Runtime Complexity (innermost) proof of /tmp/tmpDQ1Ums/PALINDROME_complete

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 (⇔, 1311 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)), 324 ms)

        ↳14 CdtProblem

        ↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 123 ms)

        ↳16 CdtProblem

        ↳17 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 94 ms)

        ↳18 CdtProblem

        ↳19 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 108 ms)

        ↳20 CdtProblem

        ↳21 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 69 ms)

        ↳22 CdtProblem

        ↳23 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 88 ms)

        ↳24 CdtProblem

        ↳25 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 78 ms)

        ↳26 CdtProblem

        ↳27 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 57 ms)

        ↳28 CdtProblem

        ↳29 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 52 ms)

        ↳30 CdtProblem

        ↳31 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 36 ms)

        ↳32 CdtProblem

        ↳33 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 49 ms)

        ↳34 CdtProblem

        ↳35 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 25 ms)

        ↳36 CdtProblem

        ↳37 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 53 ms)

        ↳38 CdtProblem

        ↳39 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 50 ms)

        ↳40 CdtProblem

        ↳41 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 48 ms)

        ↳42 CdtProblem

        ↳43 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 40 ms)

        ↳44 CdtProblem

        ↳45 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 53 ms)

        ↳46 CdtProblem

        ↳47 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 46 ms)

        ↳48 CdtProblem

        ↳49 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 7 ms)

        ↳50 CdtProblem

        ↳51 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 43 ms)

        ↳52 CdtProblem

        ↳53 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)

        ↳54 CdtProblem

        ↳55 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 37 ms)

        ↳56 CdtProblem

        ↳57 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 38 ms)

        ↳58 CdtProblem

        ↳59 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 38 ms)

        ↳60 CdtProblem

        ↳61 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)

        ↳62 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(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
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(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(and(X1, X2)) → and(active(X1), X2)
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isList(X)) → isList(proper(X))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isPalListKind(X)) → isPalListKind(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))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNeList(ok(X)) → ok(isNeList(X))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isList(ok(X)) → ok(isList(X))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
isQid(ok(X)) → ok(isQid(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isNePal(ok(X)) → ok(isNePal(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isPalListKind(ok(X)) → ok(isPalListKind(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(U11(tt, V)) → mark(U12(isNeList(V)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1, V2)) → mark(U22(isList(V1), V2))
active(U22(tt, V2)) → mark(U23(isList(V2)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isQid(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isList(V1), V2))
active(U42(tt, V2)) → mark(U43(isNeList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) → mark(U53(isList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V)) → mark(U62(isQid(V)))
active(U62(tt)) → mark(tt)
active(U71(tt, V)) → mark(U72(isNePal(V)))
active(U72(tt)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(U11(isPalListKind(V), V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) → mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) → mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) → mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) → mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) → mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) → mark(U71(isPalListKind(V), V))
active(isPal(nil)) → mark(tt)
active(isPalListKind(a)) → mark(tt)
active(isPalListKind(e)) → mark(tt)
active(isPalListKind(i)) → mark(tt)
active(isPalListKind(nil)) → mark(tt)
active(isPalListKind(o)) → mark(tt)
active(isPalListKind(u)) → mark(tt)
active(isPalListKind(__(V1, V2))) → mark(and(isPalListKind(V1), isPalListKind(V2)))
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(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(and(X1, X2)) → and(active(X1), X2)
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X)) → U12(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isList(X)) → isList(proper(X))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isPalListKind(X)) → isPalListKind(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:

U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
top(ok(X)) → top(active(X))
isNeList(ok(X)) → ok(isNeList(X))
U12(mark(X)) → mark(U12(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
proper(i) → ok(i)
and(ok(X1), ok(X2)) → ok(and(X1, X2))
U62(mark(X)) → mark(U62(X))
U43(ok(X)) → ok(U43(X))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
__(X1, mark(X2)) → mark(__(X1, X2))
proper(u) → ok(u)
proper(nil) → ok(nil)
proper(tt) → ok(tt)
isPal(ok(X)) → ok(isPal(X))
U32(mark(X)) → mark(U32(X))
U53(mark(X)) → mark(U53(X))
U72(mark(X)) → mark(U72(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U23(ok(X)) → ok(U23(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
__(mark(X1), X2) → mark(__(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U43(mark(X)) → mark(U43(X))
U12(ok(X)) → ok(U12(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U62(ok(X)) → ok(U62(X))
proper(o) → ok(o)
proper(e) → ok(e)
isList(ok(X)) → ok(isList(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U53(ok(X)) → ok(U53(X))
isPalListKind(ok(X)) → ok(isPalListKind(X))
U72(ok(X)) → ok(U72(X))
and(mark(X1), X2) → mark(and(X1, X2))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
proper(a) → ok(a)
U23(mark(X)) → mark(U23(X))
U32(ok(X)) → ok(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
top(mark(X)) → top(proper(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))

