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

0 CpxTRS
↳1 NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CpxTRS
    ↳3 CpxTrsMatchBoundsTAProof (⇔, 121 ms)
    ↳4 BOUNDS(1, n^1)
    ↳5 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 CdtProblem
        ↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳8 CdtProblem
        ↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳10 CdtProblem
        ↳11 CdtUsableRulesProof (⇔, 0 ms)
        ↳12 CdtProblem
        ↳13 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 183 ms)
        ↳14 CdtProblem
        ↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 38 ms)
        ↳16 CdtProblem
        ↳17 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 28 ms)
        ↳18 CdtProblem
        ↳19 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 31 ms)
        ↳20 CdtProblem
        ↳21 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)
        ↳22 CdtProblem
        ↳23 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 15 ms)
        ↳24 CdtProblem
        ↳25 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)
        ↳26 CdtProblem
        ↳27 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳28 BOUNDS(1, 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(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(isNeList(V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(and(isList(V1), isList(V2)))
active(isNeList(V)) → mark(isQid(V))
active(isNeList(__(V1, V2))) → mark(and(isList(V1), isNeList(V2)))
active(isNeList(__(V1, V2))) → mark(and(isNeList(V1), isList(V2)))
active(isNePal(V)) → mark(isQid(V))
active(isNePal(__(I, __(P, I)))) → mark(and(isQid(I), isPal(P)))
active(isPal(V)) → mark(isNePal(V))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isList(X)) → isList(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isList(ok(X)) → ok(isList(X))
isNeList(ok(X)) → ok(isNeList(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
isPal(ok(X)) → ok(isPal(X))
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 top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(isNeList(V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(and(isList(V1), isList(V2)))
active(isNeList(V)) → mark(isQid(V))
active(isNeList(__(V1, V2))) → mark(and(isList(V1), isNeList(V2)))
active(isNeList(__(V1, V2))) → mark(and(isNeList(V1), isList(V2)))
active(isNePal(V)) → mark(isQid(V))
active(isNePal(__(I, __(P, I)))) → mark(and(isQid(I), isPal(P)))
active(isPal(V)) → mark(isNePal(V))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isList(X)) → isList(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(isPal(X)) → isPal(proper(X))

(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(nil) → ok(nil)
proper(tt) → ok(tt)
isNeList(ok(X)) → ok(isNeList(X))
isQid(ok(X)) → ok(isQid(X))
isPal(ok(X)) → ok(isPal(X))
isNePal(ok(X)) → ok(isNePal(X))
and(mark(X1), X2) → mark(and(X1, X2))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
__(mark(X1), X2) → mark(__(X1, X2))
proper(a) → ok(a)
proper(i) → ok(i)
and(ok(X1), ok(X2)) → ok(and(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
proper(o) → ok(o)
proper(u) → ok(u)
top(mark(X)) → top(proper(X))
proper(e) → ok(e)
isList(ok(X)) → ok(isList(X))

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 2.

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, 9]
transitions:
ok0(0) → 0
active0(0) → 0
nil0() → 0
tt0() → 0
mark0(0) → 0
a0() → 0
i0() → 0
o0() → 0
u0() → 0
e0() → 0
top0(0) → 1
proper0(0) → 2
isNeList0(0) → 3
isQid0(0) → 4
isPal0(0) → 5
isNePal0(0) → 6
and0(0, 0) → 7
__0(0, 0) → 8
isList0(0) → 9
active1(0) → 10
top1(10) → 1
nil1() → 11
ok1(11) → 2
tt1() → 12
ok1(12) → 2
isNeList1(0) → 13
ok1(13) → 3
isQid1(0) → 14
ok1(14) → 4
isPal1(0) → 15
ok1(15) → 5
isNePal1(0) → 16
ok1(16) → 6
and1(0, 0) → 17
mark1(17) → 7
__1(0, 0) → 18
ok1(18) → 8
__1(0, 0) → 19
mark1(19) → 8
a1() → 20
ok1(20) → 2
i1() → 21
ok1(21) → 2
and1(0, 0) → 22
ok1(22) → 7
o1() → 23
ok1(23) → 2
u1() → 24
ok1(24) → 2
proper1(0) → 25
top1(25) → 1
e1() → 26
ok1(26) → 2
isList1(0) → 27
ok1(27) → 9
ok1(11) → 25
ok1(12) → 25
ok1(13) → 13
ok1(14) → 14
ok1(15) → 15
ok1(16) → 16
mark1(17) → 17
mark1(17) → 22
ok1(18) → 18
ok1(18) → 19
mark1(19) → 18
mark1(19) → 19
ok1(20) → 25
ok1(21) → 25
ok1(22) → 17
ok1(22) → 22
ok1(23) → 25
ok1(24) → 25
ok1(26) → 25
ok1(27) → 27
active2(11) → 28
top2(28) → 1
active2(12) → 28
active2(20) → 28
active2(21) → 28
active2(23) → 28
active2(24) → 28
active2(26) → 28

