(y0))), ok(mark(ok(mark(ok(y1)))))) → c9(TAKE(ok(ok(y0)), mark(ok(mark(ok(y1))))))
TAKE(ok(ok(mark(mark(y0)))), ok(mark(ok(z1)))) → c9(TAKE(ok(mark(mark(y0))), mark(ok(z1))))
TAKE(ok(ok(mark(ok(y0)))), ok(mark(ok(ok(y1))))) → c9(TAKE(ok(mark(ok(y0))), mark(ok(ok(y1)))))
TAKE(ok(ok(mark(y0))), ok(mark(ok(mark(mark(y1)))))) → c9(TAKE(ok(mark(y0)), mark(ok(mark(mark(y1))))))
TAKE(ok(ok(mark(mark(y0)))), ok(mark(ok(mark(y1))))) → c9(TAKE(ok(mark(mark(y0))), mark(ok(mark(y1)))))
TAKE(ok(ok(mark(ok(y0)))), ok(mark(ok(mark(ok(y1)))))) → c9(TAKE(ok(mark(ok(y0))), mark(ok(mark(ok(y1))))))

Defined Rule Symbols:

active, cons, proper

Defined Pair Symbols:

TOP, PROPER, ACTIVE, TAKE, CONS, AND, S, LENGTH

Compound Symbols:

c, c1, c, c1, c6, c6, c15, c9, c10, c6, c6, c11, c12, c13, c16, c17, c18, c19, c7, c8