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, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]
transitions:
ok0(0) → 0
active0(0) → 0
mark0(0) → 0
i0() → 0
u0() → 0
nil0() → 0
tt0() → 0
o0() → 0
e0() → 0
a0() → 0
U110(0, 0) → 1
top0(0) → 2
isNeList0(0) → 3
U120(0) → 4
U610(0, 0) → 5
proper0(0) → 6
and0(0, 0) → 7
U620(0) → 8
U430(0) → 9
U520(0, 0) → 10
U510(0, 0, 0) → 11
__0(0, 0) → 12
isPal0(0) → 13
U320(0) → 14
U530(0) → 15
U720(0) → 16
U310(0, 0) → 17
U230(0) → 18
U710(0, 0) → 19
U210(0, 0, 0) → 20
U420(0, 0) → 21
isQid0(0) → 22
isNePal0(0) → 23
U220(0, 0) → 24
isList0(0) → 25
isPalListKind0(0) → 26
U410(0, 0, 0) → 27
U111(0, 0) → 28
ok1(28) → 1
active1(0) → 29
top1(29) → 2
isNeList1(0) → 30
ok1(30) → 3
U121(0) → 31
mark1(31) → 4
U611(0, 0) → 32
ok1(32) → 5
i1() → 33
ok1(33) → 6
and1(0, 0) → 34
ok1(34) → 7
U621(0) → 35
mark1(35) → 8
U431(0) → 36
ok1(36) → 9
U521(0, 0) → 37
ok1(37) → 10
U511(0, 0, 0) → 38
mark1(38) → 11
__1(0, 0) → 39
mark1(39) → 12
u1() → 40
ok1(40) → 6
nil1() → 41
ok1(41) → 6
tt1() → 42
ok1(42) → 6
isPal1(0) → 43
ok1(43) → 13
U321(0) → 44
mark1(44) → 14
U531(0) → 45
mark1(45) → 15
U721(0) → 46
mark1(46) → 16
U311(0, 0) → 47
ok1(47) → 17
U611(0, 0) → 48
mark1(48) → 5
U231(0) → 49
ok1(49) → 18
U111(0, 0) → 50
mark1(50) → 1
U711(0, 0) → 51
ok1(51) → 19
U211(0, 0, 0) → 52
ok1(52) → 20
U421(0, 0) → 53
ok1(53) → 21
U211(0, 0, 0) → 54
mark1(54) → 20
isQid1(0) → 55
ok1(55) → 22
isNePal1(0) → 56
ok1(56) → 23
U221(0, 0) → 57
mark1(57) → 24
U431(0) → 58
mark1(58) → 9
U121(0) → 59
ok1(59) → 4
U421(0, 0) → 60
mark1(60) → 21
U621(0) → 61
ok1(61) → 8
o1() → 62
ok1(62) → 6
e1() → 63
ok1(63) → 6
isList1(0) → 64
ok1(64) → 25
U521(0, 0) → 65
mark1(65) → 10
U511(0, 0, 0) → 66
ok1(66) → 11
U531(0) → 67
ok1(67) → 15
isPalListKind1(0) → 68
ok1(68) → 26
U721(0) → 69
ok1(69) → 16
and1(0, 0) → 70
mark1(70) → 7
__1(0, 0) → 71
ok1(71) → 12
U711(0, 0) → 72
mark1(72) → 19
a1() → 73
ok1(73) → 6
U231(0) → 74
mark1(74) → 18
U321(0) → 75
ok1(75) → 14
U411(0, 0, 0) → 76
mark1(76) → 27
U221(0, 0) → 77
ok1(77) → 24
U311(0, 0) → 78
mark1(78) → 17
proper1(0) → 79
top1(79) → 2
U411(0, 0, 0) → 80
ok1(80) → 27
ok1(28) → 28
ok1(28) → 50
ok1(30) → 30
mark1(31) → 31
mark1(31) → 59
ok1(32) → 32
ok1(32) → 48
ok1(33) → 79
ok1(34) → 34
ok1(34) → 70
mark1(35) → 35
mark1(35) → 61
ok1(36) → 36
ok1(36) → 58
ok1(37) → 37
ok1(37) → 65
mark1(38) → 38
mark1(38) → 66
mark1(39) → 39
mark1(39) → 71
ok1(40) → 79
ok1(41) → 79
ok1(42) → 79
ok1(43) → 43
mark1(44) → 44
mark1(44) → 75
mark1(45) → 45
mark1(45) → 67
mark1(46) → 46
mark1(46) → 69
ok1(47) → 47
ok1(47) → 78
mark1(48) → 32
mark1(48) → 48
ok1(49) → 49
ok1(49) → 74
mark1(50) → 28
mark1(50) → 50
ok1(51) → 51
ok1(51) → 72
ok1(52) → 52
ok1(52) → 54
ok1(53) → 53
ok1(53) → 60
mark1(54) → 52
mark1(54) → 54
ok1(55) → 55
ok1(56) → 56
mark1(57) → 57
mark1(57) → 77
mark1(58) → 36
mark1(58) → 58
ok1(59) → 31
ok1(59) → 59
mark1(60) → 53
mark1(60) → 60
ok1(61) → 35
ok1(61) → 61
ok1(62) → 79
ok1(63) → 79
ok1(64) → 64
mark1(65) → 37
mark1(65) → 65
ok1(66) → 38
ok1(66) → 66
ok1(67) → 45
ok1(67) → 67
ok1(68) → 68
ok1(69) → 46
ok1(69) → 69
mark1(70) → 34
mark1(70) → 70
ok1(71) → 39
ok1(71) → 71
mark1(72) → 51
mark1(72) → 72
ok1(73) → 79
mark1(74) → 49
mark1(74) → 74
ok1(75) → 44
ok1(75) → 75
mark1(76) → 76
mark1(76) → 80
ok1(77) → 57
ok1(77) → 77
mark1(78) → 47
mark1(78) → 78
ok1(80) → 76
ok1(80) → 80
active2(33) → 81
top2(81) → 2
active2(40) → 81
active2(41) → 81
active2(42) → 81
active2(62) → 81
active2(63) → 81
active2(73) → 81