(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(nil) → ok(nil)
proper(tt) → ok(tt)
proper(a) → ok(a)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
proper(e) → ok(e)
isNeList(ok(z0)) → ok(isNeList(z0))
isQid(ok(z0)) → ok(isQid(z0))
isPal(ok(z0)) → ok(isPal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
__(z0, mark(z1)) → mark(__(z0, z1))
isList(ok(z0)) → ok(isList(z0))
Tuples:

TOP(ok(z0)) → c(TOP(active(z0)))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(nil) → c2
PROPER(tt) → c3
PROPER(a) → c4
PROPER(i) → c5
PROPER(o) → c6
PROPER(u) → c7
PROPER(e) → c8
ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
S tuples:

TOP(ok(z0)) → c(TOP(active(z0)))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(nil) → c2
PROPER(tt) → c3
PROPER(a) → c4
PROPER(i) → c5
PROPER(o) → c6
PROPER(u) → c7
PROPER(e) → c8
ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
K tuples:none
Defined Rule Symbols:

top, proper, isNeList, isQid, isPal, isNePal, and, __, isList

Defined Pair Symbols:

TOP, PROPER, ISNELIST, ISQID, ISPAL, ISNEPAL, AND, __', ISLIST

Compound Symbols:

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

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

Removed 8 trailing nodes:

PROPER(u) → c7
PROPER(o) → c6
PROPER(e) → c8
PROPER(i) → c5
PROPER(a) → c4
PROPER(nil) → c2
PROPER(tt) → c3
TOP(ok(z0)) → c(TOP(active(z0)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(a) → ok(a)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
proper(e) → ok(e)
isNeList(ok(z0)) → ok(isNeList(z0))
isQid(ok(z0)) → ok(isQid(z0))
isPal(ok(z0)) → ok(isPal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
__(z0, mark(z1)) → mark(__(z0, z1))
isList(ok(z0)) → ok(isList(z0))
Tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
S tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
K tuples:none
Defined Rule Symbols:

top, proper, isNeList, isQid, isPal, isNePal, and, __, isList

Defined Pair Symbols:

TOP, ISNELIST, ISQID, ISPAL, ISNEPAL, AND, __', ISLIST

Compound Symbols:

c1, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18

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

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(a) → ok(a)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
proper(e) → ok(e)
isNeList(ok(z0)) → ok(isNeList(z0))
isQid(ok(z0)) → ok(isQid(z0))
isPal(ok(z0)) → ok(isPal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
__(z0, mark(z1)) → mark(__(z0, z1))
isList(ok(z0)) → ok(isList(z0))
Tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
K tuples:none
Defined Rule Symbols:

top, proper, isNeList, isQid, isPal, isNePal, and, __, isList

Defined Pair Symbols:

ISNELIST, ISQID, ISPAL, ISNEPAL, AND, __', ISLIST, TOP

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c1

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
isNeList(ok(z0)) → ok(isNeList(z0))
isQid(ok(z0)) → ok(isQid(z0))
isPal(ok(z0)) → ok(isPal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
__(z0, mark(z1)) → mark(__(z0, z1))
isList(ok(z0)) → ok(isList(z0))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(a) → ok(a)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
proper(e) → ok(e)
Tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
K tuples:none
Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNELIST, ISQID, ISPAL, ISNEPAL, AND, __', ISLIST, TOP

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c1

(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.

