Runtime Complexity (innermost) proof of /tmp/tmpNSBqFi/PALINDROME_nokinds_noand

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 (⇔, 530 ms)

    ↳4 BOUNDS(1, n^1)

    ↳5 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 16 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)), 249 ms)

        ↳14 CdtProblem

        ↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 62 ms)

        ↳16 CdtProblem

        ↳17 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 44 ms)

        ↳18 CdtProblem

        ↳19 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 58 ms)

        ↳20 CdtProblem

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

        ↳22 CdtProblem

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

        ↳24 CdtProblem

        ↳25 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 62 ms)

        ↳26 CdtProblem

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

        ↳28 CdtProblem

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

        ↳30 CdtProblem

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

        ↳32 CdtProblem

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

        ↳34 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)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(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(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(U11(X)) → U11(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(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:

U52(mark(X)) → mark(U52(X))
top(ok(X)) → top(active(X))
isNeList(ok(X)) → ok(isNeList(X))
isQid(ok(X)) → ok(isQid(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
__(mark(X1), X2) → mark(__(X1, X2))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
proper(i) → ok(i)
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
proper(o) → ok(o)
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U42(mark(X)) → mark(U42(X))
proper(u) → ok(u)
proper(e) → ok(e)
isList(ok(X)) → ok(isList(X))
proper(nil) → ok(nil)
proper(tt) → ok(tt)
isPal(ok(X)) → ok(isPal(X))
U72(mark(X)) → mark(U72(X))
U72(ok(X)) → ok(U72(X))
U22(mark(X)) → mark(U22(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U22(ok(X)) → ok(U22(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
proper(a) → ok(a)
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X)) → ok(U31(X))
U31(mark(X)) → mark(U31(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U81(ok(X)) → ok(U81(X))
U81(mark(X)) → mark(U81(X))
top(mark(X)) → top(proper(X))
U52(ok(X)) → ok(U52(X))