(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:

U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U11(mark(z0), z1) → mark(U11(z0, z1))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
isNeList(ok(z0)) → ok(isNeList(z0))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
U61(ok(z0), ok(z1)) → ok(U61(z0, z1))
U61(mark(z0), z1) → mark(U61(z0, z1))
proper(i) → ok(i)
proper(u) → ok(u)
proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(o) → ok(o)
proper(e) → ok(e)
proper(a) → ok(a)
and(ok(z0), ok(z1)) → ok(and(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
U62(mark(z0)) → mark(U62(z0))
U62(ok(z0)) → ok(U62(z0))
U43(ok(z0)) → ok(U43(z0))
U43(mark(z0)) → mark(U43(z0))
U52(ok(z0), ok(z1)) → ok(U52(z0, z1))
U52(mark(z0), z1) → mark(U52(z0, z1))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
__(z0, mark(z1)) → mark(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
isPal(ok(z0)) → ok(isPal(z0))
U32(mark(z0)) → mark(U32(z0))
U32(ok(z0)) → ok(U32(z0))
U53(mark(z0)) → mark(U53(z0))
U53(ok(z0)) → ok(U53(z0))
U72(mark(z0)) → mark(U72(z0))
U72(ok(z0)) → ok(U72(z0))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U23(ok(z0)) → ok(U23(z0))
U23(mark(z0)) → mark(U23(z0))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U42(ok(z0), ok(z1)) → ok(U42(z0, z1))
U42(mark(z0), z1) → mark(U42(z0, z1))
isQid(ok(z0)) → ok(isQid(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
U22(mark(z0), z1) → mark(U22(z0, z1))
U22(ok(z0), ok(z1)) → ok(U22(z0, z1))
isList(ok(z0)) → ok(isList(z0))
isPalListKind(ok(z0)) → ok(isPalListKind(z0))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
TOP(ok(z0)) → c2(TOP(active(z0)))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
PROPER(i) → c9
PROPER(u) → c10
PROPER(nil) → c11
PROPER(tt) → c12
PROPER(o) → c13
PROPER(e) → c14
PROPER(a) → c15
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))

S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
TOP(ok(z0)) → c2(TOP(active(z0)))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
PROPER(i) → c9
PROPER(u) → c10
PROPER(nil) → c11
PROPER(tt) → c12
PROPER(o) → c13
PROPER(e) → c14
PROPER(a) → c15
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))

K tuples:none
Defined Rule Symbols:

U11, top, isNeList, U12, U61, proper, and, U62, U43, U52, U51, __, isPal, U32, U53, U72, U31, U23, U71, U21, U42, isQid, isNePal, U22, isList, isPalListKind, U41