TOP(mark(z0)) → c1(TOP(proper(z0)))
We considered the (Usable) Rules:

proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(o) → ok(o)
proper(u) → ok(u)
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
And the Tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x1, x2)) = 0   
POL(ISLIST(x1)) = 0   
POL(ISNELIST(x1)) = 0   
POL(ISNEPAL(x1)) = 0   
POL(ISPAL(x1)) = 0   
POL(ISQID(x1)) = 0   
POL(TOP(x1)) = x1   
POL(__'(x1, x2)) = 0   
POL(a) = 0   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c9(x1)) = x1   
POL(e) = 0   
POL(i) = 0   
POL(mark(x1)) = [1] + x1   
POL(nil) = 0   
POL(o) = 0   
POL(ok(x1)) = 0   
POL(proper(x1)) = 0   
POL(tt) = 0   
POL(u) = 0   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(a) → ok(a)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
proper(e) → ok(e)
Tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNELIST, ISQID, ISPAL, ISNEPAL, AND, __', ISLIST, TOP

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c1

(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.

AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x1, x2)) = x2   
POL(ISLIST(x1)) = 0   
POL(ISNELIST(x1)) = 0   
POL(ISNEPAL(x1)) = 0   
POL(ISPAL(x1)) = 0   
POL(ISQID(x1)) = 0   
POL(TOP(x1)) = 0   
POL(__'(x1, x2)) = 0   
POL(a) = 0   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c9(x1)) = x1   
POL(e) = 0   
POL(i) = 0   
POL(mark(x1)) = 0   
POL(nil) = 0   
POL(o) = 0   
POL(ok(x1)) = [1] + x1   
POL(proper(x1)) = 0   
POL(tt) = 0   
POL(u) = 0   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(a) → ok(a)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
proper(e) → ok(e)
Tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNELIST, ISQID, ISPAL, ISNEPAL, AND, __', ISLIST, TOP

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c1

(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.

AND(mark(z0), z1) → c13(AND(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x1, x2)) = x1   
POL(ISLIST(x1)) = 0   
POL(ISNELIST(x1)) = 0   
POL(ISNEPAL(x1)) = 0   
POL(ISPAL(x1)) = 0   
POL(ISQID(x1)) = 0   
POL(TOP(x1)) = 0   
POL(__'(x1, x2)) = x2   
POL(a) = 0   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c9(x1)) = x1   
POL(e) = 0   
POL(i) = 0   
POL(mark(x1)) = [1] + x1   
POL(nil) = 0   
POL(o) = 0   
POL(ok(x1)) = x1   
POL(proper(x1)) = 0   
POL(tt) = 0   
POL(u) = 0   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(a) → ok(a)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
proper(e) → ok(e)
Tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
AND(mark(z0), z1) → c13(AND(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNELIST, ISQID, ISPAL, ISNEPAL, AND, __', ISLIST, TOP

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c1

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

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

ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
We considered the (Usable) Rules:none
And the Tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x1, x2)) = x1 + [2]x2   
POL(ISLIST(x1)) = [2]x1   
POL(ISNELIST(x1)) = 0   
POL(ISNEPAL(x1)) = 0   
POL(ISPAL(x1)) = x1   
POL(ISQID(x1)) = x1   
POL(TOP(x1)) = 0   
POL(__'(x1, x2)) = x1   
POL(a) = 0   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c9(x1)) = x1   
POL(e) = [1]   
POL(i) = 0   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(o) = 0   
POL(ok(x1)) = [1] + x1   
POL(proper(x1)) = [1] + [2]x1   
POL(tt) = 0   
POL(u) = [2]   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(a) → ok(a)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
proper(e) → ok(e)
Tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
__'(mark(z0), z1) → c16(__'(z0, z1))
K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
AND(mark(z0), z1) → c13(AND(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNELIST, ISQID, ISPAL, ISNEPAL, AND, __', ISLIST, TOP