Rewrite Strategy: INNERMOST

(3) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]
transitions:
mark0(0) → 0
ok0(0) → 0
active0(0) → 0
i0() → 0
o0() → 0
u0() → 0
e0() → 0
nil0() → 0
tt0() → 0
a0() → 0
U520(0) → 1
top0(0) → 2
isNeList0(0) → 3
isQid0(0) → 4
U510(0, 0) → 5
isNePal0(0) → 6
__0(0, 0) → 7
U610(0) → 8
proper0(0) → 9
U410(0, 0) → 10
U420(0) → 11
U210(0, 0) → 12
isList0(0) → 13
isPal0(0) → 14
U720(0) → 15
U220(0) → 16
U710(0, 0) → 17
U110(0) → 18
U310(0) → 19
U810(0) → 20
U521(0) → 21
mark1(21) → 1
active1(0) → 22
top1(22) → 2
isNeList1(0) → 23
ok1(23) → 3
isQid1(0) → 24
ok1(24) → 4
U511(0, 0) → 25
mark1(25) → 5
isNePal1(0) → 26
ok1(26) → 6
__1(0, 0) → 27
mark1(27) → 7
U611(0) → 28
mark1(28) → 8
U611(0) → 29
ok1(29) → 8
i1() → 30
ok1(30) → 9
U511(0, 0) → 31
ok1(31) → 5
o1() → 32
ok1(32) → 9
U411(0, 0) → 33
mark1(33) → 10
U421(0) → 34
ok1(34) → 11
U211(0, 0) → 35
mark1(35) → 12
U421(0) → 36
mark1(36) → 11
u1() → 37
ok1(37) → 9
e1() → 38
ok1(38) → 9
isList1(0) → 39
ok1(39) → 13
nil1() → 40
ok1(40) → 9
tt1() → 41
ok1(41) → 9
isPal1(0) → 42
ok1(42) → 14
U721(0) → 43
mark1(43) → 15
U721(0) → 44
ok1(44) → 15
U221(0) → 45
mark1(45) → 16
__1(0, 0) → 46
ok1(46) → 7
U221(0) → 47
ok1(47) → 16
U411(0, 0) → 48
ok1(48) → 10
U711(0, 0) → 49
mark1(49) → 17
a1() → 50
ok1(50) → 9
U111(0) → 51
mark1(51) → 18
U111(0) → 52
ok1(52) → 18
U211(0, 0) → 53
ok1(53) → 12
U311(0) → 54
ok1(54) → 19
U311(0) → 55
mark1(55) → 19
U711(0, 0) → 56
ok1(56) → 17
U811(0) → 57
ok1(57) → 20
U811(0) → 58
mark1(58) → 20
proper1(0) → 59
top1(59) → 2
U521(0) → 60
ok1(60) → 1
mark1(21) → 21
mark1(21) → 60
ok1(23) → 23
ok1(24) → 24
mark1(25) → 25
mark1(25) → 31
ok1(26) → 26
mark1(27) → 27
mark1(27) → 46
mark1(28) → 28
mark1(28) → 29
ok1(29) → 28
ok1(29) → 29
ok1(30) → 59
ok1(31) → 25
ok1(31) → 31
ok1(32) → 59
mark1(33) → 33
mark1(33) → 48
ok1(34) → 34
ok1(34) → 36
mark1(35) → 35
mark1(35) → 53
mark1(36) → 34
mark1(36) → 36
ok1(37) → 59
ok1(38) → 59
ok1(39) → 39
ok1(40) → 59
ok1(41) → 59
ok1(42) → 42
mark1(43) → 43
mark1(43) → 44
ok1(44) → 43
ok1(44) → 44
mark1(45) → 45
mark1(45) → 47
ok1(46) → 27
ok1(46) → 46
ok1(47) → 45
ok1(47) → 47
ok1(48) → 33
ok1(48) → 48
mark1(49) → 49
mark1(49) → 56
ok1(50) → 59
mark1(51) → 51
mark1(51) → 52
ok1(52) → 51
ok1(52) → 52
ok1(53) → 35
ok1(53) → 53
ok1(54) → 54
ok1(54) → 55
mark1(55) → 54
mark1(55) → 55
ok1(56) → 49
ok1(56) → 56
ok1(57) → 57
ok1(57) → 58
mark1(58) → 57
mark1(58) → 58
ok1(60) → 21
ok1(60) → 60
active2(30) → 61
top2(61) → 2
active2(32) → 61
active2(37) → 61
active2(38) → 61
active2(40) → 61
active2(41) → 61
active2(50) → 61

(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:

U52(mark(z0)) → mark(U52(z0))
U52(ok(z0)) → ok(U52(z0))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
isNeList(ok(z0)) → ok(isNeList(z0))
isQid(ok(z0)) → ok(isQid(z0))
U51(mark(z0), z1) → mark(U51(z0, z1))
U51(ok(z0), ok(z1)) → ok(U51(z0, z1))
isNePal(ok(z0)) → ok(isNePal(z0))
__(mark(z0), z1) → mark(__(z0, z1))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
U61(mark(z0)) → mark(U61(z0))
U61(ok(z0)) → ok(U61(z0))
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
proper(e) → ok(e)
proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(a) → ok(a)
U41(mark(z0), z1) → mark(U41(z0, z1))
U41(ok(z0), ok(z1)) → ok(U41(z0, z1))
U42(ok(z0)) → ok(U42(z0))
U42(mark(z0)) → mark(U42(z0))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
isList(ok(z0)) → ok(isList(z0))
isPal(ok(z0)) → ok(isPal(z0))
U72(mark(z0)) → mark(U72(z0))
U72(ok(z0)) → ok(U72(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
U71(mark(z0), z1) → mark(U71(z0, z1))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U11(mark(z0)) → mark(U11(z0))
U11(ok(z0)) → ok(U11(z0))
U31(ok(z0)) → ok(U31(z0))
U31(mark(z0)) → mark(U31(z0))
U81(ok(z0)) → ok(U81(z0))
U81(mark(z0)) → mark(U81(z0))

Tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
TOP(ok(z0)) → c2(TOP(active(z0)))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
PROPER(i) → c14
PROPER(o) → c15
PROPER(u) → c16
PROPER(e) → c17
PROPER(nil) → c18
PROPER(tt) → c19
PROPER(a) → c20
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))

S tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
TOP(ok(z0)) → c2(TOP(active(z0)))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
PROPER(i) → c14
PROPER(o) → c15
PROPER(u) → c16
PROPER(e) → c17
PROPER(nil) → c18
PROPER(tt) → c19
PROPER(a) → c20
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))

K tuples:none
Defined Rule Symbols:

U52, top, isNeList, isQid, U51, isNePal, __, U61, proper, U41, U42, U21, isList, isPal, U72, U22, U71, U11, U31, U81

Defined Pair Symbols:

U52', TOP, ISNELIST, ISQID, U51', ISNEPAL, __', U61', PROPER, U41', U42', U21', ISLIST, ISPAL, U72', U22', U71', U11', U31', U81'

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

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

Removed 8 trailing nodes:

PROPER(nil) → c18
PROPER(o) → c15
PROPER(a) → c20
PROPER(tt) → c19
PROPER(e) → c17
PROPER(i) → c14
TOP(ok(z0)) → c2(TOP(active(z0)))
PROPER(u) → c16

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

U52(mark(z0)) → mark(U52(z0))
U52(ok(z0)) → ok(U52(z0))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
isNeList(ok(z0)) → ok(isNeList(z0))
isQid(ok(z0)) → ok(isQid(z0))
U51(mark(z0), z1) → mark(U51(z0, z1))
U51(ok(z0), ok(z1)) → ok(U51(z0, z1))
isNePal(ok(z0)) → ok(isNePal(z0))
__(mark(z0), z1) → mark(__(z0, z1))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
U61(mark(z0)) → mark(U61(z0))
U61(ok(z0)) → ok(U61(z0))
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
proper(e) → ok(e)
proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(a) → ok(a)
U41(mark(z0), z1) → mark(U41(z0, z1))
U41(ok(z0), ok(z1)) → ok(U41(z0, z1))
U42(ok(z0)) → ok(U42(z0))
U42(mark(z0)) → mark(U42(z0))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
isList(ok(z0)) → ok(isList(z0))
isPal(ok(z0)) → ok(isPal(z0))
U72(mark(z0)) → mark(U72(z0))
U72(ok(z0)) → ok(U72(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
U71(mark(z0), z1) → mark(U71(z0, z1))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U11(mark(z0)) → mark(U11(z0))
U11(ok(z0)) → ok(U11(z0))
U31(ok(z0)) → ok(U31(z0))
U31(mark(z0)) → mark(U31(z0))
U81(ok(z0)) → ok(U81(z0))
U81(mark(z0)) → mark(U81(z0))

Tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))

S tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))

K tuples:none
Defined Rule Symbols:

U52, top, isNeList, isQid, U51, isNePal, __, U61, proper, U41, U42, U21, isList, isPal, U72, U22, U71, U11, U31, U81

Defined Pair Symbols:

U52', TOP, ISNELIST, ISQID, U51', ISNEPAL, __', U61', U41', U42', U21', ISLIST, ISPAL, U72', U22', U71', U11', U31', U81'

Compound Symbols:

c, c1, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40

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

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

U52(mark(z0)) → mark(U52(z0))
U52(ok(z0)) → ok(U52(z0))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
isNeList(ok(z0)) → ok(isNeList(z0))
isQid(ok(z0)) → ok(isQid(z0))
U51(mark(z0), z1) → mark(U51(z0, z1))
U51(ok(z0), ok(z1)) → ok(U51(z0, z1))
isNePal(ok(z0)) → ok(isNePal(z0))
__(mark(z0), z1) → mark(__(z0, z1))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
U61(mark(z0)) → mark(U61(z0))
U61(ok(z0)) → ok(U61(z0))
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
proper(e) → ok(e)
proper(nil) → ok(nil)
proper(tt) → ok(tt)
proper(a) → ok(a)
U41(mark(z0), z1) → mark(U41(z0, z1))
U41(ok(z0), ok(z1)) → ok(U41(z0, z1))
U42(ok(z0)) → ok(U42(z0))
U42(mark(z0)) → mark(U42(z0))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
isList(ok(z0)) → ok(isList(z0))
isPal(ok(z0)) → ok(isPal(z0))
U72(mark(z0)) → mark(U72(z0))
U72(ok(z0)) → ok(U72(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
U71(mark(z0), z1) → mark(U71(z0, z1))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U11(mark(z0)) → mark(U11(z0))
U11(ok(z0)) → ok(U11(z0))
U31(ok(z0)) → ok(U31(z0))
U31(mark(z0)) → mark(U31(z0))
U81(ok(z0)) → ok(U81(z0))
U81(mark(z0)) → mark(U81(z0))

Tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

U52, top, isNeList, isQid, U51, isNePal, __, U61, proper, U41, U42, U21, isList, isPal, U72, U22, U71, U11, U31, U81

Defined Pair Symbols:

U52', ISNELIST, ISQID, U51', ISNEPAL, __', U61', U41', U42', U21', ISLIST, ISPAL, U72', U22', U71', U11', U31', U81', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c3

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

U52(mark(z0)) → mark(U52(z0))
U52(ok(z0)) → ok(U52(z0))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
isNeList(ok(z0)) → ok(isNeList(z0))
isQid(ok(z0)) → ok(isQid(z0))
U51(mark(z0), z1) → mark(U51(z0, z1))
U51(ok(z0), ok(z1)) → ok(U51(z0, z1))
isNePal(ok(z0)) → ok(isNePal(z0))
__(mark(z0), z1) → mark(__(z0, z1))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
U61(mark(z0)) → mark(U61(z0))
U61(ok(z0)) → ok(U61(z0))
U41(mark(z0), z1) → mark(U41(z0, z1))
U41(ok(z0), ok(z1)) → ok(U41(z0, z1))
U42(ok(z0)) → ok(U42(z0))
U42(mark(z0)) → mark(U42(z0))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
isList(ok(z0)) → ok(isList(z0))
isPal(ok(z0)) → ok(isPal(z0))
U72(mark(z0)) → mark(U72(z0))
U72(ok(z0)) → ok(U72(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
U71(mark(z0), z1) → mark(U71(z0, z1))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U11(mark(z0)) → mark(U11(z0))
U11(ok(z0)) → ok(U11(z0))
U31(ok(z0)) → ok(U31(z0))
U31(mark(z0)) → mark(U31(z0))
U81(ok(z0)) → ok(U81(z0))
U81(mark(z0)) → mark(U81(z0))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

proper

Defined Pair Symbols:

U52', ISNELIST, ISQID, U51', ISNEPAL, __', U61', U41', U42', U21', ISLIST, ISPAL, U72', U22', U71', U11', U31', U81', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, 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.

ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))

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

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(ISLIST(x₁)) = 0
POL(ISNELIST(x₁)) = x₁
POL(ISNEPAL(x₁)) = x₁
POL(ISPAL(x₁)) = x₁
POL(ISQID(x₁)) = x₁
POL(TOP(x₁)) = 0
POL(U11'(x₁)) = x₁
POL(U21'(x₁, x₂)) = x₂
POL(U22'(x₁)) = 0
POL(U31'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁, x₂)) = 0
POL(U52'(x₁)) = 0
POL(U61'(x₁)) = x₁
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(U81'(x₁)) = 0
POL(__'(x₁, x₂)) = x₂
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(e) = 0
POL(i) = 0
POL(mark(x₁)) = [1] + x₁
POL(nil) = 0
POL(o) = 0
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0
POL(tt) = 0
POL(u) = 0

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

K tuples:

ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U52', ISNELIST, ISQID, U51', ISNEPAL, __', U61', U41', U42', U21', ISLIST, ISPAL, U72', U22', U71', U11', U31', U81', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, 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.