Defined Pair Symbols:

U11', TOP, ISNELIST, U12', U61', PROPER, AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41'

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53

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

Removed 8 trailing nodes:

PROPER(u) → c10
PROPER(a) → c15
PROPER(o) → c13
PROPER(e) → c14
PROPER(i) → c9
PROPER(nil) → c11
PROPER(tt) → c12
TOP(ok(z0)) → c2(TOP(active(z0)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U11(mark(z0), z1) → mark(U11(z0, z1))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
isNeList(ok(z0)) → ok(isNeList(z0))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
U61(ok(z0), ok(z1)) → ok(U61(z0, z1))
U61(mark(z0), z1) → mark(U61(z0, z1))
proper(i) → ok(i)
proper(u) → ok(u)
proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(o) → ok(o)
proper(e) → ok(e)
proper(a) → ok(a)
and(ok(z0), ok(z1)) → ok(and(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
U62(mark(z0)) → mark(U62(z0))
U62(ok(z0)) → ok(U62(z0))
U43(ok(z0)) → ok(U43(z0))
U43(mark(z0)) → mark(U43(z0))
U52(ok(z0), ok(z1)) → ok(U52(z0, z1))
U52(mark(z0), z1) → mark(U52(z0, z1))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
__(z0, mark(z1)) → mark(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
isPal(ok(z0)) → ok(isPal(z0))
U32(mark(z0)) → mark(U32(z0))
U32(ok(z0)) → ok(U32(z0))
U53(mark(z0)) → mark(U53(z0))
U53(ok(z0)) → ok(U53(z0))
U72(mark(z0)) → mark(U72(z0))
U72(ok(z0)) → ok(U72(z0))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U23(ok(z0)) → ok(U23(z0))
U23(mark(z0)) → mark(U23(z0))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U42(ok(z0), ok(z1)) → ok(U42(z0, z1))
U42(mark(z0), z1) → mark(U42(z0, z1))
isQid(ok(z0)) → ok(isQid(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
U22(mark(z0), z1) → mark(U22(z0, z1))
U22(ok(z0), ok(z1)) → ok(U22(z0, z1))
isList(ok(z0)) → ok(isList(z0))
isPalListKind(ok(z0)) → ok(isPalListKind(z0))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))

S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))

K tuples:none
Defined Rule Symbols:

U11, top, isNeList, U12, U61, proper, and, U62, U43, U52, U51, __, isPal, U32, U53, U72, U31, U23, U71, U21, U42, isQid, isNePal, U22, isList, isPalListKind, U41

Defined Pair Symbols:

U11', TOP, ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41'

Compound Symbols:

c, c1, c3, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53

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

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U11(mark(z0), z1) → mark(U11(z0, z1))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
isNeList(ok(z0)) → ok(isNeList(z0))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
U61(ok(z0), ok(z1)) → ok(U61(z0, z1))
U61(mark(z0), z1) → mark(U61(z0, z1))
proper(i) → ok(i)
proper(u) → ok(u)
proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(o) → ok(o)
proper(e) → ok(e)
proper(a) → ok(a)
and(ok(z0), ok(z1)) → ok(and(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
U62(mark(z0)) → mark(U62(z0))
U62(ok(z0)) → ok(U62(z0))
U43(ok(z0)) → ok(U43(z0))
U43(mark(z0)) → mark(U43(z0))
U52(ok(z0), ok(z1)) → ok(U52(z0, z1))
U52(mark(z0), z1) → mark(U52(z0, z1))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
__(z0, mark(z1)) → mark(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
isPal(ok(z0)) → ok(isPal(z0))
U32(mark(z0)) → mark(U32(z0))
U32(ok(z0)) → ok(U32(z0))
U53(mark(z0)) → mark(U53(z0))
U53(ok(z0)) → ok(U53(z0))
U72(mark(z0)) → mark(U72(z0))
U72(ok(z0)) → ok(U72(z0))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U23(ok(z0)) → ok(U23(z0))
U23(mark(z0)) → mark(U23(z0))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U42(ok(z0), ok(z1)) → ok(U42(z0, z1))
U42(mark(z0), z1) → mark(U42(z0, z1))
isQid(ok(z0)) → ok(isQid(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
U22(mark(z0), z1) → mark(U22(z0, z1))
U22(ok(z0), ok(z1)) → ok(U22(z0, z1))
isList(ok(z0)) → ok(isList(z0))
isPalListKind(ok(z0)) → ok(isPalListKind(z0))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