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c1

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

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

ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
We considered the (Usable) Rules:none
And the Tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x1, x2)) = 0   
POL(ISLIST(x1)) = 0   
POL(ISNELIST(x1)) = 0   
POL(ISNEPAL(x1)) = [2]x1   
POL(ISPAL(x1)) = 0   
POL(ISQID(x1)) = 0   
POL(TOP(x1)) = 0   
POL(__'(x1, x2)) = 0   
POL(a) = 0   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c9(x1)) = x1   
POL(e) = 0   
POL(i) = 0   
POL(mark(x1)) = 0   
POL(nil) = 0   
POL(o) = 0   
POL(ok(x1)) = [2] + x1   
POL(proper(x1)) = 0   
POL(tt) = 0   
POL(u) = 0   

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(a) → ok(a)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
proper(e) → ok(e)
Tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
__'(mark(z0), z1) → c16(__'(z0, z1))
K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
AND(mark(z0), z1) → c13(AND(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNELIST, ISQID, ISPAL, ISNEPAL, AND, __', ISLIST, TOP

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c1

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

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

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
We considered the (Usable) Rules:none
And the Tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x1, x2)) = 0   
POL(ISLIST(x1)) = 0   
POL(ISNELIST(x1)) = x1   
POL(ISNEPAL(x1)) = 0   
POL(ISPAL(x1)) = 0   
POL(ISQID(x1)) = 0   
POL(TOP(x1)) = 0   
POL(__'(x1, x2)) = 0   
POL(a) = 0   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c9(x1)) = x1   
POL(e) = 0   
POL(i) = 0   
POL(mark(x1)) = 0   
POL(nil) = 0   
POL(o) = 0   
POL(ok(x1)) = [1] + x1   
POL(proper(x1)) = 0   
POL(tt) = 0   
POL(u) = 0   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(a) → ok(a)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
proper(e) → ok(e)
Tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

__'(mark(z0), z1) → c16(__'(z0, z1))
K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
AND(mark(z0), z1) → c13(AND(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
ISNELIST(ok(z0)) → c9(ISNELIST(z0))
Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNELIST, ISQID, ISPAL, ISNEPAL, AND, __', ISLIST, TOP

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c1

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

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

__'(mark(z0), z1) → c16(__'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x1, x2)) = 0   
POL(ISLIST(x1)) = 0   
POL(ISNELIST(x1)) = [2]x1   
POL(ISNEPAL(x1)) = [2]x1   
POL(ISPAL(x1)) = 0   
POL(ISQID(x1)) = 0   
POL(TOP(x1)) = 0   
POL(__'(x1, x2)) = [2]x1 + x2   
POL(a) = 0   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c9(x1)) = x1   
POL(e) = 0   
POL(i) = [3]   
POL(mark(x1)) = [1] + x1   
POL(nil) = [2]   
POL(o) = [1]   
POL(ok(x1)) = x1   
POL(proper(x1)) = [2]x1   
POL(tt) = 0   
POL(u) = [2]   

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(a) → ok(a)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
proper(e) → ok(e)
Tuples:

ISNELIST(ok(z0)) → c9(ISNELIST(z0))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
AND(mark(z0), z1) → c13(AND(z0, z1))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
__'(mark(z0), z1) → c16(__'(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:none
K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
AND(ok(z0), ok(z1)) → c14(AND(z0, z1))
AND(mark(z0), z1) → c13(AND(z0, z1))
__'(z0, mark(z1)) → c17(__'(z0, z1))
ISQID(ok(z0)) → c10(ISQID(z0))
ISPAL(ok(z0)) → c11(ISPAL(z0))
__'(ok(z0), ok(z1)) → c15(__'(z0, z1))
ISLIST(ok(z0)) → c18(ISLIST(z0))
ISNEPAL(ok(z0)) → c12(ISNEPAL(z0))
ISNELIST(ok(z0)) → c9(ISNELIST(z0))
__'(mark(z0), z1) → c16(__'(z0, z1))
Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNELIST, ISQID, ISPAL, ISNEPAL, AND, __', ISLIST, TOP

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c1

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

The set S is empty

(28) BOUNDS(1, 1)