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:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(ISLIST(x₁)) = 0
POL(ISNELIST(x₁)) = 0
POL(ISNEPAL(x₁)) = 0
POL(ISPAL(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = x₁
POL(U11'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = 0
POL(U31'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁, x₂)) = 0
POL(U52'(x₁)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(U81'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(e) = 0
POL(i) = 0
POL(mark(x₁)) = [1] + x₁
POL(nil) = 0
POL(o) = 0
POL(ok(x₁)) = 0
POL(proper(x₁)) = 0
POL(tt) = 0
POL(u) = 0

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))

K tuples:

ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U52', ISNELIST, ISQID, U51', ISNEPAL, __', U61', U41', U42', U21', ISLIST, ISPAL, U72', U22', U71', U11', U31', U81', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, 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.

U52'(mark(z0)) → c(U52'(z0))
U72'(mark(z0)) → c29(U72'(z0))

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

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(ISLIST(x₁)) = 0
POL(ISNELIST(x₁)) = x₁
POL(ISNEPAL(x₁)) = 0
POL(ISPAL(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = 0
POL(U31'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁, x₂)) = 0
POL(U52'(x₁)) = x₁
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = x₁
POL(U81'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(e) = 0
POL(i) = 0
POL(mark(x₁)) = [1] + x₁
POL(nil) = 0
POL(o) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(tt) = 0
POL(u) = 0

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U52'(ok(z0)) → c1(U52'(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))

K tuples:

ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U52'(mark(z0)) → c(U52'(z0))
U72'(mark(z0)) → c29(U72'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U52', ISNELIST, ISQID, U51', ISNEPAL, __', U61', U41', U42', U21', ISLIST, ISPAL, U72', U22', U71', U11', U31', U81', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, 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.

U51'(mark(z0), z1) → c6(U51'(z0, z1))

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

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(ISLIST(x₁)) = 0
POL(ISNELIST(x₁)) = 0
POL(ISNEPAL(x₁)) = 0
POL(ISPAL(x₁)) = 0
POL(ISQID(x₁)) = x₁
POL(TOP(x₁)) = 0
POL(U11'(x₁)) = 0
POL(U21'(x₁, x₂)) = x₂
POL(U22'(x₁)) = 0
POL(U31'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁, x₂)) = x₁
POL(U52'(x₁)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(U81'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(e) = 0
POL(i) = 0
POL(mark(x₁)) = [1] + x₁
POL(nil) = 0
POL(o) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(tt) = 0
POL(u) = 0

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U52'(ok(z0)) → c1(U52'(z0))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))

K tuples:

ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U52'(mark(z0)) → c(U52'(z0))
U72'(mark(z0)) → c29(U72'(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U52', ISNELIST, ISQID, U51', ISNEPAL, __', U61', U41', U42', U21', ISLIST, ISPAL, U72', U22', U71', U11', U31', U81', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, 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.

U41'(mark(z0), z1) → c21(U41'(z0, z1))
U42'(mark(z0)) → c24(U42'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))

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

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(ISLIST(x₁)) = 0
POL(ISNELIST(x₁)) = 0
POL(ISNEPAL(x₁)) = 0
POL(ISPAL(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = 0
POL(U31'(x₁)) = 0
POL(U41'(x₁, x₂)) = x₁
POL(U42'(x₁)) = x₁
POL(U51'(x₁, x₂)) = 0
POL(U52'(x₁)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂)) = x₁
POL(U72'(x₁)) = 0
POL(U81'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(e) = 0
POL(i) = 0
POL(mark(x₁)) = [1] + x₁
POL(nil) = 0
POL(o) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(tt) = 0
POL(u) = 0

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U52'(ok(z0)) → c1(U52'(z0))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))

K tuples:

ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U52'(mark(z0)) → c(U52'(z0))
U72'(mark(z0)) → c29(U72'(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U42'(mark(z0)) → c24(U42'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U52', ISNELIST, ISQID, U51', ISNEPAL, __', U61', U41', U42', U21', ISLIST, ISPAL, U72', U22', U71', U11', U31', U81', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, 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.