U11, top, isNeList, U12, U61, proper, and, U62, U43, U52, U51, __, isPal, U32, U53, U72, U31, U23, U71, U21, U42, isQid, isNePal, U22, isList, isPalListKind, U41

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U11(mark(z0), z1) → mark(U11(z0, z1))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
isNeList(ok(z0)) → ok(isNeList(z0))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
U61(ok(z0), ok(z1)) → ok(U61(z0, z1))
U61(mark(z0), z1) → mark(U61(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
U62(mark(z0)) → mark(U62(z0))
U62(ok(z0)) → ok(U62(z0))
U43(ok(z0)) → ok(U43(z0))
U43(mark(z0)) → mark(U43(z0))
U52(ok(z0), ok(z1)) → ok(U52(z0, z1))
U52(mark(z0), z1) → mark(U52(z0, z1))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
__(z0, mark(z1)) → mark(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
isPal(ok(z0)) → ok(isPal(z0))
U32(mark(z0)) → mark(U32(z0))
U32(ok(z0)) → ok(U32(z0))
U53(mark(z0)) → mark(U53(z0))
U53(ok(z0)) → ok(U53(z0))
U72(mark(z0)) → mark(U72(z0))
U72(ok(z0)) → ok(U72(z0))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U23(ok(z0)) → ok(U23(z0))
U23(mark(z0)) → mark(U23(z0))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U42(ok(z0), ok(z1)) → ok(U42(z0, z1))
U42(mark(z0), z1) → mark(U42(z0, z1))
isQid(ok(z0)) → ok(isQid(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
U22(mark(z0), z1) → mark(U22(z0, z1))
U22(ok(z0), ok(z1)) → ok(U22(z0, z1))
isList(ok(z0)) → ok(isList(z0))
isPalListKind(ok(z0)) → ok(isPalListKind(z0))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

(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)) → c3(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(e) → ok(e)
proper(a) → ok(a)
proper(i) → ok(i)

And the Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(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(ISPALLISTKIND(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = x₁
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = 0
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(e) = 0
POL(i) = 0
POL(mark(x₁)) = [1]
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(i) → ok(i)
proper(u) → ok(u)
proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(o) → ok(o)
proper(e) → ok(e)
proper(a) → ok(a)

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

(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.

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(AND(x₁, x₂)) = x₁
POL(ISLIST(x₁)) = x₁
POL(ISNELIST(x₁)) = 0
POL(ISNEPAL(x₁)) = 0
POL(ISPAL(x₁)) = x₁
POL(ISPALLISTKIND(x₁)) = x₁
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = x₂
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = x₃
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = x₃
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = x₁
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(e) = 0
POL(i) = 0
POL(mark(x₁)) = x₁
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(i) → ok(i)
proper(u) → ok(u)
proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(o) → ok(o)
proper(e) → ok(e)
proper(a) → ok(a)

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

(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) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(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(ISPALLISTKIND(x₁)) = 0
POL(ISQID(x₁)) = x₁
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = 0
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = x₁
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(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(i) → ok(i)
proper(u) → ok(u)
proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(o) → ok(o)
proper(e) → ok(e)
proper(a) → ok(a)

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

(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.

U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(AND(x₁, x₂)) = 0
POL(ISLIST(x₁)) = x₁
POL(ISNELIST(x₁)) = 0
POL(ISNEPAL(x₁)) = 0
POL(ISPAL(x₁)) = 0
POL(ISPALLISTKIND(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂)) = x₁
POL(U43'(x₁)) = x₁
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = x₂
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(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

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

(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.

U32'(mark(z0)) → c30(U32'(z0))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(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(ISPALLISTKIND(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = x₁
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = 0
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(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

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

(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.

U61'(mark(z0), z1) → c8(U61'(z0, z1))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(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(ISPALLISTKIND(x₁)) = x₁
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = x₂
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = 0
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = x₁
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(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

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

(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.

U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(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(ISPALLISTKIND(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = x₂
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = x₁
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = x₂
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = x₁
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(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

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

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

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

U12'(mark(z0)) → c5(U12'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U72'(mark(z0)) → c34(U72'(z0))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(AND(x₁, x₂)) = 0
POL(ISLIST(x₁)) = x₁
POL(ISNELIST(x₁)) = 0
POL(ISNEPAL(x₁)) = 0
POL(ISPAL(x₁)) = x₁
POL(ISPALLISTKIND(x₁)) = x₁
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = x₁
POL(U21'(x₁, x₂, x₃)) = x₃
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = x₂
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = x₁
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = x₁
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(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

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U12'(mark(z0)) → c5(U12'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U72'(mark(z0)) → c34(U72'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

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

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

U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(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(ISPALLISTKIND(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = x₂
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = 0
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(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

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U12'(mark(z0)) → c5(U12'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U72'(mark(z0)) → c34(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

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

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

U11'(mark(z0), z1) → c1(U11'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(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(ISPALLISTKIND(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = x₁
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = 0
POL(U22'(x₁, x₂)) = x₁
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = x₁
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = 0
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(e) = 0
POL(i) = 0
POL(mark(x₁)) = [1] + x₁
POL(nil) = 0
POL(o) = 0
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0
POL(tt) = 0
POL(u) = 0

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U12'(mark(z0)) → c5(U12'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U72'(mark(z0)) → c34(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

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

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

__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U72'(ok(z0)) → c35(U72'(z0))
U23'(ok(z0)) → c38(U23'(z0))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(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(ISPALLISTKIND(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = x₃
POL(U22'(x₁, x₂)) = x₂
POL(U23'(x₁)) = x₁
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = 0
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = x₁
POL(__'(x₁, x₂)) = x₁
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(e) = 0
POL(i) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0
POL(tt) = 0
POL(u) = 0

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U12'(mark(z0)) → c5(U12'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U72'(mark(z0)) → c34(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U72'(ok(z0)) → c35(U72'(z0))
U23'(ok(z0)) → c38(U23'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

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

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

U23'(mark(z0)) → c39(U23'(z0))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(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(ISPALLISTKIND(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = x₁
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = 0
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(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

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U12'(mark(z0)) → c5(U12'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U72'(mark(z0)) → c34(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U72'(ok(z0)) → c35(U72'(z0))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

(37) 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)) → c4(ISNELIST(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(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)) = [2]x₁
POL(ISPALLISTKIND(x₁)) = [2]x₁
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = [2]x₃
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = [2]x₂
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = x₂
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(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

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U12'(ok(z0)) → c6(U12'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U12'(mark(z0)) → c5(U12'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U72'(mark(z0)) → c34(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U72'(ok(z0)) → c35(U72'(z0))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

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

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

U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(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(ISPALLISTKIND(x₁)) = x₁
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = x₁ + x₂
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = x₂
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(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

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U12'(ok(z0)) → c6(U12'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U12'(mark(z0)) → c5(U12'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U72'(mark(z0)) → c34(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U72'(ok(z0)) → c35(U72'(z0))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

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

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

U62'(ok(z0)) → c19(U62'(z0))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(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(ISPALLISTKIND(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = 0
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = x₁
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(e) = 0
POL(i) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0
POL(tt) = 0
POL(u) = 0

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U12'(ok(z0)) → c6(U12'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U12'(mark(z0)) → c5(U12'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U72'(mark(z0)) → c34(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U72'(ok(z0)) → c35(U72'(z0))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U62'(ok(z0)) → c19(U62'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

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

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

U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(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(ISPALLISTKIND(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = x₁
POL(U52'(x₁, x₂)) = 0
POL(U53'(x₁)) = x₁
POL(U61'(x₁, x₂)) = x₂
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(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

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U12'(ok(z0)) → c6(U12'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
U53'(ok(z0)) → c33(U53'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U12'(mark(z0)) → c5(U12'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U72'(mark(z0)) → c34(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U72'(ok(z0)) → c35(U72'(z0))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U62'(ok(z0)) → c19(U62'(z0))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

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

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

U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(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(ISPALLISTKIND(x₁)) = 0
POL(ISQID(x₁)) = x₁
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = [3]x₃
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = [2]x₂
POL(U42'(x₁, x₂)) = x₂
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = x₂
POL(U52'(x₁, x₂)) = 0
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = x₂
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(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

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U12'(ok(z0)) → c6(U12'(z0))
U43'(ok(z0)) → c20(U43'(z0))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
U53'(ok(z0)) → c33(U53'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U12'(mark(z0)) → c5(U12'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U72'(mark(z0)) → c34(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U72'(ok(z0)) → c35(U72'(z0))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U62'(ok(z0)) → c19(U62'(z0))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

(47) 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)) → c47(ISNEPAL(z0))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(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(ISPALLISTKIND(x₁)) = [2]x₁
POL(ISQID(x₁)) = [2]x₁
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = [2]x₂
POL(U42'(x₁, x₂)) = [2]x₂
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = [2]x₂
POL(U52'(x₁, x₂)) = 0
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(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