U81'(mark(z0)) → c40(U81'(z0))

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

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(ISLIST(x₁)) = 0
POL(ISNELIST(x₁)) = x₁
POL(ISNEPAL(x₁)) = x₁
POL(ISPAL(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁)) = 0
POL(U21'(x₁, x₂)) = x₂
POL(U22'(x₁)) = 0
POL(U31'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁, x₂)) = 0
POL(U52'(x₁)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(U81'(x₁)) = x₁
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(e) = 0
POL(i) = 0
POL(mark(x₁)) = [1] + x₁
POL(nil) = 0
POL(o) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(tt) = 0
POL(u) = 0

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U52'(ok(z0)) → c1(U52'(z0))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))

K tuples:

ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U52'(mark(z0)) → c(U52'(z0))
U72'(mark(z0)) → c29(U72'(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U42'(mark(z0)) → c24(U42'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U81'(mark(z0)) → c40(U81'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U52', ISNELIST, ISQID, U51', ISNEPAL, __', U61', U41', U42', U21', ISLIST, ISPAL, U72', U22', U71', U11', U31', U81', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, 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.

__'(mark(z0), z1) → c9(__'(z0, z1))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U22'(mark(z0)) → c31(U22'(z0))
U31'(mark(z0)) → c38(U31'(z0))

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

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(ISLIST(x₁)) = 0
POL(ISNELIST(x₁)) = 0
POL(ISNEPAL(x₁)) = x₁
POL(ISPAL(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁)) = 0
POL(U21'(x₁, x₂)) = x₁
POL(U22'(x₁)) = x₁
POL(U31'(x₁)) = x₁
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁, x₂)) = 0
POL(U52'(x₁)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(U81'(x₁)) = 0
POL(__'(x₁, x₂)) = x₁
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(e) = 0
POL(i) = 0
POL(mark(x₁)) = [1] + x₁
POL(nil) = 0
POL(o) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(tt) = 0
POL(u) = 0

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U52'(ok(z0)) → c1(U52'(z0))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
ISLIST(ok(z0)) → c27(ISLIST(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U31'(ok(z0)) → c37(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))

K tuples:

ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U52'(mark(z0)) → c(U52'(z0))
U72'(mark(z0)) → c29(U72'(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U42'(mark(z0)) → c24(U42'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U81'(mark(z0)) → c40(U81'(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U22'(mark(z0)) → c31(U22'(z0))
U31'(mark(z0)) → c38(U31'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U52', ISNELIST, ISQID, U51', ISNEPAL, __', U61', U41', U42', U21', ISLIST, ISPAL, U72', U22', U71', U11', U31', U81', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, 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.

U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U81'(ok(z0)) → c39(U81'(z0))

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

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(ISLIST(x₁)) = 0
POL(ISNELIST(x₁)) = 0
POL(ISNEPAL(x₁)) = 0
POL(ISPAL(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = x₁
POL(U31'(x₁)) = 0
POL(U41'(x₁, x₂)) = x₂
POL(U42'(x₁)) = x₁
POL(U51'(x₁, x₂)) = x₂
POL(U52'(x₁)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(U81'(x₁)) = x₁
POL(__'(x₁, x₂)) = x₂
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(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

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U52'(ok(z0)) → c1(U52'(z0))
ISLIST(ok(z0)) → c27(ISLIST(z0))
U72'(ok(z0)) → c30(U72'(z0))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U31'(ok(z0)) → c37(U31'(z0))

K tuples:

ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U52'(mark(z0)) → c(U52'(z0))
U72'(mark(z0)) → c29(U72'(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U42'(mark(z0)) → c24(U42'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U81'(mark(z0)) → c40(U81'(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U22'(mark(z0)) → c31(U22'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U81'(ok(z0)) → c39(U81'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U52', ISNELIST, ISQID, U51', ISNEPAL, __', U61', U41', U42', U21', ISLIST, ISPAL, U72', U22', U71', U11', U31', U81', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, 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.