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U12'(ok(z0)) → c6(U12'(z0))
U43'(ok(z0)) → c20(U43'(z0))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
U53'(ok(z0)) → c33(U53'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U12'(mark(z0)) → c5(U12'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U72'(mark(z0)) → c34(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U72'(ok(z0)) → c35(U72'(z0))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U62'(ok(z0)) → c19(U62'(z0))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

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

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

U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(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(ISPALLISTKIND(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = [2]x₂ + [2]x₃
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = [2]x₂
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = [2]x₃
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = 0
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = x₂
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(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

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U12'(ok(z0)) → c6(U12'(z0))
U43'(ok(z0)) → c20(U43'(z0))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
U53'(ok(z0)) → c33(U53'(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U12'(mark(z0)) → c5(U12'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U72'(mark(z0)) → c34(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U72'(ok(z0)) → c35(U72'(z0))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U62'(ok(z0)) → c19(U62'(z0))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

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

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

U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(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(ISPALLISTKIND(x₁)) = x₁
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = x₁
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = 0
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(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

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U12'(ok(z0)) → c6(U12'(z0))
U43'(ok(z0)) → c20(U43'(z0))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
U53'(ok(z0)) → c33(U53'(z0))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U12'(mark(z0)) → c5(U12'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U72'(mark(z0)) → c34(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U72'(ok(z0)) → c35(U72'(z0))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U62'(ok(z0)) → c19(U62'(z0))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

(53) 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) → c27(__'(z0, z1))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(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(ISPALLISTKIND(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = x₃
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = x₂
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = x₂
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = x₂
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = x₁
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(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

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U12'(ok(z0)) → c6(U12'(z0))
U43'(ok(z0)) → c20(U43'(z0))
__'(z0, mark(z1)) → c26(__'(z0, z1))
U53'(ok(z0)) → c33(U53'(z0))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U12'(mark(z0)) → c5(U12'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U72'(mark(z0)) → c34(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U72'(ok(z0)) → c35(U72'(z0))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U62'(ok(z0)) → c19(U62'(z0))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
__'(mark(z0), z1) → c27(__'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

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

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

__'(z0, mark(z1)) → c26(__'(z0, z1))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(AND(x₁, x₂)) = 0
POL(ISLIST(x₁)) = x₁
POL(ISNELIST(x₁)) = 0
POL(ISNEPAL(x₁)) = 0
POL(ISPAL(x₁)) = 0
POL(ISPALLISTKIND(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = x₂
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = x₁
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = 0
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = x₂
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = x₂
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(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

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U12'(ok(z0)) → c6(U12'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U53'(ok(z0)) → c33(U53'(z0))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U12'(mark(z0)) → c5(U12'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U72'(mark(z0)) → c34(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U72'(ok(z0)) → c35(U72'(z0))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U62'(ok(z0)) → c19(U62'(z0))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(z0, mark(z1)) → c26(__'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

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

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

U12'(ok(z0)) → c6(U12'(z0))
U43'(ok(z0)) → c20(U43'(z0))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(AND(x₁, x₂)) = 0
POL(ISLIST(x₁)) = x₁
POL(ISNELIST(x₁)) = 0
POL(ISNEPAL(x₁)) = 0
POL(ISPAL(x₁)) = 0
POL(ISPALLISTKIND(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = x₁
POL(U21'(x₁, x₂, x₃)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = x₁
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = x₂
POL(U53'(x₁)) = 0
POL(U61'(x₁, x₂)) = x₂
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(e) = 0
POL(i) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0
POL(tt) = 0
POL(u) = 0

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U53'(ok(z0)) → c33(U53'(z0))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U12'(mark(z0)) → c5(U12'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U72'(mark(z0)) → c34(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U72'(ok(z0)) → c35(U72'(z0))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U62'(ok(z0)) → c19(U62'(z0))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(z0, mark(z1)) → c26(__'(z0, z1))
U12'(ok(z0)) → c6(U12'(z0))
U43'(ok(z0)) → c20(U43'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

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

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

U53'(ok(z0)) → c33(U53'(z0))

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

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(AND(x₁, x₂)) = [2]x₂
POL(ISLIST(x₁)) = 0
POL(ISNELIST(x₁)) = x₁
POL(ISNEPAL(x₁)) = 0
POL(ISPAL(x₁)) = 0
POL(ISPALLISTKIND(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = [2]x₂
POL(U12'(x₁)) = 0
POL(U21'(x₁, x₂, x₃)) = 0
POL(U22'(x₁, x₂)) = [2]x₂
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = [2]x₁ + [2]x₂
POL(U42'(x₁, x₂)) = 0
POL(U43'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂)) = 0
POL(U53'(x₁)) = [2]x₁
POL(U61'(x₁, x₂)) = [2]x₁ + [2]x₂
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = [2]x₁
POL(U72'(x₁)) = 0
POL(__'(x₁, x₂)) = [2]x₁
POL(a) = [2]
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(e) = [2]
POL(i) = [3]
POL(mark(x₁)) = x₁
POL(nil) = [3]
POL(o) = [2]
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = [2]x₁
POL(tt) = [2]
POL(u) = [2]

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U12'(mark(z0)) → c5(U12'(z0))
U12'(ok(z0)) → c6(U12'(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U62'(ok(z0)) → c19(U62'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
__'(z0, mark(z1)) → c26(__'(z0, z1))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U32'(mark(z0)) → c30(U32'(z0))
U32'(ok(z0)) → c31(U32'(z0))
U53'(mark(z0)) → c32(U53'(z0))
U53'(ok(z0)) → c33(U53'(z0))
U72'(mark(z0)) → c34(U72'(z0))
U72'(ok(z0)) → c35(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:none
K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c16(AND(z0, z1))
U52'(ok(z0), ok(z1)) → c22(U52'(z0, z1))
ISPAL(ok(z0)) → c29(ISPAL(z0))
U21'(ok(z0), ok(z1), ok(z2)) → c42(U21'(z0, z1, z2))
ISLIST(ok(z0)) → c50(ISLIST(z0))
ISPALLISTKIND(ok(z0)) → c51(ISPALLISTKIND(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c53(U41'(z0, z1, z2))
AND(mark(z0), z1) → c17(AND(z0, z1))
U62'(mark(z0)) → c18(U62'(z0))
U43'(mark(z0)) → c21(U43'(z0))
U42'(mark(z0), z1) → c45(U42'(z0, z1))
U32'(mark(z0)) → c30(U32'(z0))
U61'(mark(z0), z1) → c8(U61'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
U71'(mark(z0), z1) → c41(U71'(z0, z1))
U12'(mark(z0)) → c5(U12'(z0))
U52'(mark(z0), z1) → c23(U52'(z0, z1))
U72'(mark(z0)) → c34(U72'(z0))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U32'(ok(z0)) → c31(U32'(z0))
U22'(mark(z0), z1) → c48(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c49(U22'(z0, z1))
__'(ok(z0), ok(z1)) → c28(__'(z0, z1))
U72'(ok(z0)) → c35(U72'(z0))
U23'(ok(z0)) → c38(U23'(z0))
U23'(mark(z0)) → c39(U23'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
U61'(ok(z0), ok(z1)) → c7(U61'(z0, z1))
U21'(mark(z0), z1, z2) → c43(U21'(z0, z1, z2))
U62'(ok(z0)) → c19(U62'(z0))
U51'(mark(z0), z1, z2) → c24(U51'(z0, z1, z2))
U53'(mark(z0)) → c32(U53'(z0))
U51'(ok(z0), ok(z1), ok(z2)) → c25(U51'(z0, z1, z2))
U42'(ok(z0), ok(z1)) → c44(U42'(z0, z1))
ISQID(ok(z0)) → c46(ISQID(z0))
ISNEPAL(ok(z0)) → c47(ISNEPAL(z0))
U71'(ok(z0), ok(z1)) → c40(U71'(z0, z1))
U41'(mark(z0), z1, z2) → c52(U41'(z0, z1, z2))
__'(mark(z0), z1) → c27(__'(z0, z1))
__'(z0, mark(z1)) → c26(__'(z0, z1))
U12'(ok(z0)) → c6(U12'(z0))
U43'(ok(z0)) → c20(U43'(z0))
U53'(ok(z0)) → c33(U53'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', ISNELIST, U12', U61', AND, U62', U43', U52', U51', __', ISPAL, U32', U53', U72', U31', U23', U71', U21', U42', ISQID, ISNEPAL, U22', ISLIST, ISPALLISTKIND, U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c3

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

The set S is empty

(0) Obligation:

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

(2) Obligation:

(3) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

(4) BOUNDS(1, n^1)

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

(6) Obligation:

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

(8) Obligation:

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

(10) Obligation:

(11) CdtUsableRulesProof (EQUIVALENT transformation)

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

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

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

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

(38) Obligation:

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

(40) Obligation:

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

(42) Obligation:

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

(44) Obligation:

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

(46) Obligation:

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

(48) Obligation:

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

(50) Obligation:

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

(52) Obligation:

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

(54) Obligation:

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

(56) Obligation:

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

(58) Obligation:

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

(60) Obligation:

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

(62) BOUNDS(1, 1)