U72'(ok(z0)) → c30(U72'(z0))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U31'(ok(z0)) → c37(U31'(z0))

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

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(ISLIST(x₁)) = 0
POL(ISNELIST(x₁)) = 0
POL(ISNEPAL(x₁)) = 0
POL(ISPAL(x₁)) = 0
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = 0
POL(U31'(x₁)) = x₁
POL(U41'(x₁, x₂)) = x₂
POL(U42'(x₁)) = x₁
POL(U51'(x₁, x₂)) = 0
POL(U52'(x₁)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂)) = x₂
POL(U72'(x₁)) = x₁
POL(U81'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(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

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U52'(ok(z0)) → c1(U52'(z0))
ISLIST(ok(z0)) → c27(ISLIST(z0))

K tuples:

ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U52'(mark(z0)) → c(U52'(z0))
U72'(mark(z0)) → c29(U72'(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U42'(mark(z0)) → c24(U42'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U81'(mark(z0)) → c40(U81'(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U22'(mark(z0)) → c31(U22'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U31'(ok(z0)) → c37(U31'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U52', ISNELIST, ISQID, U51', ISNEPAL, __', U61', U41', U42', U21', ISLIST, ISPAL, U72', U22', U71', U11', U31', U81', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, 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.

U52'(ok(z0)) → c1(U52'(z0))
ISLIST(ok(z0)) → c27(ISLIST(z0))

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

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(ISLIST(x₁)) = x₁
POL(ISNELIST(x₁)) = 0
POL(ISNEPAL(x₁)) = x₁
POL(ISPAL(x₁)) = x₁
POL(ISQID(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = 0
POL(U31'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = x₁
POL(U51'(x₁, x₂)) = 0
POL(U52'(x₁)) = x₁
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁)) = 0
POL(U81'(x₁)) = 0
POL(__'(x₁, x₂)) = 0
POL(a) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(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

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U52'(mark(z0)) → c(U52'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U42'(mark(z0)) → c24(U42'(z0))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISLIST(ok(z0)) → c27(ISLIST(z0))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U72'(mark(z0)) → c29(U72'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
U31'(ok(z0)) → c37(U31'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U81'(mark(z0)) → c40(U81'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:none
K tuples:

ISNELIST(ok(z0)) → c4(ISNELIST(z0))
ISQID(ok(z0)) → c5(ISQID(z0))
ISNEPAL(ok(z0)) → c8(ISNEPAL(z0))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
U61'(mark(z0)) → c12(U61'(z0))
U61'(ok(z0)) → c13(U61'(z0))
U21'(ok(z0), ok(z1)) → c26(U21'(z0, z1))
ISPAL(ok(z0)) → c28(ISPAL(z0))
U11'(mark(z0)) → c35(U11'(z0))
U11'(ok(z0)) → c36(U11'(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U52'(mark(z0)) → c(U52'(z0))
U72'(mark(z0)) → c29(U72'(z0))
U51'(mark(z0), z1) → c6(U51'(z0, z1))
U41'(mark(z0), z1) → c21(U41'(z0, z1))
U42'(mark(z0)) → c24(U42'(z0))
U71'(mark(z0), z1) → c33(U71'(z0, z1))
U81'(mark(z0)) → c40(U81'(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
U21'(mark(z0), z1) → c25(U21'(z0, z1))
U22'(mark(z0)) → c31(U22'(z0))
U31'(mark(z0)) → c38(U31'(z0))
U51'(ok(z0), ok(z1)) → c7(U51'(z0, z1))
U41'(ok(z0), ok(z1)) → c22(U41'(z0, z1))
U42'(ok(z0)) → c23(U42'(z0))
U22'(ok(z0)) → c32(U22'(z0))
U81'(ok(z0)) → c39(U81'(z0))
U72'(ok(z0)) → c30(U72'(z0))
U71'(ok(z0), ok(z1)) → c34(U71'(z0, z1))
U31'(ok(z0)) → c37(U31'(z0))
U52'(ok(z0)) → c1(U52'(z0))
ISLIST(ok(z0)) → c27(ISLIST(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U52', ISNELIST, ISQID, U51', ISNEPAL, __', U61', U41', U42', U21', ISLIST, ISPAL, U72', U22', U71', U11', U31', U81', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c3

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

(34) BOUNDS(1, 1)